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Practical Algorithms for Selection on Coarse-Grained Parallel Computers.

AbstractIn this paper, we consider the problem of selection on coarse-grained distributed memory parallel computers. We discuss several deterministic and randomized algorithms for parallel selection. We also consider several algorithms for load balancing needed to keep a balanced distribution of data across processors during the execution of the selection algorithms. We have carried out detailed implementations of all the algorithms discussed on the CM-5 and report on the experimental results. The results clearly demonstrate the role of randomization in reducing communication overhead.

better in practice than its deterministic counterpart due to the low constant associated with the algorithm. Parallel selection algorithms are useful in such practical applications as dynamic distribution of multidimensional data sets, parallel graph partitioning and parallel construction of multidimensional binary search trees. Many parallel algorithms for selection have been designed for the PRAM model [2, 3, 4, 9, 14] and for various network models including trees, meshes, hypercubes and re-configurable architectures [6, 7, 13, 16, 22]. More recently, Bader et.al. [5] implement a parallel deterministic selection algorithm on several distributed memory machines including CM-5, IBM SP-2 and INTEL Paragon. In this paper, we consider and evaluate parallel selection algorithms for coarse-grained distributed memory parallel computers. A coarse-grained parallel computer consists of several relatively powerful processors connected by an interconnection network. Most of the commercially available parallel computers belong to this category. Examples of such machines include CM-5, IBM SP-1 and SP-2, nCUBE 2, INTEL Paragon and Cray T3D. The rest of the paper is organized as follows: In Section 2, we describe our model of parallel computation and outline some primitives used by the algorithms. In Section 3, we present two deterministic and two randomized algorithms for parallel selection. Selection algorithms are iterative and work by reducing the number of elements to consider from iteration to iteration. Since we can not guarantee that the same number of elements are removed on every processor, this leads to load imbalance. In Section 4, we present several algorithms to perform such a load balancing. Each of the load balancing algorithms can be used by any selection algorithm that requires load balancing. In Section 5, we report and analyze the results we have obtained on the CM-5 by detailed implementation of the selection and load balancing algorithms presented. In section 6, we analyze the selection algorithms for meshes and hypercubes. Section 7 discusses parallel weighted selection. We conclude the paper in Section 8. Preliminaries 2.1 Model of Parallel Computation We model a coarse-grained parallel machine as follows: A coarse-grained machine consists of several relatively powerful processors connected by an interconnection network. Rather than making specific assumptions about the underlying network, we assume a two-level model of computation. The two-level model assumes a fixed cost for an off-processor access independent of the distance between the communicating processors. Communication between processors has a start-up overhead of - , while the data transfer rate is 1 - . For our complexity analysis we assume that - and - are constant and independent of the link congestion and distance between two processors. With new techniques, such as wormhole routing and randomized routing, the distance between communicating processors seems to be less of a determining factor on the amount of time needed to complete the communica- tion. Furthermore, the effect of link contention is eased due to the presence of virtual channels and the fact that link bandwidth is much higher than the bandwidth of node interface. This permits us to use the two-level model and view the underlying interconnection network as a virtual crossbar network connecting the processors. These assumptions closely model the behavior of the CM-5 on which our experimental results are presented. A discussion on other architectures is presented in Section 6. 2.2 Parallel Primitives In the following, we describe some important parallel primitives that are repeatedly used in our algorithms and implementations. We state the running time required for each of these primitives under our model of parallel computation. The analysis of the run times for the primitives described is fairly simple and is omitted in the interest of brevity. The interested reader is referred to [15]. In what follows, p refers to the number of processors. 1. Broadcast In a Broadcast operation, one processor has an element of data to be broadcast to all other processors. This operation can be performed in O((-) log p) time. 2. Combine Given an element of data on each processor and a binary associative and commutative op- eration, the Combine operation computes the result of combining the elements stored on all the processors using the operation and stores the result on every processor. This operation can also be performed in O((-) log p) time. 3. Parallel Prefix Suppose that x are p data elements with processor P i containing x i . Let\Omega be a binary associative operation. The Parallel Prefix operation stores the value of x on processor P i . This operation can be be performed in O((-) log p) time. 4. Gather Given an element of data on each processor, the Gather operation collects all the data and stores it in one of the processors. This can be accomplished in O(- log 5. Global Concatenate This is same as Gather except that the collected data should be stored on all the processors. This operation can also be performed in O(- log 6. Transportation Primitive The transportation primitive performs many-to-many personalized communication with possibly high variance in message size. If the total length of the messages being sent out or received at any processor is bounded by t, the time taken for the communication is 2-t (+ lower order terms) when t - O(p p-). If the outgoing and incoming traffic bounds are r and c instead, the communication takes time c) (+ lower order terms) when either 3 Parallel Algorithms for Selection Parallel algorithms for selection are also iterative and work by reducing the number of elements to be considered from iteration to iteration. The elements are distributed across processors and each iteration is performed in parallel by all the processors. Let n be the number of elements and p be the number of processors. To begin with, each processor is given d n Otherwise, this can be easily achieved by using one of the load balancing techniques to be described in Section 4. Let n (j) i be the number of elements in processor P i at the beginning of iteration j. Algorithm 1 Median of Medians selection algorithm Total number of elements Total number of processors labeled from 0 to List of elements on processor P i , where jL desired rank among the total elements On each processor P i Step 1. Use sequential selection to find median m i of list L i [l; r] 2. Step 3. On P0 Find median of M , say MoM , and broadcast it to all processors. Step 4. Partition L i into - MoM and ? MoM to give index i , the split index Step 5. count = Combine(index i , add) calculates the number of elements Step 6. If (rank - count ) else Step 7. LoadBalance(L Step 8. Step 9. On P0 Perform sequential selection to find element q of rank in L Figure 1: Median on Medians selection Algorithm . Let k (j) be the rank of the element we need to identify among these n (j) elements. We use this notation to describe all the selection algorithms presented in this paper. 3.1 Median of Medians Algorithm The median of medians algorithm is a straightforward parallelization of the deterministic sequential algorithm [8] and has recently been suggested and implemented by Bader et. al. [5]. This algorithm Figure load balancing at the beginning of each iteration. At the beginning of iteration j, each processor finds the median of its n (j) elements using the sequential deterministic algorithm. All such medians are gathered on one pro- cessor, which then finds the median of these medians. The median of medians is then estimated to be the median of all the n (j) elements. The estimated median is broadcast to all the processors. Each processor scans through its set of points and splits them into two subsets containing elements less than or equal to and greater than the estimated median, respectively. A Combine operation and a comparison with k (j) determines which of these two subsets to be discarded and the value of k (j+1) needed for the next iteration. Selecting the median of medians as the estimated median ensures that the estimated median will have at least a guaranteed fraction of the number of elements below it and at least a guaranteed fraction of the elements above it, just as in the sequential algorithm. This ensures that the worst case number of iterations required by the algorithm is O(log n). Let n (j) . Thus, finding the local median and splitting the set of points into two subsets based on the estimated median each requires O(n (j) in the j th iteration. The remaining work is one Gather, one Broadcast and one Combine operation. Therefore, the worst-case running time of this algorithm is log p ), the running time is O( n log n+ - log p log n+ -p log n). This algorithm requires the use of load balancing between iterations. With load balancing, . Assuming load balancing and ignoring the cost of load balancing itself, the running time of the algorithm reduces to P log 3.2 Bucket-Based Algorithm The bucket-based algorithm [17] attempts to reduce the worst-case running time of the above algorithm without requiring load balance. This algorithm is shown in Figure 2. First, in order to keep the algorithm deterministic without a balanced number of elements on each processor, the median of medians is replaced by the weighted median of medians. As before, local medians are computed on each processor. However, the estimated median is taken to be the weighted median of the local medians, with each median weighted by the number of elements on the corresponding processor. This will again guarantee that a fixed fraction of the elements is dropped from consideration every iteration. The number of iterations of the algorithm remains O(log n). The dominant computational work in the median of medians algorithm is the computation of the local median and scanning through the local elements to split them into two sets based on the estimated median. In order to reduce this work which is repeated every iteration, the bucket-based approach preprocesses the local data into O(log p) buckets such that for any 0 - every element in bucket i is smaller than any element in bucket j. This can be accomplished by finding the median of the local elements, splitting them into two buckets based on this median and recursively splitting each of these buckets into log pbuckets using the same procedure. Thus, preprocessing the local data into O(log p) buckets requires O( n log log p) time. Bucketing the data simplifies the task of finding the local median and the task of splitting the local data into two sets based on the estimated median. To find the local median, identify the bucket containing the median and find the rank of the median in the bucket containing the median Algorithm 2 Bucket-based selection algorithm Total number of elements Total number of processors labeled from 0 to List of elements on processor P i , where jL desired rank among the total elements On each processor P i Step 0. Partition L i on P i into log p buckets of equal size such that if r 2 bucket j , and s 2 bucketk , then r ! s if whilen ? C (a constant) Step 1. Find the bucket bktk containing the median element using a binary search on the remaining buckets. This is followed by finding the appropriate rank in bktk to find the median m i . Let N i be the number of remaining keys on P i . 2. Step 3. On P0 Find the weighted median of M , say WM and broadcast it. Step 4. Partition L i into - WM and ? WM using the buckets to give index i ; the split index Step 5. count = Combine(index i , add) calculates the number of elements less than WM Step 6. If (rank - count ) else Step 7. Step 8. On P0 Perform sequential selection to find element q of rank in L Figure 2: Bucket-based selection algorithm in O(log log p) time using binary search. The local median can be located in the bucket by the sequential selection algorithm in O( n time. The cost of finding the local median reduces from O( n log p ). To split the local data into two sets based on the estimated median, first identify the bucket that should contain the estimated median. Only the elements in this bucket need to be split. Thus, this operation also requires only O(log log log p time. After preprocessing, the worst-case run time for selection is O(log log p log log p log log p log n+ -p log n) = O( n log p log log log p. Therefore, the worst-case run time of the bucket-based approach is O( n log without any load balancing. Algorithm 3 Randomized selection algorithm Total number of elements Total number of processors labeled from 0 to List of elements on processor P i , where jL desired rank among the total elements On each processor P i whilen ? C (a constant) Step Step 1. Step 2. Generate a random number nr (same on all processors) between 0 and Step 3. On Pk where (nr Step 4. Partition L i into - mguess and ? mguess to give index i , the split index Step 5. count = Combine(index i , add) calculates the number of elements less than mguess Step 6. If (rank - count ) else Step 7. Step 8. On P0 Perform sequential selection to find element q of rank in L Figure 3: Randomized selection algorithm 3.3 Randomized Selection Algorithm The randomized median finding algorithm (Figure 3) is a straightforward parallelization of the randomized sequential algorithm described in [12]. All processors use the same random number generator with the same seed so that they can produce identical random numbers. Consider the behavior of the algorithm in iteration j. First, a parallel prefix operation is performed on the 's. All processors generate a random number between 1 and n (j) to pick an element at random, which is taken to be the estimate median. From the parallel prefix operation, each processor can determine if it has the estimated median and if so broadcasts it. Each processor scans through its set of points and splits them into two subsets containing elements less than or equal to and greater than the estimated median, respectively. A Combine operation and a comparison with k (j) determines which of these two subsets to be discarded and the value of k (j+1) needed for the next iteration. Since in each iteration approximately half the remaining points are discarded, the expected number of iterations is O(log n) [12]. Let n (j) . Thus, splitting the set of points into two subsets based on the median requires O(n (j) in the j th iteration. The remaining work is one Parallel Prefix, one Broadcast and one Combine operation. Therefore, the total expected running time of the algorithm is P log p ), the expected running time is O( n log n). In practice, one can expect that n (j) max reduces from iteration to iteration, perhaps by half. This is especially true if the data is randomly distributed to the processors, eliminating any order present in the input. In fact, by a load balancing operation at the end of every iteration, we can ensure that for every iteration j, n (j) . With load balancing and ignoring the cost of it, the running time of the algorithm reduces to P log log n). Even without this load balancing, assuming that the initial data is randomly distributed, the running time is expected to be O( n log n). 3.4 Fast Randomized Selection Algorithm The expected number of iterations required for the randomized median finding algorithm is O(log n). In this section we discuss an approach due to Rajasekharan et. al. [17] that requires only O(log log n) iterations for convergence with high probability (Figure 4). Suppose we want to find the k th smallest element among a given set of n elements. Sample a set S of o(n) keys at random and sort S. The element with rank e in S will have an expected rank of k in the set of all points. Identify two keys l 1 and l 2 in S with ranks ffi is a small integer such that with high probability the rank of l 1 is ! k and the rank of l 2 is ? k in the given set of points. With this, all the elements that are either ! l 1 or ? l 2 can be eliminated. Recursively find the element with rank in the remaining elements. If the number of elements is sufficiently small, they can be directly sorted to find the required element. If the ranks of l 1 and l 2 are both ! k or both ? k, the iteration is repeated with a different sample set. We make the following modification that may help improve the running time of the algorithm in practice. Suppose that the ranks of l 1 and l 2 are both ! k. Instead of repeating the iteration to find element of rank k among the n elements, we discard all the elements less than l 2 and find the element of rank in the remaining elements. If the ranks of l 1 and l 2 are both ? k, elements greater than l 1 can be discarded. Rajasekharan et. al. show that the expected number of iterations of this median finding algorithm is O(log log n) and that the expected number of points decreases geometrically after each iteration. If n (j) is the number of points at the start of the j th iteration, only a sample of o(n (j) ) keys is sorted. Thus, the cost of sorting, o(n (j) log n (j) ) is dominated by the O(n (j) ) work involved in scanning the points. Algorithm 4 Fast randomized selection algorithm Total number of elements Total number of processors labeled from 0 to List of elements on processor P i , where jL desired rank among the total elements On each processor P i whilen ? C (a constant) Step Step 1. Collect a sample S i from L i [l; r] by picking n i n ffl n elements at random on P i between l and r. Step 2. On P0 Step 3. Pick k1 , k2 from S with ranks d ijSj jSjlogne and d ijSj jSjlogne Step 4. Broadcast k1 and k2.The rank to be found will be in [k1 , k2 ] with high probability. Step 5. Partition L i between l and r into ! k1 , [k1 , k2 ] and ? k2 to give counts less, middle and high and splitters s1 and s2 . Step 6. Step 7. cless = Combine(less, add) Step 8. If (rank 2 (cless ; cmid ]) else else Step 9. Step 10. On P0 Perform sequential selection to find element q of rank in L Figure 4: Fast Randomized selection Algorithm In iteration j, Processor P (j) randomly selects n (j) n (j) of its n (j) elements. The selected elements are sorted using a parallel sorting algorithm. Once sorted, the processors containing the elements l (j)and l (j)broadcast them. Each processor finds the number of elements less than l (j)and greater than l (j) contained by it. Using Combine operations, the ranks of l (j) 1 and l (j) are computed and the appropriate action of discarding elements is undertaken by each processor. A large value of ffl increases the overhead due to sorting. A small value of ffl increases the probability that both the selected elements (l (j) 1 and l (j) lie on one side of the element with rank k (j) , thus causing an unsuccessful iteration. By experimentation, we found a value of 0:6 to be appropriate. As in the randomized median finding algorithm, one iteration of the median finding algorithm takes O(n (j) log log n iterations are required, median finding requires O( n log log n + (-) log p log log n) time. As before, we can do load balancing to ensure that n (j) reduces by half in every iteration. Assuming this and ignoring the cost of load balancing, the running time of median finding reduces to log log log log n). Even without this load balancing, the running time is expected to be O( n log log n). 4 Algorithms for load balancing In order to ensure that the computational load on each processor is approximately the same during every iteration of a selection algorithm, we need to dynamically redistribute the data such that every processor has nearly equal number of elements. We present three algorithms for performing such a load balancing. The algorithms can also be used in other problems that require dynamic redistribution of data and where there is no restriction on the assignment of data to processors. We use the following notation to describe the algorithms for load balancing: Initially, processor is the total number of elements on all the processors, i.e. 4.1 Order Maintaining Load Balance Suppose that each processor has its set of elements stored in an array. We can view the n elements as if they were globally sorted based on processor and array indices. For any i ! j, any element in processor P i appears earlier in this sorted order than any element in processor P j . The order maintaining load balance algorithm is a parallel prefix based algorithm that preserves this global order of data after load balancing. The algorithm first performs a Parallel Prefix operation to find the position of the elements it contains in the global order. The objective is to redistribute data such that processor P i contains Algorithm 5 Modified order maintaining load balance - Number of total elements Total number of processors labeled from 0 to List of elements on processor P i of size n i On each processor P i Step increment navg Step 1. Step 2. diff Step 3. If diff [j] is positive P j is labeled as a source. If diff [j] is negative P j is labeled as a sink. Step 4. If P i is a source calculate the prefix sum of the positive diff [ ] in an array p src, else calculate the prefix sums for sinks using negative diff [ ] in p snk. Step 5. l Step 7. Calculate the range of destination processors [P l ; Pr ] using a binary search on p snk. Step 8. while(l - r) elements to P l and increment l Step 5. l Step 7. Calculate the range of source processors [P l ; Pr ] using a binary search on src. Step 8. while( l - r) Receive elements from P l and increment l Figure 5: Modified order maintaining load balance the elements with positions n avg in the global order. Using the parallel prefix operation, each processor can figure out the processors to which it should send data and the amount of data to send to each processor. Similarly, each processor can figure out the amount of data it should receive, if any, from each processor. Communication is generated according to this and the data is redistributed. In our model of computation, the running time of this algorithm only depends on the maximum communication generated/received by a processor. The maximum number of messages sent out by a processor is d nmax navg e+1 and the maximum number of elements sent is n max . The maximum number of elements received by a processor is n avg . Therefore, the running time is O(- nmax The order maintaining load balance algorithm may generate much more communication than necessary. For example, consider the case where all processors have n avg elements except that P 0 has one element less and P p\Gamma1 has one element more than n avg . The optimal strategy is to transfer the one extra element from P p to P 0 . However, this algorithm transfers one element from P i to messages. Since preserving the order of data is not important for the selection algorithm, the following modification is done to the algorithm: Every processor retains minfn of its original elements. the processor has (n excess and is labeled a source. Otherwise, the processor needs (n avg \Gamma n i ) elements and is labeled a sink. The excessive elements in the source processors and the number of elements needed by the sink processors are ranked separately using two Parallel Prefix operations. The data is transferred from sources to sinks using a strategy similar to the order maintaining load balance algorithm. This algorithm (Figure 5), which we call modified order maintaining load balance algorithm (modified OMLB), is implemented in [5]. The maximum number of messages sent out by a processor in modified OMLB is O(p) and the maximum number of elements sent is (n The maximum number of elements received by a processor is n avg . The worst-case running time is O(-p 4.2 Dimension Exchange Method The dimension exchange method (Figure 6) is a load balancing technique originally proposed for hypercubes [11][21]. In each iteration of this method, processors are paired to balance the load locally among themselves which eventually leads to global load balance. The algorithm runs in log p iterations. In iteration i (0 processors that differ in the i th least significant bit position of their id's exchange and balance the load. After iteration i, for any 0 - processors have the same number of elements. In each iteration, ppairs of processors communicate in parallel. No processor communicates more than nmaxelements in an iteration. Therefore, the running time is O(- log However, since 2 j processors hold the maximum number of elements in iteration j, it is likely that either n max is small or far fewer elements than nmaxare communicated. Therefore, the running time in practice is expected to be much better than what is predicated by the worst-case. 4.3 Global Exchange This algorithm is similar to the modified order maintaining load balance algorithm except that processors with large amounts of data are directly paired with processor with small amounts of data to minimize the number of messages (Figure 7). As in the modified order maintaining load balance algorithm, every processor retains minfn of its original elements. If the processor has (n excess and is la- Algorithm 6 Dimension exchange method - Number of total elements Total number of processors labeled from 0 to List of elements on processor P i of size n i On each processor P i Step 1. P Step 2. Exchange the count of elements between P Step 3. Step 4. Send elements from L i [navg ] to processor P l Step 5. n else Step 4. Receive n l \Gamma navg elements from processor P l at Step 5. Increment n i by n l \Gamma navg Figure exchange method for load balancing beled a source. Otherwise, the processor needs (n avg \Gamma n i ) elements and is labeled a sink. All the source processors are sorted in non-increasing order of the number of excess elements each processor holds. Similarly, all the sink processors are sorted in non-increasing order of the number of elements each processor needs. The information on the number of excessive elements in each source processor is collected using a Global Concatenate operation. Each processor locally ranks the excessive elements using a prefix operation according to the order of the processors obtained by the sorting. Another Global Concatenate operation collects the number of elements needed by each sink processor. These elements are then ranked locally by each processor using a prefix operation performed using the ordering of the sink processors obtained by sorting. Using the results of the prefix operation, each source processor can find the sink processors to which its excessive elements should be sent and the number of element that should be sent to each such processor. The sink processors can similarly compute information on the number of elements to be received from each source processor. The data is transferred from sources to sinks. Since the sources containing large number of excessive elements send data to sinks requiring large number of elements, this may reduce the total number of messages sent. In the worst-case, there may be only one processor containing all the excessive elements and thus the total number of messages sent out by the algorithm is O(p). No processor will send more than data and the maximum number of elements received by any processor is n avg . The worst-case run time is O(-p Algorithm 7 Global Exchange load balance - Number of total elements Total number of processors labeled from 0 to List of elements on processor P i of size n i On each processor P i Step increment navg Step 1. for j /0 to Step 2. diff Step 3. If diff [j] is positive P j is labeled as a source. If diff [j] is negative P j is labeled as a sink. Step 4. For k 2 [0; sources in descending order maintaining appropriate processor indices. Also sort diff [k] for sinks in ascending order. Step 5. If P i is a source calculate the prefix sum of the positive diff [ ] in an array p src, else calculate the prefix sums for sinks using negative diff [ ] in p snk. Step 6. If P i is a source calculate the prefix sum of the positive diff [ ] in an array p src, else calculate the prefix sums for sinks using negative diff [ ] in p snk. Step 7. l 8. r Step 9. Calculate the range of destination processors [P l ; Pr ] using a binary search on p snk. Step 10. while(l - r) elements to P l and increment l Step 7. l 8. r Step 9. Calculate the range of source processors [P l ; Pr ] using a binary search on src. Step 10. while( l - r) Receive elements from P l and increment l Figure 7: Global exchange method for load balancing Selection Algorithm Run-time Median of Medians O( n Randomized O( n log n) Fast randomized O( n log log n) Table 1: The running times of various selection algorithm assuming but not including the cost of load balancing Selection Algorithm Run-time Median of Medians O( n Bucket-based O( n log Randomized O( n log log n) Fast randomized O( n log log n + (-) log p log log n) Table 2: The worst-case running times of various selection algorithms 5 Implementation Results The estimated running times of various selection algorithms are summarized in Table 1 and Table 2. Table 1 shows the estimated running times assuming that each processor contains approximately the same number of elements at the end of each iteration of the selection algorithm. This can be expected to hold for random data even without performing any load balancing and we also observe this experimentally. Table 2 shows the worst-case running times in the absence of load balancing. We have implemented all the selection algorithms and the load balancing techniques on the CM- 5. To experimentally evaluate the algorithms, we have chosen the problem of finding the median of a given set of numbers. We ran each selection algorithm without any load balancing and with each of the load balancing algorithms described (except for the bucket-based approach which does not use load balancing). We have run all the resulting algorithms on 32k, 64k, 128k, 256k, 512k, 1024k and 2048k numbers using 2, 4, 8, 16, 32, 64 and 128 processors. The algorithms are run until the total number of elements falls below p 2 , at which point the elements are gathered on one processor and the problem is solved by sequential selection. We found this to be appropriate by experimentation, to avoid the overhead of communication when each processor contains only a small number of elements. For each value of the total number of elements, we have run each of the algorithms on two types of inputs - random and sorted. In the random case, n p elements are randomly generated on each processor. To eliminate peculiar cases while using the random data, we ran each experiment on five different random sets of data and used the average running time. Random data sets constitute close to the best case input for the selection algorithms. In the sorted case, the n numbers are chosen to be the numbers containing the numbers i n The sorted input is a close to the worst-case input for the selection algorithms. For example, after the first iteration of a selection algorithm using this input, approximately half of the processors lose all their data while the other half retains all of their data. Without load balancing, the number of active processors is cut down by about half every iteration. The same is true even if modified order maintaining load balance and global exchange load balancing algorithms are used. After every iteration, about half the processors contain zero elements leading to severe load imbalance for the load balancing algorithm to rectify. Only some of the data we have collected is illustrated in order to save space. The execution times of the four different selection algorithms without using load balancing for random data (except for median of medians algorithm requiring load balancing for which global exchange is used) with 128k, 512k and 2048k numbers is shown in Figure 8. The graphs clearly demonstrate that all four selection algorithms scale well with the number of processors. An immediate observation is that the randomized algorithms are superior to the deterministic algorithms by an order of magnitude. For example, with the median of medians algorithm ran at least 16 times slower and the bucket-based selection algorithm ran at least 9 times slower than either of the randomized algorithms. Such an order of magnitude difference is uniformly observed even using any of the load balancing techniques and also in the case of sorted data. This is not surprising since the constants involved in the deterministic algorithms are higher due to recursively finding the estimated median. Among the deterministic algorithms, the bucket-based approach consistently performed better than the median of medians approach by about a factor of two for random data. For sorted data, the bucket-based approach which does not use any load balancing ran only about 25% slower than median of medians approach with load balancing. In each iteration of the parallel selection algorithm, each processor also performs a local selection algorithm. Thus the algorithm can be split into a parallel part where the processors combine the results of their local selections and a sequential part involving executing the sequential selection locally on each processor. In order to convince ourselves that randomized algorithms are superior in either part, we ran the following hybrid experiment. We ran both the deterministic parallel selection algorithms replacing the sequential selection parts by randomized sequential selection. The running time of the hybrid algorithms was in between the deterministic and randomized parallel selection algorithms. We made the following observation: The factor of improvement in randomized parallel selection algorithms over deterministic parallel selection is due to improvements in both the sequential and parallel parts. For large n, much of the improvement is due to the sequential part. For large p, the improvement is due to the parallel part. We conclude that randomized algorithms are faster in practice and drop the deterministic algorithms from further consideration. Time (in seconds) Number of Processors Median of Medians Bucket Based Randomized Fast Randomized0.0150.0250.0350.0450.0550.065 Time (in seconds) Number of Processors Randomized Fast Randomized0.51.52.53.5 Time (in seconds) Number of Processors Median of Medians Bucket Based Randomized Fast Randomized0.040.080.120.160.20.24 Time (in seconds) Number of Processors Randomized Fast Randomized261014 Time (in seconds) Number of Processors Median of Medians Bucket Based Randomized Fast Randomized0.10.30.50.7 Time (in seconds) Number of Processors Randomized Fast Randomized Figure 8: Performance of different selection algorithms without load balancing (except for median of medians selection algorithm for which global exchange is used) on random data sets. Time (in seconds) Number of Processors Random data, n=512k Balance Mod. order Maintaining Load Balance Dimension Exchange Global Exchange0.10.30.50.7 Time (in seconds) Number of Processors Random data, n=2M Balance Mod. order Maintaining Load Balance Dimension Exchange Global Exchange0.050.150.250.350.45 Time (in seconds) Number of Processors Sorted data, n=512k Balance Mod. order Maintaining Load Balance Dimension Exchange Global Exchange0.20.611.41.8 Time (in seconds) Number of Processors Sorted data, n=2m Balance Mod. order Maintaining Load Balance Dimension Exchange Global Exchange Figure 9: Performance of randomized selection algorithm with different load balancing strategies on random and sorted data sets. To facilitate an easier comparison of the two randomized algorithms, we show their performance separately in Figure 8. Fast randomized selection is asymptotically superior to randomized selection for worst-case data. For random data, the expected running times of randomized and fast randomized algorithms are O( n log n) and O( n log log n), respectively. Consider the effect of increasing n for a fixed p. Initially, the difference in log n and log log n is not significant enough to offset the overhead due to sorting in fast randomized selection and randomized selection performs better. As n is increased, fast randomized selection begins to outperform randomized selection. For large n, both the algorithms converge to the same execution time since the O( n dominates. Reversing this point view, we find that for any fixed n, as we increase p, randomized selection will eventually perform better and this can be readily observed in the graphs. The effect of the various load balancing techniques on the randomized algorithms for random data is shown in Figure 9 and Figure 10. The execution times are consistently better without using any load balancing than using any of the three load balancing techniques. Load balancing Time (in seconds) Number of Processors Random data, n=512k Balance Mod. order Maintaining Load Balance Dimension Exchange Global Exchange0.10.30.50.7 Time (in seconds) Number of Processors Random data, n=2m Balance Mod. order Maintaining Load Balance Dimension Exchange Global Time (in seconds) Number of Processors Sorted data, n=512k Balance Mod. order Maintaining Load Balance Dimension Exchange Global Time (in seconds) Number of Processors Sorted data, n=2m Balance Mod. order Maintaining Load Balance Dimension Exchange Global Exchange Figure 10: Performance of fast randomized selection algorithm with different load balancing strategies on random and sorted data sets. for random data almost always had a negative effect on the total execution time and this effect is more pronounced in randomized selection than in fast randomized selection. This is explained by the fact that fast randomized selection has fewer iterations (O(log log n) vs. O(log n)) and less data in each iteration. The observation that load balancing has a negative effect on the running time for random data can be easily explained: In load balancing, a processor with more elements sends some of its elements to another processor. The time taken to send the data is justified only if the time taken to process this data in future iterations is more than the time for sending it. Suppose that a processor sends m elements to another processor. The processing of this data involves scanning it in each iteration based on an estimated median and discarding part of the data. For random data, it is expected that half the data is discarded in every iteration. Thus, the estimated total time to process this data is O(m). The time for sending the data is (-m), which is also O(m). By observation, the constants involved are such that load balancing is taking more time than the reduction in running time caused by it. Time (in seconds) Number of Processors Comparing two randomized selection algorithms using sorted data for n=512k Randomized Fast Randomized0.20.611.41.8 Time (in seconds) Number of Processors Comparing two randomized selection algorithm using sorted datas for n=2M Randomized Fast Randomized Figure 11: Performance of the two randomized selection algorithms on sorted data sets using the best load balancing strategies for each algorithm \Gamma no load balancing for randomized selection and modified order maintaining load balancing for fast randomized selection. Consider the effect of the various load balancing techniques on the randomized algorithms for sorted data (see Figure 9 and Figure 10). Even in this case, the cost of load balancing more than offset the benefit of it for randomized selection. However, load balancing significantly improved the performance of fast randomized selection. In Figure 11, we see a comparison of the two randomized algorithms for sorted data with the best load balancing strategies for each algorithm \Gamma no load balancing for randomized selection and modified order maintaining load balancing for fast randomized algorithm (which performed slightly better than other strategies). We see that, for large n, fast randomized selection is superior. We also observe (see Figure 11 and Figure 8) that the fast randomized selection has better comparative advantage over randomized selection for sorted data. Finally, we consider the time spent in load balancing itself for the randomized algorithms on both random and sorted data (see Figure 12 and Figure 13). For both types of data inputs, fast randomized selection spends much less time than randomized selection in balancing the load. This is reflective of the number of times the load balancing algorithms are utilized (O(log log n) vs. O(log n)). Clearly, the cost of load balancing increases with the amount of imbalance and the number of processors. For random data, the overhead due to load balancing is quite tolerable for the range of n and p used in our experiments. For sorted data, a significant fraction of the execution time of randomized selection is spent in load balancing. Load balancing never improved the running time of randomized selection. Fast randomized selection benefited from load balancing for sorted data. The choice of the load balancing algorithm did not make a significant difference in the running time. Consider the variance in the running times between random and sorted data for both the Number of Processors0.10.30.50.7Time (in seconds) Randomized selection , random data load balancing time O Number of Processors0.20.61.01.41.8 Time (in seconds) Randomized selection , sorted data load balancing time O Figure 12: Performance of randomized selection algorithm with different load balancing strategies balancing (N), Order maintaining load balancing (O), Dimension exchange method (D) and Global exchange (G). Number of Processors0.10.30.5Time (in seconds) Fast Randomized selection , random data load balancing time O Number of Processors0.20.61.0Time (in seconds) Fast Randomized selection , sorted data load balancing time OD G Figure 13: Performance of fast randomized selection algorithm with different load balancing strategies balancing (N), Order maintaining load balancing (O), Dimension exchange method (D) and Global exchange (G). Primitive Two-level model Hypercube Mesh Broadcast O((-) log p) O((-) log p) O(- log Combine O((-) log p) O((-) log p) O(- log Parallel Prefix O((-) log p) O((-) log p) O(- log Gather O(- log Global Concatenate O(- log Transportation O(-p Table 3: Running time for basic communication primitives on meshes and hypercubes using cut-through routing. For the transportation primitive, t refers to the maximum of the total size of messages sent out or received by any processor. randomized algorithms. The randomized selection algorithm ran 2 to 2.5 times faster for random data than for sorted data (see Figure 12). Using any of the load balancing strategies, there is very little variance in the running time of fast randomized selection (Figure 13). The algorithm performs equally well on both best and worst-case data. For the case of 128 processors the stopping criterion results in execution of one iteration in most runs. Thus, load balancing has a detrimental effect on the overall cost. We had decided to choose the same stopping criterion to provide a fair comparison between the different algorithms. However, an appropriate fine tuning of this stopping criterion and corresponding increase in the number of iterations should provide time improvements with load balancing for 2M data size on 128 processors. 6 Selection on Meshes and Hypercubes Consider the analysis of the algorithms presented for cut-through routed hypercubes and square meshes with p processors. The running time of the various algorithms on meshes and hypercubes is easily obtained by substituting the corresponding running times for the basic parallel communication primitives used by the algorithms. Table 3 shows the time required for each parallel primitive on the two-level model of computation, a hypercube of p processors and a p p \Theta mesh. The analysis is omitted to save space and similar analysis can be found in [15]. Load balancing can be achieved by using the communication pattern of the transportation primitive [24] which involves two all-to-all personalized communications. Each processor has O( n elements to be sent out. The worst-case time of order maintaining load balance is O(-p for the hypercube and mesh respectively when n ? O(-p exchange load balancing algorithm on the hypercube has worst-case run time of O(- log p+- n log p) and on the mesh it is O(- log The global exchange load balancing algorithmm has the same time complexities as the modified order maintaining load balancing algorithm, both on the hypercube and the mesh. These costs must be added to the selection algorithms if analysis of the algorithms with load balancing is desired. From the table, the running times of all the primitives remain the same on a hypercube and hence the analysis and the experimental results obtained for the two-level model will be valid for hypercubes. Thus, the time complexity of all the selection algorithms is the same on the hypercube as the two-level model discussed in this paper. If the ratio of unit computation cost to the unit communication cost is large, i.e the processor is much faster than the underlying communication network, cost of load balancing will offset its advantages and fast randomized algorithm without load balancing will have superior performance for practical scenarios. Load balancing on the mesh results in asymptotically worse time requirements. We would expect load balancing to be useful for small number of processors. For large number of processors, even one step of load balancing would dominate the overall time and hence would not be effective. In the following we present results for the performance for best case and worst case data on a mesh. 1. Deterministic Algorithms: The communication primitives used in the deterministic selection algorithms are Gather, Broadcast and Combine. Even though Broadcast and Combine require more time than the two-level model, their cost is absorbed by the time required for the Gather operation which is identical on the mesh and the two-level model. Hence, the complexity of the deterministic algorithms on the mesh remains the same as the two-level model. The total time requirements for the median of medians algorithm are O( n log n) for the best case and O( n log n) for the worst case. The bucket-based deterministic algorithm runs in to O( n log time in the worst case without load balancing. 2. Randomized Algorithms: The communication for the the randomized algorithm includes one PrefixSum, one Broadcast and one Combine. The communication time on a mesh for one iteration of the randomized algorithm is O(- log p+- p p), making its overall time complexity O( n log n) for the best case and O( n log log n) for the worst case data. The fast randomized algorithm involves a parallel sort of the sample for which we use bitonic sort. A sample of n ffl (0 chosen from the n elements and sorted. On the mesh, sorting a sample of n ffl elements using bitonic sort takes O( log should be acceptably small to keep the sorting phase from dominating every iteration. The run-time of the fast randomized selection on the mesh is O( n for the best case data. For the worst case data the time requirement would be O( n log log p+ 7 Weighted selection having a corresponding weight w i attached to it. The problem of weighted selection is to find an element x i such that any x l As an example, the weighted median will be the element that divides the data set S with sum of weights W , into two sets S 1 and S 2 with approximately equal sum of weights. Simple modifications can be made to the deterministic algorithms to adapt them for weighted selection. In iteration j of the selection algorithms, a set S (j) of elements is split into two subsets S (j) 1 and S (j) 2 and a count of elements is used to choose the subset in which the desired element can be found. Weighted selection is performed as follows: First, the elements of S (j) are divided into two subsets S (j) 1 and S (j) 2 as in the selection algorithm. The sum of weights of all the elements in subset S (j) 1 is computed. Let k j be the weight metric in iteration j. If k j is greater than the sum of weights of S (j) 1 , the problem reduces to performing weighted selection with k (j). Otherwise, we need to perform weighted selection with k 1 . This method retains the property that a guaranteed fraction of elements will be discarded at each iteration keeping the worst case number of iterations to be O(log n). Therefore, both the median of medians selection algorithm and the bucket-based selection algorithm can be used for weighted selection without any change in their run time complexities. The randomized selection algorithm can also be modified in the same way. However, the same modification to the fast randomized selection will not work. This algorithm works by sorting a sample of the data set and picking up two elements that with high probability lie on either side of the element with rank k in sorted order. In weighted selection, the weights determine the position of the desired element in the sorted order. Thus, one may be tempted to select a sample of weights. However, this does not work since the weights of the elements should be considered in the order of the sorted data and a list of the elements sorted according to the weights does not make sense. Hence, randomized selection without load balancing is the best choice for parallel weighted selection. Conclusions In this paper, we have tried to identify the selection algorithms that are most suited for fast execution on coarse-grained distributed memory parallel computers. After surveying various algorithms, we have identified four algorithms and have described and analyzed them in detail. We also considered three load balancing strategies that can be used for balancing data during the execution of the selection algorithms. Based on the analysis and experimental results, we conclude that randomized algorithms are faster by an order of magnitude. If determinism is desired, the bucket-based approach is superior to the median of medians algorithm. Of the two randomized algorithms, fast randomized selection with load balancing delivers good performance for all types of input distributions with very little variation in the running time. The overhead of using load balancing with well-behaved data is insignificant. Any of the load balancing techniques described can be used without significant variation in the running time. Randomized selection performs well for well-behaved data. There is a large variation in the running time between best and worst-case data. Load balancing does not improve the performance of randomized selection irrespective of the input data distribution. 9 Acknowledgements We are grateful to Northeast Parallel Architectures Center and Minnesota Supercomputing Center for allowing us to use their CM-5. We would like to thank David Bader for providing us a copy of his paper and the corresponding code. --R Deterministic selection in O(log log N) parallel time The design and analysis of parallel algorithms Parallel selection in O(log log n) time using O(n An optimal algorithm for parallel selection Practical parallel algorithms for dynamic data redistribution Technical Report CMU-CS-90-190 Time bounds for selection A parallel median algorithm Introduction to algorithms. Dynamic load balancing for distributed memory multiprocessors Expected time bounds for selection Selection on the Reconfigurable Mesh An introduction to parallel algorithms Introduction to Parallel Computing: Design and Analysis of Algorithms Efficient computation on sparse interconnection networks Unifying themes for parallel selection Derivation of randomized sorting and selection algorithms Randomized parallel selection Programming a Hypercube Multicomputer Efficient parallel algorithms for selection and searching on sorted matrices Finding the median Random Data Accesses on a Coarse-grained Parallel Machine II Load balancing on a hypercube --TR --CTR Ibraheem Al-Furaih , Srinivas Aluru , Sanjay Goil , Sanjay Ranka, Parallel Construction of Multidimensional Binary Search Trees, IEEE Transactions on Parallel and Distributed Systems, v.11 n.2, p.136-148, February 2000 David A. Bader, An improved, randomized algorithm for parallel selection with an experimental study, Journal of Parallel and Distributed Computing, v.64 n.9, p.1051-1059, September 2004 Marc Daumas , Paraskevas Evripidou, Parallel Implementations of the Selection Problem: A Case Study, International Journal of Parallel Programming, v.28 n.1, p.103-131, February 2000

parallel algorithms;selection;parallel computers;coarse-grained;median finding;randomized algorithms;load balancing;meshes;hypercubes

262003

Parallel Incremental Graph Partitioning.

AbstractPartitioning graphs into equally large groups of nodes while minimizing the number of edges between different groups is an extremely important problem in parallel computing. For instance, efficiently parallelizing several scientific and engineering applications requires the partitioning of data or tasks among processors such that the computational load on each node is roughly the same, while communication is minimized. Obtaining exact solutions is computationally intractable, since graph partitioning is an NP-complete.For a large class of irregular and adaptive data parallel applications (such as adaptive graphs), the computational structure changes from one phase to another in an incremental fashion. In incremental graph-partitioning problems the partitioning of the graph needs to be updated as the graph changes over time; a small number of nodes or edges may be added or deleted at any given instant.In this paper, we use a linear programming-based method to solve the incremental graph-partitioning problem. All the steps used by our method are inherently parallel and hence our approach can be easily parallelized. By using an initial solution for the graph partitions derived from recursive spectral bisection-based methods, our methods can achieve repartitioning at considerably lower cost than can be obtained by applying recursive spectral bisection. Further, the quality of the partitioning achieved is comparable to that achieved by applying recursive spectral bisection to the incremental graphs from scratch.

Introduction Graph partitioning is a well-known problem for which fast solutions are extremely important in parallel computing and in research areas such as circuit partitioning for VLSI design. For instance, parallelization of many scientific and engineering problems requires partitioning data among the processors in such a fashion that the computation load on each node is balanced, while communication is minimized. This is a graph-partitioning problem, where nodes of the graph represent computational tasks, and edges describe the communication between tasks with each partition corresponding to one processor. Optimal partitioning would allow optimal parallelization of the computations with the load balanced over various processors and with minimized communication time. For many applications, the computational graph can be derived only at runtime and requires that graph partitioning also be done in parallel. Since graph partitioning is NP-complete, obtaining suboptimal solutions quickly is desirable and often satisfactory. For a large class of irregular and adaptive data parallel applications such as adaptive meshes [2], the computational structure changes from one phase to another in an incremental fashion. In incremental graph-partitioning problems, the partitioning of the graph needs to be updated as the graph changes over time; a small number of nodes or edges may be added or deleted at any given instant. A solution of the previous graph-partitioning problem can be utilized to partition the updated graph, such that the time required will be much less than the time required to reapply a partitioning algorithm to the entire updated graph. If the graph is not repartitioned, it may lead to imbalance in the time required for computation on each node and cause considerable deterioration in the overall performance. For many of these problems the graph may be modified after every few iterations (albeit incrementally), and so the remapping must have a lower cost relative to the computational cost of executing the few iterations for which the computational structure remains fixed. Unless this incremental partitioning can itself be performed in parallel, it may become a bottleneck. Several suboptimal methods have been suggested for finding good solutions to the graph-partitioning problem. For many applications, the computational graph is such that the vertices correspond to two- or three-dimensional coordinates and the interaction between computations is limited to vertices that are physically proximate. This information can be exploited to achieve the partitioning relatively quickly by clustering physically proximate points in two or three dimensions. Important heuristics include recursive coordinate bisection, inertial bisection, scattered decomposition, and index based partitioners [3, 6, 12, 11, 14, 16]. There are a number of methods which use explicit graph information to achieve partitioning. Important heuristics include simulated annealing, mean field annealing, recursive spectral bisection, recursive spectral multisection, mincut-based methods, and genetic algorithms [1, 4, 5, 7, 8, 9, 10, 13]. Since, the methods use explicit graph information, they have wider applicability and produce better quality partitioning. In this paper we develop methods which use explicit graph information to perform incremental graph- partitioning. Using recursive spectral bisection, which is regarded as one of the best-known methods for graph partitioning, our methods can partition the new graph at considerably lower cost. The quality of partitioning achieved is close to that achieved by applying recursive spectral bisection from scratch. Further, our algorithms are inherently parallel. The rest of the paper is outlined as follows. Section 2 defines the incremental graph-partitioning problem. Section 3 describes linear programming-based incremental graph partitioning. Section 4 describes a multilevel approach to solve the linear programming-based incremental graph partitioning. Experimental results of our methods on sample meshes are described in Section 5, and conclusions are given in Section 6. Problem definition Consider a graph represents a set of vertices, E represents a set of undirected edges, the number of vertices is given by and the number of edges is given by jEj. The graph-partitioning problem can be defined as an assignment scheme that maps vertices to partitions. We denote by B(q) the set of vertices assigned to a partition q, i.e., qg. The weight w i corresponds to the computation cost (or weight) of the vertex v i . The cost of an edge given by the amount of interaction between vertices v 1 and v 2 . The weight of every partition can be defined as The cost of all the outgoing edges from a partition represent the total amount of communication cost and is given by We would like to make an assignment such that the time spent by every node is minimized, i.e., represents the ratio of cost of unit computation/cost of unit communication on a machine. Assuming computational loads are nearly balanced (W (0) - W (1) - the second term needs to be minimized. In the literature P C(q) has also been used to represent the communication. Assume that a solution is available for a graph G(V; E) by using one of the many available methods in the literature, e.g., the mapping function M is available such that and the communication cost is close to optimal. Let G 0 be an incremental graph of G(V; E) i.e., some vertices are added and some vertices are deleted. Similarly, i.e., some edges are added and some are deleted. We would like to find a new mapping that the new partitioning is as load balanced as possible and the communication cost is minimized. The methods described in this paper assume that G 0 sufficiently similar to G(V; E) that this can be achieved, i.e., the number of vertices and edges added/deleted are a small fraction of the original number of vertices and edges. 3 Incremental partitioning In this section we formulate incremental graph partitioning in terms of linear programming. A high-level overview of the four phases of our incremental graph-partitioning algorithm is shown in Figure 1. Some notation is in order. Let 1. P be the number of partitions. 2. represent the set of vertices in partition i. 3. - represent the average load for each partition The four steps are described in detail in the following sections. 1: Assign the new vertices to one of the partitions (given by M 0 ). Step 2: Layer each partition to find the closest partition for each vertex (given by L 0 ). Step 3: Formulate the linear programming problem based on the mapping of Step 1 and balance loads (i.e., modify M 0 ) minimizing the total number of changes in M 0 . Step 4: Refine the mapping in Step 2 to reduce the communication cost. Figure 1: The different steps used in our incremental graph-partitioning algorithm. 3.1 Assigning an initial partition to the new nodes The first step of the algorithm is to assign an initial partition to the nodes of the new graph (given by simple method for initializing M 0 (V ) is given as follows. Let For all the vertices d(v; x) is the shortest distance in the graph G 0 For the examples considered in this paper we assume that G 0 is connected. If this is not the case, several other strategies can be used. is connected, this graph can be used instead of G for calculation of M 0 (V ). connected, then the new nodes that are not connected to any of the old nodes can be clustered together (into potentially disjoint clusters) and assigned to the partition that has the least number of vertices. For the rest of the paper we will assume that M 0 (v) can be calculated using the definition in (7), although the strategies developed in this paper are, in general, independent of this mapping. Further, for ease of presentation, we will assume that the edge and the vertex weights are of unit value. All of our algorithms can be easily modified if this is not the case. Figure 2 (a) describes the mapping of each the vertices of a graph. Figure (b) describes the mapping of the additional vertices using the above strategy. 3.2 Layering each partition The above mapping would ordinarily generate partitions of unequal size. We would like to move vertices from one partition to another to achieve load balancing, while keeping the communication cost as small as possible. This is achieved by making sure that the vertices transferred between two partitions are close to the boundary of the two partitions. We assign each vertex of a given partition to a different partition it is close to (ties are broken arbitrarily). (b) Figure 2: (a) Initial Graph (b) Incremental Graph (New vertices are shown by "*"). where x is such that min is is the shortest distance in the graph between v and x. A simple algorithm to perform the layering is given in Figure 3. It assumes the graph is connected. Let the number of such vertices of partition i that can be moved to partition j. For the example case of Figure 3, labels of all the vertices are given in Figure 4. A label 2 of vertex in partition 1 corresponds to the fact that this vertex belongs to the set that contributed to ff 12 . 3.3 Load balancing the number of vertices to be moved from partition i to partition j to achieve load balance. There are several ways of achieving load balancing. However, since one of our goals is to minimize communication cost, we would like to minimize l ij , because this would correspond to a minimization of the amount of vertex movement (or "deformity") in the original partitions. Thus the load-balancing step can be formally defined as the following linear programming problem. Minimize X subject to Constraint 12 corresponds to the load balance condition. The above formulation is based on the assumption that changes to the original graph are small and the initial partitioning is well balanced, hence moving the boundaries by a small amount will give balanced partitioning with low communication cost. f map[v[j]] represents the mapping of vertex j. g represents the j th element of the local adjacent list in partition i. g represents the starting address of vertex j in the local adjacent list of partition i. g i represents the set of vertices of partition i at a distance k from a node in partition j. f Neighbor i represents the set of partitions which have common boundaries with partition i. g For each partition i do For vertex do For do Count if l Add v[j] into S (tag;0) f where level := 0 repeat For do For vertex v[j] 2 S (k;level) do For l /\Gamma xadj i [v[j]] to xadj do count Add v[j] into tmpS level For vertex v[j] 2 tmpS do Add v[j] into S (tag;level) f where count i until For do 0-k!level Figure 3: Layering Algorithm (b) Figure 4: Labeling the nodes of a graph to the closest outside partition; (a) a microscopic view of the layering for a graph near the boundary of three partitions; (b) layering of the graph in Figure 2 (b); no edges are shown. Constraints in (11): l Constraints in (12): \Gammal \Gammal Solution using the Simplex Method l all other values are zero. Figure 5: Linear programming formulation and its solution, based on the mapping of the graph in Figure 2; (b) using the labeling information in Figure 4 (b). There are several approaches to solving the above linear programming problem. We decided to use the simplex method because it has been shown to work well in practice and because it can be easily parallelized. 1 The simplex formulation of the example in Figure 2 is given in Figure 5. The corresponding solution is l and l 1. The new partitioning is given in Figure 6. 20Initial partitions Incremental partitions Figure The new partition of the graph in Figure 2 (b) after the Load Balancing step. The above set of constraints may not have a feasible solution. One approach is to relax the constraint in (11) and not have l ij constraint. Clearly, this would achieve load balance but may lead to major modifications in the mapping. Another approach is to replace the constraint in (12) by Assuming would not achieve load balancing in one step, but several such steps can be applied to do so. If a feasible solution cannot be found with a reasonable value of \Delta (within an upper bound C), it would be better to start partitioning from scratch or solve the problem by adding only a fraction of the nodes at a given time, i.e., solve the problem in multiple stages. Typically, such cases arise when all the new nodes correspond to a few partitions and the amount of incremental change is greater than the size of one partition. 3.4 Refinement of partitions The formulation in the previous section achieves load balance but does not try explicitly to reduce the number of cross-edges. The minimization term in (10) and the constraint in (11) indirectly keep the cross-edges to a minimum under the assumption that the initial partition is good. In this section we describe a linear programming-based strategy to reduce the number of cross-edges, while still maintaining the load balance. This is achieved by finding all the vertices of partitions i on the boundary of partition i and j such that the cost of edges to the vertices in j are larger than the cost of edges to local vertices (Figure 7), i.e., the total cost of cross-edges will decrease by moving the vertex from partition i to j, which will affect the load We have used a dense version of simplex algorithm. The total time can potentially be reduced by using sparse representation. local non-local edge to partition non-local edge to partition = 3 (a) Figure 7: Choosing vertices for refinement. (a) Microscopic view of a vertex which can be moved from partition P i to P j , reducing the number of cross edges; (b) the set of vertices with the above property in the partition of Figure 6. balance. In the following a linear programming formulation is given that moves the vertices while keeping the load balance. mapping of each vertex after the load-balancing step. Let out(k; represent the number of edges of vertex k in partition M 00 (k) connected to partition j(j 6= M 00 (k)), and let represent the number of vertices a vertex k is connected to in partition M 00 (k). Let b ij represent the number of vertices in partition i which have more outgoing edges to partition j than local edges. We would like to maximize the number of vertices moved so that moving a vertex will not increase the cost of cross-edges. The inequality in the above definition can be changed to a strict inequality. We leave the equality, however, since by including such vertices the number of points that can be moved can be larger (because these vertices can be moved to satisfy load balance constraints without affecting the number of cross-edges). The refinement problem can now be posed as the following linear programming problem: Maximize X such that This refinement step can be applied iteratively until the effective gain by the movement of vertices is small. After a few steps, the inequalities (l ij need to be replaced by strict inequalities (l ij Constraint (15) l Load Balancing Constraint (16) \Gammal \Gammal Solution using Simplex Method l Figure 8: Formulation of the refinement step using linear programming and its solution. otherwise, vertices having an equal number of local and nonlocal vertices may move between boundaries without reducing the total cost. The simplex formulation of the example in Figure 6 is given in Figure 8, and the new partitioning after refinement is given in Figure 9. 20Incremental partitions Refined partitions Figure 9: The new partition of the graph in Figure 6 after the Refinement step. 3.5 Time complexity Let the number of vertices and the number of edges in a graph be given by V and E, respectively. The time for layering is O(V +E). Let the number of partitions be P and the number of edges in the partition graph 2 Each node of this graph represents a partition. An edge in the super graph is present whenever there are any cross edges from a node of one partition to a node of another partition. be R. The number of constraints and variables generated for linear programming are O(P +R) and O(2R), respectively. Thus the time required for the linear programming is O((P +R)R). Assuming R is O(P ), this reduces to The number of iterations required for linear programming is problem dependent. We will use f(P to denote the number of iterations. Thus the time required for the linear programming is O(P 2 f(P )). This gives the total time for repartitioning as O(E The parallel time is considerably more difficult to analyze. We will analyze the complexity of neglecting the setup overhead of coarse-grained machines. The parallel time complexity of the layering step depends on the maximum number of edges assigned to any processor. This could be approximated by O(E=P ) for each level, assuming the changes to the graph are incremental and that the graph is much larger than the number of processors. The parallelization of the linear programming requires a broadcast of length proportional to O(P ). Assuming that a broadcast of size P requires b(P ) amount of time on a parallel machine with P processors, the time complexity can be approximated by O( E 4 A multilevel approach For small graphs a large fraction of the total time spent in the algorithm described in the previous section will be on the linear programming formulation and its solution. Since the time required for one iteration of the linear programming formulation is proportional to the square of the number of partitions, it can be substantially reduced by using a multilevel approach. Consider the partitioning of an incremental graph for partitions. This can be completed in two stages: partitioning the graph into 4 super partitions and partitioning each of the 4 super partitions into 4 partitions each. Clearly, more than two stages can be used. The advantage of this algorithm is that the time required for applying linear programming to each stage would be much less than the time required for linear programming using only one stage. This is due to a substantial reduction in the number of variables as well as in the constraints, which are directly dependent on the number of of partitions. However, the mapping initialization and the layering needs to be performed from scratch for each level. Thus the decrease in cost of linear programming leads to a potential increase in the time spent in layering. The multilevel algorithm requires combining the partitions of the original graph into super partitions. For our implementations, recursive spectral bisection was used as an ab initio partitioning algorithm. Due to its recursive property it creates a natural hierarchy of partitions. Figure 10 shows a two-level hierarchy of partitions. After the linear programming-based algorithm has been applied for repartitioning a graph that has been adapted several times, it is possible that some of the partitions corresponding to a lower level subtree have a small number of boundary edges between them. Since the multilevel approach results in repartitioning with a small number of partitions at the lower levels, the linear programming formulations may produce infeasible solutions at the lower levels. This problem can be partially addressed by reconfiguring the partitioning hierarchy. A simple algorithm can be used to achieve reconfiguration. It tries to group proximate partitions to form a multilevel hierarchy. At each level it tries to combine two partitions into one larger parti- tion. Thus the number of partitions is reduced by a factor of two at every level by using a procedure FIND UNIQUE NEIGHBOR(P ) (Figure 11), which finds a unique neighbor for each partition such that the number of cross-edges between them is as large as possible. This is achieved by applying a simple heuristic Figure 12) that uses a list of all the partitions in a random order (each processor has a different order). If more than one processor is successful in generating a feasible solution, ties are broken based on the weight and the processor number. The result of the merging is broadcast to all the processors. In case none of the Figure 10: A two-level hierarchy of 16 partitions partitions g represents the number of edges from partition i to partition j. g global success := FALSE trial := 0 While (not global success) and (trial ! T ) do For each processor i do list of all partitions in a random order Weight := 0 FIND PAIR(success;Mark;Weight; Edge) global success := GLOBAL OR(success) if (not global success) then FIX PAIR(success;Mark;Weight; Edge) global success := GLOBAL OR(success) if (global success) then winner := FIND WINNER(success;Weight) f Return the processor number of maximum Weight g f Processor winner broadcast Mark to all the processors g return(global success) else trial := trial+1 Figure Reconstruction Algorithm FIND PAIR(success; Mark;W eight; Edge) success := TRUE for Find a neighbor k of j where (Mark[k] ! if k exists then Weight else success := FALSE FIX PAIR(success; Mark;W eight; Edge) success := TRUE While (j ! P ) and (success) do if a x exists such that (Mark[x] ! 0), (x is a neighbor of l), is a neighbor of Mark[x] := l Mark[l] := x Weight else success := FALSE else Figure 12: A high level description of the procedures used in FIND UNIQUE NEIGHBOR. processors are successful, another heuristic (Figure 12) is applied that tries to modify the partial assignments made by heuristic 1 to find a neighbor for each partition. If none of the processors are able to find a feasible solution, each processor starts with another random solution and the above step is iterated a constant number (L) times. 3 Figure 11 shows the partition reconfiguration for a simple example. If the reconfiguration algorithm fails, the multilevel algorithm can be applied with a lower number of levels (or only one level). Random_list Random_list Random_list Random_list (a) (b) (a) (d) (e) (f) Figure 13: A working example of the reconstruction algorithm. (a) Graph with 4 partitions; (b) partition (c) adjacency lists; (d) random order lists; (e) partition rearrangement; (f) processor 3 broadcasts the result to the other processors; (g) hierarchy after reconfiguration. 3 In practice, we found that the algorithm never requires more than one iteration. 4.1 Time complexity In the following we provide an analysis assuming that reconfiguration is not required. The complexity of reconfiguration will be discussed later. For the multilevel approach we assume that at each level the number of partitions done is equal and given by k. Thus the number of levels generated is log k P . The time required for layering increases to O(Elog k P ). The number of linear programming formulations can be given by O( P Thus the total time for linear programming can be given by O( P f(k)). The total time required for repartitioning is given by O(Elog k P value of k would minimize the sum of the cost of layering and the cost of the linear programming formulation. The choice of k also depends on the quality of partitioning achieved; increasing the number of layers would, in general, have a deteriorating effect on the quality of partitioning. Thus values of k have to be chosen based on the above tradeoffs. However, the analysis suggests that for reasonably sized graphs the layering time would dominate the total time. Since the layering time is bounded by O(ElogP ), this time is considerably lower than applying spectral bisection-based methods from scratch. Parallel time is considerably more difficult to analyze. The parallel time complexity of the layering step depends on the maximum number of edges any processor has for each level. This can be approximated by each level, assuming the changes to the graph are incremental and that the graph is much larger than the number of processors. As discussed earlier, the parallelization of linear programming requires a broadcast of length proportional to O(k). For small values of k, each linear programming formulation has to be executed on only one processor, else the communication will dominate the total time. Thus the parallel time is proportional to O( E The above analysis did not take reconfiguration into account. The cost of reconfiguration requires O(kd 2 ) time in parallel for every iteration, where d is the average number of partitions to which every partition is connected. The total time is O(kd 2 log P ) for the reconfiguration. This time should not dominate the total time required by the linear programming algorithm. 5 Experimental results In this section we present experimental results of the linear programming-based incremental partitioning methods presented in the previous section. We will use the term "incremental graph partitioner" (IGP) to refer to the linear programming based algorithm. All our experiments were conducted on the 32-node CM-5 available at NPAC at Syracuse University. Meshes We used two sets of adaptive meshes for our experiments. These meshes were generated using the DIME environment [15]. The initial mesh of Set A is given in Figure 14 (a). The other incremental meshes are generated by making refinements in a localized area of the initial mesh. These meshes represent a sequence of refinements in a localized area. The number of nodes in the meshes are 1071, 1096, 1121, 1152, and 1192, respectively. The partitioning of the initial mesh (1071 nodes) was determined using recursive spectral bisection. This was the partitioning used by algorithm IGP to determine the partitioning of the incremental mesh (1096 nodes). The repartitioning of the next set of refinement (1121, 1152, and 1192 nodes, respectively) was achieved using the partitioning obtained by using the IGP for the previous mesh in the sequence. These meshes are used to test whether IGP is suitable for repartitioning a mesh after several refinements. Figure 14: Test graphs set A (a) an irregular graph with 1071 nodes and 3185 edges; (b) graph in (a) with additional nodes; (c) graph in (b) with 25 additional nodes; (d) graph in (c) with 31 additional nodes; graph in (d) with 40 additional nodes. Figure 15: Test graphs Set B (a) a mesh with 10166 nodes and 30471 edges; (b) mesh a with 48 additional nodes; (c) mesh a with 139 additional nodes; (d) mesh a with 229 additional nodes; (e) mesh a with 672 Results Initial Graph - Figure 14 (a) Total Cutset Max Cutset Min Cutset Figure 14 (b) Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 31.71 - 733 56 33 IGP 14.75 0.68 747 55 34 IGP with Refinement 16.87 0.88 730 54 34 Figure 14 (c) Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 34.05 - 732 56 34 IGP 13.63 0.73 752 54 33 IGP with Refinement 16.42 1.05 727 54 33 Figure 14 (d) Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 34.96 - 716 57 34 IGP 15.89 0.92 757 56 33 IGP with Refinement 18.32 1.28 741 56 33 Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 38.20 - 774 63 34 IGP 15.69 0.94 815 63 34 IGP with Refinement 18.43 1.26 779 59 34 Time unit in seconds. p - parallel timing on a 32-node CM-5. s - timing on 1-node CM-5. Figure Incremental graph partitioning using linear programming and its comparison with spectral bisection from scratch for meshes in Figure 14 (Set A). The next data set (Set B) corresponds to highly irregular meshes with 10166 nodes and 30471 edges. This data set was generated to study the effect of different amounts of new data added to the original mesh. Figures 17 (b), 17 (c), 17 (d), and 17 (e) correspond to meshes with 68, 139, 229, and 672 additional nodes over the mesh in Figure 15. The results of the one-level IGP for Set A meshes are presented in Figure 16. The results show that, even after multiple refinements, the quality of partitioning achieved is comparable to that achieved by recursive spectral bisection from scratch, thus this method can be used for repartitioning several stages. The time required by repartitioning is about half the time required for partitioning using RSB. The algorithm provides speedup of around 15 to 20 on a 32-node CM-5. Most of the time spent by our algorithm is in the solution of the linear programming formulation using the simplex method. The number of variables and constraints generated by the one-level linear programming algorithm for the load-balancing step for meshes in Figure partitions are 188 and 126, respectively. For the multilevel approach, the linear programming formulation for each subproblem at a given level Initial Graph - Figure 15 (a) Total Cutset Max Cutset Min Cutset (b) Initial assignment by IGP using the partition of Figure 15 (a') Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 800.05 - 2137 178 90 IGP before Refinement 13.90 1.01 2139 186 84 IGP after Refinement 24.07 1.83 2040 172 82 (c) Initial assignment by IGP using the partition of Figure 15 (a') Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 814.36 - 2099 166 87 IGP before Refinement 18.89 1.08 2295 219 93 IGP after Refinement 29.33 2.01 2162 206 85 (d) Initial assignment by IGP using the partition of Figure 15 (a') Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 853.35 - 2057 169 94 IGP before Refinement (2) 35.98 2.08 2418 256 92 IGP after Refinement 43.86 2.76 2139 190 85 Initial assignment by IGP using the partition of Figure 15 (a') Partitioner Time-s Time-p Total Cutset Max Cutset Min Cutset Spectral Bisection 904.81 - 2158 158 94 IGP before Refinement (3) 76.78 3.66 2572 301 102 IGP after Refinement 89.48 4.39 2270 237 96 Time unit in seconds. p - parallel timing on a 32-node CM-5. s - timing on 1-node CM-5. Figure 17: Incremental graph partitioning using linear programming and its comparison with spectral bisection from scratch for meshes in Figure 15 (Set B). was solved by assigning a subset of processors. Table 19 gives the time required for different algorithms and the quality of partitioning achieved for different numbers of levels. A 4 \Theta 4 \Theta 2-based repartitioning implies that the repartitioning was performed in three stages with decomposition into 4, 4, 2 partitions, respectively. The results are presented in Figure 19. The solution qualities of multilevel algorithms show an insignificant deterioration in number of cross edges and a considerable reduction in total time. The partitioning achieved by algorithm IGP for Set B meshes in Figure 18 using the partition of mesh in Figure 15 (a) is given in Figure 17. The number of stages required (by choosing an appropriate value of \Delta, as described in section 2.3) were 1, 1, 2, and 3, respectively. 4 It is worth noting that, although the load imbalance created by the additional nodes was severe, the quality of partitioning achieved for each case was close to that of applying recursive spectral bisection from scratch. Further, the sequential time is at least an order of magnitude better than that of recursive spectral bisection. The CM-5 implementation improved the time required by a factor of 15 to 20. The time required for repartitioning Figure 17 (b) and Figure 17 (c) is close to that required for meshes in Figure 14. The timings for meshes in Figure 17 (d) and 17 (e) are larger because they use multiple stages. The time can be reduced by using a multilevel approach (Figure 20). However, the time reduction is relatively small (from 24.07 seconds to 6.70 seconds for a two-level approach). Increasing the number of levels increases the total time as the cost of layering increases. The time reduction for the last mesh (10838 nodes) is largely due to the reduction of the number of stages used in the multilevel algorithm (Section 3.3). For almost all cases a speedup of 15 to 25 was achieved on a 32-node CM-5. Figure 21 and Figure 22 show the detailed timing for different steps for the mesh in Figure 14 (d) and mesh in Figure 15 (b) of the sequential and parallel versions of the repartitioning algorithm, respectively. Clearly, the time spent in reconfiguration is negligible compared to the total execution time. Also, the time spent for linear programming in a multilevel algorithm is much less than that in a single-level algorithm. The results also show that the time for the linear programming remains approximately the same for both meshes, while the time for layering is proportionally larger. For the multilevel parallel algorithm, the time for layering is comparable with the time spent on linear programming for the smaller mesh, while it dominates the time for the larger mesh. Since the layering term is O(levels E results support the complexity analysis in the previous section. The time spent on reconfiguration is negligible compared to the total time. 6 Conclusions In this paper we have presented novel linear programming-based formulations for solving incremental graph-partitioning problems. The quality of partitioning produced by our methods is close to that achieved by applying the best partitioning methods from scratch. Further, the time needed is a small fraction of the latter and our algorithms are inherently parallel. We believe the methods described in this paper are of critical importance in the parallelization of adaptive and incremental problems. 4 The number of stages chosen were by trial and error, but can be determined by the load imbalance. (b Figure Partitions using RSB; (b 0 ) partitions using IGP starting from a using IGP starting from a 0 ; (d 0 ) partitions using IGP starting from a 0 ; using IGP starting from a 0 . Graph Level Description Time-s Time-p Total Cutset Time unit in seconds on CM-5. Figure 19: Incremental multilevel graph partitioning using linear programming and its comparison with single-level graph partitioning for the sequence of graphs in Figure 14. Graph Level Description Time-s Time-p Total Cutset Time unit in seconds on CM-5. Figure 20: Incremental multilevel graph partitioning using linear programming and its comparison with single-level graph partitioning for the sequence of meshes in Figure 15. in Figure 14 (d) Level Reconfiguration Layering Linear programming Total Figure 15 (b) Level Reconfiguration Layering Linear programming Total Time in seconds Balancing. R - Refinement. T - Total. Figure 21: Time required for different steps in the sequential repartitioning algorithm. in Figure 14 (d) Level Reconfiguration Layering Linear programming Data movement Total Figure 15 (b) Level Reconfiguration Layering Linear programming Data movement Total Time in seconds Balancing. R - Refinement. T - Total. Figure 22: Time required for different steps in the parallel repartitioning algorithm (on a 32-node CM-5). --R Solving Problems on Concurrent Processors Software Support for Irregular and Loosely Synchronous Problems. Heuristic Approaches to Task Allocation for Parallel Computing. Load Balancing Loosely Synchronous Problems with a Neural Network. Solving Problems on Concurrent Processors Graphical Approach to Load Balancing and Sparse Matrix Vector Multiplication on the Hypercube. An Improved Spectral Graph Partitioning Algorithm for Mapping Parallel Computations. Multidimensional Spectral Load Balancing. Genetic Algorithms for Graph Partitioning and Incremental Graph Partitioning. Physical Optimization Algorithms for Mapping Data to Distributed-Memory Multi- processors Solving Finite Element Equations on Current Computers. Fast Mapping And Remapping Algorithm For Irregular and Adaptive Problems. Partitioning Sparse Matrices with Eigenvectors of Graphs. Partitioning of Unstructured Mesh Problems for Parallel Processing. DIME: Distributed Irregular Mesh Enviroment. Performance of Dynamic Load-Balancing Algorithm for Unstructured Mesh Calcula- tions --TR --CTR Sung-Ho Woo , Sung-Bong Yang, An improved network clustering method for I/O-efficient query processing, Proceedings of the 8th ACM international symposium on Advances in geographic information systems, p.62-68, November 06-11, 2000, Washington, D.C., United States Mark J. Clement , Glenn M. Judd , Bryan S. Morse , J. Kelly Flanagan, Performance Surface Prediction for WAN-Based Clusters, The Journal of Supercomputing, v.13 n.3, p.267-281, May 1999 Don-Lin Yang , Yeh-Ching Chung , Chih-Chang Chen , Ching-Jung Liao, A Dynamic Diffusion Optimization Method for Irregular Finite Element Graph Partitioning, The Journal of Supercomputing, v.17 n.1, p.91-110, Aug. 2000 Ching-Jung Liao , Yeh-Ching Chung, Tree-Based Parallel Load-Balancing Methods for Solution-Adaptive Finite Element Graphs on Distributed Memory Multicomputers, IEEE Transactions on Parallel and Distributed Systems, v.10 n.4, p.360-370, April 1999 John C. S. Lui , M. F. Chan, An Efficient Partitioning Algorithm for Distributed Virtual Environment Systems, IEEE Transactions on Parallel and Distributed Systems, v.13 n.3, p.193-211, March 2002 Yeh-Ching Chung , Ching-Jung Liao , Don-Lin Yang, A Prefix Code Matching Parallel Load-Balancing Method for Solution-Adaptive Unstructured Finite Element Graphs on Distributed Memory Multicomputers, The Journal of Supercomputing, v.15 n.1, p.25-49, Jan. 2000 Umit Catalyurek , Cevdet Aykanat, A hypergraph-partitioning approach for coarse-grain decomposition, Proceedings of the 2001 ACM/IEEE conference on Supercomputing (CDROM), p.28-28, November 10-16, 2001, Denver, Colorado Umit Catalyurek , Cevdet Aykanat, Hypergraph-Partitioning-Based Decomposition for Parallel Sparse-Matrix Vector Multiplication, IEEE Transactions on Parallel and Distributed Systems, v.10 n.7, p.673-693, July 1999 Cevdet Aykanat , B. Barla Cambazoglu , Ferit Findik , Tahsin Kurc, Adaptive decomposition and remapping algorithms for object-space-parallel direct volume rendering of unstructured grids, Journal of Parallel and Distributed Computing, v.67 n.1, p.77-99, January, 2007 Y. F. Hu , R. J. Blake, Load balancing for unstructured mesh applications, Progress in computer research, Nova Science Publishers, Inc., Commack, NY, 2001

remapping;mapping;refinement;parallel;linear-programming

262369

Computing Accumulated Delays in Real-time Systems.

We present a verification algorithm for duration properties of real-time systems. While simple real-time properties constrain the total elapsed time between events, duration properties constrain the accumulated satisfaction time of state predicates. We formalize the concept of durations by introducing duration measures for timed automata. A duration measure assigns to each finite run of a timed automaton a real number the duration of the run which may be the accumulated satisfaction time of a state predicate along the run. Given a timed automaton with a duration measure, an initial and a final state, and an arithmetic constraint, the duration-bounded reachability problem asks if there is a run of the automaton from the initial state to the final state such that the duration of the run satisfies the constraint. Our main result is an (optimal) PSPACE decision procedure for the duration-bounded reachability problem.

Introduction Over the past decade, model checking [CE81, QS81] has emerged as a powerful tool for the automatic verification of finite-state systems. Recently the model-checking paradigm has been extended to real-time systems [ACD93, HNSY94, AFH96]. Thus, given the description of a finite-state system together with its timing assumptions, there are algorithms that test whether the system satisfies a specification written in a real-time temporal logic. A typical property that can be specified in real-time temporal logics is the following time-bounded causality property: A response is obtained whenever a ringer has been pressed continuously for 2 seconds. Standard real-time temporal logics [AH92], however, have limited expressiveness and cannot specify some timing properties we may want to verify of a given system. In particular, they do not allow us to constrain the accumulated satisfaction times of state predicates. As an example, consider the following duration-bounded causality property: A response is obtained whenever a ringer has been pressed, possibly intermittently, for a total duration of 2 seconds. ( ) A preliminary version of this paper appeared in the Proceedings of the Fifth International Conference on Computer-Aided Verification (CAV 93), Springer-Verlag LNCS 818, pp. 181-193, 1993. y Bell Laboratories, Murray Hill, New Jersey, U.S.A. z Department of Computer Science, University of Crete, and Institute of Computer Science, FORTH, Greece. Partially supported by the BRA ESPRIT project REACT. x Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, U.S.A. Partially supported by the ONR YIP award N00014-95-1-0520, by the NSF CAREER award CCR-9501708, by the NSF grants CCR-9200794 and CCR-9504469, by the AFOSR contract F49620-93-1-0056, and by the ARPA grant NAG2-892. To specify this duration property, we need to measure the accumulated time spent in the state that models "the ringer is pressed." For this purpose, the concept of duration operators on state predicates is introduced in the Calculus of Durations [CHR91]. There, an axiom system is given for proving duration properties of real-time systems. Here we address the algorithmic verification problem for duration properties of real-time sys- tems. We use the formalism of timed automata [AD94] for representing real-time systems. A timed automaton operates with a finite control and a finite number of fictitious time gauges called clocks, which allow the annotation of the control graph with timing constraints. The state of a timed automaton includes, apart from the location of the control, also the real-numbered values of all clocks. Consequently, the state space of a timed automaton is infinite, and this complicates its analysis. The basic question about a timed automaton is the following time-bounded reachability problem: Given an initial state oe, a final state - , and an interval I , is there a run of the automaton starting in state oe and ending in state - such that the total elapsed time of the run is in the interval I? (y) The solution to this problem relies on a partition of the infinite state space into finitely many regions, which are connected with transition and time edges to form the region graph of the timed automaton [AD94]. The states within a region are equivalent with respect to many standard questions. In particular, the region graph can be used for testing the emptiness of a timed automaton [AD94], for checking time-bounded branching properties [ACD93], for testing the bisimilarity of states [Cer92], and for computing lower and upper bounds on time delays [CY91]. Unfortunately, the region graph is not adequate for checking duration properties such as the duration-bounded causality property ( ); that is, of two runs that start in different states within the same region, one may satisfy the duration-bounded causality property, whereas the other one does not. Hence a new technique is needed for analyzing duration properties. To introduce the concept of durations in a timed automaton, we associate with every finite run a nonnegative real number, which is called the duration of the run. The duration of a run is defined inductively using a duration measure, which is a function that maps the control locations to nonnegative integers: the duration of an empty run is 0; and the duration measure of a location gives the rate at which the duration of a run increases while the automaton control resides in that location. For example, a duration measure of 0 means that the duration of the run stays unchanged (i.e., the time spent in the location is not accumulated), a duration measure of 1 means that the duration of the run increases at the rate of time (i.e., the time spent in the location is accumulated), and a duration measure of 2 means that the duration of the run increases at twice the rate of time. The time-bounded reachability problem (y) can now be generalized to the duration-bounded reachability problem: Given an initial state oe, a final state - , a duration measure, and an interval I , is there a run of the automaton starting in state oe and ending in state - such that the duration of the run is in the interval I? We show that the duration-bounded reachability problem is Pspace-complete, and we provide an optimal solution. Our algorithm can be used to verify duration properties of real-time systems that are modeled as timed automata, such as the duration-bounded causality property ( ). Let us briefly outline our construction. Given a region R, a final state - , and a path in the region graph from R to - , we show that the lower and upper bounds on the durations of all runs that start at some state in R and follow the chosen path can be written as linear expressions over the variables that represent the clock values of the start state. In a first step, we provide a recipe for computing these so-called bound expressions. In the next step, we define an infinite graph, the bounds graph, whose vertices are regions tagged with bound expressions that specify the set of possible durations for any path to the final state. In the final step, we show that the infinite bounds graph can be collapsed into a finite graph for solving the duration-bounded reachability problem. 2 The Duration-bounded Reachability Problem Timed automata Timed automata are a formal model for real-time systems [Dil89, AD94]. Each automaton has a finite set of control locations and a finite set of real-valued clocks. All clocks proceed at the same rate, and thus each clock measures the amount of time that has elapsed since it was started. A transition of a timed automaton can be taken only if the current clock values satisfy the constraint that is associated with the transition. When taken, the transition changes the control location of the automaton and restarts one of the clocks. Formally, a timed automaton A is a triple (S; X; E) with the following components: ffl S is a finite set of locations ; ffl X is a finite set of clocks ; ffl E is a finite set of transitions of the form (s; t; '; x), for a source location s 2 S, a target location t 2 S, a clock constraint ', and a clock x 2 X . Each clock constraint is a positive boolean combination of atomic formulas of the form y - k or y ! k or k - y or k ! y, for a clock y 2 X and a nonnegative integer constant k 2 N. A configuration of the timed automaton A is fully described by specifying the location of the control and the values of all clocks. A clock valuation c 2 R X is an assignment of nonnegative reals to the clocks in X . A state oe of A is a pair (s; c) consisting of a location s 2 S and a clock valuation c. We write \Sigma for the (infinite) set of states of A. As time elapses, the values of all clocks increase uniformly with time, thereby changing the state of A. Thus, if the state of A is (s; c), then after time assuming that no transition occurs, the state of A is (s; c is the clock valuation that assigns c(x) + ffi to each clock x. The state of A may also change because of a transition (s; t; '; x) in E. Such a transition can be taken only in a state whose location is s and whose clock valuation satisfies the constraint '. The transition is instantaneous. After the transition, the automaton is in a state with location t and the new clock valuation is c[x := 0]; that is, the clock x associated with the transition is reset to the value 0, and all other clocks remain unchanged. The possible behaviors of the timed automaton A are defined through a successor relation on the states of A: Transition successor For all states (s; c) 2 \Sigma and transitions (s; t; '; x) 2 E, if c satisfies ', then (s; c) 0 Time successor For all states (s; c) 2 \Sigma and time increments A state (t; d) is a successor of the state (s; c), written (s; c) ! (t; d), iff there exists a nonnegative real ffi such that (s; c) ffi d). The successor relation defines an infinite graph K(A) on the state space \Sigma of A. The transitive closure ! of the successor relation ! is called the reachability relation of A. s (y - 2; y) Figure 1: Sample timed automaton Example 1 A sample timed automaton is shown in Figure 1. The automaton has four locations and two clocks. Each edge is labeled with a clock constraint and the clock to be reset. A state of the automaton contains a location and real-numbered values for the clocks x and y. Some sample pairs in the successor relation are shown below: (s; 0; 0):Depending on the application, a timed automaton may be augmented with additional components such as initial locations, accepting locations, transition labels for synchronization with other timed automata, and atomic propositions as location labels. It is also useful to label each location with a clock constraint that limits the amount of time spent in that location [HNSY94]. We have chosen a very simple definition of timed automata to illustrate the essential computational aspects of solving reachability problems. Also, the standard definition of a timed automaton allows a (possibly empty) set of clocks to be reset with each transition. Our requirement that precisely one clock is reset with each transition does not affect the expressiveness of timed automata. Clock regions and the region graph Let us review the standard method for analyzing timed automata. The key to solving many verification problems for a timed automaton is the construction of the so-called region graph [AD94]. The region graph of a timed automaton is a finite quotient of the infinite state graph that retains enough information for answering certain reachability questions. Suppose that we are given a timed automaton A and an equivalence relation - = on the states \Sigma of A. For oe 2 \Sigma, we write for the equivalence class of states that contains the state oe. The successor relation ! is extended to -equivalence classes as follows: define there is a state oe 0 2 nonnegative real ffi such that oe 0 ffi nonnegative reals " ! ffi, we have (oe The quotient graph of A with respect to the equivalence relation - =, written is a graph whose vertices are the -equivalence classes and whose edges are given by the successor relation ! . The equivalence relation - = is stable iff there is a state - 0 2 is back stable iff whenever oe ! - , then for all states - there is a state oe 0 2 . The quotient graph with respect to a (back) stable equivalence relation can be used for solving the reachability problem between equivalence classes: given two -equivalence classes R 0 and R f , is there a state oe 2 R 0 and a state - 2 R f such that oe ! -? If the equivalence relation is (back) stable, then the answer to the reachability problem is affirmative iff there is a path from R 0 to R f in the quotient graph The region graph of the timed automaton A is a quotient graph of A with respect to the equivalence relation defined below. For x 2 X , let m x be the largest constant that the clock x is compared to in any clock constraint of A. For denote the integral part of ffi, and let - denote the fractional part of ffi (thus, We freely use constraints like - for a clock x and a nonnegative integer constant k (e.g., a clock valuation c satisfies the constraint bxc - k iff bc(x)c - k). Two states (s; c) and (t; d) of A are region-equivalent, written (s; c) - (t; d), iff the following four conditions hold: 1. 2. for each clock x 2 X , either 3. for all clocks x; y 2 X , the valuation c satisfies - the valuation d satisfies - 4. for each clock x 2 X , the valuation c satisfies - the valuation d satisfies - A (clock) region R ' \Sigma is a -equivalence class of states. Hence, a region is fully specified by a location, the integral parts of all clock values, and the ordering of the fractional parts of the clock values. For instance, if X contains three clocks, x, y, and z, then the region contains all states (s; c) such that c satisfies z ! 1. For the region R, we write [s; z], and we say that R has the location s and satisfies the constraints etc. There are only finitely many regions, because the exact value of the integral part of a clock x is recorded only if it is smaller than . The number of regions is bounded by jSj is the number of clocks. The region graph R(A) of the timed automaton A is the (finite) quotient graph of A with respect to the region equivalence relation -. The region equivalence relation - is stable as well as back-stable [AD94]. Hence the region graph can be used for solving reachability problems between regions, and also for solving time-bounded reachability problems [ACD93]. It is useful to define the edges of the region graph explicitly. A region R is a boundary region iff there is some clock x such that R satisfies the constraint - region that is not a boundary region is called an open region. For a boundary region R, we define its predecessor region pred(R) to be the open region Q such that for all states (s; c) 2 Q, there is a time increment ffi 2 R such that (s; c and for all nonnegative reals Similarly, we define the successor region succ(R) of R to be the open region Q 0 such that for all states (s; c) 2 Q there is a time increment ffi 2 R such that (s; and for all nonnegative reals we have (s; . The state of a timed automaton belongs to a boundary region R only instantaneously. Just before that instant the state belongs to pred(R), and just after that instant the state belongs to succ(R). For example, for the boundary region R given by pred(R) is the open region and succ(R) is the open region z]: The edges of the region graph R(A) fall into two categories: Transition edges If (s; c) 0 then there is an edge from the region [s; c] - to the region [t; d] - . Time edges For each boundary region R, there is an edge from pred(R) to R, and an edge from R to succ(R). In addition, each region has a self-loop, which can be ignored for solving reachability problems. Duration measures and duration-bounded reachability A duration measure for the timed automaton A is a function p from the locations of A to the nonnegative integers. A duration constraint for A is of the form R is a duration measure for A and I is a bounded interval of the nonnegative real line whose endpoints are integers may be open, half-open, or closed). Let p be a duration measure for A. We extend the state space of A to evaluate the integral R along the runs of A. An extended state of A is a pair (oe; ") consisting of a state oe of A and a nonnegative real number ". The successor relation on states is extended as follows: Transition successor For all extended states (s; c; ") and all transitions (s; t; '; x) such that c satisfies ', define (s; c; '') 0 Time successor For all extended states (s; c; ") and all time increments (s; c; ") ffi We consider the duration-bounded reachability problem between regions: given two regions R 0 and R f of a timed automaton A, and a duration constraint R I for A, is there a state oe 2 R 0 , a state nonnegative real I such that (oe; We refer to this duration-bounded reachability problem using the tuple R Example 2 Recall the sample timed automaton from Figure 1. Suppose that the duration measure p is defined by 1. Let the initial region R 0 be the singleton let the final region R f be f(s; 0)g. For the duration constraint R the answer to the duration-bounded reachability problem is in the affirmative, and the following sequence of successor pairs is a justification (the last component denotes the value of the integral R (s; 0; 0; On the other hand, for the duration constraint R the answer to the duration-bounded reachability problem is negative. The reader can verify that if (s; 0; 0; 2. 2 If the duration measure p is the constant function 1 (i.e., locations s), then the integral R measures the total elapsed time, and the duration-bounded reachability problem between regions is called a time-bounded reachability problem. In this case, if (oe; some I , then for all states oe 0 2 [oe] - there is a state - 0 2 [- and a real number I such that Hence, the region graph suffices to solve the time-bounded reachability problem. This, however, is not true for general duration measures. 3 A Solution to the Duration-bounded Reachability Problem Bound-labeled regions and the bounds graph Consider a timed automaton A, two regions R 0 and R f , and a duration measure p. We determine the set I of possible values of ffi such that (oe; To compute the lower and upper bounds on the integral R along a path of the region graph, we refine the graph by labeling all regions with expressions that specify the extremal values of the integral. We define an infinite graph with vertices of the form (R; L; l; U; u), where R is a region, L and U are linear expressions over the clock variables, and l and u are boolean values. The intended meaning of the bound expressions L and U is that in moving from a state (s; c) 2 R to a state in the final region R f , the set of possible values of the integral R p has the infimum L and the supremum U , both of which are functions of the current clock values c. If the bit l is 0, then the infimum L is included in the set of possible values of the integral, and if l is 1, then L is excluded. Similarly, if the bit u is 0, then the supremum U is included in the set of possible values of R and if u is 1, then U is excluded. For example, if then the left-closed right-open interval [L; U) gives the possible values of the integral R p. The bound expressions L and U associated with the region R have a special form. Suppose that is the set of clocks and that for all states (s; c) 2 R, the clock valuation c is the clock with the smallest fractional part in R, and x n is the clock with the largest fractional part. The fractional parts of all n clocks partition the unit interval into represented by the expressions e x x n . A bound expression for R is a positive linear combination of the expressions e that is, a bound expression for R has the form a 0 are nonnegative integer constants. We denote bound expressions by (n + 1)-tuples of coefficients and write (a the bound expression a . For a bound expression e and a clock valuation c, we to denote the result of evaluating e using the clock values given by c. When time advances, the value of a bound expression changes at the rate a 0 \Gamma a n . If the region R satisfies the constraint - is a boundary region), then the coefficient a 0 is irrelevant, and if R then the coefficient a i is irrelevant. Henceforth, we assume that all irrelevant coefficients are set to 0. A bound-labeled region (R; L; l; U; u) of the timed automaton A consists of a clock region R of A, two bound expressions L and U for R, and two bits l; u 2 f0; 1g. We construct B p;R f (A), the bounds graph of A for the duration measure p and the final region R f . The vertices of B p;R f (A) are the bound-labeled regions of A and the special vertex R f , which has no outgoing edges. We first define the edges with the target R f , and then the edges between bound-labeled regions. The edges with the target R f correspond to runs of the automaton that reach a state in R f without passing through other regions. Suppose that R f is an open region with the duration measure a (i.e., a for the location s of R f ). The final region R f is reachable from a state (s; c) 2 R f by remaining in R f for at least 0 and at most units. Since the integral R p increases at the rate a, the lower bound on the integral value over all states (s; c) 2 R f is 0, z - x a 1 a 2 a 3 a 1 a 2 a 3 a 1 + a Figure 2: and the upper bound is a While the lower bound 0 is a possible value of the integral, if a ? 0, then the upper bound is only a supremum of all possible values. Hence, we add an edge in the bounds graph to R f from (R f ; L; 0; U; u) for if If R f is a boundary region, no time can be spent in R f , and both bounds are 0. In this case, we add an edge to R f from (R f ; L; 0; U; Now let us look at paths that reach the final region R f by passing through other regions. For each edge from R to R 0 in the region graph R(A), the bounds graph B p;R f (A) has exactly one edge to each bound-labeled region of the form (R bound-labeled region of the form (R; L; l; U; u). First, let us consider an example to understand the determination of the lower bound L and the corresponding bit l (the upper bound U and the bit u are determined similarly). Suppose that and that the boundary region R 1 , which satisfies labeled with the lower bound L and the bit l 1 . This means that starting from a state (s; c) 2 R 1 , the lower bound on the integral R reaching some state in R f is Consider the open predecessor region R 2 of R 1 , which satisfies x. Let a be the duration measure of R 2 . There is a time edge from R 2 to R 1 in the region graph. We want to compute the lower-bound label L 2 for R 2 from the lower-bound label L 1 of R 1 . Starting in a state (s; c) 2 R 2 , the state (s; c reached after time Furthermore, from the state (s; c) 2 R 2 the integral R p has the value [[a before entering the region R 1 . Hence, the new lower bound is and the label L 2 is (a 1 ; a 2 ; a 3 ; a 1 +a). See Figure 2. Whether the lower bound L 2 is a possible value of the integral depends on whether the original lower bound L 1 is a possible value of the integral starting in R 1 . Thus, the bit l 2 labeling R 2 is the same as the bit l 1 labeling R 1 . Next, consider the boundary region R 3 such that R 2 is the successor region of R 3 . The region x, and there is a time edge from R 3 to R 2 in the region graph. The reader can verify that the updated lower-bound label L 3 of R 3 is the same as L 2 , namely (a 1 ; a 2 ; a 3 ; a 1 +a), which can be simplified to (0; a region. See Figure 3. The updated bit l 3 of R 3 is the same as l 2 . z - x y a 1 a 2 a 1 + a a 3 a 2 a 3 a 1 + a Figure 3: (0 ! - z - x immediate: delayed: a 2 a 3 a 1 + a a 2 a 2 a 3 a 3 a 3 a 3 Figure 4: The process repeats if we consider further time edges, so let us consider a transition edge from region R 4 to region R 3 , which resets the clock y. We assume that the region R 4 is open with duration measure b, and that R 4 satisfies x. Consider a state (t; d) 2 R 4 . Suppose that the transition happens after time ffi; then . If the state after the transition is (s; c) 2 R 3 , then ffi. The lower bound L 4 corresponding to this scenario is the value of the integral before the transition, which is b \Delta ffi, added to the value of the lower bound L 3 at the state (s; c), which is z To obtain the value of the lower bound L 4 at the state (t; d), we need to compute the infimum over all choices of ffi , for . Hence, the desired lower bound is z After substituting simplifies to z The infimum of the monotonic function in ffi is reached at one of the two extreme points. When (i.e., the transition occurs immediately), then the value of L 4 at (t; d) is z When d (i.e., the transition occurs as late as possible), then the value of L 4 at (t; d) is z y), the lower-bound label L 4 for R 4 is (a 2 ; a 3 ; a 3 ; a 4 ), where a 4 is the minimum of a 1 + a and a 2 Figure 4. Finally, we need to x a 2 a 3 a 3 a 4 a 2 a 3 a 4 Figure 5: deduce the bit l 4 , which indicates whether the lower bound L 4 is a possible value of the integral. If a 1 then the lower bound is obtained with possible for R 4 iff L 3 is possible for R 3 ; so l 4 is the same as l 3 . Otherwise, if a 1 then the lower bound is obtained with ffi approaching d , and L 4 is possible iff both l 3 is possible for R 3 ; so l We now formally define the edges between bound-labeled regions of the bounds graph B p;R f (A). Suppose that the region graph R(A) has an edge from R to R 0 , and let a be the duration measure of R. Then the bounds graph has an edge from (R; L; l; U; u) to (R iff the bound expressions and the bits l, u, l 0 , and u 0 are related as follows. There are various cases to consider, depending on whether the edge from R to R 0 is a time edge or a transition edge: Time edge 1 R 0 is a boundary region and is an open region: let 1 - k - n be the largest index such that R 0 satisfies - x for all we have a i+k and b for all a Time edge 2 R is a boundary region and R is an open region: a for all Transition edge 1 R 0 is a boundary region, R is an open region, and the clock with the k-th smallest fractional part in R is reset: for all we have a if a 0 a then a if a 0 a and a ? 0 then a and a ? 0 then Transition edge 2 Both R and R 0 are boundary regions, and the clock with the k-th smallest fractional part in R is reset: for all we have a for all k - i - n, we have a This case is illustrated in Figure 5. This completes the definition of the bounds graph B p;R f (A). Reachability in the bounds graph Given a state oe = (s; c), two bound expressions L and U , and two bits l and u, we define the (bounded) interval I(oe; L; l; U; u) of the nonnegative real line as follows: the left endpoint is the right endpoint is [[U then the interval is left-closed, else it is left-open; if then the interval is right-closed, else it is right-open. The following lemma states the fundamental property of the bounds graph B p;R f (A). A be a timed automaton, let p be a duration measure for A, and let R f be a region of A. For every state oe of A and every nonnegative real ffi , there is a state - 2 R f such that in the bounds graph B p;R f (A), there is path to R f from a bound-labeled region (R; Proof. Consider a state oe of A and a nonnegative real ffi. Suppose (oe; Then, by the definition of the region graph R(A), we have a sequence of successors of extended states with oe region graph contains an edge from the region R i+1 containing oe i+1 to the region R i containing oe i . We claim that there exist bound-labeled regions such that (1) for all the region component of B i is R i , (2) the bounds graph B p;R f (A) has an edge from B 0 to R f and from B i+1 to B i for all This claim is proved by induction on i, using the definition of the edges in the bounds graph. Conversely, consider a sequence of bound-labeled regions B such that the bounds graph has an edge from B 0 to R f and from B i+1 to B i for all (R We claim that for all there exists - 2 R f with (oe; This is again proved by induction on i, using the definition of the edges in the bounds graph. 2 For a bound-labeled region denote the union S oe2R I(oe; L; l; U; u) of intervals. It is not difficult to check that the set I(B) is a bounded interval of the nonnegative real line with integer endpoints. The left endpoint ' of I(B) is the infimum of all choices of clock valuations c that are consistent with R; that is, Rg. Since all irrelevant coefficients in the bound expression L are 0, the infimum ' is equal to the smallest nonzero coefficient in L (the left end-point is 0 if all coefficients are 0). Similarly, the right endpoint of I(B) is the supremum of [[U all choices of c that are consistent with R, and this supremum is equal to the largest coefficient in U . The type of the interval I(B) can be determined as follows. Let ffl If a then I(B) is left-closed, and otherwise I(B) is left-open. then I(B) is right-closed, and otherwise I(B) is right-open. For instance, consider the region R that satisfies z. Let is the open interval (1; 5), irrespective of the values of l and u. A be a timed automaton, let R I be a duration constraint for A, and let R 0 be two regions of A. There are two states oe 2 R 0 and - 2 R f and a real number I such that in the bounds graph B p;R f (A), there is path to R f from a bound-labeled region B with region component R 0 and I(B) " I 6= ;. Hence, to solve the duration-bounded reachability problem R we construct the portion of the bounds graph B p;R f (A) from which the special vertex R f is reachable. This can be done in a backward breadth-first fashion starting from the final region R f . On a particular path through the bounds graph, the same region may appear with different bound expressions. Although there are infinitely many distinct bound expressions, the backward search can be terminated within finitely many steps, because when the coefficients of the bound expressions become sufficiently large relative to I , then their actual values become irrelevant. This is shown in the following section. Collapsing the bounds graph Given a nonnegative integer constant m, we define an equivalence relation - =m over bound-labeled regions as follows. For two nonnegative integers a and b, define a - =m b iff either a = b, or both m. For two bound expressions e = (a iff for all . For two bound-labeled regions iff the following four conditions hold: 2. 3. either l some coefficient in L 1 is greater than m; 4. either some coefficient in U 1 is greater than m. The following lemma states that the equivalence relation - =m on bound-labeled regions is back stable. Lemma 3 If the bounds graph B p;R f (A) contains an edge from a bound-labeled region B 1 to a bound- labeled region B 0 , then there exists a bound-labeled region B 2 such that B 1 and the bounds graph contains an edge from B 2 to B 0 . Proof. Consider two bound-labeled regions B 0 2 such that B 0- =m B 0 . Let R 0 be the clock region of B 0 1 and of B 0 R be a clock region such that the region graph R(A) has an edge from R to R 0 . Then there is a unique bound-labeled region such that the bounds graph B p;R f (A) has an edge from B 1 to B 0 1 , and there is a unique bound-labeled region such that the bounds graph has an edge from B 2 to B 0 2 . It remains to be shown that B 1 There are 4 cases to consider according to the rules for edges of the bounds graph. We consider only the case corresponding to Transition edge 2. This corresponds to the case when R 0 is a boundary region, R is an open region, and the clock with the k-th smallest fractional part in R is reset. Let the duration measure be a in R. We will establish that L 1 some coefficient in L 1 is greater than m; the case of upper bounds is similar. According to the rule, for all 2 , we have a 0 It follows that for We have a a). We have 4 cases to consider. (i) a n and n . Since a 0 n , we have a n . In this case, l 1 and l 2 . If l 0 2 , we have boundary region). Each coefficient a 0 or a j , and thus some coefficient of L 1 also exceeds m. (ii) a a. In this case, we have a 0 It follows that all the values a 0 exceed m. Hence, a and b n ? m. Since at least one coefficient of L 1 is at least m, there is no requirement that l (indeed, they can be different). The cases (iii) a n , and (iv) a a and a have similar analysis. 2 Since the equivalence relation - =m is back stable, for checking reachability between bound-labeled regions in the bounds graph B p;R f (A), it suffices to look at the quotient graph [B p;R f (A)]- =m . The following lemma indicates a suitable choice for the constant m for solving a duration-bounded reachability problem. Lemma 4 Consider two bound-labeled regions B 1 and B 2 and a bounded interval I ' R with integer endpoints. If B 1 for the right endpoint m of I, then I " Proof. Consider bound-labeled regions that . It is easy to check that the left end-points of I(B 1 ) and I(B 2 ) are either equal or both exceed m (see the rules for determining the left end-point). We need to show that when the left end-points are at most m, either both I(B 1 ) and I(B 2 ) are left-open or both are left-closed. If this is trivially true. Suppose l 1 we know that some coefficient of L 1 and of L 2 exceeds m. Since the left end-point is m or smaller, we know that both L 1 and L 2 have at least two nonzero coefficients. In this case, both the intervals are left-open irrespective of the bits l 1 and l 2 . A similar analysis of right end-points shows that either both the right end-points exceed m, or both are at most m, are equal, and both the intervals are either right-open or right-closed. 2 A bound expression e is m-constrained, for a nonnegative integer m, iff all coefficients in e are at most m + 1. Clearly, for every bound expression e, there exists a unique m-constrained bound expression fl(e) such that e - =m fl(e). A bound-labeled region m-constrained iff (1) both L and U are m-constrained, (2) if some coefficient of L is m+ 1, then l = 0, and (3) if some coefficient of U is m for every bound-labeled region B, there exists a unique m-constrained bound-labeled region fl(B) such that B - =m fl(B). Since no two distinct m-constrained bound-labeled regions are - =m -equivalent, it follows that every - =m -equivalence class contains precisely one m-constrained bound-labeled region. We use the m-constrained bound- labeled regions to represent the - =m -equivalence classes. The number of m-constrained expressions over n clocks is (m+2) n+1 . Hence, for a given region R, the number of m-constrained bound-labeled regions of the form (R; L; l; U; u) is 4 \Delta (m+2) 2(n+1) . From the bound on the number of clock regions, we obtain a bound on the number of m-constrained bound-labeled regions of A, or equivalently, on the number of - =m -equivalence classes of bound- labeled regions. Lemma 5 Let A be a timed automaton with location set S and clock set X such that n is the number of clocks, and no clock x is compared to a constant larger than m x . For every nonnegative integer m, the number of m-constrained bound-labeled regions of A is at most Consider the duration-bounded reachability problem R be the right endpoint of the interval I . By Lemma 5, the number of m-constrained bound-labeled regions is exponential in the length of the problem description. By combining Lemmas 2, 3, and 4, we obtain the following exponential-time decision procedure for solving the given duration-bounded reachability problem. Theorem be the right endpoint of the interval I ' R. The answer to the duration- bounded reachability problem R affirmative iff there exists a finite sequence of m-constrained bound-labeled regions of A such that 1. the bounds graph B p;R f (A) contains an edge to R f from some bound-labeled region B with 2. for all the bounds graph B p;R f (A) contains an edge to B i from some bound-labeled region B with 3. and the clock region of B k is R 0 . Hence, when constructing, in a backward breadth-first fashion, the portion of the bounds graph (A) from which the special vertex R f is reachable, we need to explore only m-constrained bound-labeled regions. For each m-constrained bound-labeled region B i , we first construct all predecessors of B i . The number of predecessors of B i is finite, and corresponds to the number of predecessors of the clock region of B i in the region graph R(A). Each predecessor B of B i that is not an m-constrained bound-labeled region is replaced by the - =m -equivalent m-constrained region fl(B). The duration-bounded reachability property holds if a bound-labeled region B with found. If the search terminates otherwise, by generating no new m-constrained bound-labeled regions, then the answer to the duration-bounded reachability problem is negative. The time complexity of the search is proportional to the number of m-constrained bound-labeled regions, which is given in Lemma 5. The space complexity of the search is Pspace, because the representation of an m-constrained bound-labeled region and its predecessor computation requires only space polynomial in the length of the problem description. Corollary 1 The duration-bounded reachability problem for timed automata can be decided in Pspace. The duration-bounded reachability problem for timed automata is Pspace-hard, because already the (unbounded) reachability problem between clock regions is Pspace-hard [AD94]. We solved the duration-bounded reachability problem between two specified clock regions. Our construction can be used for solving many related problems. First, it should be clear that the initial and/or final region can be replaced either by a specific state with rational clock values, or by a specific location (i.e., a set of clock regions). For instance, suppose that we are given an initial state oe, a final state - , a duration constraint R I , and we are asked to decide whether I . Assuming oe and - assign rational values to all clocks, we can choose an appropriate time unit so that the regions [oe] - and [- are singletons. It follows that the duration-bounded reachability problem between rational states is also solvable in Pspace. A second example of a duration property we can decide is the following. Given a real-time system modeled as a timed automaton, and nonnegative integers m, a, and b, we sometimes want to verify that in every time interval of length m, the system spends at least a and at most b accumulated time units in a given set of locations. For instance, for a railroad crossing similar to the one that appears in various papers on real-time verification [AHH96], our algorithm can be used to check that "in every interval of 1 hour, the gate is closed for at most 5 minutes." The verification of this duration property, which depends on various gate delays and on the minimum separation time between consecutive trains, requires the accumulation of the time during which the gate is closed. As a third, and final, example, recall the duration-bounded causality property ( ) from the introduction. Assume that each location of the timed automaton is labeled with atomic propositions such as q, denoting that the ringer is pressed, and r, denoting the response. The duration measure is defined so that is a label of s, and otherwise. The labeling of the locations with atomic propositions is extended to regions and bound-labeled regions. The desired duration- bounded causality property, then, does not hold iff there is an initial region R 0 , a final region R f labeled with r, and a bound-labeled region (R ;, and in the bounds graph B p;R f , there is a path from B to R f that passes only through regions that are not labeled with r. The duration-bounded reachability problem has been studied, independently, in [KPSY93] also. The approach taken there is quite different from ours. First, the problem is solved in the case of discrete time, where all transitions of a timed automaton occur at integer time values. Next, it is shown that the cases of discrete (integer-valued) time and dense (real-valued) time have the same answer, provided the following two conditions are met: (1) the clock constraints of timed automata use only positive boolean combinations of non-strict inequalities (i.e., inequalities involving - and - and (2) the duration constraint is one-sided (i.e., it has the form R N). The first requirement ensures that the runs of a timed automaton are closed under digitization (i.e., rounding of real-numbered transition times relative to an arbitrary, but fixed fractional part ffl 2 [0; 1) [HMP92]). The second requirement rules out duration constraints such as R R 3. The approach of proving that the discrete-time and the dense-time answers to the duration-bounded reachability problem coincide gives a simpler solution than ours, and it also admits duration measures that assign negative integers to some locations. However, both requirements (1) and (2) are essential for this approach. We also note that for timed automata with a single clock, [KPSY93] gives an algorithm for checking more complex duration constraints, such as R R different duration measures p and p 0 . Instead of equipping timed automata with duration measures, a more general approach extends timed automata with variables that measure accumulated durations. Such variables, which are called integrators or stop watches, may advance in any given location either with time derivative 1 (like a clock) or with time derivative 0 (not changing in value). Like clocks, integrators can be reset with transitions of the automaton, and the constraints guarding the automaton transitions can test integrator values. The reachability problem between the locations of a timed automaton with integrators, however, is undecidable [ACH single integrator can cause undecidability [HKPV95]. Still, in many cases of practical interest, the reachability problem for timed automata with integrators can be answered by a symbolic execution of the automaton In contrast to the use of integrators, whose real-numbered values are part of the automaton state, we achieved decidability by separating duration constraints from the system and treating them as properties. This distinction between strengthening the model and strengthening the specification language with the duration constraints is essential for the decidability of the resulting verification problem. The expressiveness of specification languages can be increased further. For example, it is possible to define temporal logics with duration constraints or integrators. The decidability of the model-checking problem for such logics remains an open problem. For model checking a given formula, we need to compute the characteristic set, which contains the states that satisfy the formula. In particular, given an initial region R 0 , a final state - , and a duration constraint R we need to compute the set Q 0 ' R 0 of states oe 2 R 0 for which there exists a real number such that (oe; Each bound-labeled region (R u) from which R f is reachable in the bounds graph B p;R f contributes the subset foe 2 R 0 j I(oe; L; l; U; u) " I 6= ;g to Q 0 . In general, there are infinitely many such contributions, possibly all singletons, and we know of no description of Q 0 that can be used to decide the model-checking problem. By contrast, over discrete time, the characteristic sets for formulas with integrators can be computed [BES93]. Also, over dense time, the characteristic sets can be approximated symbolically [AHH96]. Acknowledgements . We thank Sergio Yovine for a careful reading of the manuscript. --R Model checking in dense real time. A theory of timed automata. The benefits of relaxing punctuality. Logics and models of real time: a survey. Automatic symbolic verification of embedded systems. On model checking for real-time properties with durations Design and synthesis of synchronization skeletons using branching-time temporal logic Decidability of bisimulation equivalence for parallel timer processes. A calculus of durations. Minimum and maximum delay problems in real-time systems Timing assumptions and verification of finite-state concurrent systems What's decidable about hybrid automata? What good are digital clocks? Symbolic model checking for real-time systems Integration graphs: a class of decidable hybrid systems. Specification and verification of concurrent systems in CESAR. --TR --CTR Nicolas Markey , Jean-Franois Raskin, Model checking restricted sets of timed paths, Theoretical Computer Science, v.358 n.2, p.273-292, 7 August 2006 Yasmina Abdeddam , Eugene Asarin , Oded Maler, Scheduling with timed automata, Theoretical Computer Science, v.354 n.2, p.272-300, 28 March 2006

real-time systems;duration properties;formal verification;model checking

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Compile-Time Scheduling of Dynamic Constructs in Dataflow Program Graphs.

AbstractScheduling dataflow graphs onto processors consists of assigning actors to processors, ordering their execution within the processors, and specifying their firing time. While all scheduling decisions can be made at runtime, the overhead is excessive for most real systems. To reduce this overhead, compile-time decisions can be made for assigning and/or ordering actors on processors. Compile-time decisions are based on known profiles available for each actor at compile time. The profile of an actor is the information necessary for scheduling, such as the execution time and the communication patterns. However, a dynamic construct within a macro actor, such as a conditional and a data-dependent iteration, makes the profile of the actor unpredictable at compile time. For those constructs, we propose to assume some profile at compile-time and define a cost to be minimized when deciding on the profile under the assumption that the runtime statistics are available at compile-time. Our decisions on the profiles of dynamic constructs are shown to be optimal under some bold assumptions, and expected to be near-optimal in most cases. The proposed scheduling technique has been implemented as one of the rapid prototyping facilities in Ptolemy. This paper presents the preliminary results on the performance with synthetic examples.

Introduction A D ataflow graph representation, either as a programming language or as an intermediate representation during compilation, is suitable for programming multiprocessors because parallelism can be extracted automatically from the representation [1], [2] Each node, or actor, in a dataflow graph represents either an individual program instruction or a group thereof to be executed according to the precedence constraints represented by arcs, which also represent the flow of data. A dataflow graph is usually made hierarchical. In a hierarchical graph, an actor itself may represent another dataflow graph: it is called a macro actor. Particularly, we define a data-dependent macro actor, or data-dependent actor, as a macro actor of which the execution sequence of the internal dataflow graph is data dependent (cannot be predicted at compile time). Some examples are macro actors that contain dynamic constructs such as data-dependent iteration, and recursion. Actors are said to be data-independent if not data-dependent. The scheduling task consists of assigning actors to pro- cessors, specifying the order in which actors are executed on each processor, and specifying the time at which they are S. Ha is with the Department of Computer Engineering, Seoul National University, Seoul, 151-742, Korea. e-mail: [email protected] E. Lee is with the Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA 94720, USA. e-mail: [email protected] executed. These tasks can be performed either at compile time or at run time [3]. In the fully-dynamic scheduling, all scheduling decisions are made at run time. It has the flexibility to balance the computational load of processors in response to changing conditions in the program. In case a program has a large amount of non-deterministic behav- ior, any static assignment of actors may result in very poor load balancing or poor scheduling performance. Then, the fully dynamic scheduling would be desirable. However, the run-time overhead may be excessive; for example it may be necessary to monitor the computational loads of processors and ship the program code between processors via networks at run time. Furthermore, it is not usually practical to make globally optimal scheduling decision at run time. In this paper, we focus on the applications with a moderate amount of non-deterministic behavior such as DSP applications and graphics applications. Then, the more scheduling decisions are made at compile time the better in order to reduce the implementation costs and to make it possible to reliably meet any timing constraints. While compile-time processor scheduling has a very rich and distinguished history [4], [5], most efforts have been focused on deterministic models: the execution time of each actor T i on a processor P k is fixed and there are no data-dependent actors in the program graph. Even in this restricted domain of applications, algorithms that accomplish an optimal scheduling have combinatorial complexity, except in certain trivial cases. Therefore, good heuristic methods have been developed over the years [4], [6], [7], [8]. Also, most of the scheduling techniques are applied to a completely expanded dataflow graph and assume that an actor is assigned to a processor as an indivisible unit. It is simpler, however, to treat a data-dependent actor as a schedulable indivisible unit. Regarding a macro actor as a schedulable unit greatly simplifies the scheduling task. Prasanna et al [9] schedule the macro dataflow graphs hierarchically to treat macro actors of matrix operations as schedulable units. Then, a macro actor may be assigned to more than one processor. Therefore, new scheduling techniques to treat a macro actor as a schedulable unit was devised. Compile-time scheduling assumes that static information about each actor is known. We define the profile of an actor as the static information about the actor necessary for a given scheduling technique. For example, if we use a list scheduling technique, the profile of an actor is simply the computation time of the actor on a processor. The communication requirements of an actor with other actors are included in the profile if the scheduling tech- HA AND LEE: COMPILE-TIME SCHEDULING OF DYNAMIC CONSTRUCTS IN DATAFLOW PROGRAM GRAPHS 769 nique requires that information. The profile of a macro actor would be the number of the assigned processors and the local schedule of the actor on the assigned processors. For a data-independent macro actor such as a matrix op- eration, the profile is deterministic. However, the profile of a data-dependent actor of dynamic construct cannot be determined at compile time since the execution sequence of the internal dataflow subgraph varies at run time. For those constructs, we have to assume the profiles somehow at compile-time. The main purpose of this paper is to show how we can define the profiles of dynamic constructs at compile-time. A crucial assumption we rely on is that we can approximate the runtime statistics of the dynamic behavior at compile- time. Simulation may be a proper method to gather these statistics if the program is to be run on an embedded DSP system. Sometimes, the runtime statistics could be given by the programmer for graphics applications or scientific applications. By optimally choosing the profile of the dynamic con- structs, we will minimize the expected schedule length of a program assuming the quasi-static scheduling. In figure 1, actor A is a data-dependent actor. The scheduling result is shown with a gantt chart, in which the horizontal axis indicates the scheduling time and the vertical axis indicates the processors. At compile time, the profile of actor A is assumed. At run time, the schedule length of the program varies depending on the actual behavior of actor A. Note that the pattern of processor availability before actor B starts execution is preserved at run time by inserting idle time. Then, after actor A is executed, the remaining static schedule can be followed. This scheduling strategy is called quasi-static scheduling that was first proposed by Lee [10] for DSP applications. The strict application of the quasi-static scheduling requires that the synchronization between actors is guaranteed at compile time so that no run-time synchronization is necessary as long as the pattern of processor availability is consistent with the scheduled one. It is generally impractical to assume that the exact run-time behaviors of actors are known at compile time. Therefore, synchronization between actors is usually performed at run time. In this case, it is not necessary to enforce the pattern of processor availability by inserting idle time. Instead, idle time will be inserted when synchronization is required to execute actors. When the execution order of the actors is not changed from the scheduled order, the actual schedule length obtained from run-time synchronization is proven to be not much different from what the quasi-static scheduling would produce [3]. Hence, our optimality criterion for the profile of dynamic constructs is based on the quasi-static scheduling strategy, which makes analysis simpler. II. Previous Work All of the deterministic scheduling heuristics assume that static information about the actors is known. But almost none have addressed how to define the static information of data-dependent actors. The pioneering work on this issue was done by Martin and Estrin [11]. They calculated A A A (b) (c) (d) (a) Fig. 1. (a) A dataflow graph consists of five actors among which actor A is a data-dependent actor. (b) Gantt chart for compile-time scheduling assuming a certain execution time for actor A. (c) At run time, if actor A takes longer, the second processor is padded with no-ops and (d) if actor A takes less, the first processor is idled to make the pattern of processor availability same as the scheduled one (dark line) in the quasi-static scheduling. the mean path length from each actor to a dummy terminal actor as the level of the actor for list scheduling. For exam- ple, if there are two possible paths divided by a conditional construct from an actor to the dummy terminal actor, the level of the actor is a sum of the path lengths weighted by the probability with which the path is taken. Thus, the levels of actors are based on statistical distribution of dynamic behavior of data-dependent actors. Since this is expensive to compute, the mean execution times instead are usually used as the static information of data-dependent actors [12]. Even though the mean execution time seems a reasonable choice, it is by no means optimal. In addition, both approaches have the common drawback that a data-dependent actor is assigned to a single processor, which is a severe limitation for a multiprocessor system. Two groups of researchers have proposed quasi-static scheduling techniques independently: Lee [10] and Loeffler et al [13]. They developed methods to schedule conditional and data-dependent iteration constructs respectively. Both approaches allow more than one processor to be assigned to dynamic constructs. Figure 2 shows a conditional and compares three scheduling methods. In figure 2 (b), the local schedules of both branches are shown, where two branches are scheduled on three processors while the total number of processors is 4 In Lee's method, we overlap the local schedules of both branches and choose the maximum termination for each processor. For hard real-time systems, it is the proper choice. Otherwise, it may be inefficient if either one branch is more likely to be taken and the size of the likely branch is much smaller. On the other hand, Loeffler takes the local schedule of more likely branch as the profile of the conditional. This strategy is inefficient if both branches are equally likely to be taken and the size of the assumed branch is much larger. Finally, a conditional evaluation can be replaced with a conditional assignment to make the construct static; the graph is modified as illustrated in figure (c). In this scheme, both true and false branches are sched- 770 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 7, JULY 1997 A (a) if-then-else construct (b) local schedule of two (d) Lee's method (e) Leoffler's method (c) a fully-static schedule Fig. 2. Three different schedules of a conditional construct. (a) An example of a conditional construct that forms a data-dependent actor as a whole. (b) Local deterministic schedules of the two branches. (c) A static schedule by modifying the graph to use conditional assignment. (d) Lee's method to overlap the local schedules of both branches and to choose the maximum for each processor. (e) Loeffler's method to take the local schedule of the branch which is more likely to be executed. uled and the result from one branch is selected depending on the control boolean. An immediate drawback is inefficiency which becomes severe when one of the two branches is not a small actor. Another problem occurs when the unselected branch generates an exception condition such as divide-by-zero error. All these methods on conditionals are ad-hoc and not appropriate as a general solution. Quasi-static scheduling is very effective for a data-dependent iteration construct if the construct can make effective use of all processors in each cycle of the iteration. It schedules one iteration and pads with no-ops to make the pattern of processor availability at the termination the same as the pattern of the start (figure (Equivalently, all processors are occupied for the same amount of time in each iteration). Then, the pattern of processor availability after the iteration construct is independent of the number of iteration cycles. This scheme breaks down if the construct cannot utilize all processors effectively. one iteration cycle Fig. 3. A quasi-static scheduling of a data-dependent iteration con- struct. The pattern of processor availability is independent of the number of iteration cycles. The recursion construct has not yet been treated successfully in any statically scheduled data flow paradigm. Recently, a proper representation of the recursion construct has been proposed [14]. But, it is not explained how to schedule the recursion construct onto multiproces- sors. With finite resources, careless exploitation of the parallelism of the recursion construct may cause the system to deadlock. In summary, dynamic constructs such as conditionals, data-dependent iterations, and recursions, have not been treated properly in past scheduling efforts, either for static scheduling or dynamic scheduling. Some ad-hoc methods have been introduced but proven unsuitable as general so- lutions. Our earlier result with data-dependent iteration [3] demonstrated that a systematic approach to determine the profile of data-dependent iteration actor could minimize the expected schedule length. In this paper, we extend our analysis to general dynamic constructs. In the next section, we will show how dynamic constructs are assigned their profiles at compile-time. We also prove the given profiles are optimal under some unrealistic as- sumptions. Our experiments enable us to expect that our decisions are near-optimal in most cases. Section 4,5 and 6 contains an example with data-dependent iteration, recur- sion, and conditionals respectively to show how the profiles of dynamic constructs can be determined with known runtime statistics. We implement our technique in the Ptolemy framework [15]. The preliminary simulation results will be discussed in section 7. Finally, we discuss the limits of our method and mention the future work. III. Compile-Time Profile of Dynamic Constructs Each actor should be assigned its compile-time profile for static scheduling. Assuming a quasi-static scheduling strategy, the proposed scheme is to decide the profile of a construct so that the average schedule length is minimized assuming that all actors except the dynamic construct are data-independent. This objective is not suitable for a hard real-time system as it does not bound the worst case be- havior. We also assume that all dynamic constructs are uncorrelated. With this assumption, we may isolate the effect of each dynamic construct on the schedule length sep- arately. In case there are inter-dependent actors, we may group those actors as another macro actor, and decide the optimal profile of the large actor. Even though the decision of the profile of the new macro actor would be complicated in this case, the approach is still valid. For nested dynamic constructs, we apply the proposed scheme from the inner dynamic construct first. For simplicity, all examples in this paper will have only one dynamic construct in the dataflow graph. The run-time cost of an actor i, C i , is the sum of the total computation time devoted to the actor and the idle time due to the quasi-static scheduling strategy over all proces- sors. In figure 1, the run-time cost of a data-dependent actor A is the sum of the lightly (computation time) and darkly shaded areas after actor A or C (immediate idle time after the dynamic construct). The schedule length of a certain iteration can be written as schedule where T is the total number of processors in the system, and R is the rest of the computation including all idle time that may result both within the schedule and at the end. Therefore, we can minimize the expected schedule length by minimizing the expected cost of the data-dependent acHA AND LEE: COMPILE-TIME SCHEDULING OF DYNAMIC CONSTRUCTS IN DATAFLOW PROGRAM GRAPHS 771 tor or dynamic construct if we assume that R is independent of our decisions for the profile of actor i. This assumption is unreasonable when precedence constraints make R dependent on our choice of profile. Consider, for example, a situation where the dynamic construct is always on the critical path and there are more processors than we can effectively use. Then, our decision on the profile of the construct will directly affect the idle time at the end of the schedule, which is included in R. On the other hand, if there is enough parallelism to make effective use of the unassigned processors and the execution times of all actors are small relative to the schedule length, the assumption is valid. Realistic situations are likely to fall between these two extremes. To select the optimal compile-time profile of actor i, we assume that the statistics of the runtime behavior is known at compile-time. The validity of this assumption varies to large extent depending on the application. In digital signal processing applications where a given program is repeatedly executed with data stream, simulation can be useful to obtain the necessary information. In general, however, we may use a well-known distribution, for example uniform or geometric distribution, which makes the analysis simple. Using the statistical information, we choose the profile to give the least expected cost at runtime as the compile-time profile. The profile of a data-dependent actor is a local schedule which determines the number of assigned processors and computation times taken on the assigned processors. The overall algorithm of profile decision is as follows. We assume that the dynamic behavior of actor i is expressed with parameter k and its distribution p(k). // T is the total number of processors. // N is the number of processors assigned to the actor. for // A(N,k) is the actor cost with parameter N, k // p(k) is the probability of parameter k In the next section, we will illustrate the proposed scheme with data-dependent iteration, recursion, and conditionals respectively to show how profiles are decided with runtime statistics. IV. Data Dependent Iteration In a data-dependent iteration, the number of iteration cycles is determined at runtime and cannot be known at compile-time. Two possible dataflow representations for data-dependent iteration are shown in figure 4 [10]. The numbers adjacent to the arcs indicate the number of tokens produced or consumed when an actor fires [2]. In figure 4 (a), since the upsample actor produces M tokens each time it fires, and the iteration body consumes only one token when it fires, the iteration body must fire M times for each firing of the upsample actor. In figure 4 (b), the f Iteration body source1 of M M (a) (b) Fig. 4. Data-dependent iteration can be represented using the either of the dataflow graphs shown. The graph in (a) is used when the number of iterations is known prior to the commencement of the iteration, and (b) is used otherwise. number of iterations need not be known prior to the commencement of the iteration. Here, a token coming in from above is routed through a "select" actor into the iteration body. The "D" on the arc connected to the control input of the "select" actor indicates an initial token on that arc with value "false". This ensures that the data coming into the "F" input will be consumed the first time the "select" actor fires. After this first input token is consumed, the control input to the "select" actor will have value "true" until function t() indicates that the iteration is finished by producing a token with value "false". During the itera- tion, the output of the iteration function f() will be routed around by the "switch" actor, again until the test function t() produces a token with value "false". There are many variations on these two basic models for data-dependent iteration. The previous work [3] considered a subset of data-dependent iterations, in which simultaneous execution of successive cycles is prohibited as in figure 4 (b). In figure 4 (a), there is no such restriction, unless the iteration body itself contains a recurrence. Therefore, we generalize the previous method to permit overlapped cycles when successive iteration cycles are invokable before the completion of an iteration cycle. Detection of the intercycle dependency from a sequential language is the main task of the parallel compiler to maximize the parallelism. A dataflow represen- tation, however, reveals the dependency rather easily with the presence of delay on a feedback arc. We assume that the probability distribution of the number of iteration cycles is known or can be approximated at compile time. Let the number of iteration cycles be a random variable I with known probability mass function p(i). For simplicity, we set the minimum possible value of I to be 0. We let the number of assigned processors be N and the total number of processors be T . We assume a blocked schedule as the local schedule of the iteration body to remove the unnecessary complexity in all illustrations, although the proposed technique can be applicable to the overlap execution schedule [16]. In a blocked schedule, all assigned processors are assumed to be available, or synchronized at the beginning. Thus, the execution time of one iteration cycle with N assigned processors is t N as displayed in figure 5 (a). We denote by s N the time that must 772 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 7, JULY 1997 elapse in one iteration before the next iteration is enabled. This time could be zero, if there is no data dependency between iterations. Given the local schedule of one iteration cycle, we decide on the assumed number of iteration cycles, xN , and the number of overlapped cycles kN . Once the two parameters, xN and kN , are chosen, the profile of the data-dependent iteration actor is determined as shown in figure 5 (b). The subscript N of t N , s N , xN and kN represents that they are functions of N , the number of the assigned processors. For brevity, we will omit the subscript N for the variables without confusion. Using this profile of the data-dependent macro actor, global scheduling is performed to make a hierarchical compilation. Note that the pattern of processor availability after execution of the construct is different from that before execution. We do not address how to schedule the iteration body in this paper since it is the standard problem of static scheduling.2 x2 x next iteration cycle is executable s (a) Fig. 5. (a) A blocked schedule of one iteration cycle of a data-dependent iteration actor. A quasi-static schedule is constructed using a fixed assumed number x of cycles in the iteration. The cost of the actor is the sum of the dotted area (execution time) and the dark area (idle time due to the iteration). There displays 3 possible cases depending on the actual number of cycles i in (b) According to the quasi-static scheduling policy, three cases can happen at runtime. If the actual number of cycles coincides with the assumed number of iteration cycles, the iteration actor causes no idle time and the cost of the actor consists only of the execution time of the actor. Oth- erwise, some of the assigned processors will be idled if the iteration takes fewer than x cycles (figure 5 (c)), or else the other processors as well will be idled (figure 5 (d)). The expected cost of the iteration actor, C(N; k; x), is a sum of the individual costs weighted by the probability mass function of the number of iteration cycles. The expected cost becomes x p(i)Ntx +X (2) By combining the first term with the first element of the second term, this reduces to e: (3) Our method is to choose three parameters (N , k, and x) in order to minimize the expected cost in equation (3). First, we assume that N is fixed. Since C(N; k; x) is a decreasing function of k with fixed N , we select the maximum possible number for k. The number k is bounded by two ratios: T N and t s . The latter constraint is necessary to avoid any idle time between iteration cycles on a processor. As a result, k is set to be The next step is to determine the optimal x. If a value x is optimal, the expected cost is not decreased if we vary x by or \Gamma1. Therefore, we obtain the following inequalities, Since t is positive, from inequality (5),X If k is equal to 1, the above inequality becomes same as inequality (5) in [3], which shows that the previous work is a special case of this more general method. Up to now, we decided the optimal value for x and k for a given number N . How to choose optimal N is the next question we have to answer. Since t is not a simple function of N , no closed form for N minimizing C(N; k; x) exists, unfortunately. However, we may use exhaustive search through all possible values N and select the value minimizing the cost in polynomial time. Moreover, our experiments show that the search space for N is often reduced significantly using some criteria. Our experiments show that the method is relatively insensitive to the approximated probability mass function for Using some well-known distributions which have nice mathematical properties for the approximation, we can reduce the summation terms in (3) and (6) to closed forms. Let us consider a geometric probability mass function with parameter q as the approximated distribution of the number of iteration cycles. This models a class of asymmetric bell-shaped distributions. The geometric probability mass function means that for any non-negative integer r, To use inequality (6), we findX HA AND LEE: COMPILE-TIME SCHEDULING OF DYNAMIC CONSTRUCTS IN DATAFLOW PROGRAM GRAPHS 773 Therefore, from the inequality (6), the optimal value of x satisfies Using floor notation, we can obtain the closed form for the optimal value as follows: Furthermore, equation (3) is simplified by using the factX getting Now, we have all simplified formulas for the optimal profile of the iteration actor. Similar simplification is possible also with uniform distributions [17]. If k equals to 1, our results coincide with the previous result reported in [3]. V. Recursion Recursion is a construct which instantiates itself as a part of the computation if some termination condition is not sat- isfied. Most high level programming languages support this construct since it makes a program compact and easy to understand. However, the number of self-instantiations, called the depth of recursion, is usually not known at compile-time since the termination condition is calculated at run-time. In the dataflow paradigm, recursion can be represented as a macro actor that contains a SELF actor (figure 6). A SELF actor simply represents an instance of a subgraph within which it sits. If the recursion actor has only one SELF actor, the function of the actor can be identically represented by a data-dependent iteration actor as shown in figure 4 (b) in the previous section. This includes as a special case all tail recursive constructs. Accordingly, the scheduling decision for the recursion actor will be same as that of the translated data-dependent iteration actor. In a generalized recursion construct, we may have more than one SELF actor. The number of SELF actors in a recursion construct is called the width of the recursion. In most real applications, the width of the recursion is no more than two. A recursion construct with width 2 and depth 2 is illustrated in figure 6 (b) and (c). We assume that all nodes of the same depth in the computation tree have the same termination condition. We will discuss the limitation of this assumption later. We also assume that the run-time probability mass function of the depth of the recursion is known or can be approximated at compile-time. The potential parallelism of the computation tree of a generalized recursion construct may be huge, since all nodes at the same depth can be executed concurrently. The maximum degree of parallelism, however, is usually not known at compile-time. When we exploit the parallelism of the construct, we should consider the resource limitations. We may have to restrict the parallelism in order not to deadlock the system. Restricting the parallelism in case the maximum degree of parallelism is too large has been recognized as a difficult problem to be solved in a dynamic dataflow system. Our approach proposes an efficient solution by taking the degree of parallelism as an additional component of the profile of the recursion construct. Suppose that the width of the recursion construct is k. Let the depth of the recursion be a random variable I with known probability mass function p(i). We denote the degree of parallelism by d, which means that the descendents at depth d in the computation graph are assigned to different processor groups. A descendent recursion construct at depth d is called a ground construct (figure 7 (a)). If we denote the size of each processor group by N , the total number of processors devoted to the recursion becomes Nk d . Then, the profile of a recursion construct is defined by three parameters: the assumed depth of recursion x, the degree of parallelism d, and the size of a processor group N . Our approach optimizes the parameters to minimize the expected cost of the recursion construct. An example of the profile of a recursion construct is displayed in figure 7 (b). Let - be the sum of the execution times of actors a,c, and h in figure 6. And, let - o be the sum of the execution times of actors a and b. Then, the schedule length l x of a ground construct becomes l when x - d. At run time, some processors will be idled if the actual depth of recursion is different from the assumed depth of recursion, which is illustrated in figure 7 (c) and (d). When the actual depth of recursion i is smaller than the assumed depth x, the assigned processors are idled. Otherwise, the other processors as well are idled. Let R be the sum of the execution times of the recursion besides the ground constructs. This basic cost R is equal to N-(k d \Gamma1) . For the runtime cost, C 1 , becomes assuming that x is not less than d. For i ? x, the cost C 2 becomes Therefore, the expected cost of the recursion construct, d) is the sum of the run-time cost weighted by the probability mass function. x 774 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 7, JULY 1997 a,c a,c a,c a a a a a test f (b) (c) depth112 SELF actor function f(x) if test(x) is TRUE else return (a) return h(f(c1(x)),f(c2(x))); c Fig. 6. (a) An example of a recursion construct and (b) its dataflow representation. The SELF actor represents the recursive call. (c) The computation tree of a recursion construct with two SELF actors when the depth of the recursion is two. a,c a,c a,c24 (b) Nk d l x a,c a,c a,c construct ground (a) a,c a,c a,c (c) a,c a,c a,c (d) Fig. 7. (a) The reduced computation graph of a recursion construct of width 2 when the degree of parallelism is 2. (b) The profile of the recursion construct. The schedule length of the ground construct is a function of the assumed depth of recursion x and the degree of parallelism d. A quasi-static schedule is constructed depending on the actual depth i of the recursion in (c) for i ! x and in (d) for i ? x. After a few manipulations, First, we assume that N is fixed. Since the expected cost is a decreasing function of d, we select the maximum possible number for d. The number d is bounded by the processor constraint: Nk d - T . Since we assume that the assumed depth of recursion x is greater than the degree of parallelism d, the optimal value for d is Next, we decide the optimal value for x from the observation that if x is optimal, the expected cost is not decreased when x is varied by +1 and \Gamma1. Therefore, we get HA AND LEE: COMPILE-TIME SCHEDULING OF DYNAMIC CONSTRUCTS IN DATAFLOW PROGRAM GRAPHS 775 Rearranging the inequalities, we get the following,X Nk d Note the similarity of inequality (20) with that for data-dependent iterations (6). In particular, if k is 1, the two formulas are equivalent as expected. The optimal values d and x depend on each other as shown in (18) and (20). We may need to use iterative computations to obtain the optimal values of d and x starting from Let us consider an example in which the probability mass function for the depth of the recursion is geometric with parameter q. At each execution of depth i of the recursion, we proceed to depth to depth From the inequality (20), the optimal x satisfies Nk d As a result, x becomes Nk d Up to now, we assume that N is fixed. Since - is a transcendental function of N , the dependency of the expected cost upon the size of a processor group N is not clear. In- stead, we examine the all possible values for N , calculate the expected cost from equation (3) and choose the optimal N giving the minimum cost. The complexity of this procedure is still polynomial and usually reduced significantly since the search space of N can be reduced by some criteria. In case of geometric distribution for the depth of the recursion, the expected cost is simplified to In case the number of child functions is one our simplified formulas with a geometric distribution coincide with those for data-dependent iterations, except for an overhead term to detect the loop termination. Recall that our analysis is based on the assumption that all nodes of the same depth in the computation tree have the same termination condition. This assumption roughly approximates a more realistic assumption, which we call the independence assumption, that all nodes of the same depth have equal probability of terminating the recursion, and that they are independent each other. This equal probability is considered as the probability that all nodes of the same depth terminate the recursion in our assumption. The expected number of nodes at a certain depth is the same in both assumptions even though they describe different behaviors. Under the independence assumption, the shape of the profile would be the same as shown in figure 7: the degree of parallelism d is maximized. Moreover, all recursion processors have the same schedule length for the ground constructs. However, the optimal schedule length l x of the ground construct would be different. The length l x is proportional to the number of executions of the recursion constructs inside a ground construct. This number can be any integer under the independence assumptions, while it belongs to a subset f0; our assumption. Since the probability mass function for this number is likely to be too complicated under the independence assumption, we sacrifice performance by choosing a sub-optimal schedule length under a simpler assumption. VI. Conditionals Decision making capability is an indispensable requirement of a programming language for general applications, and even for signal processing applications. A dataflow representation for an if-then-else and the local schedules of both branches are shown in figure 2 (a) and (b). We assume that the probability p 1 with which the "TRUE" branch (branch 1) is selected is known. The "FALSE" branch (branch 2) is selected with probability ij be the finishing time of the local schedule of the i-th branch on the j-th processor. And let - t j be the finishing time on the j-th processor in the optimal profile of the conditional construct. We determine the optimal values f - t j g to minimize the expected runtime cost of the construct. When the i-th branch is selected, the cost becomes Therefore, the expected cost C(N) is It is not feasible to obtain the closed form solutions for - t j because the max function is non-linear and discontinuous. Instead, a numerical algorithm is developed. 1. Initially, take the maximum finish time of both branch schedules for each processor according to Lee's method [10]. 2. Define ff Initially, all The variable ff i represents the excessive cost per processor over the expected cost when branch i is selected at run time. We define the bottleneck processors of branch i as the processors fjg that satisfy the . For all branches fig, repeat the next step. 3. Choose the set of bottleneck processors, \Theta i , of branch only. If we decrease - t j by ffi for all j 2 \Theta i , the variation of the expected cost becomes \DeltaC until the set \Theta i needs to be updated. Update \Theta i and repeat step 3. Now, we consider the example shown in figure 2. Suppose 0:7. The initial profile in our algorithm is same as Lee's profile as shown in figure 8 (a), which 776 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 7, JULY 1997 happens to be same as Loeffler's profile in this specific ex- ample. The optimal profile determined by our algorithm is displayed in figure 8 (b).1 (a) initial profile (b) optimal profile Fig. 8. Generation of the optimal profile for the conditional construct in figure 2. (a) initial profile (b) optimal profile. We generalized the proposed algorithm to the M-way branch construct by case construct. To realize an M-way branch, we prefer to using case construct to using a nested if-then-else constructs. Generalization of the proposed algorithm and proof of optimality is beyond the scope of this paper. For the detailed discussion, refer to [17]. For a given number of assigned processors, the proposed algorithm determines the optimal profile. To obtain the optimal number of assigned processors, we compute the total expected cost for each number and choose the minimum. VII. An Example The proposed technique to schedule data-dependent actors has been implemented in Ptolemy, which is a heterogeneous simulation and prototyping environment being developed in U.C.Berkeley, U.S.A. [15]. One of the key objectives of Ptolemy is to allow many different computational models to coexist in the same system. A domain is a set of blocks that obey a common computational model. An example of mixed domain system is shown in figure 9. The synchronous dataflow (SDF) domain contains all data-independent actors and performs compile-time scheduling. Two branches of the conditional constructs are represented as SDF subsystems, so their local schedules are generated by a static scheduler. Using the local schedules of both branches, the dynamic dataflow(DDF) domain executes the proposed algorithm to obtain the optimal profile of the conditional construct. The topmost SDF domain system regards the DDF domain as a macro actor with the assumed profile when it performs the global static scheduling. DDF Fig. 9. An example of mixed domain system. The topmost level of the system is a SDF domain. A dynamic construct(if-then-else) is in the DDF domain, which in turn contains two subsystems in the SDF domain for its branches. We apply the proposed scheduling technique to several synthetic examples as preliminary experiments. These experiments do not serve as a full test or proof of generality of the technique. However, they verify that the proposed technique can make better scheduling decisions than other simple but ad-hoc decisions on dynamic constructs in many applications. The target architecture is assumed to be a shared bus architecture with 4 processors, in which communication can be overlapped with computation. To test the effectiveness of the proposed technique, we compare it with the following scheduling alternatives for the dynamic constructs. Method 1. Assign all processors to each dynamic con- struct Method 2. Assign only one processor to each dynamic construct Method 3. Apply a fully dynamic scheduling ignoring all overhead Method 4. Apply a fully static scheduling Method 1 corresponds to the previous research on quasi-static scheduling technique made by Lee [10] and by Loeffler et. al. [13] for data dependent iterations. Method 2 approximately models the situation when we implement each dynamic construct as a single big atomic actor. To simulate the third model, we list all possible outcomes, each of which can be represented as a data-independent macro actor. With each possible outcome, we replace the dynamic construct with a data-independent macro actor and perform fully-static scheduling. The scheduling result from Method 3 is non-realistic since it ignores all the overhead of the fully dynamic scheduling strategy. Nonetheless, it will give a yardstick to measure the relative performance of other scheduling decisions. By modifying the dataflow graphs, we may use fully static scheduling in Method 4. For a conditional construct, we may evaluate both branches and select one by a multiplexor actor. For a data-dependent iteration construct, we always perform the worst case number of iterations. For comparison, we use the average schedule length of the program as the performance measure. As an example, consider a For construct of data-dependent iteration as shown in figure 10. The number inside each actor represents the execution length. To increase the parallelism, we pipelined the graph at the beginning of the For construct. The scheduling decisions to be made for the For construct are how many processors to be assigned to the iteration body and how many iteration cycles to be scheduled explicitly. We assume that the number of iteration cycles is uniformly distributed between 1 and 7. To determine the optimal number of assigned processors, we compare the expected total cost as shown in table I. Since the iteration body can utilize two processors effectively, the expected total cost of the first two columns are very close. How- ever, the schedule determines that assigning one processor is slightly better. Rather than parallelizing the iteration body, the scheduler automatically parallelizes the iteration cycles. If we change the parameters, we may want to parallelize the iteration body first and the iteration cycles next. HA AND LEE: COMPILE-TIME SCHEDULING OF DYNAMIC CONSTRUCTS IN DATAFLOW PROGRAM GRAPHS 777 14 52176 UP- body554 Fig. 10. An example with a For construct at the top level. The subsystems associated with the For construct are also displayed. The number inside an actor represents the execution length of the actor. The proposed technique considers the tradeoffs of parallelizing inner loops or parallelizing outer loops in a nested loop construct, which has been the main problem of parallelizing compilers for sequential programs. The resulting Gantt chart for this example is shown in figure 11. 9 Fig. 11. A Gantt chart disply of the scheduling result over 4 processors from the proposed scheduling technique for the example in figure 10. The profile of the For construct is identified. If the number of iteration cycles at run time is less than or equal to 3, the schedule length of the example is same as the schedule period 66. If it is greater than 3, the schedule length will increase. Therefore, the average schedule length of the example becomes 79.9. The average schedule length from other scheduling decisions are compared in table II. The proposed technique outperforms other realistic methods and achieves 85% of the ideal schedule length by Method 3. In this example, assigning 4 processors to the iteration body (Method 1) worsens the performance since it fails to exploit the intercycle parallelism. Confining the dynamic construct in a single actor (Method 2) gives the worst performance as expected since it fails to exploit both intercycle parallelism and the parallelism of the iteration body. Since the range of the number of iteration cycles is not big, assuming the worst case iteration (Method 4) is not bad. This example, however, reveals a shortcoming of the proposed technique. If we assign 2 processors to the iteration body to exploit the parallelism of the iteration body as well as the intercycle parallelism, the average schedule length becomes 77.7, which is slightly better than the scheduling result by the proposed technique. When we calculate the expected total cost to decide the optimal number of processors to assign to the iteration body, we do not account for the global effect of the decision. Since the difference of the expected total costs between the proposed technique and the best scheduling was not significant, as shown in table I, this non-optimality of the proposed technique could be anticipated. To improve the performance of the proposed technique, we can add a heuristic that if the expected total cost is not significantly greater than the optimal one, we perform scheduling with that assigned number and compare the performance with the proposed technique to choose the best scheduling result. The search for the assumed number of iteration cycles for the optimal profile is not faultless either, since the proposed technique finds a local optimum. The proposed technique selects 3 as the assumed number of iteration cycles. It is proved, however, that the best assumed number is 2 in this example even though the performance difference is negligible. Although the proposed technique is not always optimal, it is certainly better than any of the other scheduling methods demonstrated in table II. Experiments with other dynamic constructs as well as nested constructs have been performed to obtain the similar results that the proposed technique outperforms other ad-hoc decisions. The resulting quasi-static schedule could be at least 10% faster than other scheduling decisions currently existent, while it is as little as 15 % slower than an ideal (and highly unrealistic) fully-dynamic schedule. In a nested dynamic construct, the compile-time profile of the inner dynamic construct affects that of the outer dynamic construct. In general, there is a trade-off between exploiting parallelism of the inner dynamic construct first and that of the outer construct first. The proposed technique resolves this conflict automatically. Refer to [17] for detailed discussion. Let us assess the complexity of the proposed scheme. If the number of dynamic constructs including all nested ones is D and the number of processors is N , the total number of profile decision steps is order of ND, O(ND). To determine the optimal profile also consumes O(ND) time units. Therefore, the overall complexity is order of ND. The memory requirements are the same order od magnitude as the number of profiles to be maintained, which is also order of ND. VIII. Conclusion As long as the data-dependent behavior is not dominating in a dataflow program, the more scheduling decisions are made at compile time the better, since we can reduce the hardware and software overhead for scheduling at run time. For compile-time decision of task assignment and/or ordering, we need the static information, called profiles, of all actors. Most heuristics for compile-time decisions assume the static information of all tasks, or use ad-hoc approximations. In this paper, we propose a systematic method to decide on profiles for each dynamic construct. We define the compile-time profile of a dynamic construct as an assumed local schedule of the body of the dynamic 778 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 7, JULY 1997 I The expected total cost of the For construct as a function of the number of assigned processors Number of assigned processors 1 2 3 4 Expected total cost 129.9 135.9 177.9 N/A II Performance comparison among several scheduling decisions Average schedule length 79.7 90.9 104.3 68.1 90 % of ideal 0.85 0.75 0.65 1.0 0.76 construct. We define the cost of a dynamic construct and choose its compile-time profile in order to minimize the expected cost. The cost of a dynamic construct is the sum of execution length of the construct and the idle time on all processors at run-time due to the difference between the compile-time profile and the actual run-time profile. We discussed in detail how to compute the profile of three kinds of common dynamic constructs: conditionals, data-dependent iterations, and recursion. To compute the expected cost, we require that the statistical distribution of the dynamic behavior, for example the distribution of the number of iteration cycles for a data-dependent iteration, must be known or approximated at compile-time. For the particular examples we used for ex- periments, the performance does not degrade rapidly as the stochastic model deviates from the actual program be- havior, suggesting that a compiler can use fairly simple techniques to estimate the model. We implemented the technique in Ptolemy as a part of a rapid prototyping environment. We illustrated the effectiveness of the proposed technique with a synthetic example in this paper and with many other examples in [17]. The results are only a preliminary indication of the potential in practical applications, but they are very promising. While the proposed technique makes locally optimal decisions for each dynamic construct, it is shown that the proposed technique is effective when the amount of data dependency from a dynamic construct is small. But, we admittedly cannot quantify at what level the technique breaks down. Acknowledgments The authors would like to gratefully thank the anonymous reviewers for their helpful suggestions. This research is part of the Ptolemy project, which is supported by the Advanced Research Projects Agency and the U.S. Air Force (under the RASSP program, contract F33615-93-C-1317), the State of California MICRO program, and the following companies: Bell Northern Research, Cadence, Dolby, Hi- tachi, Lucky-Goldstar, Mentor Graphics, Mitsubishi, Mo- torola, NEC, Philips, and, Rockwell. --R "Data Flow Languages" "Synchronous Data Flow" "Compile-Time Scheduling and Assignment of Dataflow Program Graphs with Data-Dependent Iteration" "Deterministic Processor Scheduling" "Multiprocessor Scheduling to Account for Interprocessor Communications" "A General Approach to Mapping of Parallel Computations Upon Multiprocessor Architecture" "Task Allocation and Scheduling Models for Multiprocessor Digital Signal Processing" "Hierarchical Compilation of Macro Dataflow Graphs for Multiprocessors with Local Memory" "Recurrences, Iteration, and Conditionals in Statically Scheduled Block Diagram Languages" "Path Length Computation on Graph Models of Computations" "The Effect of Operation Scheduling on the Performance of a Data Flow Computer" "Hierar- chical Scheduling Systems for Parallel Architectures" "TDFL: A Task-Level Dataflow Language" "Ptolemy: A Framework for Simulating and Prototyping Heterogeneous Sys- tems" "Program Partitioning for a Reconfigurable Multiprocessor System" "Compile-Time Scheduling of Dataflow Program Graphs with Dynamic Constructs," --TR --CTR D. Ziegenbein , K. Richter , R. Ernst , J. Teich , L. Thiele, Representation of process mode correlation for scheduling, Proceedings of the 1998 IEEE/ACM international conference on Computer-aided design, p.54-61, November 08-12, 1998, San Jose, California, United States Karsten Strehl , Lothar Thiele , Dirk Ziegenbein , Rolf Ernst , Jrgen Teich, Scheduling hardware/software systems using symbolic techniques, Proceedings of the seventh international workshop on Hardware/software codesign, p.173-177, March 1999, Rome, Italy Jack S. N. Jean , Karen Tomko , Vikram Yavagal , Jignesh Shah , Robert Cook, Dynamic Reconfiguration to Support Concurrent Applications, IEEE Transactions on Computers, v.48 n.6, p.591-602, June 1999 Yury Markovskiy , Eylon Caspi , Randy Huang , Joseph Yeh , Michael Chu , John Wawrzynek , Andr DeHon, Analysis of quasi-static scheduling techniques in a virtualized reconfigurable machine, Proceedings of the 2002 ACM/SIGDA tenth international symposium on Field-programmable gate arrays, February 24-26, 2002, Monterey, California, USA Chanik Park , Sungchan Kim , Soonhoi Ha, A dataflow specification for system level synthesis of 3D graphics applications, Proceedings of the 2001 conference on Asia South Pacific design automation, p.78-84, January 2001, Yokohama, Japan Thies , Michal Karczmarek , Janis Sermulins , Rodric Rabbah , Saman Amarasinghe, Teleport messaging for distributed stream programs, Proceedings of the tenth ACM SIGPLAN symposium on Principles and practice of parallel programming, June 15-17, 2005, Chicago, IL, USA Jin Hwan Park , H. K. Dai, Reconfigurable hardware solution to parallel prefix computation, The Journal of Supercomputing, v.43 n.1, p.43-58, January 2008 Praveen K. Murthy , Etan G. Cohen , Steve Rowland, System canvas: a new design environment for embedded DSP and telecommunication systems, Proceedings of the ninth international symposium on Hardware/software codesign, p.54-59, April 2001, Copenhagen, Denmark L. Thiele , K. Strehl , D. Ziegenbein , R. Ernst , J. Teich, FunState

macro actor;dynamic constructs;dataflow program graphs;profile;multiprocessor scheduling

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Singular and Plural Nondeterministic Parameters.

The article defines algebraic semantics of singular (call-time-choice) and plural (run-time-choice) nondeterministic parameter passing and presents a specification language in which operations with both kinds of parameters can be defined simultaneously. Sound and complete calculi for both semantics are introduced. We study the relations between the two semantics and point out that axioms for operations with plural arguments may be considered as axiom schemata for operations with singular arguments.

Introduction The notion of nondeterminism arises naturally in describing concurrent systems. Various approaches to the theory and specification of such systems, for instance, CCS [16], CSP [9], process algebras [1], event structures [26], include the phenomenon of nondeterminism. But nondeterminism is also a natural concept in describing sequential programs, either as a means of indicating a "don't care'' attitude as to which among a number of computational paths will actually be utilized in a particular computation (e.g., [3]) or as a means of increasing the level of abstraction [14,25]. The present work proceeds from the theory of algebraic specifications [4, 27] and generalizes it so that it can be applied to describing nondeterministic operations. In deterministic programming the distinction between call-by-value and call-by-name semantics of parameter passing is well known. The former corresponds to the situation where the actual parameters to function calls are evaluated and passed as values. The latter allows parameters which are function expressions, passed by a kind of Algol copy rule [21], and which are evaluated whenever a need for their value arises. Thus call-by-name will terminate in many cases when the value of a function may be determined without looking at (some of) the actual parameters, i.e., even if these parameters are undefined. Call-by-value will, in such cases, always lead to undefined result of the call. Nevertheless, the call-by-value semantics is usually preferred in the actual programming languages since it leads to clearer and more tractable programs. *) This work has been partially supported by the Architectural Abstraction project under NFR (Norway), by CEC under ESPRIT-II Basic Reearch Working Group No. 6112 COMPASS, by the US DARPA under ONR contract N00014-92-J-1928, N00014-93-1-1335 and by the US Air Force Office of Scientific Research under Grant AFOSR-91-0354. Following [20], we call the nondeterministic counterparts of these two notions singular (call-by-value) and plural (call-by-name) parameter passing. Other names applied to this, or closely related distinction, are call-time-choice vs. run-time-choice [2, 8], or inside-out (IO) vs . outside-in (OI) which reflect the substitution order corresponding to the respective semantics [5, 6]. In the context where one allows nondeterministic parameters the difference between the two semantics becomes quite apparent even without looking at their termination properties. Let us suppose that we have defined operation g(x) as "if x=0 then a else (if x=0 then b else c)", and that we have a nondeterministic choice operation #. returning an arbitrary element from the argument set. The singular interpretation will satisfy the formula f: then a else c), while the plural interpretation need not satisfy this formula. For instance, under the singular interpretation g(#.{0,1}) will yield either a or c, while the set of possible results of g(#.{0,1}) under the plural interpretation will be {a,b,c}. (Notice that in a deterministic environment both semantics would yield the same results.) The fact that the difference between the two semantics occurs already in very trivial examples of terminating nondeterministic operations motivates our investigation. We discuss the distinction between the singular and plural passing of nondeterministic parameters in the context of algebraic semantics focusing on the associated reasoning systems. The singular semantics is given by multialgebras, that is, algebras where functions are set-valued and where these values correspond to the sets of possibile results returned by nondeterministic operations. Thus, if f is a nondeterministic operation, f(t) will denote the set of possible results returned by f when applied to t. We introduce the calculus NEQ which is sound and complete with respect to this semantics. Although terms may denote sets the variables in the language range only over individuals. This is motivated by the interest in describing unique results returned by each particular application of an operation (execution of the program). It gives us the possibility of writing, instead of a formula F(f(t)) which expresses something about the whole set of possible results of f(t), the formula corresponding to x# f(t) # F(x) which express something about each particular result x returned by f(t). Unfortunately, this poses the main problem of reasoning in the context of nondeterminism - the lack of general substitutivity. From the fact that h(x) is deterministic (for each x, has a unique value) we cannot conclude that so is h(t) for an arbitrary term t. If t is nondeteministic, h(t) may have several possible results. The calculus NEQ is designed so that it appropriately restricts the substitution of terms for singular variables. Although operations in multialgebras are set-valued their carriers are usual sets. Thus operations map individuals to sets. This is not sufficient to model plural arguments. Such arguments can be understood as sets being passed to the operation. The fact that, under plural interpretation, g(x) as defined above need not satisfy f results from the two occurrences of x in the body of g. Each of these occurrences corresponds to a repeated application of choice from the argument set x, that is, potentially, to a different value. I n order to model such operations we take as the carrier of the algebra a (subset of the) power set - operations map sets to sets. In this way we obtain power algebra semantics. The extension of the semantics is reflected at the syntactic level by introduction of plural variables ranging over sets rather than over individuals. The sound and complete extension of NEQ is obtained by adding one new rule which allows for usual substitution of arbitrary terms for plural variables. The structure of the paper is as follows. In sections 2-3 we introduce the language for specifying nondeterministic operations and explain the intuition behind its main features. I n section 4 we define multialgebraic semantics for singular specifications and introduce a sound and complete calculus for such specifications. In section 5 the semantics is generalized to power algebras capable of modeling plural parameters, and the sound and complete extension of the calculus is obtained by introducing one additional rule. A comparison of both semantics in section 6 is guided by the similarity of the respective calculi. We identify the subclasses of multimodels and power models which may serve as equivalent semantics of one specification. We also highlight the increased complexity of the power algebra semantics reflecting the problems with intuitive understanding of plural arguments. Proofs of the theorems are merely indicated in this presentation. It reports some of the results from [24] where the full proofs and other details can be found. 2. The specification language A specification is a pair ((,P) where the signature ( is a pair (S,F) of sorts S and operation (with argument and result sorts in S). The set of terms over a signature ( and variable set X is denoted by W (,X . We always assume that, for every sort S, the set of ground words of sort S, S W ( , is not empty. 1 P is a set of sequents of atomic formulae written as a 1 ,.,a n a e 1 ,.,e m . The left hand side (LHS) of a is called the antecedent and the right hand side (RHS) the consequent, and both are to be understood as sets of atomic formulae (i.e., the ordering and multiplicity of the atomic formulae do not matter). In general, we allow either antecedent or consequent to be empty, though - is usually dropped in the notation. A sequent with exactly one formula in the consequent (m=1) is called a Horn formula, and a Horn formula with empty antecedent (n=0) is a simple formula (or a simple sequent). This restriction is motivated by the fact (pointed out in [7]) that admitting empty carriers requires additional mechanisms (explicit quantification) in order to obtain sound logic. We conjecture that s i m i l ar solution can be applied in our case. Singular and Plural Nondeterministic ParametersAll variables occurring in a sequent are implicitly universally quantified over the whole sequent. A sequent is satisfied if, for every assignment to the variables, one of the antecedents is false or one of the consequents is true (it is valid iff the formula a 1 #a n # For any term (formula, set of formulae) j, V[j] will denote the set of variables in j. If the variable set is not mentioned explicitly, we may also write x#V to indicate that x is a variable. An atomic formula in the consequent is either an equation, t=s, or an inclusion, t#s, of terms t, s#W (,X . An atomic formula in the antecedent, written tas, will be interpreted as non-empty intersection of the (result) sets corresponding to t and s. For a given specification SP=((,P), L(SP) will denote the above language over the signature (. The above conventions will be used throughout the paper. The distinction between the singular and plural parameters (introduced in the section 5) will be reflected in the notation by the superscript * : a plural variable will be denoted by x * , the set of plural variables in a term t by V * [t], a specification with plural arguments SP * , the corresponding extension of the language L by L * etc. 3. A note on the intuitive interpretation Multialgebraic semantics [10, 13] interprets specifications in some form of power structures where the (nondeterministic) operations correspond to set-valued functions. This means that a (ground) term is interpreted as a set of possibilities - it denotes the set of possible results of the corresponding operation. We, on the other hand, want our formulae to express necessary i.e., facts which have to hold in every evaluation of a program (specification). This is achieved by interpreting terms as applications of the respective operations. Every two syntactic occurrences of a term t will refer to possibly distinct applications of t. For nondeterministic terms this means that they may denote two distinct values. Typically, equality is interpreted in a multialgebra as set equality [13, 23, 12]. For instance, the formula a t=s means that the sets corresponding to all possible results of the operations t and s are equal. This gives a model which is mathematically plausible, but which does not correspond to our operational intuition. The (set) equality a t=s does not guarantee that the result returned by some particular application of t will actually be equal to the result returned by an application of s. It merely tells us that in principle (in all possible executions) any result produced by t can also be produced by s and vice versa. Equality in our view should be a necessary equality which must hold in every evaluation of a program (specification). It does not correspond to set equality, but to identity of 1-element sets. Thus the simple formula a t=s will hold in a multistructure M iff both t and s are interpreted in M as one and the same set which, in addition, has only one element. Equality is then a partial equivalence relation and terms t for which a t=t holds are exactly the deterministic terms, denoted by D SP ,X . This last equality indicates that arbitrary two applications of t have to return the same result. If it is possible to produce a computation where t and s return different results - and this is possible when they are nondeterministic - then the terms are not equal but, at best, equivalent. They are equivalent if they are capable of returning the same results, i.e., if they are interpreted as the same set. This may be expressed using the inclusion relation: s#t holds iff the set of possible results of s is included in the set of possible results of t, and s#t if each is included in the other. Having introduced inclusion one might expect that a nondeterministic operation can be specified by a series of inclusions - each defining one of its possible results. However, such a specification gives only a "lower bound" on the admitted nondeterminism. Consider the following example: Example 3.1 S: { Nat }, F: 0: # Nat (zero) _#_: Nat-Nat # Nat (binary nondeterministic choice) P: 1. a 0=0 2. a s(x)=s(x) 3. 1a0 a (As usual, we abbreviate s n (0) as n.) 4. a 0 # 0#1 a 1 # 0#1 The first two axioms make zero and successor deterministic. A limited form of negation is present in L in the form of sequents with empty consequent. Axiom 3. makes 0 distinct from Axioms 4. make then # a nondeterministic choice with 0 and 1 among its possible results. This, however, ensures only that in every model both 0 and 1 can be returned by 0#1. I n most models all other kinds of elements may be among its possible results as well, since no extension of the result set of 0#1 will violate the inclusions of 4. If we are satisfied with this degree of precision, we may stop here and use only Horn formula. All the results in the rest of the paper apply to this special case. But to specify an "upper bound" of nondeterministic operations we need disjunction - the multiple formulae in the consequents. Now, if we write the axiom: 5. a 0#1=0, 0#1=1 the two occurrences of 0#1 refer to two arbitrary applications and, consequently, we obtain Singular and Plural Nondeterministic Parametersthat either any application of 0#1 equals 0 or else it equals 1, i.e., that # is not really nondeterministic, but merely underspecified. Since axioms 4. require that both 0 and 1 be among the results of t, the addition of 5. will actually make the specification inconsistent. What we are trying to say with the disjunction of 5. is that every application of 0#1 returns either 0 or 1, i.e., we need a means of identifying two occurrences of a nondeterministic term as referring to one and the same application. This can be done b y binding both occurrences to a variable.The appropriate axiom will be: 59. xa0#1 a x=0, x=1 The axiom says: whenever 0#1 returns x, then x equals 0 or x equals 1. Notice that such an interpretation presupposes that the variable x refers to a unique, individual value. Thus bindings have the intended function only if they involve singular variables. (Plural variables, on the other hand, will refer to sets and not individuals, and so the axiom 599. x * a0#1 a x * =0, x * =1 would have a completely different meaning.) The singular semantics is the most common in the literature on algebraic semantics of nondeterministic specification languages [2, 8, 11], in spite of the fact that it prohibits unrestricted substitution of terms for variables. Any substitution must now be guarded by the check that the substituted term yields a unique value, i.e., is deterministic. We return to this point in the subsection on reasoning where we introduce a calculus which does not allow one, for instance, to conclude 0#1=0#1 a 0#1=0, 0#1=1 from the axiom 59 (though it could be obtained from 599). 4. The singular case: semantics and calculus This section defines the multialgebraic semantics of specifications with singular arguments and introduces a sound and complete calculus. 4.1. Multistructures and multimodels. Definition 4.2 (Multistructures). Let ( be a signature. M is a (-multistructure if (1) its carrier _M_ is an S-sorted set and (2) for every f: S 1 -S n #S in F, there is a corresponding fuction A function F: A#B (i.e., a family of functions for every S#S) is a multihomomorphism from a (-multistructure A to B if for each constant symbol c#F, F(c A (H2) for every f: S 1 -S n #S in F and a 1 .a n #S 1 A F(f A (a 1 .a n If all inclusions in H1 and H2 are (set) equalities the homomorphism is tight, otherwise it is strictly loose (or just loose). denotes the set of non-empty subsets of the set S. Operations applied to sets refer to their unique pointwise extensions. Notice that for a constant c: #S, 2. indicates that c M can be a set of several elements of sort S. Since multihomomorphisms are defined on individuals and not sets they preserve singletons and are #-monotonic. We denote the class of (-multistructures by MStr((). It has the distinguished word structure MW ( defined in the obvious way, where each ground term is interpreted as a singleton set. We will treat such singleton sets as terms rather than 1- element sets (i.e., we do not take special pains to distinguish MW ( and W ( ). MW ( is not an initial (-structure since it is deterministic and there can exist several homomorphisms from it to a given multistructure. We do not focus on the aspect of initiality and merely register the useful fact from [11]: 4.3. M is a (-multistructure iff, for every set of variables X and assignment b: X#_M_, there exists a unique function b[_]: W (,X #P + (_M_) such that: 1. 2. 3. b[f(t In particular, for X=-, there is a unique interpretation function (not a multihomomorphism) satisfying the last two points of this definition. As a consequence of the definition of multistructures, all operations are #-monotonic, i.e., b[s]#b[t] # b[f(s)]#b[f(t)]. Notice also that assignment in the lemma (and in general, whenever it is an assignment of elements from a multistructure) means assignment of individuals, not sets. Next we define the class of multimodels of a specification. Definition 4.4 (Satisfiability). A (-multistructure M satisfies an L(() sequent p for every b: X#M we have where A#B iff A and B are the same 1-element set. An SP-multimodel is a (-multistructure which satisfies all the axioms of SP. We denote the class of multimodels of SP by MMod(SP). The reason for using nonempty intersection (and not set equality) as the interpretation of a in the antecedents is the same as using "elementwise" equality # in the consequents. Since we Singular and Plural Nondeterministic Parametersavoid set equality in the positive sense (in the consequents), the most natural negative form seems to be the one we have chosen. For deterministic terms this is the same as equality, i.e., deterministic antecedents correspond exactly to the usual (deterministic) conditions. For nondeterministic terms this reflects our interest in binding such terms: the sequent ".sat.a." is equivalent to ".xas, xat.a. A binding ".xat.a." is also equivalent to the more familiar ".x#t.a.", so the notation sat may be read as an abbreviation for the more elaborate formula with two # and a new variable x not occurring in the rest of the sequent. For a justification of this, as well as other choices we have made here, the reader is referred to [24]. 4.2. The calculus for singular semantics In [24] we have introduced the calculus NEQ which is sound and complete with respect to the class MMod(SP). Its rules are: R1: a x=x x#V R2: G D G D G G D D x x x x a a a R3: G D G D G G D D x x x x a a a - x not in a RHS of # R4: a) xay a x=y b) xat a x#t x,y#V R5: G D G D G G D D a a a a (CUT) (# stands for either = or #) a) G D G D a a , e G D a a R7: G D G D x x a a a x#V-V[t], at most one x in G a D (ELIM) a denotes G with b substituted for a. Short comments on each of the rules may be in order. The fact that '=' is a partial equivalence relation is expressed in R1. It applies only to variables and is sound because all assignments assign individual values to the (singular) variables. is a paramodulation rule allowing replacement of terms which may be deterministic (in the case when t 1 =t 2 holds in the second assumption). In particular, it allows derivation of the standard substitution rule when the substituted terms are deterministic, and prevents substitution of nondeterministic terms for variables. R3 allows "specialization" of a sequent by substituting for a term t 2 another term t 1 which is included in t 2 . The restriction that the occurrences of t 2 which are substituted for don't occur in the RHS of # is needed to prevent, for instance, the unsound conclusion a t 3 #t 1 from the premises a t 3 #t 2 and a t 1 #t 2 . R4 and R5 express the relation between a in the antecedent and the equality and inclusion in the consequent. The axiom of standard sequent calculus, e a e, (i.e., sat a s#t) does not hold in general here because the antecedent corresponds to non-empty intersection of the result sets while the consequent to the inclusion (#) or identity of 1-element (=) result sets. Only for deterministic terms s, t, do we have that sa t a s=t holds. R5 allows us to cut both a s=t and a s#t with sa t a D. R7 eliminates redundant bindings, namely those that bind an application of a term occurring at most once in the rest of the sequent. We will write P # CAL p to indicate that p is provable from P with the calculus CAL. When we need to write the sequent p explicitly this notation becomes sometimes awkward, and so we will optionally write it as P The counterpart of soundness/completeness of the equational calculus is [24]: Theorem 4.5. NEQ is sound and complete wrt. MMod(SP): Proof idea: Soundness is proved by induction on the length of the proof P # NEQ p. The proof of the completeness part is a standard, albeit rather involved, Henkin-style argument. The axiom set P of SP is extended by adding all L(SP) formulae p which are consistent with P (and the previously added formulae). If the addition of p leads to inconsistency, one adds the negation of p. Since empty consequents provide only a restricted form of negation, the general negation operation is defined as a set of formulae over the original signature extended with new constants. One shows then that the construction yields a consistent specification with a deterministic basis from which a model can be constructed. We also register an easy lemma that the set-equivalent terms, t#s satisfy the same formulae: Lemma 4.6. t#s iff, for any sequent p, P# NEQ p t z iff P# NEQ p s z . Singular and Plural Nondeterministic Parameters5. The plural case: semantics and calculus The singular semantics for passing nondeterminate arguments is the most common notion to be found in the literature. Nevertheless, the plural semantics has also received some attention. In the denotational tradition most approaches considered both possibilities [18, 19, 20, 22]. Engelfriet and Schmidt gave a detailed study of both - in their language, IO and OI - semantics based on tree languages [5], and continuous algebras of relations and power sets [6]. The unified algebras of Mosses and the rewriting logic of Meseguer [15] represent other algebraic approaches distinguishing these aspects. We will define the semantics for specifications where operations may have both singular and plural arguments. The next subsection gives the necessary extension of the calculus NEQ to handle this generalized situation. 5.1. Power structures and power models Singular arguments (such as the variables in L) have the usual algebraic property that they refer to a unique value. This reflects the fact that they are evaluated at the moment of substitution and the result is passed to the following computation. Plural arguments, on the other hand, are best understood as textual parameters. They are not passed as a single value, but every occurrence of the formal parameter denotes a distinct application of the operation. We will allow both singular and plural parameter passing in one specification. The corresponding semantic distinction is between power set functions which are merely #- monotonic and those which also are #-additive. In the language we merely introduce a notational device for distinguishing the singular and plural arguments. We allow annotating the sorts in the profiles of the operation by a superscript, like S * , to indicate that an argument is plural. Furthermore, we partition the set of variables into two disjoint subsets of singular, X, and plural, X * , variables. x and x * are to be understood as distinct symbols. We will say that an operation f is singular in the i-th argument iff the i-th argument (in its signature) is singular. The specification language extended with such annotations of the signatures will be referred to as L * . These are the only extensions of the language we need. We may, optionally, use superscripts t * at any (sub)term to indicate that it is passed as a plural argument. The outermost applications, e.g. f in f(.), are always to be understood plurally, and no superscripting will be used at such places. Definition 5.7. Let ( be a L * -signature. A is a (-power structure, A#PStr((), iff A is a (deterministic) structure such that: 1. for every sort S, the carrier S A is a (subset of the) power set P of some basis set S - 2. for every f: S 1 -S n #S in (, f A is a # -monotonic function S 1 A A #S A such that, if the i-th argument is S i (singular) then f A is singular in the i-th argument. The singularity in the i-th argument in this definition refers not to the syntactic notion but to its semantic counterpart: Definition 5.8. A function f A A A #S A in a power structure A is singular in the i-th argument iff if it is #-additive in the i-th argument, i.e., iff for all x i #S i A and all A (for k-i), f A (.x 1 .x i .x n Thus, the definition of power structures requires that syntactic singularity be modeled b y the semantic one. Note the unorthodox point in the definition - we do not require the carrier to be the whole power set, but allow it to be a subset of some power set. Usually one assumes a primitive nondeterministic operation with the predefined semantics as set union. Then all finite subsets are needed for the interpretation of this primitive operator. Also, the join operation (under the set inclusion as partial order) corresponds exactly to set union only if all sets are present (see example 6.18). None of these assumptions seem necessary. Consequently, we do not assume any predefined (choice) operation but, instead, give the user means of specifying any nondeterministic operation (including choice) directly. Let ( be a signature, A a (-power structure, X a set of singular and X * a set of plural variables, and b an assignment X#X * # _A_ such that for all x#X (Saying "assignment" we will from now on mean only assignments satisfying this last condition.) Then, every term t(x,x * )#W (,X,X * has a unique set interpretation b[t(x,x * )] in A defined as t A (b(x),b(x * )). Definition 5.9 (Satisfiability). Let A be a (-power structure and p: t i as i a be a sequent over L * ((,X,X * ). A satisfies p, A-p, iff for every assignment b: # _A_, we have that: A is a power model of the specification SP=((,P), A#PMod(SP), iff A#PStr(() and A satisfies all axioms from P. Except for the change in the notion of an assignment, this is identical to the definition 4.4, which is the reason for retaining the same notation for the satisfiability relation. Singular and Plural Nondeterministic Parameters5.2. The calculus for plural parameters The calculus NEQ is extended with one additional rule: R8: G D G D a a x x Rules R1-R7 remain unchanged, but now all terms t i belong to W (,X,X * . In particular, any t i may be a plural variable. We let NEQ * denote the calculus NEQ+R8. The new rule R8 expresses the semantics of plural variables. It allows us to substitute an arbitrary term t for a plural variable x * . Taking t to be a singular variable x, we can thus exchange plural variables in a provable sequent p with singular ones. The opposite is, in general, not possible because rule applies only to singular variables. For instance, a plural variable x * will satisfy a x * #x * but this is not sufficient for performing a general substitution for a singular variable. The main result concerning PMod and NEQ * is: Theorem 5.10. For any L * -specification SP and L * (SP) sequent p: Proof idea: The proof is a straightforward extension of the proof of theorem 4.5. 6. Comparison Since plural and singular semantics are certainly not one and the same thing, it may seem surprising that essentially the same calculus can be used for reasoning about both. One would perhaps expect that PMod, being a richer class than MMod, will satisfy fewer formulae than the latter, and that some additional restrictions of the calculus would be needed to reflect the increased generality of the model class. In this section we describe precisely the relation between the L and L * specifications (6.1) and emphasize some points of difference (6.2). 6.1. The "equivalence" of both semantics The following example illustrates a strong sense of equivalence of L and L * . Example 6.11 Consider the following plural definition: a It is "equivalent" to the collection of definitions a f(t) # if t=t then 0 else 1 for all terms t. In the rest of this section we will clarify the meaning of this "equivalence". Since the partial order of functions from a set A to the power set of a set B is isomorphic to the partial order of additive (and strict, if we take P (all subsets) instead of from the power set of A to the power set of B, [A#P(B)] # [P(A) # P(B)], we may consider every multistructure A to be a power structure A * by taking _A * extending all operations in A pointwise. We then have the obvious Lemma 6.12. Let SP be a singular specification (i.e., all operations are singular in all arguments), let A#MStr(SP), and p be a sequent in L(SP). Then A-p iff A * -p, and so #PMod(SP). Call an L * sequent p p-ground (for plurally ground) if it does not contain any plural variables. Theorem 6.13. Let SP * ,P * ) be an L * specification. There exists a (usually infinite) specification SP=((,P) such that 1. 2. for any p-ground p#L * (SP * ) . PMod(SP * )-p iff MMod(SP)-p. Proof: Let ( be ( * with all " * " symbols removed. This makes 1. true. Any p-ground p as in 2. is then a p over the language L((,X). The axioms P are obtained from P * as in the example 6.11. For every p * #P * with plural variables x 1 Obviously, for any p#L(SP) if P# NEQ p then P * then the proof can be simulated in NEQ. Let p9(x * ) be the last sequent used in the NEQ * -proof which contains plural variables x * , and the sequent p9 be the next one obtained by R8. Build the analogous NEQ-proof tree with all plural variables replaced by the terms which occupy their place in p9. The leaves of this tree will be instances of the P * axioms with plural variables replaced by the appropriate terms, and all such axioms are in P. Then soundness and completness of NEQ and NEQ * imply the conclusion of the theorem. Singular and Plural Nondeterministic ParametersWe now ask whether, or under which conditions, the classes PMod and MMod are interchangeable as the models of a specification. Let SP * , SP be as in the theorem. The one way transition is trivial. Axioms of SP are p-ground so PMod(SP * ) will satisfy all these axioms by the theorem. The subclass #PMod(SP * )#PMod(SP * ) where, for every P#PMod(SP * ), all operations are singular, will yield a subclass of MMod(SP). For the other direction, we have to observe that the restriction to p-ground sequents in the theorem is crucial because plural variables range over arbitrary - also undenotable - sets. Let MMod * (SP) denote the class of power structures obtained as in lemma 6.12. It is not necessarily the case that MMod * (SP)-P * as the following argument illustrates. Example 6.14 Let M * #MMod * (SP) have infinite carrier, p * #P * be t i as i a and let b: X#X * # _M * _ be an assignment such that b(x * )={m 1 .m l .} is a set which is not denoted by any term in W (,X . Let b l be an assignment equal to b except that b l (x * )={m l }, i.e., b=U l b l . Then (a) M * - U l l b l [s l l l l l since operations in M * are defined by pointwise extension. M * #MMod * (SP) implies that, for all l But (b) does not necessarily imply (a). In particular, even if for all l, all intersections in the antecedent of (b) are empty, those in (a) may be non-empty. So we are not guaranteed that M * #PMod(SP * ). Thus, the intuition that the multimodels are contained in the power models is not quite correct. To ensure that no undenotable sets from M * can be assigned to the plural variables we redefine the lifting operator * : MMod(SP)# PMod(SP) from 6.12. Definition 6.15. Given a singular specification SP, and M#MMod(SP), we denote b y -M the following power structure is such that a) for every n#_M_: {n}#_-M_, b) for every m#_-M_ there exist a t#W (,X , n#_M_, such 2) the operations in -M can be then defined by: f(m) Then, for any assignment b: X * #_-M_ there exists an assignment u: X * (1b), and an assignment a: X#_M_ (1a) such that b(x * i.e., such that the following diagram commutes: x * -M a x Since M#MMod(SP) it satisfies all the axioms P obtained from P * and the commutativity of the diagram gives us the second part of: Corollary 6.16. Let SP * and SP be as in the theorem 6.13. Then The corollary makes precise the claim that the class of power models of a plural specification SP * may be seen as a class of multimodels of some singular specification SP, and vice versa. The reasoning about both semantics is essentially the same because the only difference concerns the (arbitrary) undenotable sets which can be referred to by plural variables. 6.2. Plural specification of choice Plural variables provide access to arbitrary sets. In the following example we attempt to utilize this fact to give a more concise form to the specification of choice. Example 6.17 The specification S: { S } F: { #._ : S * P: { a#.x * #x * } defines #. as the choice operator - for any argument t, #.t is capable of returning any element belonging to the set interpreting t. The specification may seem plausible but there are several difficulties. Obviously, such a Singular and Plural Nondeterministic Parameterschoice operation would be redundant in any specification since the axiom makes #.t observationally equivalent to t, and lemma 4.6 allows us to remove any occurrences of #. from the (derivable) formulae. Furthermore, observe how such a specification confuses the issue of nondeterministic choice. Choice is supposed to take a set as an argument and return one element from the set, or, perhaps, to convert an argument of type "set" to a result of type "individual". This is the intention of writing the specification above. But power algebras model all operations as functions between power sets and such a "conversion" simply does not make sense. The only points where conversion of a set to an individual takes place is when a term is passed as a singular argument to another operation. If we have an operation with a singular argument f: S#S, then f(t) will make (implicitly) the choice from t. This might be particularly confusing because one tends to think of plural arguments as sets and mix up the semantic sets (i.e., the elements of the carrier of a power algebra) and the syntactic ones (as expressed by the profiles of the operations in the signature). As a matter of fact, the above specification does not at all express the intention of choosing an element from the set. In order to do that it would have to give choice the signature Set(S)#S. Semantically, this would be then a function from P + (Set(S)) to P Assuming that semantics of Set(S) will somehow correspond to the power set construction, this makes things rather complicated, forcing us to work with a power set of a power set. Furthermore, since Set(S) and S are different sorts we cannot let the same variable range over both as was done in the example above. The above example and remarks illustrate some of the problems with the intuitive understanding of plural parameters. Power algebras - needed for modelling such parameters significantly complicate the model of nondeterminism as compared to multialgebras. On the other hand, plural variables allow us to specify the "upper bound" of nondeterministic choice without using disjunction. The choice operation can be specified as the join which under the partial ordering # interpreted as set inclusion will correspond to set union (cf. [17]). Example 6.18 The following specification makes binary choice the join operation wrt. # : S: { S } F: { _#_ : S-S # S } P: { 1. a x * #y * a y * #y * 2. xaz * , yaz * a x#y # z * } Axiom 2, although using singular variables x, y, does specify the minimality of # with respect to all terms. (Notice that the axiom x * az * , y * az * a x * #y * # z * would have a different, and in this context unintended, meaning.) We can show that whenever a t#p and a s#p hold (for arbitrary terms) then so does a t#s#p. x z z z x a a a a a a a a a a a a a a a Violating our formalism a bit, we may say that the above proof shows the validity of the formula stating the expected minimality of join: t#p, s#p a t#s#p. Thus, in any model of the specification from 6.18 # will be a join. It is then natural to consider # as the basic (primitive) operation used for defining other nondeterministic operations. Observe also that in order to ensure that join is the same as set union, we have to require the presence of all (finite) subsets in the carrier of the model. For instance, the power structure A with the carrier { {1},{2},{3},{1,2,3} } and # A defined as x A will be a model of the specification although # A is not the same as set union. 7. Conclusion We have defined the algebraic semantics for singular (call-time-choice) and plural (run-time- choice) passing of nondeterministic parameters. One of the central results reported in the paper is soundness and completeness of two new reasoning systems, NEQ and NEQ * , respectively, for singular and plural semantics. The plural calculus NEQ * is a minimal extension of NEQ which merely allows unrestricted substitution for plural variables. This indicated a close relationship between the two semantics. We have shown that plural specifications have equivalent (modulo undenotable sets) singular formulations if one considers the plural axioms as singular axiom schemata. Acknowledgments We are grateful to Manfred Broy for pointing out the inadequacy of our original notation and to Peter D. Mosses for the observation that in the presence of plural variables choice may be specified as join with Horn formulae. --R "Algebra of communicating processes" "Nondeterministic call by need is neither lazy nor by name" A Discipline of Programming Fundamentals of Algebraic Specification "IO and OI. 1" "IO and OI. 2" "Completeness of Many-Sorted Equational Logic" "The semantics of call-by-value and call-by-name in a nondeterministic environment" Nondeterminism in Algebraic Specifications and Algebraic Programs "Rewriting with a Nondeterministic Choice Operator" Towards a theory of abstract data types "An Abstract Axiomatization of Pointer Types" "Conditional rewriting logic as a unified model of concurrency" Calculi for Communicating Systems "Unified Algebras and Institutions" Introducing Girard's quantitative domains "Domains" "An axiomatic treatment of ALGOL 68 routines" "Power domains" "Nondeterminism in Abstract Data Types" Algebraic Specifications of Nondeterminism "An introduction to event structures" "Algebraic Specification" --TR

many-sorted algebra;sequent calculus;nondeterminism;algebraic specification

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Adaptive Multilevel Techniques for Mixed Finite Element Discretizations of Elliptic Boundary Value Problems.

We consider mixed finite element discretizations of linear second-order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. By a well-known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned CG iterations using a multilevel preconditioner in the spirit of Bramble, Pasciak, and Xu designed for standard nonconforming approximations. Local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results. The performance of the preconditioner and the error estimator is illustrated by several numerical examples.

Introduction . In this work, we are concerned with adaptive multilevel techniques for the efficient solution of mixed finite element discretizations of linear second order elliptic boundary value problems. In recent years, mixed finite element methods have been increasingly used in applications, in particular for such problems where instead of the primal variable its gradient is of major interest. As examples we mention the flux in stationary flow problems or neutron diffusion and the current in semiconductor device simulation (cf. e.g. [4], [13], [14], [22], [27], [36], [42] and [44]). An excellent treatment of mixed methods and further references can be found in the monography of Brezzi and Fortin [12]. Mixed discretization give rise to linear systems associated with saddle point problems whose characteristic feature is a symmetric but indefinite coefficient matrix. Since the systems typically become large for discretized partial differential equations, there is a need for fast iterative solvers. We note that preconditioned iterative methods for saddle point problems have been considered by Bank, Welfert and Yserentant [8] based on a modification on Uzawa's method leading to an outer/inner iterative scheme and by Rusten and Winther [43] relying on the minimum residual method. Moreover, there are several approaches using domain decomposition techniques and related multilevel Schwarz iterations (cf. e.g. Cowsar [15], Ewing and Wang [23, 24, 25], Mathew [32, 33] and Vassilevski and Wang [46]). A further important aspect is to increase efficiency by using adaptively generated triangulations. In contrast to the existing concepts for standard conforming finite element discretizations as realized for example in the finite element codes PLTMG [5] and KASKADE [19, 20], not much work has been done concerning local refinement of the triangulations in mixed discretizations. There is some work by Ewing et al. [21] in case of quadrilateral mixed elements but the emphasis is more on the appropriate treatment of the slave nodes then on efficient and reliable indicators for local refinement. It is the purpose of this paper to develop a fully adaptive algorithm for mixed discretizations based on the lowest order Raviart-Thomas elements featuring a multilevel iterative solver and an a posteriori error estimator as indicator for local refinement. The paper is organized as follows: In section 2 we will present the mixed discretization and a postprocessing technique due to Fraeijs de Veubeke [26] and Arnold and Brezzi [1]. This technique is based on the elimination of the continuity constraints for the normal components of the flux on the interelement boundaries from the conforming Raviart-Thomas ansatz space. Instead, the continuity constraints are taken care of by appropriate Lagrangian multipliers resulting in an extended saddle point problem. Static condensation of the flux leads to a linear system which is equivalent to a modified nonconforming approach involving the lowest order Crouzeix-Raviart elements augmented by cubic bubble functions. Section 3 is devoted to the numerical solution of that nonconforming discretization by a multilevel preconditioned cg-iteration using a BPX-type preconditioner. This preconditioner has been designed by the authors [30, 49] for standard non-conforming approaches and is closely related to that of Oswald [39]. By an application of Nepomnyaschikh's fictitious domain lemma [34, 35] it can be verified that the spectral condition number of the preconditioned stiffness matrix behaves like O(1). In section 4 we present an a posteriori error estimator in terms of the L 2 -norm which can be easily derived from a superconvergence result for mixed discretizations due to Arnold and Brezzi [1]. It will be shown that the error estimator is equivalent to a weighted sum of the squares of the jumps of the approximation of the primal variable across the interelement boundaries. Finally, in section 5 some numerical results are given illustrating both the performance of the preconditioner and the error estimator. Mixed discretization and postprocessing. We consider linear, second order elliptic boundary value problems of the form \Gammadiv(a @\Omega stands for a bounded, polygonal domain in the Euclidean space IR 2 with boundary \Gamma and f is a given function in L 2(\Omega\Gamma4 We further assume that i;j=1 is a symmetric 2 \Theta 2 matrix-valued function with a ij 2 L and b is a function in L satisfying for almost all x 2 \Omega\Gamma We note that only for simplicity we have chosen homogeneous Dirichlet boundary conditions in (2.1). Other boundary conditions of Neumann type or mixed boundary conditions can be treated as well. Introducing the Hilbert space ae oe and the flux as an additional unknown, the standard mixed formulation of (2.1) is given as follows: Find (j; u) 2 H(div; \Omega\Gamma \Theta L 2(\Omega\Gamma such that where the bilinear forms a : H(div; \Omega\Gamma \Theta are given by R \Omega R \Omega divq R \Omega bu and (\Delta; \Delta) 0 stands for the usual L 2 -inner product. Note that under the above assumption on the data of the problem the existence and uniqueness of a solution to (2.3) is well established (cf. e.g. [12]). For the mixed discretization of (2.3) we suppose that a regular simplicial triangulation T h of\Omega is given. In particular, for an element K 2 T h we refer to e i its edges and we denote by E h the set of edges of T h and by E 0 the subsets of interior and boundary edges, respectively. Further, for D '\Omega we refer to jDj as the measure of D and we denote by P k (D), k - 0, the linear space of polynomials of degree - k on D. Then, a conforming approximation of the flux space H(div;\Omega\Gamma is given by V h := RT RT and RT 0 (K) stands for the lowest order Raviart-Thomas element Note that any q h 2 RT 0 (K) is uniquely determined by its normal components on the edges e i denotes the outer normal vector of K. In particular, the conformity of the approximation is guaranteed by specifying the basis in such a way that continuity of the normal components is satisfied across interelement boundaries. Consequently, we have dimV the standard mixed discretization of (2.3) is given by: Find (j h ; u h h such that For D '\Omega we denote by (\Delta; \Delta) k;D , k - 0, the standard inner products and by k \Delta k k;D the associated norms on the Sobolev spaces H k (D) and respectively. For simplicity, the lower index D will be omitted if it is well known that assuming u the following a priori error estimates hold true where h stands as usual for the maximum diameter of the elements of T h and C is a positive constant independent of h, u and j (cf. e.g. [1]; Thm. 1.1). We further observe that the algebraic formulation of (2.5) gives rise to a linear system with coefficient matrix which is symmetric but indefinite. There exist several efficient iterative solvers for such systems, for example those proposed by Bank et al. [8], Cowsar [15], Ewing and Wang [23, 24, 25], Mathew [32], Rusten and Winther [43] and Vassilevski and Wang [46]. However, we will follow an idea suggested by Fraeijs de Veubeke [26] and further analyzed by Arnold and Brezzi in [1] (cf. also [12]). Eliminating the continuity constraints (2.4) from V h results in the nonconforming Raviart-Thomas space - ae oe Since there are now two basic vector fields associated with each e h , we have - h . Instead, the continuity constraints are taken care of by Lagrangian multipliers living in M h := M 0 and Then the nonconforming mixed discretization of (2.3) is to find (j h ; u h \Theta W h \Theta M h such that are given by R R R As shown in [1] the above multiplier technique has two significant advantages. The first one is some sort of a superconvergence result concerning the approximation of the solution u in (2.1) in the L 2 -norm while the second one is related to the specific structure of (2.7) and has an important impact on the efficiency of the solution process. To begin with the first one we denote by \Pi h the L 2 -projection onto M h . Then it is easy to see that there exists a unique (cf. [1] Lemma 2.1). The function - represents a nonconforming interpolation of - h which can be shown to provide a more accurate approximation of u in the L 2 -norm. In particular, if u 1(\Omega\Gamma then there exists a constant c ? 0 independent of h, u and j such that (cf. [12] Theorem 3.1, Chap. 5). The preceding result will be used for the construction of a local a posteriori error estimator to be developed in Section 4. As far as the efficient solution of (2.7) is concerned we note that the algebraic formulation leads to a linear system with a coefficient matrix of the formB @ In particular, - A stands for a block-diagonal matrix, each block being a 3 \Theta 3 matrix corresponding to an element K 2 T h . Hence, - A is easily invertible which suggests block elimination of the unknown flux (also known as static resulting in a 2 \Theta 2 block system with a symmetric, positive definite coefficient matrix. This linear system is equivalent to a modified nonconforming approximation involving the lowest order Crouzeix-Raviart elements augmented by cubic bubble functions. Denoting by m e the midpoint of an and we set Note, that dimCR Further, we denote by P h and - P c the L 2 -projections onto W h and - the latter with respect to the weighted L 2 -inner product (\Delta; \Delta) As shown in [1], (Lemma 2.3 and Lemma 2.4) there exists a unique \Psi h 2 N h such that Originally, Lemma 2.4 is only proved for b j 0 but the result can be easily generalized for functions b - 0. Moreover, \Psi h is the unique solution of the variational problem where the bilinear form aN h is given by Z We will solve (2.11) numerically by preconditioned cg-iterations using a multilevel preconditioner of BPX-type. The construction of that preconditioner will be dealt with in the following section. 3 Iterative solution by multilevel preconditioned cg-iterations. We assume a hierarchy (T k k=0 of possibly highly nonuniform triangulations of\Omega obtained by the refinement process due to Bank et al. [6] based on regular refinements (partition into four congruent subtriangles) and irregular refinements (bisection). For a detailed description including the refinement rules we refer to [5] and [17]. We remark that the refinement rules are such that each K 2 geometrically similar either to an element of or to an irregular refinement of a triangle in T 0 . Consequently, there exist constants depending only on the local geometry of T 0 such that for all K 2 its edges e ae @K Moreover, the refinement rules imply the property of local quasiuniformity, i.e., there exists a constant depending only on the local geometry of T 0 such that for all K;K where hK := diamK. We consider the modified nonconforming approximation (2.11) on the highest level j and we attempt to solve (3.3) by preconditioned cg-iterations. The preconditioner will be constructed by means of the natural splitting of N j into the standard nonconforming part CR j := CR h j and the "bubble" part B and a further multilevel preconditioning of BPX-type for the nonconforming part. For that purpose we introduce the bilinear form a CR j a CR j (u CR aj K (u CR is the standard bilinear form associated with the primal variational formulation (2.1) Z \Omega (aru In the sequel we will refer to A : H 1 0(\Omega\Gamma as the operator associated with the bilinear form a. Further, we define the bilinear form a (w B Z a - for all w B . Denoting by AD j , D g, the operators associated with aD j , we will prove the spectral equivalence of AN j and ACR . To this end we need the following technical lemmas: Lemma 3.1 For all u CR there holds Proof. For the reference triangle - K with vertices (0; 0), (1; 0) and (0; 1) it is easy to establish (3.7) can be deduced by the affine equivalence of the Crouzeix-Raviart elements Lemma 3.2 For all w B there holds Proof. Since are the barycentric coordinates of K, we have Denoting by - i the local basis of - h and by ( - the matrix representation of -aj K in case stands for the vertex opposite to e i , by Green's 3: (3.11) If we consider the reference triangle - K where the vertices are given by (0; 0), refers to the usual partial order on the set of symmetric, positive definite matrices. Moreover, taking advantage of the affine equivalence of the Raviart-Thomas elements it is easy to show Using (3.1), (3.11) and (3.12) in (3.10) and observing (3.9) it follows that We assume a and b to be locally constant, i.e., a ij and we denote by ff 0;K and ff 1;K the lower and upper bounds arising in (2.2) when restricting a to K. We further suppose that a and b are such that min - 0: (3.13) Note, that only for simplicity we have chosen the strong inequality (3.13). All results can be extended to the more general case that a constant c ? 0, independent of K exists such that for all K 2 ch 2 holds. Under the assumption (3.13) there holds: Theorem 3.3 Under the assumption (3.13) there exist constants depending on the local bounds ff l;K , l 2 f0; 1g, K 2 T j , such that for all with a CR j (w B a CR j (u CR Proof. For the proof of the preceding result we use the following lemma which can easily established. Lemma 3.4 For all there hold ff 0;K (aru CR (3.15 a) Proof. Using the Cauchy-Schwarz inequality we obtain j as well as the orthogonality (ru CR we obtain ff 0;K (aru CR The following inequality deduces (3.15 b) On the other hand, in view of kP h j u CR 0;k we have (bu CR Combining (3.15 a) and (3.16 a) gives the upper bound in (3.14) with c ff 1;K ff 0;K . Further, by Young's inequality, (bu CR (bu CR Consequently, using (3.15 b), (3.16 b), (3.13) and ff 1;K (aru CR (bu CR which yields the lower bound in (3.14) with c ff 0;K ff 1;K We note that the bilinear form a B j gives rise to a diagonal matrix which thus can be easily used in the preconditioning process. On the other hand, the bi-linear form a CR j corresponds to the standard nonconforming approximation of (2.1) by the lowest order Crouzeix-Raviart elements. Multilevel preconditioner for such nonconforming finite element discretizations have been developed by Oswald [39, 40], Zhang [53] and the authors [30, 49]. Here we will use a BPX- type preconditioner based on the use of a pseudo-interpolant which allows to identify with a closed subspace of the standard conforming ansatz space with respect to the next higher level. More precisely, we denote by T j+1 , the triangulation obtained from T h by regular refinement of all elements and we refer to S k ae H 1 as the standard conforming ansatz space generated by continuous, piecewise linear finite elements with respect to the triangulation T k . Denoting by N 0 j+1 the set of interior vertices of T j+1 and recalling that the midpoints m e of interior edges j correspond to vertices j+1 , we define a mapping P CR are the midpoints of those interior edges having j+1 as a common vertex. We note that this pseudo-interpolant has been originally proposed by Cowsar [15] in the framework of related domain decomposition techniques. The following result will lay the basis for the construction of the multilevel preconditioner: Lemma j be the pseudo-interpolant given by (3.17). Then there exist constants depending only on the local geometry of T 0 such that for all Proof. The assertion follows by arguing literally in the same way as in [15] (Theorem 2) and taking advantage of the local quasiuniformity of the triangulations It follows from (3.18) that ~ represents a closed subspace of being isomorphic to CR j . Based on this observation we may now use the well known BPX-preconditioner for conforming discretizations with respect to the hierarchy (S k ) j+1 k=0 of finite element spaces (cf. e.g. [10], [11], [16], [41], [50], [52], and [53]). We remark that for a nonvanishing Helmholtz term in (2.1) the initial triangulation T 0 should be chosen in such a way that the magnitude of the coefficients of the principal part of the elliptic operator is not dominated by the magnitude of the Helmholtz coefficient times the square of the maximal diameter of the elements in T 0 (cf. e.g. [37], [51]). Denoting by \Gamma k := fOE (k) , the set of nodal basis functions of the BPX-preconditioner is based on the following structuring of the nodal bases of varying index k: We introduce the Hilbert space Y equipped with the inner product where , and we consider the bilinear form denoting by ~ the operator associated with ~ b. We further define a mapping and refer to R V as its adjoint in the sense that (R V u; v) . Then the BPX-preconditioner is given by satisfying with constants depending only on the local geometry of T 0 and on the bounds for the data a, b in (2.2). The condition number estimates (3.23) have been established by various authors (cf. [10], [16], [38]). They can be derived using the powerful Dryja- theory [18] of additive Schwarz iterations. Another approach due to Oswald [41] is based on Nepomnyaschikh's fictitious domain lemma: Lemma 3.6 Let S and V be two Hilbert spaces with inner products (\Delta; \Delta) S and generated by symmetric, positive definite operators A S : S 7\Gamma! S and ~ . Assume that there exist a linear operator necessarily linear) operator R a S (Rv; Rv) - c 1 Then there holds c 0 a S (u; u) - a S (R ~ is the adjoint of R in the sense that (Rv; u) Proof. See e.g. [35]. In the framework of BPX-preconditioner with a S being the bilinear form in (3.5) while V , ~ b and R are given by (3.19), (3.20) and (3.21), respectively. The estimate (3.24 b) is usually established by means of a strengthened Cauchy-Schwarz inequality. Further, is an appropriately chosen decomposition operator such that the P.L. Lions type estimate (3.24 c) holds true (cf. e.g. [41] Chapter 4). Now, returning to the nonconforming approximation we define I S CR j by I S . Note that in view of (3.17) the operators I S j corresponds to the identity on CR j . Then, with C as in (3.22) the operator is an appropriate BPX-preconditioner for the nonconforming discretization of (2.1). In particular, we have: Theorem 3.7 Let CNC be given by (3.25). Then there exist positive constants depending only on the local geometry of T 0 and on the bounds for the coefficients a, b in (2.2) such that for all u 2 CR j Proof. In view of the fictitious domain lemma we choose as in (3.4) and V and ~ b according to (3.19), (3.20). Furthermore, we specify with T S as the decomposition operator in the conforming setting. Obviously Moreover, using the obvious inequality a CR j and (3.24 b), for all v 2 V we have a CR j (Rv; Finally, using again (3.18) and (3.24 c) for get ~ 1 a CR j (P CR (3.27 c) In terms of (3.27 a-c) we have verified the hypotheses of the fictitious domain lemma which gives the assertion. 4 A posteriori error estimation. Efficient and reliable error estimators for the total error providing indicators for local refinement of the triangulations are an indispensable tool for efficient adaptive algorithms. Concerning the finite element solution of elliptic boundary value problems we mention the pioneering work done by Babuska and Rheinboldt [2, 3] which has been extended among others by Bank and Weiser [7] and Deuflhard, Leinen Yserentant [17] to derive element-oriented and edge-oriented local error estimators for standard conforming approxima- tions. We remark that these concepts have been adapted to nonconforming discretizations by the authors in [29, 30] and [49]. The basic idea is to discretize the defect problem for the available approximation with respect to a finite element space of higher accuracy. For a detailed representation of the different concepts and further references we refer to the monographs of Johnson [31], Szabo and Babuska [45] and Zienkiewicz and Taylor [54] (cf. also the recent survey articles by Bornemann et al. [9] and Verf?rth [47, 48]). In this section we will derive an error estimator for the L 2 -norm of the total error in the primal variable u based on the superconvergence result (2.9). As we shall see this estimator does not require the solution of an additional defect problem and hence is much more cheaper than the estimators mentioned above. We note, however, that an error estimator for the total error in the flux based on the solution of localized defect problems has been developed by the first author in [28]. We suppose that ~ is an approximation of the solution / h 2 N h of (2.11) obtained, for example, by the multilevel iterative solution process described in the preceding section. Then, in view of (2.7) and (2.10) we get an approximation ( ~ h \Theta W h \Theta M h of the unique solution (j h \Theta W h \Theta M h of (2.7) by means of ~ and The last equality is obtained due to: R R R R Further, we denote by - ~ u h 2 CR h the nonconforming extension of ~ - h . In lights of the superconvergence result (2.9) we assume the existence of a constant In other words, (4.3) states that the nonconforming extension - u h of - h does provide a better approximation of the primal variable u than the piecewise constant approximation u h . It is easy to see that (4.3) yields (4. Observing (2.10) and (4.1), we have K2T h3 K2T h3 ii q3 Using (4.5), (4.6) in (4.4) we get q3 We note that k/ h \Gamma ~ represents the L 2 - norm of the iteration error whose actual size can be controlled by the iterative solution process. Therefore, the provides an efficient and reliable error estimator for the L 2 - norm of the total error whose local contributions k~u be used as indicators for local refinement of T h . Moreover, the estimator can be cheaply computed, since it only requires the evaluation of the available approximations ~ u h 2 W h and ~ For a better understanding of the estimator the rest of this section will be devoted to show that it is equivalent to a weighted sum of the squares of the jumps of ~ across the edges e 2 E h . For that purpose we introduce the jump and the average of piecewise continuous functions v along edges In particular, for e h we denote by K in and K out the two adjacent triangles and by n e the unit normal outward from K in . On the other hand, for we refer to n e as the usual outward normal. Then, we define the average [v] A of v on e 2 E h and the jump [v] J of v on e 2 E h according to (vj K in It is easy to see that for piecewise continuous functions u, v there holds Z Z e (uj K in \Delta vj K in Z e (uj K in \Delta vj K in \Gamma uj Kout \Delta vj Kout Z e Further, we observe that for vector fields q the quantity [n \Delta q] J is independent of the choice of K in and K out . In terms of the averages [n e \Delta q h ] A and the jumps [n e \Delta q h ] J we may decompose the nonconforming Raviart-Thomas space - into the sum where the subspaces - h and - H are given by Obviously, we have - g. As the main result of this section we will prove: Theorem 4.1 Let ( ~ j h h \Theta W h \Theta M h be an approximation of the unique solution of (2.6) obtained according to (4.1), (4.2) and let - ~ be the nonconforming extension of ~ - h . Then there exist constants depending only on the shape regularity of T h and the ellipticity constants in (2.2) such that The proof of the preceding result will be provided in several steps. Firstly, due to the shape regularity of T h we have: Lemma 4.2 Under the assumptions of Theorem 4.1 there holds3 'i 'i Proof. By straightforward computation ~ jK in j ~ ~ which easily gives (4.12) by taking advantage of (3.1). As a direct consequence of Lemma 4.1 we obtain the lower bound in (4.11) with oe However, the proof of the upper bound is more elaborate. In view of (4.12) it is sufficient to show that ii holds true with an appropriate positive constant c. As a first step in this direction we will establish the following relationship between ~ - h and the averages and jumps of ~ Lemma 4.3 Under the assumptions of Theorem 4.1 for all q h V h there holds ii where P c denotes the projection onto V h with respect to the weighted L 2 -inner product (\Delta; \Delta) 0;c . Proof. We denote by ~ OE h the unique element in B h satisfying Z ~ Z ~ In view of (4.2) we thus have K2T h@ Z Z By Green's formula, observing ~ Z Z and hence Z \Omega c ar ~ which shows that Consequently, for q h Z Z cP c (ar ~ Z r ~ Z ~ Z ~ Z ~ It follows from (4.2) that Z ~ which by (4.8 b) is clearly equivalent to the assertion. For a particular choice of q h 2 - V h in Lemma 4.2 we obtain an explicit representation of ~ - h on e h . We choose q (- K in and - K in e are the standard basis vector fields in - h with support in K in resp. K out given by Corollary 4.4 Let the assumptions of Lemma 4.2 be satisfied and let - e 2 - h , be given by (4.15). Then there holds ~ Proof. Observing [n e 0 the assertion is a direct consequence of (4.14). Moreover, with regard to (4.13) we get: Corollary 4.5 Under the assumption of Lemma 4.2 there holds@ X ii Proof. Since for each - h 2 M h (E h ) there exists a unique q h 2 - A satisfying by means of (4.14) we get ii A A which gives (4.17) by the Schwarz inequality. The preceding result tells us that for the proof of (4.13) we have to verify A Since (4.18) obviously holds true for q h 2 - , it is sufficient to show: Lemma 4.6 Let the assumptions of Lemma 4.2 satisfied. Then there holds A Proof. We refer to A, - A and P c as the matrix representations of the operators A and P c . With respect to the standard bases of V h and - we may identify vectors q respectively. We remark that q h 2 - A iff q K in h , and q K h . It follows that for q h A (q K in Obviously stands for the spectral radius of P c \Delta P T c . Denoting by S the natural embedding of V h into - h and by S its matrix representation, it is easy to see that A whence ASA ASA We further refer to AK and - AK as the local stiffness matrices. Using (2.2) and (3.12), we get AKg with . Consequently, introducing the local vectors it follows that (jK in e (jK in Using (4.24), (4.25) in (4.23) we find which gives (4.19) in view of (4.20), (4.21) and (4.22). Summarizing the preceding results it follows that the upper estimate in (4.11) holds true with oe s 5 Numerical results. In this section, we will present the numerical results obtained by the application of the adaptive multilevel algorithm to some selected second order elliptic boundary problems. In particular, we will illustrate the refinement process as well as the performance of both the multilevel preconditioner and the a posteriori error estimator. The following model problems from [5] and [20] have been chosen as test examples: Problem 1. Equation (2.1) with on the unit square with the right-hand side f and the Dirichlet boundary conditions according to the solution u(x; which has a boundary layer along the lines Fig. 5.1). Problem 2. Equation (2.1) with the right-hand side f j 0 and a hexagon \Omega with corners (\Sigma1; 0), . The coefficients are chosen according to b j 0 and a(x; y) being piecewise constant with the values 1 and 100 on alternate triangles of the initial triangulation (cf. Fig. 5.2). The solution given by u(x; continuous with a jump discontinuity of the first derivatives at the interfaces. Starting from the initial coarse triangulations depicted in Figures 5.1 and 5.2, on each refinement level l the discretized problems are solved by preconditioned cg-iterations with a BPX-type preconditioner as described in Section 3. The iteration on level l stopped when the estimated iteration error " l+1 is l l , with the safety factor l denotes the estimated error on level l, the number of nodes on level l and l +1 are given by N l and N l+1 , respectively. Denoting by ( ~ j l ; ~ resulting approximation and by - ~ u l the nonconforming extension of ~ - l , for the local refinement of T l the error contributions ffl 2 ~ and the weighted mean value K2T l K are computed. Then, an element K 2 T l is marked for refinement if jKj oe is a safety factor which is chosen as 0:95. The interpolated values of the level l approximation are used as startiterates on the next refinement level. For the global refinement process we use -ffl 2 0;\Omega as stopping criteria, where ff is a safety factor which is chosen as and tol is the required accuracy, Level 0, Figure 5.1: Initial triangulation T 0 and final triangulation T 6 (Problem 1) Level 0, Figure 5.2: Initial triangulation T 0 and final triangulation T 5 (Problem 2) Figures 5.1 and 5.2 represent the initial triangulations T 0 and the final triangulations T 6 and T 5 for Problems 1 and 2, respectively. For Problem 1 we observe a pronounced refinement in the boundary layer (cf. Fig. 5.1). For Problem 2 there is a significant refinement in the areas where the diffusion coefficient is large with a sharp resolution of the interfaces between the areas of large and small diffusion coefficient (cf. Fig. 5.2).1.11.31.51.710 100 1000 10000 estimated error/true error Number of nodes Boundary layer Discontinuous coefficients Figure 5.3: Error Estimation for Problem 1 and 25152535450 10000 20000 30000 40000 50000 Number of cg-iterations Number of nodes Boundary layer Discontinuous coefficients Figure 5.4: Preconditioner for Problem 1 and 2 The behaviour of the a posteriori L 2 -error estimator is illustrated in Figure 5.3 where the ratio of the estimated error and the true error is shown as a function of the total number of nodes. The straight and the dashed lines refer to Problem 1 (boundary layer) and Problem 2 (discontinuous coefficients), respectively. In both cases we observe a slight overestimation at the very beginning of the refinement process, but the estimated error rapidly approaches the true error with increasing refinement level. Finally, the performance of the preconditioner is depicted in Figure 5.4 displaying the number of preconditioned cg-iterations as a function of the total number of nodal points. Note that for an adequate representation of the performance we use zero as initial iterates on each refinement level and iterate until the relative iteration error is less than In both cases, we observe an increase of the number of iterations at the beginning of the refinement process until we get into the asymptotic regime where the numerical results confirm the theoretically predicted O(1) behaviour. --R Mixed and nonconforming finite element methods: im- plementation estimates for adaptive finite element compu- tations A posteriori error estimates for the finite element method. Refinement algorithm and data structures for regular local mesh refinement. Some a posteriori error estimators for elliptic partial differential equations. A class of iterative methods for solving saddle point problems. A basic norm equivalence for the theory of multilevel methods. Parallel multilevel preconditioners. Mixed and Hybrid Finite Element Methods. Two dimensional exponential fitting and application to drift-diffusion models Domain decomposition and mixed finite elements for the neutron diffusion equation. Domain decomposition methods for nonconforming finite element spaces of Lagrange-type Concepts of an adaptive hierarchical finite element code. Towards a unified theory of domain decomposition alogrithms for elliptic problems. Version 2.0. Local refinement via domain decomposition techniques for mixed finite element methods with rectangular Raviart-Thomas elements Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. The Schwarz algorithm and multilevel decomposition iterative techniques for mixed finite element methods. Analysis of multilevel decomposition iterative methods for mixed finite element methods. Analysis of the Schwarz algorithm for mixed finite element methods. Mixed finite element discretization of continuity equations arising in semiconductor device simulation. Adaptive mixed finite elements methods using flux- based a posteriori error estimators Adaptive multilevel iterative techniques for nonconforming finite elements discretizations. Numerical Solutions of Partial Differential Equations by the Finite Element Method. Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems. Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems. Decomposition and fictitious domain methods for elliptic boundary value problems. Some aspects of mixed finite elements methods for semiconductor simulation. Two remarks on multilevel preconditioners. On a hierarchical basis multilevel method with nonconforming P1 elements. On discrete norm estimates related to multilevel preconditioner in the finite elements methods. On a BPX-preconditioner for P1 elements Multilevel finite element approximation: Theory and Application. Multigrid applied to mixed finite elements schemes for current continuity equations. A preconditioned iterative method for saddle point problems. Mixed finite elements methods for flow through unstructed porous media. Finite Element Analysis. Multilevel approaches to nonconforming finite elements discretizations of linear second order elliptic boundary value problems. Iterative methods by space decomposition and subspace correction. Hierarchical bases in the numerical solution of parabolic problems. Old and new convergence proofs for multigrid methods. The Finite Element Method --TR

mixed finite elements;multilevel preconditioned CG iterations;a posteriori error estimator

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Decomposition of Gray-Scale Morphological Templates Using the Rank Method.

AbstractConvolutions are a fundamental tool in image processing. Classical examples of two dimensional linear convolutions include image correlation, the mean filter, the discrete Fourier transform, and a multitude of edge mask filters. Nonlinear convolutions are used in such operations as the median filter, the medial axis transform, and erosion and dilation as defined in mathematical morphology. For large convolution masks or structuring elements, the computation cost resulting from implementation can be prohibitive. However, in many instances, this cost can be significantly reduced by decomposing the templates representing the masks or structuring elements into a sequence of smaller templates. In addition, such decomposition can often be made architecture specific and, thus, resulting in optimal transform performance. In this paper we provide methods for decomposing morphological templates which are analogous to decomposition methods used in the linear domain. Specifically, we define the notion of the rank of a morphological template which categorizes separable morphological templates as templates of rank one. We establish a necessary and sufficient condition for the decomposability of rank one templates into 3 3 templates. We then use the invariance of the template rank under certain transformations in order to develop template decomposition techniques for templates of rank two.

INTRODUCTION OTH linear convolution and morphological methods are widely used in image processing. One of the common characteristics among them is that they both require applying a template to a given image, pixel by pixel, to yield a new image. In the case of convolution, the template is usually called convolution window or mask; while in mathematical morphology, it is referred to as structuring element. Templates used in realizing linear convolutions are often referred to as linear templates. Templates can vary greatly in their weights, sizes, and shapes, depending on the specific applications. Intuitively, the problem of template decomposition is that given a template t, find a sequence of smaller templates K n such that applying t to an image is equivalent to applying t t sequentially to the image. In other words, t can be algebraically expressed in terms of One purpose of template decomposition is to fit the support of the template (i.e., the convolution kernel) optimally into an existing machine constrained by its hardware con- figuration. For example, ERIM's CytoComputer [1] cannot deal with templates of size larger than 3 - 3 on each pipe-line stage. Thus, a large template, intended for image processing on a CytoComputer, has to be decomposed into a sequence of 3 - 3 or smaller templates. A more important motivation for template decomposition is to speed up template operations. For large convolution masks, the computation cost resulting from implementation can be prohibitive. However, in many instances, this cost can be significantly reduced by decomposing the masks or templates into a sequence of smaller templates. For instance, the linear convolution of an image with a gray-valued multiplications and - additions to compute a new image pixel value; while the same convolution computed with an 1 - n row template followed by an n - 1 column template takes only 2n multiplications a f additions for each new image pixel value. This cost saving may still hold for parallel architectures such as mesh connected array processors [2], where the cost is proportional to the size of the template. The problem of decomposing morphological templates has been investigated by a host of researchers. Zhuang and Haralick [3] gave a heuristic algorithm based on tree search that can find an optimal two-point decomposition of a morphological template if such a decomposition exits. A two-point decomposition consists of a sequence of templates each consisting of at most two points. A two-point decomposition may be best suited for parallel architectures with a limited number of local connections since each two-point template can be applied to an entire image in a multi- ply-shift-accumulate cycle [2]. Xu [4] has developed an al- gorithm, using chain code information, for the decomposition of convex morphological templates for two-point system configurations. Again using chain-code information, Park and Chin [5] provide an optimal decomposition of convex morphological templates for four-connected meshes. However, all the above decomposition methods work only on binary morphological templates and do not extend to gray-scale morphological templates. A very successful general theory for the decomposition . The authors are with the University of Florida, Gainesville, FL 32611. E-mail: {ps0; ritter}@cis.ufl.edu. Manuscript received Nov. 9, 1995; revised Mar. 14, 1997. Recommended for acceptance by V. Nalwa. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number 104798. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 6, JUNE 1997 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 2 11 of templates, in both the linear and morphological domain, evolved from the theory of image algebra [6], [7], [8], [9], [10] which provides an algebraic foundation for image processing and computer vision tasks. In this setting, Ritter and Gader [11], [9] presented efficient methods for decomposing discrete Fourier transform templates. Zhu and Ritter [12] employ the general matrix product to provide novel computational methods for computing the fast Fourier transform, the fast Walsh transform, the generalized fast Walsh transform, as well as a fast wavelet transform. In image algebra, template decomposition problems, for both linear and morphological template operations, can be reformulated in terms of corresponding matrix or polynomial factorization. Manseur and Wilson [13] used matrix as well as polynomial factorization techniques to decompose two-dimensional linear templates of size m - n into sums and products of 3 - 3 templates. Li [14] was the first to investigate polynomial factorization methods for morphological templates. He provides a uniform representation of morphological templates in terms of polynomials, thus reducing the problem of decomposing a morphological template to the problem of factoring the corresponding poly- nomials. His approach provides for the decomposition of one-dimensional morphological templates into factors of two-point templates. Crosby [15] extends Li's method to two-dimensional morphological templates. Davidson [16] proved that any morphological template has a weak local decomposition for mesh-connected array processors. Davidson s existence theorem provides a theoretical foundation for morphological template decomposi- tion, yet the algorithm conceived in its constructive proof is not very efficient. Takriti and Gader formulate the general problem of template decomposition as optimization problems [17], [18]. Sussner, Pardalos, and Ritter [19] use a similar approach to solve the even more general problem of morphological template approximation. However, since these problems are inherently NP-complete, researchers try to exploit the special structure of certain morphological templates in order to find decomposition algorithms. For example, Li and Ritter [20] provide very simple matrix techniques for decomposing binary as well as gray-scale linear and morphological convex templates. A separable template is a template that can be expressed in terms of two one-dimensional templates consisting of a row and a column template. Gader [21] uses matrix methods for decomposing any gray-scale morphological template into a sum of a separable template and a totally nonseparable template. If the original template is separable, then Gader s decomposition yields a separable decomposition. If the original template is not separable, then his method yields the closest separable template to the original in the mean square sense. Separable templates are particularly easy to decompose and the decomposition of separable templates into a product of vertical and horizontal strip templates can be used as a first step for the decomposition into a form which matches the neighborhood configuration of a particular parallel architecture. In the linear case, separable templates are also called rank one templates since their corresponding matrices are rank one matrices. O Leary [22] showed that any linear template of rank one can be factored exactly into a product of 3 - 3 linear templates. Templates of higher rank are usually not as efficiently decomposable. However, the rank of a template determines upper bounds of worst-case scenarios. For example, a linear template of rank two always decomposes into a sum of two separable templates. In the linear domain, the notion of template rank stems from the well known concept of matrix rank in linear alge- bra. The purpose of this paper is to develop the notion of a morphological matrix rank similar to the linear matrix rank. By way of bijection, matrices correspond to certain rectangular templates. In analogy to the linear case, we define the rank of a morphological template as the rank of the corresponding matrix. We demonstrate that this notion allows for an elegant and concise formulation of some new results concerning the decomposition of gray-scale morphological templates into separable morphological templates. The paper is organized as follows. In Section 2, we introduce the image algebra notation used throughout this paper and in most of the aforementioned algebraic template decomposition methods. In Section 3, we develop the notions of linear dependence, linear independence and rank pertinent to morphological image processing. In Section 4, we establish general theorems for the separability of matrices in the morphological domain. Finally, in Section 5, we apply the result of the previous sections and establish decomposition criteria, methods, and algorithms for the decomposition of gray-scale morphological templates. Proofs of theorems are given in [23] so as not to obscure the main ideas and results of this paper. Image algebra is a heterogeneous or many-valued algebra in the sense of Birkhoff and Lipson [24], [6], with multiple sets of operands and operators. In a broad sense, image algebra is a mathematical theory concerned with the transformation and analysis of images. Although much emphasis is focused on the analysis and transformation of digital images, the main goal is the establishment of a comprehensive and unifying theory of image transformations, image analysis, and image understanding in the discrete as well as the continuous domain [6], [8], [7]. In this paper, however, we restrict our attention only to the notations and operations that are necessary for establishing the results mentioned in the introduction. Hence, our focus is on morphological image algebra operations. Henceforth, let X be a subset of the digital plane Z Z c h denotes the set of integers. For any set F, we denote the set of all functions from X into F by F X . We use the symbols / and Y- to denote the binary operations of maximum and minimum, respectively. 2.1 Images and Templates From the image algebra viewpoint, images are considered to be functions and templates are viewed as functions whose values are images. In particular, an F-valued image a over the point set X is a function a X a X i. e. , while an F-valued template t on X is a function SUSSNER AND RITTER: DECOMPOSITION OF GRAY-SCALE MORPHOLOGICAL TEMPLATES USING THE RANK METHOD 3 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 3 11 i. e. , e j . For notational convenience, we define t y as t(y) for all y OE X. Note that the image t y has representation y y a f { } (1) where the pixel values t x y a f at location x of this image are called template weights at point y. Since we are concerned with optimizing morphological convolutions, the set F of interest will be the real numbers with the symbol -. appended. More precisely, F R R -. denotes the set of real num- bers. The algebraic system associated with R -. will be the semi-lattice-ordered group R -. c h with the extended arithmetic and logic operations defined as follows: a a a a a a a -. -. a f a f a f a f R R (2) Note that the element -. acts as a null element in the system R -. c h if we view the operation + as multiplication and the operation / as addition. The dual of this system is the lattice ordered group R +. Y- c h . The algebraic system R -. c h provides the mathematical environment for the morphological operation of gray scale dilation, while R +. Y- c h provides the environment for the dual operation of gray scale erosion. Our focus will be on translation invariant R -valued templates over X since gray-scale structuring elements can be realized by these templates. A template t X X OE -. R e j is called translation invariant if and only if y z y a f a f , , Z 2 (3) are elements of X. The support of a template OE -. R e j at a point y is denoted by S(t y ) and defined as follows: y y A translation invariant template t is called rectangular, if rectangular discrete array. EXAMPLE. Let r X X OE -. R e j be the translation invariant template which is determined at each point y OE X by the following function values of x OE X: r x x y x y x y y a f -. R | | | | l l l l for some l l l if if if else we can visualize the rectangular template r as shown in Fig. 1. Fig. 1. The support of the template r at point y. The hashed cell indicates the location of the target point 2.2 Basic Operations The basic operations of addition and maximum on R -. induce pixelwise operations on R -valued images and tem- plates. For any a b X , OE -. R and any s t X X , OE -. R a b x a x b x x X a b x a x b x x X y y y y a fa f a f a f a fa f a f a f a f a f y y If c X OE -. R denotes the constant image a f c h a f for some c OE -. R , then scalar operations on images and templates can be obtained by defining c c c c c c c c a a a c a a a c y y y y y y a f a f a f a f 2.3 Additive Maximum Operations Forming the additive maximum (*) of an image a X OE -. R and a template t X X OE -. R results in the image a t X * OE -. R , which is determined by the following function values. a t y a x t x OE a fb g a f a f (8) Clearly, each template t X X OE -. R defines a function 4 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 6, JUNE 1997 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 4 11 a a t -. a The additive maximum of a template t X X OE -. R e j and a template OE -. R e j is defined as the template r X X OE -. R e j which determines f f , the composition of f t followed by f s . Specifically, OE a f a f a f a f These relationships induce the associative and distributive laws given later. Note that for any constant c OE -. R a f a f a f EXAMPLE. The following column templates r s t X X R t. Fig. 2. The template r constitutes the additive maximum of the templates s and the template t. 2.4 Some Properties of Image and Template Operations The following associative and distributive laws hold for an arbitrary image a X OE -. R and arbitrary templates t X X OE -. R and s X X OE -. R a s t a s t a s t a s a t a f a f a f a f a f (12) These results establish the importance of template decomposition 2.5 Strong Decompositions of Templates A sequence of templates t t e j in R -. e j is called a (strong) decomposition (with respect to the operation "*") of a template OE -. R OE -. R e j can be written in the form In the special case where we speak of a separable template if the support of t 1 is a one dimensional vertical array and the support of t 2 is a one dimensional horizontal array. EXAMPLE. The template r X X OE -. R e j given in Fig. 1 represents a separable template since this template decomposes into a vertical strip template s X X OE -. R e j and a horizontal strip template t X X OE -. R Fig. 3. Pictorial representation of a column template s and a row template t. 2.6 Weak Decompositions of Templates A sequence of templates t t e j in R -. with a strictly increasing sequence of natural numbers K is called a (weak) decomposition (with respect to the operation "*") of a template t X X OE -. R if the template t can be represented as follows: We say s s e j is a weak decomposition of a rectangular template OE -. R separable templates if each s i , is separable and t s s 2.7 Correspondence Between Rectangular Templates and Matrices Note that there is a natural bijection f from the space of all matrices over R -. into the space of all rectangular templates in R -. be arbitrary and x X c h be such that min , , min , The image of a matrix A OE -. R is defined to be SUSSNER AND RITTER: DECOMPOSITION OF GRAY-SCALE MORPHOLOGICAL TEMPLATES USING THE RANK METHOD 5 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 5 11 the template t X X OE -. R e j which satisfies y y y y c h c h K K Henceforth, we restrict our attention to rectangular templates whose target pixel is centered, i.e., rectangular templates of the above form. The theory of minimax algebra [25] examines the algebraic structures arising from the lattice operations "maximum," "minimum," and "addition," including the space of all matrices over R -. together with the operation additive maximum. The natural correspondence between rectangular templates in t X X OE -. R e j and matrices over R -. allows us to use a minimax algebra approach in order to study the weak decomposability of rectangular templates into separable templates. EXAMPLE. Let A OE R 3-3 be the matrix and u, v the vectors given below. A u v F I F I c h (17) The function f maps A to the square template OE -. R e j in Fig. 1, and it maps the column vector u to the column template s X X OE -. R e j and the row vector v to the row template t X X OE -. R e j in Fig. 3. In this section, we develop a new notion of matrix rank within the mathematical framework of minimax algebra. We relate this concept of matrix rank to the one given by Cuninghame-Green [25] and derive the notion of the rank of a morphological template. 3.1 Algebraic Structures and Operations in Minimax Algebra The mathematical theory of minimax algebra deals with algebraic structures such as bands, belts, and blogs. For example, R -. together with the operations of maximum ("/") and addition forms a belt. Cuninghame-Green defines the matrix rank for matrices over certain subsets of the blog R -. For our purposes it suffices to consider R, the finite elements of R -. Operations such as the maximum ("/"), the minimum ("Y-"), and the addition on R induce entrywise operations - , the set of all m - n matrices over R. Minimax algebra also defines compound operations such as "*"- pronounced "additive maximum"-from R R - into - , an operation similar to the regular matrix product known from linear algebra. (An obvious dual of this operation is provided by the "additive minimum" operation.) Given matrices A OE R m-k and B OE R k-n , the additive maximum R m n is determined by c a ik kj If A is a matrix in R m n - and if u i are column vectors in R m-1 and v are row vectors in R 1-n for then the following equivalence holds for the corresponding rectangular template f(A), the vertical strip templates f u i and the horizontal strip templates f v i A u v A u v 3.2 Linear Dependence of Vectors A vector v OE R n is said to be linearly dependent on the vectors OE R if and only if there exist scalars c OE R, ce Otherwise, the vector v OE R n is called linearly independent from the vectors v 1 , -, v k OE R n . The vectors v 1 , -, v k OE R n are linearly independent if each one of them is linearly independent from the others. EXAMPLE. Consider the following elements of R 3 F F F a f , the vector v is linearly dependent on v 1 and v 2 . 3.3 Strong Linear Independence Vectors v v OE R are called strongly linearly independent (SLI) if and only if there exists a vector v OE R n such that v has a unique representation Since this definition does not provide a suitable criterion for testing a collection of vectors in R n for strong linear in- dependence, we choose to provide an alternative equivalent definition based on the following theorem. THEOREM 1. Vectors v v OE R are SLI if and only if the following inequalities hold. where ~ v v vfor any vector 6 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 6, JUNE 1997 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 6 11 c h R COROLLARY. There are k vectors v 1 , -, v k OE R n which are SLI if and only if k OE {1, -, n}. 3.4 Rank of a Matrix Cuninghame-Green defines the rank of a matrix A OE R m-n as the maximal number of SLI row vectors or, equivalently, the maximum number of SLI column vectors. The rank of a finite matrix A OE R m-n is less than or equal to min {m, n}. 3.5 Remarks on the Rank of a Matrix in Minimax Algebra The notion of (regular) linear independence is not suited to define a rank in minimax algebra because certain dimensional abnormalities would occur. For example, it is possible to find k linearly independent vectors in R n for any number k OE N. The notion of strong linear independence gives rise to a satisfactory theory of rank and dimension (although certain equivalences known from linear algebra do not hold). 3.6 The Separable Rank of a Matrix The separable rank of a matrix A OE R m-n is denoted by defined as the minimal number r of column vectors row vectors { v 1 , - v r } OE { R} 1 - which permit a representation of A in the following form: A u v A representation of this form is called a (weak) separable decomposition of A. We say A is a separable matrix (with respect to the operation *) if rank sep 3.7 The Rank of a Rectangular Template real valued matrix A, then we define the rank of the template t X X OE -. R e j as the separable rank of A. Our interest in the rank of a morphological template is motivated by the problem of morphological template decomposition since the rank of a morphological template OE -. R represents the minimal number of separable templates whose maximum is t or, equivalently, the minimal number r of column templates r X X R e j and row templates s X X R e j such that t r s r ie j . In this section, we derive some theorems concerning the separable rank of matrices which translate directly into results about the rank of rectangular templates. These theorems greatly simplify the proofs of the decomposition results which we will present in the next section. THEOREM 2. If a matrix A OE R m-n has a representation A u v l le j in terms of column vectors u m R 1 and row vectors v l n then A can be expressed in the following form: A l l R 1 is given by w a l , . , (26) REMARK. Theorem 2 implies that, for any matrix A OE R m-n of separable rank k, it suffices to know the row vectors , , . , which permit a weak decomposition of A into k separable matrices in order to determine a representation of A in the form: A w v l l l R , . , (27) Like most of the theorems established in this paper, Theorem 2 has an obvious dual in terms of column vectors which we choose to omit. We now are going to introduce certain transforms which preserve the separable matrix rank. These transforms are suited to simplify the task of determining the separable rank of a given matrix. 4.1 Column Permutations of Matrices Let A OE R m-n and r be a permutation of {1, -, n}. The associated column permuted matrix r c (A) of A with respect to r is defined as follows: r c a a a a a a a a a A a f a f a f a f a f a f a f a f a f a f F I 4.2 Row Permutations If r is a permutation of {1, -, m}, then we define the associated row permuted matrix r r (A) of A OE R m-n with respect to r as follows: r r a a a a a a a a a A a f a f a f a f a f a f a f a f a f a f F I The multiplication of a matrix A OE R m-n by a scalar c OE R is defined as usual. In this case, -A stands for (-1) # A. THEOREM 3. The following transformations preserve the separable rank of a matrix A OE R m-n : SUSSNER AND RITTER: DECOMPOSITION OF GRAY-SCALE MORPHOLOGICAL TEMPLATES USING THE RANK METHOD 7 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 7 11 . column and row permutations; . additions of separable matrices; . scalar multiplications. REMARK. Column and row permutations as well as additions of separable matrices also preserve the rank of a matrix, as defined by Cuninghame-Green. This invariance property follows directly from the definition of this matrix rank as the minimal number of SLI row vectors or column vectors. THEOREM 4. The separable rank of a finite matrix A OE R m-n is bounded from below by the rank of A and bounded from above by the minimal number l of linearly independent row vectors or column vectors of A. At this point, we are finally ready to tackle the problem of determining weak decompositions of matrices in view of their separable ranks. The reader should bear in mind the consequences for the corresponding rectangular templates. For any matrix A OE R m-n , we use the notation a(i), -, m to denote the ith row vector of A and we use the notation the jth column vector of A. THEOREM 5 [20]. Let A OE R m-n be a separable matrix and {a(i) : be the collection of row vectors of A. For each arbitrary row vector a(i 0 ), there exist scalars l i OE R, m, such that the following equations are satisfied: l In other words, given an arbitrary index 1 - i 0 - m, each row vector a(i) is linearly dependent on the i 0 th row vector A. Clearly, Li and Ritter's theorem gives rise to the following straightforward algorithm which tests if a given matrix over R is separable. In the separable case, the algorithm computes a vector pair into which the given matrix can be decomposed. ALGORITHM 1. Let A OE R m-n be given and let a(i) denote the ith row vector of A. The algorithm proceeds as follows for all Subtract a 1j from a ij . Compare c i with a ij - a 1j . If a 1j , the matrix A is not separable and the algorithm stops. If step has been successfully completed, then A is separable and A is given by c*a(1). Note that this algorithm only involves (m - 1)n subtractions and (m - 1)(n - 1) comparisons. Hence, the number of operations adds up to 2(m - 1)n which implies that the algorithm has order O(2mn). REMARK. As mentioned earlier, the given image processing hardware often calls for the decomposition of a given template into 3 - 3 templates. The theorem below shows that, in the case of a separable square template, this problem reduces to the problem of decomposing a column template into 3 - 1 templates as well as decomposing a row template into 1 - 3 templates. Suppose that the original template is of size (2n operations per pixel are needed when applying this template to an image. If the template decomposes into n 3 - 3 templates, this number of operations reduces to 9n. However, the simple strong decomposition of the original separable template into a row and a column template of length especially when using a sequential machine since only 4n operations per pixel are required when taking advantage of this decomposition. THEOREM 6. Let t be a square morphological template of rank 1, given by r is a column template and s is a row template. The template t is decomposable into templates if and only if r is decomposable into m - 1 templates and s is decomposable into templates. EXAMPLE. Let A be the real valued 5 - 5 matrix given below. A u v F I where F I The template is not decomposable into two templates since the template r = f(u) is not decomposable into two 3 - 1 templates. REMARK. Of course, Theorem 6 does not preclude the existence of templates t of rank # 2 which are strongly decomposable into 3 - 3 templates. EXAMPLE. The following template t of rank > 1 can be written as a *-product of two 3 - 3 templates t 1 and t 2 . See Fig. 4 and Fig. 5. COROLLARY. Let t be a square morphological template of rank k, given by -, r k are column templates and s 1 , -, s k are row tem- plates. If the templates r 1 , -, r k are decomposable into templates and the templates s 1 , -, s k are decomposable then t has representation in terms of templates t 1 , -, t k which are strongly decomposable into templates. 8 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 6, JUNE 1997 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 8 11 Fig. 4. Example of a 5 - 5 template t of rank > 1 which is decomposable into 3 - 3 templates t 1 and t 2 . Fig. 5. Templates t 1 and t 2 which LEMMA 1. If a matrix A OE R m-n has separable rank two, then there exits a transform T-consisting of only row permuta- tions, column permutations, and additions of row or column vectors-as well as vectors u OE R m-1 and v OE R 1-n such that T(A) can be written in the following form: where and 0 m-n denotes the m - n zero matrix. LEMMA 2. Let T be a (separable) rank preserving transform as in Theorem 3 and A OE R m-n The transform T maps row vectors of A to row vectors of T(A), and column vectors of A to column vectors of T(A). THEOREM 7. A matrix A OE R m-n has separable rank two if and only there are two row vectors of A on which all other row vectors depend (linearly). A similar theorem does not hold for matrices of separable 3. This fact is expressed by Theorem 8. THEOREM 8. For every natural number k # 3, there are matrices over R -. which are weakly *-decomposable into a product of vector pairs, but not all of whose row vectors are linearly dependent on a single k tuple of their row vectors. REMARK. By Theorem 6, a matrix A OE R m-n has separable rank two if and only if there exist two row vectors of A-a(o), a(p) where o, p OE {1, -, m}-which allow for a weak decomposition of A. In this case, an application of Theorem 2 yields the following representation of A: A u a v a a f b g (35) where u a a i m v a a i m , . , , . , OE R 111 Hence, in order to test an arbitrary matrix A OE R m-n for weak decomposability into two vector pairs, it is enough to compare A with (b[i] * a(i)) / (b[j] * a(j)) for all indices i, j OE {1, -, m}. Here B OE R m-n is computed as follows: b a a is js , . , (37) ALGORITHM 2. Assume a matrix A OE R m-n needs to be decomposed into a weak *-product of two vector pairs if such a decomposition is possible. Considering the preceding remarks, we are able to give a polynomial time algorithm for solving this problem. For each step, we include the number of operations involved in square brackets. Compute 2: 3: and compare the result with A. If C m, then the algorithm stops yielding the following result: does not have a weak decomposition into two vector pairs. [At most 2 2 I K a f comparisons]. This algorithm involves at most a total number of m 2 (3n - 1 operations which amounts to order O(m 3 n). EXAMPLE. Let us apply Algorithm 2 to the following matrix A OE R 4-5 . SUSSNER AND RITTER: DECOMPOSITION OF GRAY-SCALE MORPHOLOGICAL TEMPLATES USING THE RANK METHOD 9 J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 1:49 PM 9 11 F I We compute matrices B OE R 4-4 and C i OE R 4-5 for all C b a C b a C b a C b a F I F I F I F I a f a f a f a f F I Comparing the matrices C i / C j with A for all EXAMPLE. Let A OE R 9-9 be the following matrix, which constitutes the maximum of a matrix in pyramid form and a matrix in paraboloid form. F G G G G G G G G I Since matrices in paraboloid and in pyramid form are separable, Algorithm 2 should yield a weak decomposition of A in the form [u * a(o)] / [v * a(p)] for some vectors u, v OE R 9 and some indices o, p OE {1, -, 9}. Indeed, if B OE R 9-9 denotes the matrix computed by Algorithm 2, where a F G G G G G G G G I F G G G G G G G G I a f c h a f c h (44) This weak decomposition of A can be used to further decompose the square templates f(b[1] * a(1)) and f(b[3] * templates. By Theorem 6, the templates are decomposable into 3 - 3 templates if and only if the column templates f(b[1]) and are decomposable into 3 - 1 templates and the row templates f(a[1]) and f(a[3]) are decomposable into 1 - 3 templates. It is fairly easy to choose 3 - 1 templates R R 4, such that s 4 . For more complicated examples, we recommend using one of the integer programming approaches suggested in [17], [18], [19]. See Fig. 6 and Fig. 7. Hence, we obtain a representation of f(b[1] * a(1)) in the form of (45). By rearranging the templates r and s i for 4, we can achieve a decomposition of f(b[1] * a(1)) into four 3 - 3 templates Fig. 6. Templates of size 3 - 1 and size 1 - 3 providing a decomposition of the template f(b[1] * a(1)). Fig. 7. The 3 - 3 templates, providing a decomposition of the template In a similar fashion, we are able to decompose the tem- J:\PRODUCTION\TPAMI\2-INPROD\104798\104798_1.DOC regularpaper97.dot AG 19,968 04/24/97 plate f(b[3] * a(3)) into four 3 - 3 templates. REMARK. The methods for decomposing rectangular morphological templates presented in this paper can be easily generalized to include arbitrary invariant morphological templates which correspond to matrices over R -. 6 CONCLUSIONS We introduced the new theory of the separable matrix rank within minimax algebra, which we compared to the theory of matrix rank provided by Cuninghame-Green. The definition of the separable rank of a matrix leads to the concept of the rank of a rectangular morphological template, a notion which has significance for the problem of morphological template decomposition. Using this terminology, the class of separable templates represents the class of templates of rank one. A separable templates can be strongly decomposed into a product of a column template and a row template. Generalizing this decomposition of separable templates, we developed a polynomial time algorithm for the weak decomposition of a rectangular template of rank two into horizontal and vertical strip templates. We are currently working on an improved version of this algorithm. In an upcoming paper, we will show that determining the rank of an arbitrary rectangular template is an NP complete problem, and we will discuss the consequences for morphological template decomposition problems in gen- eral. Moreover, we will present a heuristic algorithm for solving the rank problem and for finding an optimal weak decomposition into strip templates. ACKNOWLEDGMENT This research was partially supported by U.S. Air Force Contract F08635 89 C-0134. --R "Biomedical Image Processing," "Parallel 2-D Convolution on a Mesh Connected Array Processor," "Morphological Structuring Element Decomposition," "Decomposition of Convex Polygonal Morphological Structuring Elements Into Neighborhood Subsets," "Optimal Decomposition of Convex Morphological Structuring Elements for 4-Connected Parallel Array Processors," available via anonymous ftp from ftp. "Recent Developments in "Necessary and Sufficient Conditions for the Existence of Local Matrix Decompositions," "Classification of Lattice Transformations in Image Processing," "The P-Product and Its Applications in Signal Processing," "Decomposition Methods for Convolution Operators," "Morphological Template Decomposition With Max- Polynomials," Maxpolynomials and Morphological Template Decomposition "Nonlinear Matrix Decompositions and an Application to Parallel Processing," "Decomposition Techniques for Gray-Scale Morphological Templates," "Local Decomposition of Gray-Scale Morphological Templates," "Global Optimization Problems in Computer Vision," "Decomposition of Separable and Symmetric Convex Templates," "Separable Decompositions and Approximations for Gray-Scale Morphological Templates," "Some Algorithms for Approximating Convolu- tions," "Proofs of Decomposition Results of Gray-Scale Morphological Templates Using the Rank Method," "Heterogeneous Algebras," Lecture Notes in Economics and Mathematical Systems --TR --CTR A. Engbers , R. Van Den Boomgaard , A. W. M. Smeulders, Decomposition of Separable Concave Structuring Functions, Journal of Mathematical Imaging and Vision, v.15 n.3, p.181-195, November 2001 Ronaldo Fumio Hashimoto , Junior Barrera , Carlos Eduardo Ferreira, A Combinatorial Optimization Technique for the Sequential Decomposition of Erosions and Dilations, Journal of Mathematical Imaging and Vision, v.13 n.1, p.17-33, August 2000

structuring element;morphology;morphological template;template rank;convolution;template decomposition

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The Matrix Sign Function Method and the Computation of Invariant Subspaces.

A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results.

Introduction . If A 2 R n\Thetan has no eigenvalue on the imaginary axis, then the matrix sign function sign(A) may be defined as Z (1) where fl is any simple closed curve in the complex plane enclosing all eigenvalues of A with positive real part. The sign function is used to compute eigenvalues and invariant subspaces [2, 4, 6, 13, 14] and to solve Riccati and Sylvester equations [9, 15, 16, 28]. The matrix sign function is attractive for machine computation, because it can be efficiently evaluated by relatively simple numerical methods. Some of these are surveyed in [28]. It is particularly attractive for large dense problems to be solved on computers with advanced architectures [2, 11, 16, 33]. Beavers and Denman use the following equivalent definition [6, 13]. Let be the Jordan canonical decomposition of a matrix A having no eigenvalues on the imaginary axis. Let the diagonal part of J be given by the matrix then (A) be the invariant subspace of A corresponding to eigenvalues with positive real part, let be the invariant subspace of A corresponding to eigenvalues with negative real part, let be the skew projection onto parallel to be the skew projection onto parallel to . Using the same contour fl as in (1), the projection P + has the resolvent integral representation [23, Page 67] [2] Z (2) To appear in SIAM Journal on Matrix Analysis and Its Applications. y University of Kansas, Dept. of Mathematics, Lawrence, Kansas 66045, USA. Partial support was received from National Science Foundation grants INT-8922444 and CCR-9404425 and University of Kansas GRF Allocation 3514-20-0038. z TU-Chemnitz-Zwickau, Fak. f. Mathematik, D-09107 Chemnitz, FRG. Partial support was received from Deutsche Forschungsgemeinschaft, Projekt La 767/3-2. It follows from (1) and (2) that The matrix sign function was introduced using definition (1) by Roberts in a 1971 technical report [34] which was not published until 1980 [35]. Kato [23, Page 67] reports that the resolvent integral (2) goes back to 1946 [12] and 1949 [21, 22]. There is some concern about the numerical stability of numerical methods based upon the matrix sign function [2, 8, 19]. In this paper, we demonstrate that evaluating the matrix sign function is a more ill-conditioned computational problem than the problem of finding bases of the invariant subspaces V Section 3. Nevertheless, we also give perturbation and error analyses, which show that (at least for Newton's method for the computation of the matrix sign function [8, 9]) in most circumstances the accuracy is competitive with conventional methods for computing invariant subspaces. Our analysis improves some of the perturbation bounds in [2, 8, 18, 24]. In Section 2 we establish some notation and clarify the relationship between the matrix sign function and the Schur decomposition. The next two sections give a perturbation analysis of the matrix sign function and its invariant subspaces. Section 5 gives a posteriori bounds on the forward and backward error associated with a corrupted value of sign(S). Section 6 is a stability analysis of the Newton iteration. Throughout the paper, k \Delta k denotes the spectral norm, k \Delta k 1 the 1-norm (or column sum norm), and k \Delta k F the Frobenius norm k \Delta k . The set of eigenvalues of a matrix A is denoted by -(A). The open left half plane is denoted by C \Gamma and the open right half plane is denoted by C Borrowing some terminology from engineering, we refer to the invariant subspace of a matrix A 2 R n\Thetan corresponding to eigenvalues in C \Gamma as the stable invariant subspace and the subspace corresponding to eigenvalues in C + as the unstable invariant subspace. We use for the the skew projection onto V + parallel to V \Gamma and for the skew projection onto 2. Relationship with the Schur Decomposition. Suppose that A has the Schur form k A 11 A 12 where Y is a solution of the Sylvester equation then and The solution of (4) has the integral representation Y =-i Z where fl is a closed contour containing all eigenvalues of A with positive real part [29, 36]). The stable invariant subspace of A is the range (or column space) of is a QR factorization with column pivoting [1, 17], where Q and R are partitioned in the obvious way, then the columns of Q 1 form an orthonormal basis of this subspace. Here Q is orthogonal, \Pi is a permutation matrix, R is upper triangular, and R 1 is nonsingular. It is not difficult to use the singular value decomposition of Y to show that s It follows from (4) that where sep is defined as in [17] by 3. The Effect of Backward Errors. In this section we discuss the sensitivity of the matrix sign function subject to perturbations. For a perturbation matrix E, we give first order estimates for E) in terms of submatrices and powers of kEk. Based on Fr'echet derivatives, Kenney and Laub [24] presented a first order perturbation theory for the matrix sign function via the solution of a Sylvester equation. Mathias [30] derived an expression for the Fr'echet derivative using the Schur decom- position. Kato's encyclopedic monograph [23] includes an extensive study of series representations and of perturbation bounds for eigenprojections. In this section we derive an expression for the Fr'echet derivative using integral formulas. Let oe min where oe min is the smallest singular value of A \Gamma -iI. The quantity dA is the distance from A to the nearest complex matrix with an eigenvalue on the imaginary axis. Practical numerical techniques for calculating dA appear in [7, 10]. If then E is too small to perturb an eigenvalue of A on or across the imaginary axis. It follows that for kEk ! dA , sign(A+E), and the stable and unstable invariant subspaces of are smooth functions of E. Consider the relatively simple case in which A is block diagonal. Lemma 3.1. Suppose A is block diagonal, where . Partition the perturbation E 2 R n\Thetan conformally with A as where F 12 and F 21 satisfy the Sylvester equations A 22 F Proof. The hypothesis that implies that the eigenvalues of A 11 have negative real part and the eigenvalues of A 22 positive real part. In the definition (1) choose the contour fl to enclose neither In particular, for all complex numbers z lying on the contour E) are nonsingular and E) =-i Z Z e Z Partitioning F conformally with E and A, then we have F 11 =2-i Z Z Z F 22 =2-i Z As in (6), F 12 and F 21 are the solutions to the Sylvester equations (11) and (12) [29, 36]. The contour fl encloses no eigenvalues of A 11 , so inside fl and F We first prove that F 22 = 0 in the case that A 22 is diagonalizable, say A where Z Each component of the above integral is of the form R constant c. If then this is the integral of a residue free holomorphic function and hence it vanishes. If j 6= k, then Z c Z c The general case follows by taking limits of the diagonalizable case and using the dominated convergence theorem. The following theorem gives the general case. Theorem 3.2. Let the Schur form of A be given by (3) and partition E conformally as and Y satisfies (4), then e ~ ~ satisfies the Sylvester equation A 22 ~ and ~ ~ ~ Proof. If A 11 A 12 and It follows from Lemma 3.1 that I ~ on the left side and SQ H on the right side of the above equation, we have Y ~ It is easy to verify that Y ~ \GammaI Y I ~ \GammaI Y I The theorem follows from \GammaI Y If dA is small relative to kAk or kY k is large relative to kAk, then the hypothesis that kSES may be restrictive. However, a small value of dA indicates that A is very near a discontinuity in sign(A). A large value of kY k indicates that is small and the stable invariant subspace is ill-conditioned [37, 39]. Of course Theorem 3.2 also gives first order perturbations for the projections Corollary 3.3. Let the Schur form of A be given as in (3) and let E be as in (13). Under the hypothesis of Theorem 3.2, the projections P E) are as in the statement of Theorem 3.2. Taking norms in Theorem 3.2 gives the first order perturbation bounds of the next corollary. Corollary 3.4. Let the Schur form of A be given as in (3), E as in (13) and then the first order perturbation of the matrix sign function stated in Theorem 3.2 is bounded by On first examination, Corollary 3.4 is discouraging. It suggests that calculating the matrix sign function may be more ill-conditioned than finding bases of the stable and unstable invariant subspace. If the matrix A whose Schur decomposition appears in (3) is perturbed to A+ E, then the stable invariant subspace, range(Q 1 ), is perturbed to 3.4 and the following example show that sign(A+E) may indeed differ from sign(A) by a factor of ffi \Gamma3 which may be much larger than kEk=ffi. Example 1. Let The matrix A is already in Schur form, so have E) =p The difference is Perturbing A to A+E does indeed perturb the matrix sign function by a factor of ffi \Gamma3 . Of course there is no rounding error in Example 1, so the stable invariant subspace of A+E is also the stable invariant subspace of sign(A+E) and, in particular, evaluating exactly has done no more damage than perturbing A. The stable invariant subspace of A is "0 ); the stable invariant subspace of A E) is For a general small perturbation matrix E, the angle between is of order no larger than O(1=j) [17, 37, 39]. The matrix sign function (and the projections may be significantly more ill-conditioned than the stable and unstable invariant subspaces. Nevertheless, we argue in this paper that despite the possible poor conditioning of the matrix sign function, the invariant subspaces are usually preserved about as accurately as their native conditioning permits. However, if the perturbation E is large enough to perturb an eigenvalue across or on the imaginary axis, then the stable and unstable invariant subspaces may become confused and cannot be extracted from E). This may occur even when the invariant subspaces are well-conditioned, since the sign function is not defined in this case. In geometric terms, in this situation A is within distance kEk of a matrix with an eigenvalue with zero real part. This is a fundamental difficulty of any method that identifies the two invariant subspaces by the signs of the real parts of the corresponding eigenvalues of A. 4. Perturbation Theory for Invariant Subspaces of the Matrix Sign Func- tion. In this section we discuss the accuracy of the computation of the stable invariant subspace of A via the matrix sign function. An easy first observation is that if the computed value of sign(A) is the exact value of sign(A+E) for some perturbation matrix E, then the exact stable invariant subspace of E) is also an invariant subspace of A+ E. Let A have Schur form (3) and let E be a perturbation matrix partitioned conformally as in (13). Let Q 1 be the first k columns of Q and Q 2 be the remaining columns. If then A has stable invariant subspace has an invariant subspace where ffl 39]. The singular values of W are the tangents of the canonical angles between In particular, the canonical angles are at most of order O(1= Unfortunately, in general, we cannot apply backward error analysis, i.e. we cannot guarantee that the computed value of sign(A) is exactly the value of E) for some perturbation E. Consider instead the effect of forward errors, let where F represents the forward error in evaluating the matrix sign function. Let A have Schur form (3). Partition Q H sign(A)Q and Q H FQ as and where Q is the unitary factor from the Schur decomposition of A (3) and Y is a solution of (4). Assume that and let OE Perturbing sign(A) to changes the invariant subspace and by (8) and s A - OE 21 Since obeys the bounds - 4OE 21 Comparing (19) with (20) we see that the error bound (20) is no greater than twice the error bound (19). Loosely speaking, a small relative error in sign(A) of size ffl might perturb the stable invariant subspace by not much more than twice as much as a relative error of size ffl in A can. Therefore, the stable and unstable invariant subspaces of sign(A) may be less ill-conditioned and are never significantly more ill-conditioned than the corresponding invariant subspaces of A. There is no fundamental numerical instability in evaluating the matrix sign function as a means of extracting invariant subspaces. However, numerical methods used to evaluate the matrix sign function may or may not be numerically unstable. Example 1 continued. To illustrate the results, we give a comparison of our perturbation bounds and the bounds given in [3] for both the matrix sign function and the invariant subspaces in the case of Example 1. The distance to the nearest ill-posed problem, i.e., is the smallest singular value of leads to an overestimation of the error in [3]. Since dA - j \Gamma2 , the bounds given in [3] are, respectively, O(j \Gamma4 ) for the matrix sign function and O(j \Gamma2 ) for the invariant subspaces. 5. A Posteriori Backward and Forward Error Bounds. A priori backward and forward error bounds for evaluation of the matrix sign function remain elusive. However, it is not difficult to derive a posteriori error bounds for both backward and forward error. We will need the following lemma to estimate the distance between a matrix S and sign(S). Lemma 5.1. If S 2 R n\Thetan has no eigenvalue with zero real part and k sign(S)S Proof. Let S. The matrices F , S, and sign(S) commute, so This implies that Taking norms and using kFS and the lemma follows. It is clear from the proof of the Lemma 5.1 that asymptotically correct as k sign(S) \Gamma Sk tends to zero. The bound in the lemma tends to overestimate smaller values of k sign(S) \Gamma Sk by a factor of two. Suppose that a numerical procedure for evaluating sign(A) applied to a matrix A 2 R n\Thetan produces an approximation S 2 R n\Thetan . Consider the problem of finding small norm solutions E 2 R n\Thetan and F 2 R n\Thetan to Of course, this does not uniquely determine E and F . Common algorithms for evaluating sign(A) like Newton's method for the square root of I guarantee that S is very nearly a square root of I [19], i.e., S is a close approximation of sign(S). In the following theorem, we have arbitrarily taken Theorem 5.2. If k sign(S)S perturbation matrices E and F satisfying and Proof. The matrices S +F and A+E commute, so an underdetermined, consistent system of equations for E in terms of S, A, and E(S Let \GammaI Y I be a Schur decomposition of sign(S) whose unitary factor is U and whose triangular factor is on the right-hand-side of (24). Partition U H EU and U H AU conformally with the right-hand-side of (24) as and A 11 A 12 A 21 A 22 Multiplying (23) on the left by U H and on the right by U and partitioning gives Y A 21 \GammaA One of the infinitely many solutions for E is given by For this choice of E, we have from which the theorem follows. Lemma 5.1 and Theorem 5.2 agree well with intuition. In order to assure small forward error, S must be a good approximate square root of I and, in addition, to assure small backward error, sign(S) must nearly commute with the original data matrix A. Newton's method for a root of I tends to do a good job of both [19]. (Note that in general, Newton's method makes a poor algorithm to find a square root of a matrix. The square root of I is a special case. See [19] for details.) In particular, the hypothesis that k sign(S)S usually satisfied when the matrix sign function is computed by the Newton algorithm. When S - sign(S), the quantity k S+S \Gamma1j S+S \Gamma1j k makes a good estimate of the right-hand-side of (21). The bound (22) is easily computed or estimated from the known values of A and S. However, these expressions are prone to subtractive cancellation of significant digits. The quantity kE 21 k is related by (18) to perturbations in the stable invariant sub- space. The bounds (21) and (22) are a fortiori bounds on kE 21 k, but, as the (1; 2) block of (25) suggests, they tend to be pessimistic overestimates of kE 21 k if kSk AE 1. 6. The Newton Iteration for the Computation of the Matrix Sign Func- tion. There are several numerical methods for computing the matrix sign function [2, 25]. Among the simplest and most commonly used is the Newton-Raphson method for a root of starting with initial guess It is easily implemented using matrix inversion subroutines from widely available, high quality linear algebra packages like LAPACK [1, 2]. It has been extensively studied and many variations have been suggested [2, 4, 5, 9, 18, 27, 25, 26, 28]. Algorithm 1. Newton Iteration (without scaling) If A has no eigenvalues on the imaginary axis, then Algorithm 1 converges globally and locally quadratically in a neighborhood of sign(A) [28]. Although the iteration ultimately converges rapidly, initially convergence may be slow. However, the initial convergence rate (and numerical stability) may be improved by scaling [2, 5, 9, 18, 27, 25, 26, 28]. A common choice is to scale X k 1=j det(X k )j (1=n) [9]. Theorem 3.2 shows that the first order perturbation of sign(A) may be as large as is the relative uncertainty in A. (If there is no other uncertainty, then ffl is at least as large as the round-off unit of the finite precision arithmetic.) Thus, it is reasonable to stop the Newton iteration when The ad hoc constant C is chosen in order to avoid extreme situations, e.g., This choice of C works well in our numerical experiments up to shows furthermore that it is often advantageous to take an extra step of the iteration after the stopping criterion is satisfied. Example 2. This example demonstrates our stopping criterion. Algorithm 1 was implemented in MATLAB 4.1 on an HP 715/33 workstation with floating point relative accuracy We constructed a is a random unitary matrix and R an upper triangular matrix with diagonal elements parameter ff in the (k; k position and zero everywhere else. We chose ff such that the norm k sign(A)k 1 varies from small to large. The typical behavior of the error is that it goes down and then becomes stationary. This behavior is shown in the Figure 1 for the cases and Stopping criterion (26) is satisfied with 1000n at the 8-th step for at the 7-th step for Taking one extra step would stop at the 9-th step for and at the 8-th step for In exact arithmetic, the stable and unstable invariant subspaces of the iterates are the same as those of A. However, in finite precision arithmetic, rounding errors perturb these subspaces. The numerical stability of the Newton iteration for computing the stable invariant subspace has been analyzed in [8], we give an improved error bound here. Let X and X be, respectively, the computed k-th and 1)-st iterate of the Newton iteration starting from A 11 A 12 Fig. 1. 2. the number of iterations log10(||X_k-S||_1) alpha=0 Suppose that X and X + have the form A successful rounding error analysis must establish the relationship between E +and . To do so we assume that some stable algorithm is applied to compute the inverse in the Newton iteration. More precisely we assume that X where for some constant c. Note that this is a nontrivial assumption. Ordinarily, if Gaussian elimination with partial pivoting is used to compute the inverse, the above error bound can be shown to hold only for each column separately [8, 38]. The better inversion algorithms applied to "typical" matrices satisfy this assumption [38, p. 151], but it is difficult to determine if this is always the case [31, pp. 22-26], [20, p. 150]. Write EX and EZ as The following theorem bounds indirectly the perturbation in the stable invariant subspace. Theorem 6.1. Let X, and EZ be as in (27), (28), (31), and (32). If2 where c is as in (29) and (30), then Proof. We start with (28). In fact the relationship between E 21 and from applying the explicit formula for the inverse of (X ~ c ~ c c c Here, ~ Then ~ c ~ c Using the Neumann lemma that if kBk ! 1, then have The following inequalities are established similarly. 22 k 22 k(kE 22 k Inserting these inequalities in (33) we obtain The bound in Theorem 6.1 is stronger than the bound of Byers in [8]. It follows from (19) and Theorem 6.1, that if then rounding errors in a step of Newton corrupt the stable invariant subspace by no more than one might expect from the perturbation E 21 in (27). The term sep(X 22 is dominated by sep So to guarantee that rounding errors in the Newton iteration do little damage, the factors in the bound of Theorem 6.1, kX \Gamma1 22 k and (kXk should be small enough so that Very roughly speaking, to have numerical stability throughout the algorithm, neither 22 k nor (kXk should be much larger than 1= The following example from [4] demonstrates numerical instability that can be traced to the violation of inequality (34). Example 3. Let A 11 =6 6 6 6 4 be a real matrix, and let A 22 = \GammaA T. Form A 11 A 12 and where the orthogonal matrix Q is chosen to be the unitary factor of the QR factorization of a matrix with entries chosen randomly uniformly distributed in the interval [0; 1]. The parameter ff is taken as so that there are two eigenvalues of A close to the imaginary axis from the left and right side. The entries of A 12 are chosen randomly uniformly distributed in the interval [0; 1], too. The entries of E 21 are chosen randomly uniformly distributed in the interval [0; eps], where \Gamma16 is the machine precision. Table Evolution of kE in Example 3. A R 3 1.2093e-07 2.5501e-08 1.6948e+00 5 7.3034e-08 5.4025e-09 2.0000 6 7.3164e-08 2.7012e-09 2.0000 7 7.2020e-08 1.3506e-09 2.0000 8 7.1731e-08 6.7532e-10 2.0000 9 7.1866e-08 3.3766e-10 2.0000 13 7.1934e-08 2.1151e-11 2.0000 14 7.1938e-08 1.0646e-11 2.0000 19 7.1937e-08 1.7291e-12 2.0000 In this example, shows the evolution of kE during the Newton iteration starting with respectively, where E 21 is as in (27). The norm is taken to be the 1-norm. Because kA \Gamma1k 1 is violated in the first step of the Newton iteration for the starting matrix A, which is shown in the first column of the table. Newton's method never recovers from this. It is remarkable, however, that Newton's method applied to R directly seems to recover from the loss in accuracy in the first step. The second column shows that although \Gamma7 at the first step, it is reduced by the factor 1=2 every step until it reaches 1:7 \Theta 10 \Gamma12 which is approximately Observe that in this case the perturbation E 00 in EZ as in (28) is zero and dominated by 1(kE 22 It is surprising to see that from the second step on kX \Gamma1 22 k 1 is as small as eps, since A \Gamma1 11 and A \Gamma1 22 do not explicitly appear in the term X \Gamma1 22 after the first step. Our analysis suggests that the Newton iteration may be unstable when X k is ill- conditioned. To overcome this difficulty the Newton iteration may be carried out with a shift along the imaginary line. In this case we have to use complex arithmetic. Algorithm 2. Newton Iteration With Shift END The real parameter fi is chosen such that oe min fiiI) is not small. For Example 2, when fi is taken to be 0:8, we have Then by our analysis the computed invariant subspace is guaranteed to be accurate. 7. Conclusions. We have given a first order perturbation theory for the matrix sign function and an error analysis for Newton's method to compute it. This analysis suggests that computing the stable (or unstable) invariant subspace of a matrix with the Newton iteration in most circumstances yields results as good as those obtained from the Schur form. 8. Acknowledgments . The authors would like to express their thanks to N. Higham for valuable comments on an earlier draft of the paper and Z. Bai and P. Benner for helpful discussions. --R Design of a parallel nonsymmetric eigenroutine toolbox Design of a parallel nonsymmetric eigenroutine toolbox Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems. Accelerated convergence of the matrix sign function method of solving Lyapunov A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues. A regularity result and a quadratically convergent algorithm for computing its L1 norm. Numerical stability and instability in matrix sign function based algorithms. Solving the algebraic Riccati equation with the matrix sign function. A bisection method for measuring the distance of a stable matrix to the unstable matrices. A systolic algorithm for Riccati and Lyapunov equations. Perturbations des transformations autoadjointes dans l'espace de Hilbert. The matrix sign function and computations in systems. Spectral decomposition of a matrix using the generalized sign matrix. A generalization of the matrix-sign-function solution for algebraic Riccati equations Parallel algorithms for algebraic Riccati equations. Matrix Computations. Computing the polar decomposition - with applications Newton's method for the matrix square root. A survey of error analysis. On the convergence of the perturbation method On the convergence of the perturbation method Perturbation Theory for Linear Operators. Polar decompositions and matrix sign function condition estimates. Rational iterative methods for the matrix sign function. On scaling Newton's method for polar decompositions and the matrix sign function. The matrix sign function. The Theory of Matrices. Condition estimation for the matrix sign function via the Schur decomposition. Software for Roundoff Analysis of Matrix Algorithms. Schur complement and statistics. A parallel algorithm for the matrix sign function. Linear model reduction and solution of algebraic Riccati equation by use of the sign function. Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. and perturbation bounds for subspaces associated with certain eigenvalue problems. Introduction to Matrix Computations. Matrix Perturbation Theory. --TR --CTR Daniel Kressner, Block algorithms for reordering standard and generalized Schur forms, ACM Transactions on Mathematical Software (TOMS), v.32 n.4, p.521-532, December 2006 Peter Benner , Maribel Castillo , Enrique S. Quintana-Ort , Vicente Hernndez, Parallel Partial Stabilizing Algorithms for Large Linear Control Systems, The Journal of Supercomputing, v.15 n.2, p.193-206, Feb.1.2000

matrix sign function;invariant subspaces;perturbation theory

263207

An Analysis of Spectral Envelope Reduction via Quadratic Assignment Problems.

A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [Barnard, Pothen, and Simon, Numer. Linear Algebra Appl., 2 (1995), pp. 317--334]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related 1- and 2-sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work in an envelope Cholesky factorization. We formulate these two problems as quadratic assignment problems and then study the 2-sum problem in more detail. We obtain lower bounds on the 2-sum by considering a relaxation of the problem and then show that the spectral ordering finds a permutation matrix closest to an orthogonal matrix attaining the lower bound. This provides a stronger justification of the spectral envelope reduction algorithm than previously known. The lower bound on the 2-sum is seen to be tight for reasonably "uniform" finite element meshes. We show that problems with bounded separator sizes also have bounded envelope parameters.

Introduction . We provide a raison d'-etre for a novel spectral algorithm to reduce the envelope of a sparse, symmetric matrix, described in a companion paper [2]. The algorithm associates a discrete Laplacian matrix with the given symmetric matrix, and then computes a reordering of the matrix by sorting the components of an eigenvector corresponding to the smallest nonzero Laplacian eigenvalue. The results in [2] show that the spectral algorithm can obtain significantly smaller envelope sizes compared to other currently used algorithms. All previous envelope-reduction algorithms (known to us), such as the reverse Cuthill-McKee (RCM) algorithm and variants [3, 16, 17, 26, 37], are combinatorial in nature, employing breadth-first-search to compute the ordering. In contrast, the spectral algorithm is an algebraic algorithm whose good envelope-reduction properties are somewhat intriguing and poorly understood. We describe problems related to envelope-reduction called the 1- and 2-sum problems, and then formulate these latter problems as quadratic assignment problems (QAPs). We show that the QAP formulation of the 2-sum enables us to obtain lower bounds on the 2-sum (and related envelope parameters) based on the Laplacian eigenvalues. The lower bounds seem to be quite tight for finite element problems when the mesh points are nearly all of the same degree, and the geometries are simple. Further, a closest permutation matrix to an orthogonal matrix that attains the lower bound is obtained, to within a linear approxima- tion, by sorting the second Laplacian eigenvector components in monotonically increasing or decreasing order. This justifies the spectral envelope-reducing algorithm more strongly than earlier results. Although initially envelope-reducing orderings were developed for use in envelope schemes for sparse matrix factorization, these orderings have been used in the past few years in several other applications. The RCM ordering has been found to be an effective pre-ordering in computing incomplete factorization preconditioners for preconditioned conjugate-gradient methods [4, 6]. Envelope-reducing orderings have been used in frontal methods for sparse matrix factorization [7]. The wider applicability of envelope-reducing orderings prompts us to take a fresh look at the reordering algorithms currently available, and to develop new ordering algorithms. Spectral envelope-reduction algorithms seem to be attractive in this context, since they (i) compare favorably with existing algorithms in terms of the quality of the orderings [2], (ii) extend easily to problems with weights, e.g., finite element meshes arising from discretizations of anisotropic problems, and (iii) are fairly easily parallelizable. Spectral algorithms are more expensive than the other algorithms currently available. But since the envelope-reduction problem requires only one eigenvector computation (to low pre- cision), we believe the costs are not impractically high in computation-intensive applications, e.g., frontal methods for factorization. In contexts where many problems having the same structure must be solved, a substantial investment in finding a good ordering might be justi- fied, since the cost can be amortized over many solutions. Improved algorithms that reduce the costs are being designed as well [25]. We focus primarily on the class of finite element meshes arising from discretizations of partial differential equations. Our goals in this project are to develop efficient software implementing our algorithms, and to prove results about the quality of the orderings generated. The projection approach for obtaining lower bounds of a QAP is due to Hadley, Rendl, and Wolkowicz [19], and this approach has been applied to the graph partitioning problem by the latter two authors [35]. In earlier work a spectral approach for the graph (matrix) partitioning problem has been employed to compute a spectral nested dissection ordering for sparse matrix factorization, for partitioning computations on finite element meshes on a distributed-memory multiprocessor [21, 33, 34, 36], and for load-balancing parallel computations [22]. The spectral approach has also been used to find a pseudo-peripheral node [18]. Juvan and Mohar [23, 24] have provided a theoretical study of the spectral algorithm for reducing p-sums, where et al. [20] obtain spectral lower bounds on the bandwidth. A survey of some of these earlier results may be found in [31]. Paulino et al. [32] have also considered the use of spectral envelope-reduction for finite element problems. The following is an outline of the rest of this paper. In Section 2 we describe various parameters of a matrix associated with its envelope, introduce the envelope size and envelope work minimization problems, and the related 1- and 2-sum problems. We prove that bounds on the minimum 1-sum yield bounds on the minimum envelope size, and similarly, bounds on the minimum 2-sum yield bounds on the work in an envelope Cholesky factorization. We also show in this section that minimizing the 2-sum is NP-complete. We compute lower bounds for the envelope parameters of a sparse symmetric matrix in terms of the eigenvalues of the Laplacian matrix in Section 3. The popular RCM ordering is obtained by reversing the Cuthill-McKee (CM) ordering; the RCM ordering can never have a larger envelope size and work than the CM ordering, and is usually significantly better. We prove that reversing an ordering can improve or impair the envelope size by at most a factor \Delta, and the envelope work by at most \Delta 2 , where \Delta is the maximum degree of a vertex in the adjacency graph. In Section 4, we formulate the 2- and 1-sum problems as quadratic assignment problems. We obtain lower and upper bounds for the 2-sum problem in terms of the eigenvalues of the Laplacian matrix in Section 5 by means of a projection approach that relaxes a permutation matrix to an orthogonal matrix with row and column sums equal to one. We justify the spectral envelope-reduction algorithm in Section 6 by proving that a closest permutation matrix to an orthogonal matrix attaining the lower bound for the 2-sum is obtained, to within a linear approximation of the problem, by permuting the second Laplacian eigenvector in monotonically increasing or decreasing order. In Section 7 we show that graphs with small separators have small envelope parameters as well, by considering a modified nested dissection ordering. We present computational results in Section 8 to illustrate that the 2-sums obtained by the spectral reordering algorithm can be close to optimal for many finite element meshes. Section 9 contains our concluding remarks. The Appendix contains some lower bounds for the more general p-sum problem, where 1 2. A menagerie of envelope problems. 2.1. The envelope of a matrix. Let A be an n \Theta n symmetric matrix with elements whose diagonal elements are nonzero. Various parameters of the matrix A associated with its envelope are defined below. We denote the column indices of the nonzeros in the lower triangular part of the ith row by For the ith row of A we define Here is the column index of the first nonzero in the ith row of A (by our assumption of nonzero diagonals, 1 - f i - i), and the parameter r i (A) is the row-width of the ith row of A. The bandwidth of A is the maximum row-width The envelope of A is the set of index pairs For each row, the column indices lie in an interval beginning with the column index of the first nonzero element and ending with (but not including) the index of the diagonal nonzero element. We denote the size of the envelope by (which includes the diagonal elements) is called the profile of A [7].) The work in the Cholesky factorization of A that employs an envelope storage scheme is bounded from above by This bound is tight [29] when an ordering satisfies (1) f between 1 and n, and A 3 \Theta 3 7-point grid and the nonzero structure of the corresponding matrix A are shown in Figure 2.1. A ' ffl ' indicates a nonzero element, and a ` \Lambda ' indicates a zero element that belongs to the lower triangle of the envelope in the matrix. The row-widths given in Table 2.1 are easily verified from the structure of the matrix. The envelope size is obtained by summing the row-widths, and is equal to 24. (Column-widths c i are defined later in this section.) The values of these parameters strongly depend on the choice of an ordering of the rows and columns. Hence we consider how these parameters vary over symmetric permutations of a matrix A, where P is a permutation matrix. We define Esize min (A), the minimum envelope size of A, to be the minimum envelope size among all permutations P T AP of A. The quantities Wbound min (A) and bw min (A) are defined in similar fashion. Minimizing the envelope size and the bandwidth of a matrix are NP-complete problems [28], and minimizing Fig. 2.1. An ordering of 7-point grid and the corresponding matrix. The lower triangle of the envelope is indicated by marking zeros within it by asterisks. Table Row-widths and column-widths of the matrix in Figure 1. the work bound is likely to be intractable as well. So one has to settle for heuristic orderings to reduce these quantities. It is helpful to consider a "column-oriented" expression for the envelope size for obtaining a lower bound on this quantity in Section 3. The width of a column j of A is the number of row indices in the jth column of the envelope of A. In other words, (This is also called the jth front-width.) It is then easily seen that the envelope size is The work in an envelope factorization scheme is given by where we have ignored the linear term in c j . The column-widths of the matrix in Figure 2.1 are given in Table 2.1. These concepts and their inter-relationships are described by Liu and Sherman [29], and are also discussed in the books [5, 15]. The envelope parameters can also be defined with respect to the adjacency graph (V; E) of A. Denote In terms of the graph G and an ordering ff of its vertices, we can define Hence we can write the envelope size and work associated with an ordering ff as Esize(G; Wbound(G; The goal is to choose a vertex ordering ng to minimize one of the parameters described above. We denote by Esize min (G) (Wbound min (G)) the minimum value of Esize(G; ff) (Wbound(G; ff)) over all orderings ff. The reader can compute the envelope size of the numbered graph in Figure 2.1, using the definition given in this paragraph, to verify that The jth front-width has an especially nice interpretation if we consider the adjacency graph E) of A. Let the vertex corresponding to a column j of A be numbered v j so that g. Denote a subset of vertices X. Then c To illustrate the dependence of the envelope size on the ordering, we include in Figure 2.2 an ordering that leads to a smaller envelope size for the 7-point grid. Again, a ' ffl ' indicates a nonzero element, and a ' ' indicates a zero element that belongs to the lower triangle of the envelope in the matrix. This ordering by 'diagonals' yields the optimal envelope size for the 7-point grid [27]. 2.2. 1- and 2-sum problems. It will be helpful to consider quantities related to the envelope size and envelope work, the 1-sum and the 2-sum. For real 1 we define the p-sum to be Minimizing the 1-sum is the optimal linear arrangement problem, and the limiting case corresponds to the minimum bandwidth problem; both these are well-known NP-complete problems [13]. We show in the Section 2.3 that minimizing the 2-sum is NP-complete as well. We write the envelope size and 1-sum, and the envelope work and the 2-sum, in a way that shows their relationships: Fig. 2.2. Another ordering of a 7-point grid and the corresponding matrix. Again the lower triangle of the envelope is indicated by marking the zeros within it by asterisks. The parameters oe are the minimum values of these parameters over all symmetric permutations P T AP of A. We now consider the relationships between bounds on the envelope size and the 1-sum, and between the upper bound on the envelope work and the 2-sum. Let \Delta denote the maximum number of offdiagonal nonzeros in a row of A. (This is the maximum vertex degree in the adjacency graph of A.) Theorem 2.1. The minimum values of the envelope size, envelope work in the Cholesky factorization, 1-sum, and 2-sum of a symmetric matrix A are related by the following inequalities \DeltaEsize min (A); Proof. We begin by proving (2.8). Our strategy will be to first prove the inequalities and then to obtain the required result by considering two different permutations of A. The bound Wbound(A) - oe 2 is immediate from equations (2.5) and (2.6). If the inner sum in the latter equation is bounded from above by then we get \DeltaWbound(A) as an upper bound on the 2-sum. Now let X 1 be a permutation matrix such that f Wbound min (A). Then we have Further, let X 2 be a permutation matrix such that f Again, we have We obtain the result by putting the last two inequalities together. We omit the proof of (2.7) since it can be obtained by a similar argument, and proceed to prove (2.9). The first inequality oe holds since the p-norm of any real vector is a decreasing function of p. The second inequality is also standard, since it bounds the 1-norm of a vector by means of its 2-norm. This result was obtained earlier by Juvan and Mohar [24]; we include its proof for completeness. Applying the Cauchy-Schwarz inequality to oe 2 We obtain the result by considering two orderings that achieve the minimum 1- and 2-sums.2.3. Complexity of the 2-sum problem. We proceed to show that minimizing the 2-sum is NP-complete. In Section 8 we show that the spectral algorithm computes a 2-sum within a factor of two for the finite element problems in our test collection. This proof shows that despite the near-optimal solutions obtained by the spectral algorithm on this test set, it is unlikely that a polynomial time algorithm can be designed for computing the minimum 2-sum. Readers who are willing to accept the complexity of this problem without proof should skip this section; we recommend that everyone do so on a first reading. Given a graph E) on n vertices, MINTWOSUM is the problem of deciding if there exists a numbering of its vertices ng such that k, for a given positive integer k. This is the decision version of the problem of minimizing the 2-sum of G. Theorem 2.2. MINTWOSUM is NP-complete. Remark. This proof follows the framework for the NP-completeness of the 1-sum problem in Even [8] (Section 10.7); but the details are substantially different. Proof. The theorem will follow if we show that MAXTWOSUM, the problem of deciding whether a graph G 0 on n vertices has a vertex numbering with 2-sum greater than or equal to a given positive integer k 0 , is NP-complete. For, the 2-sum of G 0 under some ordering is at least k 0 if and only if the 2-sum of the complement of G 0 under the same ordering is at most is the 2-sum of the complete graph. We show that MAXTWOSUM is NP-complete by a reduction from MAXCUT, the problem of deciding whether a given graph E) has a partition of its vertices into two sets fS; V n Sg such that jffi(S; V n S)j, the number of edges joining S and V n S, is at least a given positive integer k. From the graph G we construct a graph G E) by adding n 4 isolated vertices to V and no edges to E. We claim that G has a cut of size at least k if and only if G 0 has a 2-sum at least k If G has a cut (S; V n S) of size at least k, define an ordering ff 0 of G 0 by interposing the n 4 isolated vertices between S and V n S: number the vertices in S first, the isolated vertices next, and the vertices in V n S last, where the ordering among the vertices in each set S and V n S is arbitrary. Every edge belonging to the cut contributes at least n 8 to the 2-sum, and hence its value is at least k \Delta n 8 . The converse is a little more involved. Suppose that G 0 has an ordering ff with 2-sum greater than or equal to k \Delta n 8 . The ordering ff 0 of G 0 induces a natural ordering ng of G, if we ignore the isolated vertices and maintain the relative ordering of the vertices in V . For each ig. Then each pair (S a cut in G. Further, each such cut in G induces a cut (S 0 in the larger graph G 0 as follows: The vertex set S 0 i is formed by augmenting S i with the isolated vertices numbered lower than the highest numbered (non-isolated) vertex in S i (with respect to the ordering We now choose a cut (S maximizes the "1-sum over the cut edges" from among the n cuts (S 0 By means of this cut and the ordering ff 0 , we define a new ordering fi 0 by moving the isolated vertices in the ordered set S 0 to the highest numbers in that set, and by moving the isolated vertices in V 0 n S 0 to the lowest numbers in that set, and preserving the relative ordering of the other vertices. The effect is to interpose the isolated vertices in "between" the two sets of the cut. Claim. The 2-sum of the graph G 0 under the ordering fi 0 is greater than that under ff 0 . To prove the claim, we examine what happens when an isolated vertex x belonging to S 0 is moved to the higher end of that ordered set. Define three sets A 0 , B 0 , C 0 as follows: The set A 0 (B 0 ) is the set of vertices in S 0 numbered lower (higher) than x in the ordering ff 0 , and C the edges joining A 0 and those joining A 0 and C 0 . Denote the contribution, with respect to the ordering ff 0 , of an edge e k to the 1-sum by a k , and that of an edge e l l . Then the change in the 2-sum due to moving x is (b l The third term on the right-hand-side is the contribution to the 1-sum made by the edges in the cut while the fourth term is the contribution made by the edges E 1 in the cut By the choice of the cut (S that the difference is positive, and hence that the 2-sum has increased in the new ordering obtained from ff 0 by moving the vertex x. We now show that after moving the vertex x, continues to be a cut that maximizes the 1-sum over the cut edges among all cuts (S 0 respect to the new ordering. For this cut, the 1-sum over cut edges has increased by jE 2 j because the number of each vertex in B has decreased by one in the new ordering. Among cuts with one set equal to an ordered subset of A 0 , the 1-sum over cut edges can only decrease when x is moved, since the set B 0 moves closer to A 0 , and C 0 does not move at all relative to A 0 . Now consider cuts of the form 1 an ordered subset of B 0 , and B 0 . The cut edges now join A 0 to B 0 . The edges joining A 0 to B 0 contribute a smaller value to the 1-sum in the new ordering relative to ff 0 , while the edges joining A 0 to C 0 contribute the same to the 1-sum in both cuts A under the new ordering. The edges joining B 0 2 do not change their contribution to the 1-sum in the new ordering. The edges that join B 0 1 and C 0 form a subset of the edges that join B 0 and hence the contribution of the former to the 1-sum is no larger than the contribution of the latter set in the new ordering. This shows that the cut continues to have a 1-sum over the cut edges larger than or equal to that of any cut Finally, any cut that includes A 0 , B 0 , and an ordered subset C 0 1 of C 0 can be shown by similar reasoning to not have a larger 1-sum than (S The reasoning in the previous paragraph permits us to move the isolated vertices in S 0 one by one to the higher end of that set without decreasing the 2-sum while simultaneously preserving the condition that the cut (S has the maximum value of the 1-sum over the cut edges. The argument that we can move the isolated vertices in V 0 nS 0 to the beginning of that ordered set follows from symmetry since both the 2-sum and the 1-sum are unchanged when we reverse an ordering. Hence by inducting over the number of isolated vertices moved, the ordering fi 0 has a 2-sum at least as large as the ordering ff 0 . This completes the proof of the claim. The rest of the proof involves computing an upper bound on the 2-sum of the graph G 0 under the ordering fi 0 to show that since G 0 has 2-sum greater than k 0 , the graph G has a cut of size at least k. )j. The cut edges contribute at most ffi \Delta (n 4 to the upper bound on the 2-sum; the uncut edges contribute at most the 2-sum of a complete graph on vertices. The latter is p(n) Thus we have, keeping only leading terms, The second term on the left hand side is less than 1 for n ? 2 since the number of cut edges ffi is at most n 2 =2; the third term is less than one for all n. The sum of these two terms is less than 1 for n ? 2. Hence we conclude that the graph G has a cut with at least k edges. This completes the proof of the theorem. 2 3. Bounds for envelope size. In this section we present lower bounds for the minimum envelope size and the minimum work involved in an envelope-Cholesky factorization in terms of the second Laplacian eigenvalue. We will require some background on the Laplacian matrix. 3.1. The Laplacian matrix. The Laplacian matrix Q(G) of a graph G is the n \Theta n is the diagonal degree matrix and M is the adjacency matrix of G. If G is the adjacency graph of a symmetric matrix A, then we could define the Laplacian matrix Q directly from A: Note that The eigenvalues of Q(G) are the Laplacian eigenvalues of G, and we list them as - 1 corresponding to - k (Q) will be denoted by x k , and will be called a kth eigenvector of Q. It is well-known that Q is a singular M-matrix, and hence its eigenvalues are nonnegative. Thus - 1 0, and the corresponding eigenvector is any nonzero constant vector c. If G is connected, then Q is irreducible, and then - smallest nonzero eigenvalues and the corresponding eigenvectors have important properties that make them useful in the solution of various partitioning and ordering problems. These properties were first investigated by Fiedler [9, 10]; as discussed in Section 1, more recently several authors have studied their application to such problems. 3.2. Laplacian bounds for envelope parameters. It will be helpful to work with the "column-oriented" definition of the envelope size. Let the vertex corresponding to a column j of A be numbered v j in the adjacency graph so that g. Recall that the column width of a vertex v j is c and that the envelope size of G (or A) is Recall also that \Delta denotes the maximum degree of a vertex. Given a set of vertices S, we denote by ffi(S) the set of edges with one endpoint in S and the other in V n S. We make use of the following elementary result, where the lower bound is due to Alon and Milman [1] and the upper bound is due to Juvan and Mohar [24]. Lemma 3.1. Let S ae V be a subset of the vertices of a graph G. Then Theorem 3.2. The envelope size of a symmetric matrix A can be bounded in terms of the eigenvalues of the associated Laplacian matrix as Proof. From Lemma 3.1, Now substituting the lower bound for jffi(V j )j, and summing this latter expression over all j, we obtain the lower bound on the envelope size. The upper bound is obtained by using the inequality c with the upper bound in Lemma 3.1. 2 A lower bound on the work in an envelope-Cholesky factorization can be obtained from the lower bound on the envelope size. Theorem 3.3. A lower bound on the work in the envelope-Cholesky factorization of a symmetric positive definite matrix A is Proof. The proof follows from Equations 2.1 and 2.2, by an application of the Cauchy-Schwarz inequality. We omit the details. 2 Cuthill and McKee [3] proposed one of the earliest ordering algorithms for reducing the envelope size of a sparse matrix. George [14] discovered that reversing this ordering leads to a significant reduction in envelope size and work. The envelope parameters obtained from the (RCM) ordering are never larger than those obtained from CM [29]. The RCM ordering has become one of the most popular envelope size reducing orderings. However, we do not know of any published quantitative results on the improvement that may be expected by reversing an ordering, and here we present the first such result. For degree-bounded finite element meshes, no asymptotic improvement is possible; the parameters are improved only by a constant factor. Of course, in practice, a reduction by a constant factor could be quite significant. Theorem 3.4. Reversing the ordering of a sparse symmetric matrix A can change (improve or impair) the envelope size by at most a factor \Delta, and the envelope work by at most Proof. Let v j denote the vertex in the adjacency graph corresponding to the jth column of A (in the original ordering) so that the jth column width c g. Let e A denote the permuted matrix obtained by reversing the column and row ordering of A. We have the inequality (A), summing this inequality over j from one to n, we obtain A). By symmetry, the inequality Esize( e holds as well. The inequality on the envelope work follows by a similar argument from the equation 4. Quadratic assignment formulation of 2- and 1-sum problems. We formulate the 2- and 1-sum problems as quadratic assignment problems in this section. 4.1. The 2-sum problem. Let the vector , and let ff be a permutation vector, i.e., a vector whose components form a permutation of may permutation matrix with elements It is easily verified that the (ff(i); ff(j)) element of the permuted matrix X T AX is the element a ij of the unpermuted matrix A. Let ij. We denote the set of all permutation vectors with n components by S n . We write the 2-sum as a quadratic form involving the Laplacian matrix Q. a The transformation from the second to the third line makes use of (3.1). This quadratic form can be expressed as a quadratic assignment problem by substituting min There is also a trace formulation of the QAP in which the variables are the elements of the permutation matrix X. We obtain this formulation by substituting Xp for ff. Thus min We may consider the last scalar expression as the trace of a 1 \Theta 1 matrix, and then use the identity tr tr NM to rewrite the right-hand-side of the last displayed equation as tr tr This is a quadratic assignment problem since it is a quadratic in the unknowns x ij , which are the elements of the permutation matrix X. The fact that B is a rank-one matrix leads to great simplifications and savings in the computation of good lower bounds for the 2-sum problem. 4.2. The 1-sum problem. Let M be the adjacency matrix of a given symmetric matrix A and let S denote a 'distance matrix' with elements both of order n. Then oe Unlike the 2-sum, the matrices involved in the QAP formulation of the 1-sum are both of rank n. Hence the bounds we obtain for this problem by this approach are considerably more involved, and will not be considered here. 5. Eigenvalue bounds for the 2-sum problem. 5.1. Orthogonal bounds. A technique for obtaining lower (upper) bounds for the QAP min permutation matrix; is to relax the requirement that the minimum (maximum) be attained over the class of permutation matrices. Let n) denote the normalized n-vector of all ones. A matrix X of order n is a permutation matrix if and only if it satisfies the following three constraints: (5. The first of these, the stochasticity constraint , expresses the fact that each row sum or column sum of a permutation matrix is one; the second states that a permutation matrix is orthogonal; and the third that its elements are non-negative. The simplest bounds for a QAP are obtained when we relax both the stochasticity and non-negativity constraints, and insist only that X be orthonormal. The following result is from [11]; see also [12]. Theorem 5.1. Let the eigenvalues of a matrix be ordered Then, as X varies over the set of orthogonal matrices, the following upper and lower bounds hold: The Laplacian matrix Q has - 1 1). Hence the lower bound in the theorem above is zero, and the upper bound is (1=6)- n (Q)n(n 5.2. Projection bounds. Stronger bounds can be obtained by a projection technique described by Hadley, Rendl, and Wolkowicz [19]. The idea here is to satisfy the stochasticity constraints in addition to the orthonormality constraints, and relax only the non-negativity constraints. This technique involves projecting a permutation matrix X into a subspace orthogonal to the stochasticity constraints (5.1) by means of an eigenprojection. Let the n \Theta be an orthonormal basis for the orthogonal complement of u. By the choice of V , it satisfies two properties: V T is an orthonormal matrix of order n. Observe that This suggests that we take Note that with this choice, the stochasticity constraints are satisfied. Furthermore, if X is an orthonormal matrix of order n satisfying is orthonormal, and this implies that Y is an orthonormal matrix of order n \Gamma 1. Conversely, if Y is orthonormal of order then the matrix X obtained by the construction above is orthonormal of order n. The non-negativity constraint X - 0 becomes, from (5.4), These facts will enable us to express the original QAP in terms of a projected QAP in the matrix of variables Y . To obtain the projected QAP, we substitute the representation of X from (5.4) into the objective function tr QXBX T . Since by the construction of the Laplacian, terms of the form Qu tr where we use the identity tr for an n \Theta k matrix M and a k \Theta n matrix N . Again this term is zero since u T Hence the only nonzero term in the objective function is tr where c is a projection of a matrix M . We have obtained the projected QAP in terms of the matrix Y of order the constraint that X be a permutation matrix now imposes the constraints that Y is orthonormal and that V Y V T - \Gammau u T . We obtain lower and upper bounds in terms of the eigenvalues of the matrices b by relaxing the non-negativity constraint again. Theorem 5.2. The following upper and lower bounds hold for the 2-sum problem: Proof. If we apply the orthogonal bounds to the projected QAP, we get The vector u is the eigenvector of Q corresponding to the zero eigenvalue, and hence eigen-vectors corresponding to higher Laplacian eigenvalues are orthogonal to it. Thus any such eigenvector x j can be expressed as x . Substituting this last equation into the eigenvalue equation pre-multiplying by V T , we obtain b Hence Q). Also, - are zero. Hence it remains to compute the largest eigenvalue of b B. From the representation I We get the result by substituting these eigenvalues into the bounds for the 2-sum. 2 For justifying the spectral algorithm for minimizing the 2-sum, we observe that the lower bound is attained by the matrix where R (S) is a matrix of eigenvectors of b B), and the eigenvectors correspond to the eigenvalues of b B) in non-decreasing (non-increasing) order. The result given above has been obtained by Juvan and Mohar [24] without using a QAP formulation of the 2-sum. We have included this proof for two reasons: First, in the next subsection, we show how the lower bound may be strengthened by diagonal perturbations of the Laplacian. Second, in the following section, we consider the problem of finding a permutation matrix closest to the orthogonal matrix attaining the lower bound. 5.3. Diagonal perturbations. The lower bound for the 2-sum can be further improved by perturbing the Laplacian matrix Q by a diagonal matrix Diag(d), where d is an n-vector, and then using an optimization routine to maximize the smallest eigenvalue of the perturbed matrix. Choosing the elements of d such that its elements sum to zero, i.e., u T simplifies the bounds we obtain, and hence we make this assumption in this subsection. We begin by denoting expressing The second term can be written as a linear assignment problem (LAP) since one of the matrices involved is diagonal. Let the permutation vector denote the n-vector formed from the diagonal elements of B. tr We now proceed, as in the previous subsection, to obtain projected bounds for the quadratic term, and thus for f(X). Note that n )d since since the elements of d sum to zero. We let the row-sum of the elements of B. With notation as in the previous subsection, we substitute in the quadratic term in f(X). The first term tr Q(d)u u T Bu u second and third terms are equal, and their sum can be transformed as follows: Note that this term is linear in the projected variables Y , and we shall find it convenient to express it in terms of X by the substitution X since the second term is equal to tr u T d r(B) T u, which is zero by the choice of d. Finally, the fourth term becomes tr b and as before Putting it all together, we obtain Observe that the first term is quadratic in the projected variables Y , and the remaining terms are linear in the original variables X. Our lower bound for the 2-sum shall be obtained by minimizing the quadratic and linear terms separately. We can simplify the linear assignment problem by noting that sq(p), the vector with ith component equal to i 2 . Hence the final expression for the linear assignment problem is tr d The minimum value of this problem, denoted by L(d) (the minimum over the permutation matrices X, for a given d), can be computed by sorting the components of d and The eigenvalues of b B can be computed as in the previous subsection. We may choose d to maximize the lower bound. Thus this discussion leads to the following result. Theorem 5.3. The minimum 2-sum of a symmetric matrix A can be bounded as d where the components of the vector d sum to zero. 2 6. Computing an approximate solution from the lower bound. Consider the problem of finding a permutation matrix Z "closest" to an orthogonal matrix X 0 that attains the lower bound in Theorem 5.2. We show in this section that sorting the second Laplacian eigenvector components in non-increasing (also non-decreasing) order yields a permutation matrix that solves a linear approximation to the problem. This justifies the spectral approach for minimizing the 2-sum. From (5.5), the orthogonal matrix X a matrix of eigenvectors of b B) corresponding to the eigenvalues of b B) in increasing (decreasing) order. We begin with a preliminary discussion of some properties of the matrix X 0 and the eigenvectors of Q. For let the jth column of R be denoted by r j , and similarly let s j denote the jth column of S. Then s c is a normalization the vector s j is orthogonal to V T p, i.e., Recall from the previous section that a second Laplacian eigenvector x Now we can formulate the "closest" permutation matrix problem more precisely. The minimum 2-sum problem may be written as min Z 2: We have chosen a positive shift ff to make the shifted matrix positive definite and hence to obtain a weighted norm by making the square root nonsingular. It can be verified that the shift has no effect on the minimizer since it adds only a constant term to the objective function. We substitute expand the 2-sum about X 0 to obtain 2: The first term on the right-hand-side is a constant since X 0 is a given orthogonal matrix; the third term is a quadratic in the difference hence we neglect it to obtain a linear approximation. It follows that we can choose a permutation matrix Z close to X 0 to approximately minimize the 2-sum by solving min Z Z Substituting for X 0 from (5.5) in this linear assignment problem and noting that we find min Z Z Z tr QV RS The second term on the right-hand-side is a constant since tr u u T BZ Here we have substituted Z T We proceed to simplify the first term in (6.4), which is tr QV RS From (6.1) we find that s j hence only the first term in the sum survives. Noting that s becomes The third term in (6.4) can be simplified in like manner, and hence ignoring the constant second term, this equation becomes c(- Z Hence we are required to choose a permutation matrix Z to minimize tr x . The solution to this problem is to choose Z to correspond to a permutation of the components of x 2 in non-increasing order, since the components of the vector p are in increasing order. Note that \Gammax 2 is also an eigenvector of the Laplacian matrix, and since the positive or negative signs of the components are chosen arbitrarily, sorting the eigenvector components into non-decreasing order also gives a permutation matrix Z closest, within a linear approximation, to a different choice for the orthogonal matrix X 0 (see 5.5). Similar techniques can be used to show that if one is interested in maximizing the 2-sum, then a closest permutation matrix to the orthogonal matrix that attains the upper bound in Theorem 5.2 is approximated by sorting the components of the Laplacian eigenvector x n (corresponding to the largest eigenvalue - n (Q)) in non-decreasing (non-increasing) order. 7. Asymptotic behavior of envelope parameters. In this section, we first prove that graphs with good separators have asymptotically small envelope parameters, and next study the asymptotic behavior of the lower bounds on the envelope parameters as a function of the problem size. 7.1. Upper bounds on envelope parameters. Let ff, fi, and fl be constants such that (1=2) - ff; class of graphs G has n fl - separators if every graph G on n ? n 0 vertices in G can be partitioned into three sets A, B, S such that no vertex in A is adjacent to any vertex in B, and the number of vertices in the sets are bounded by the relations jAj; jBj - ffn and jSj - fin fl . If n - n 0 , then we choose the separator S to consist of the entire graph. If n ? n 0 , then by the choice of n 0 , and we separate the graph into two parts A and B by means of a separator S. The assumption that fl is at least a half is not a restriction for the classes of graphs that we are interested in here: Planar graphs have n 1=2 -separators, and overlap graphs [30] embedded in d - 2 dimensions, have n (d\Gamma1)=d -separators. The latter class includes "well-shaped" finite element graphs in d dimensions, i.e., finite element graphs with elements of bounded aspect ratio. Theorem 7.1. Let G be a class of graphs that has n fl -separators and maximum vertex degree bounded by \Delta. The minimum envelope size Esize min (G) of any graph G 2 G on n vertices is O(n 1+fl ). Proof. If then we order the vertices of G arbitrarily. Otherwise, let a separator separate G into the two sets A and B, where we choose the subset B to have no more vertices than A. We consider a "modified nested dissection" ordering of G that orders the vertices in A first, the vertices in S next, and the vertices in B last. (See the ordering in Figure 2.1, where S corresponds to the set of vertices in the middle column.) The contribution to the envelope E S made by the vertices in S is bounded by the product of the maximum row-width of a vertex in S and the number of vertices in S. Thus We also consider the contribution made by vertices in B that are adjacent to nodes in S, as a consequence of numbering the nodes in S. There are at most \DeltajSj such vertices in B. Since these vertices are not adjacent to any vertex in A, the contribution EB made by them is the number of vertices in the subset A (B). Adding the contributions from the two sets of nodes in the previous paragraph, we obtain the recurrence relation We claim that for suitable constants C 1 and C 2 to be chosen later. We prove the claim by induction on n. For the claim may be satisfied by choosing C 1 to be greater than or equal to Now consider the case when n ? n 0 . Let the maximum in the recurrence relation (7.1) be attained for an and n have thus the inductive hypothesis can be applied to the subgraphs induced by A and B. Hence we substitute the bound (7.2) into the recurrence relation (7.1) to obtain log an For the claim to be satisfied, this bound must be less than the right-hand-side of the inequality (7.2). We prove this by considering the coefficients of each of the terms n 1+fl and Consider the n 1+fl term first. It is easy to see that a 1+fl positive. Furthermore, this expression attains its maximum when a is equal to ff. Denote this maximum value by ffl j ff 1+fl 1. Equating the coefficients of n 1+fl in the recurrence relation, if then the first term in the claimed asymptotic bound on E(n) would be true. Both this inequality and the condition on C 1 imposed by n 0 are satisfied if we choose We simplify the coefficient of the n 2fl term a bit before proceeding to analyze it. We have a 2fl log an - a 2fl log an log an - log ff n j ' log ff n log ff n: In the transformations we have used the following facts: 1 \Gamma a - a, since a - 1=2; the maximum of a 2fl and 2fl is greater than or equal to one, is attained for a = ff; this maximum value ' is less than one. Hence for the claim to hold, we require This last inequality is satisfied if we choose log ff \Gamma1 :A similar proof yields Wbound min which is an upper bound on the work in an envelope-Cholesky factorization. Hence good separators imply small envelope size and work. Although we have used a "modified nested dissection" ordering to prove asymptotic upper bounds, we do not advocate the use of this ordering for envelope-reduction. Other envelope-reducing algorithms considered in this paper are preferable, since they are faster and yield smaller envelope parameters. 7.2. Asymptotic behavior of lower bounds. In this subsection we consider the implications of the spectral lower bounds that we have obtained. We denote the eigenvalue for the sake of brevity in this subsection. We use the asymptotic behavior of the second eigenvalues together with the lower bounds we have obtained to predict the behavior of envelope parameters. For the envelope size, we make use of Theorem 3.2; for the envelope work, we employ Theorem 3.3. The bounds on envelope parameters are tight for dense and random graphs (matri- ces). For instance, the full matrix (the complete graph) has - Esize Similarly, the bound on the envelope work Ework min The predicted lower bound is within a factor of three of the envelope size. These bounds are also asymptotically tight for random graphs where each possible edge is present in the graph with a given constant probability p, since the second Laplacian eigenvalue satisfies [23] More interesting are the implications of these bounds for degree-bounded finite element meshes in two and three dimensions. We will employ the following result proved recently by Spielman and Teng [38]. Theorem 7.2. The second Laplacian eigenvalue of an overlap graph embedded in d- dimensions is bounded by O(n \Gamma2=d ). 2 problem separator - 2 size LB UB LB UB d-dim. O(n 1\Gamma1=d Table Asymptotic upper and lower bounds on envelope size and work for an overlap graph in d dimensions. Planar graphs are overlap graphs in 2 dimensions, and well-shaped meshes in 3 dimensions are also overlap graphs with Table 7.1 summarizes the asymptotic lower and upper bounds on the envelope parameters for a well-shaped mesh embedded in d dimensions. The most useful values are As before, the lower bound on the envelope size is from Theorem 3.2, while the lower bound on the envelope work is from Theorem 3.3. The upper bound on the envelope size follows from Theorem 7.1, and the upper bound on envelope work follows from the upper bound on Wbound(A), discussed at the end of the proof of that theorem. The lower bounds are obtained for problems where the upper bounds on the second eigenvalue are asymptotically tight. This is reasonable for many problems, for instance model problems in Partial Differential Equations. Note that the regular finite element mesh in a discretization of Laplace's equation in two dimensions (Neumann boundary conditions) has is the smallest diameter of an element (smallest mesh spacing for a finite difference mesh). The regular three-dimensional mesh in the discretized Laplace's equation with Neumann boundary conditions satisfies - For planar problems, the lower bound on the envelope size is \Omega\Gamma n), while the upper bound is O(n 1:5 ). For well-shaped three-dimensional meshes, these bounds O(n 5=3 ). The lower bounds on the envelope work are weaker since they are obtained from the corresponding bounds on the envelope size. Direct methods for solving sparse systems have storage requirements bounded by O(n log n) and work bounded by O(n 1:5 ) for a two-dimensional mesh; in well-shaped three dimensional meshes, these are O(n 4=3 ) and O(n 2 ). These results suggest that when a two-dimensional mesh possesses a small second Laplacian eigenvalue, envelope methods may be expected to work well. Similar conclusions should hold for three-dimensional problems when the number of mesh-points along the third dimension is small relative to the number in the other two dimensions, and for two-dimensional surfaces embedded in three-dimensional space. 8. Computational results. We present computational results to verify how well the spectral ordering reduces the 2-sum. We report results on two sets of problems. The first set of problems, shown in Table 8.1, is obtained from John Richardson's (Think- ing Machines Corporation) program for triangulating the sphere. The spectral lower bounds reported are from Theorem 5.2. Gap is the ratio with numerator equal to the difference between the 2-sum and the lower bound, and the denominator equal to the 2-sum. The results show that the spectral reordering algorithm computes values within a few percent of the optimal 2-sum, since the gap between the spectral 2-sum and the lower bound is within that range. 1,026 3,072 4.17e-2 3.75e+6 4.05e+6 7.4 7.3 Table 2-sums from the spectral reordering algorithm and lower bounds for triangulations of the sphere. Problem BARTH 6,691 19,748 2.62e-3 6.54e+7 6.69e+7 2.2 Table 2-sums from the spectral reordering algorithm and lower bounds for some problems from the Boeing- Harwell and NASA collections. Table 8.2 contains the second set of problems, taken from the Boeing-Harwell and NASA collections. Here the bounds are weaker than the bounds in Table 8.1. These problems have two features that distinguish them from the sphere problems. Many of them have less regular degree distributions-e.g., NASA1824 has maximum degree 41 and minimum degree 5. They also represent more complex geometries. Nevertheless, these results imply that the spectral 2-sum is within a factor of two of the optimal value for these problems. These results are somewhat surprising since we have shown that minimizing the 2-sum is NP-complete. The gap between the computed 2-sums and the lower bounds could be further reduced in two ways. First, a local reordering algorithm applied to the ordering computed by the spectral algorithm might potentially decrease the 2-sum. Second, the lower bounds could be improved by incorporating diagonal perturbations to the Laplacian. 9. Conclusions. The lower bounds on the 2-sums show that the spectral reordering algorithm can yield nearly optimal values, in spite of the fact that minimizing the 2-sum is an NP-complete problem. To the best of our knowledge, these are the first results providing reasonable bounds on the quality of the orderings generated by a reordering algorithm for minimizing envelope-related parameters. Earlier work had not addressed the issue of the quality of the orderings generated by the algorithms. Unfortunately the tight bounds on the 2-sum do not lead to tight bounds on the envelope parameters. However, we have shown that problems with bounded separator sizes have bounded envelope parameters and have obtained asymptotic lower and upper bounds on these parameters for finite element meshes. Our analysis further shows that the spectral orderings attempt to minimize the 2-sum rather than the envelope parameters. Hence a reordering algorithm could be used in a post-processing step to improve the envelope and wavefront parameters from a spectral ordering. A combinatorial reordering algorithm called the Sloan algorithm has been recently used to reduce envelope size and front-widths by Kumfert and Pothen [25]. Currently this algorithm computes the lowest values of the envelope parameters on a collection of finite element meshes. Acknowledgments . Professor Stan Eisenstat (Yale University) carefully read two drafts of this paper and pointed out several errors. Every author should be so blessed! Thanks, Stan. --R A spectral algorithm for envelope reduction of sparse matrices Reducing the bandwidth of sparse symmetric matrices Ordering methods for preconditioned conjugate gradients methods applied to unstructured grid problems Direct Methods for Sparse Matrices The effect of ordering on preconditioned conjugate gradients The use of profile reduction algorithms with a frontal code Graph Algorithms Algebraic connectivity of graphs in Surveys in Combinatorial Optimization Matrix inequalities in the L-owner orderings Computers and Intractability: A Guide to the Theory of NP- Completeness Computer implementation of the finite element method Computer Solution of Large Sparse Positive Definite Systems Algorithm 509: A hybrid profile reduction algorithm A new algorithm for finding a pseudoperipheral node in a graph A new lower bound via projection for the quadratic assignment problem A spectral approach to bandwidth and separator problems in graphs. An improved spectral graph partitioning algorithm for mapping parallel computations Laplace eigenvalues and bandwidth-type invariants of graphs A refined spectral algorithm to reduce the envelope and wavefront of sparse matrices. Implementations of the Gibbs-Poole-Stockmeyer and Gibbs-King algorithms Minimum profile of grid networks in structure analysis. Comparative analysis of the Cuthill-Mckee and the reverse Cuthill-Mckee ordering algorithms for sparse matrices in Graph Theory and Sparse Matrix Computation Eigenvalues in combinatorial optimization. Node and element resequencing using the Laplacian of a finite element graph Partitioning sparse matrices with eigenvectors of graphs A projection technique for partitioning the nodes of a graph Partitioning of unstructured problems for parallel processing An algorithm for profile and wavefront reduction of sparse matrices --TR --CTR Desmond J. Higham, Unravelling small world networks, Journal of Computational and Applied Mathematics, v.158 n.1, p.61-74, 1 September Shashi Shekhar , Chang-Tien Lu , Sanjay Chawla , Sivakumar Ravada, Efficient Join-Index-Based Spatial-Join Processing: A Clustering Approach, IEEE Transactions on Knowledge and Data Engineering, v.14 n.6, p.1400-1421, November 2002

sparse matrices;quadratic assignment problems;reordering algorithms;2-sum problem;envelope reduction;eigenvalues of graphs;1-sum problem;laplacian matrices

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Perturbation Analyses for the QR Factorization.

This paper gives perturbation analyses for $Q_1$ and $R$ in the QR factorization $A=Q_1R$, $Q_1^TQ_1=I$ for a given real $m\times n$ matrix $A$ of rank $n$ and general perturbations in $A$ which are sufficiently small in norm. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition numbers here are altered by any column pivoting used in $AP=Q_1R$, and the condition number for $R$ is bounded for a fixed $n$ when the standard column pivoting strategy is used. This strategy also tends to improve the condition of $Q_1$, so the computed $Q_1$ and $R$ will probably both have greatest accuracy when we use the standard column pivoting strategy.First-order perturbation analyses are given for both $Q_1$ and $R$. It is seen that the analysis for $R$ may be approached in two ways---a detailed "matrix--vector equation" analysis which provides a tight bound and corresponding condition number, which unfortunately is costly to compute and not very intuitive, and a simpler "matrix equation" analysis which provides results that are usually weaker but easier to interpret and which allows the efficient computation of satisfactory estimates for the actual condition number. These approaches are powerful general tools and appear to be applicable to the perturbation analysis of any matrix factorization.

Introduction . The QR factorization is an important tool in matrix computations (see for example [4]): given an m \Theta n real matrix A with full column rank, there exists a unique m \Theta n real matrix Q 1 with orthonormal columns, and a unique nonsingular upper triangular n \Theta n real matrix R with positive diagonal entries such that The matrix Q 1 is referred to as the orthogonal factor, and R the triangular factor. Suppose tG has the unique QR factorization differentiate R(t) T with respect to t and set which with given A and G is a linear equation for the upper triangular matrix - R(0). determines the sensitivity of R(t) at and so the core of any perturbation analysis for the QR factorization effectively involves the use of (1.1) to determine bound - R(0). There are essentially two main ways of approaching this problem. We now discuss these, together with some very recent history. Xiao-Wen Chang [1, 2, 3] was the first to realize that most of the published results on the sensitivity of factorizations, such as LU, Cholesky, and QR, were extremely School of Computer Science, McGill University, Montreal, Quebec, Canada, H3A 2A7, ([email protected]), ([email protected]). The research of the first two authors was supported by NSERC of Canada Grant OGP0009236. y Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, U.S.A., ([email protected]). G. W. Stewart's research was supported in part by the US National Science Foundation under grant CCR 95503126. G. W. STEWART weak for many matrices. He originated a general approach to obtaining provably tight results and sharp condition numbers for these problems. We will call this the "matrix- vector equation" approach. For the QR factorization this involves expressing (1.1) as a matrix-vector equation of the form where WR and ZR are matrices involving the elements of R, and vec(\Delta) transforms its argument into a vector of its elements. The notation uvec(\Delta) denotes a variant of vec(\Delta) defined in Section 5.2. Chang also realized that the condition of many such problems was significantly improved by pivoting, and provided the first firmly based theoretical explanations as to why this was so. G. W. Stewart [11, 3] was stimulated by Chang's work to understand this more deeply, and present simple explanations for what was going on. Before Chang's work, the most used approach to perturbation analyses of factorizations was what we will call the "matrix equation" approach, which keeps equations like (1.1) in their matrix-matrix form. Stewart [11] (also see [3]) used an elegant construct, partly illustrated by the "up" and "low" notation in Section 2, which makes the matrix equation approach a far more usable and intuitive tool. He combined this with deep insights on scaling to produce new matrix equation analyses which are appealingly clear, and provide excellent insight into the sensitivities of the problems. These new matrix equation analyses do not in general provide tight results like the matrix-vector equation analyses do, but they are usually more simple, and provide practical estimates for the true condition numbers obtained from the latter. It was the third author, Chris Paige, who insisted on writing this brief history, and delineating these exceptional contributions of Chang and Stewart. It requires a combination of the two analyses to provide a full understanding of the cases we have examined. The interplay of the two approaches was first revealed in [3], and we hope to convey it even more clearly here for the QR factorization. The perturbation analysis for the QR factorization has been considered by several authors. The first norm-based result for R was presented by Stewart [9]. That was further modified and improved by Sun [13]. Using different approaches Sun [13] and Stewart [10] gave first order normwise perturbation analyses for R. A first order componentwise perturbation analysis for R has been given by Zha [17], and a strict componentwise analysis for R has been given by Sun [14]. These papers also gave analyses for Q 1 . More recently Sun [15] gave strict perturbation bounds for Q 1 alone. The purpose of this paper is to establish new first order perturbation bounds which are generally sharper than the equivalent results for the R factor in [10, 13], and more straightforward than the sharp result in [15] for the Q 1 factor. In Section 2 we define some notation and give a result we will use throughout the paper. In Section 3 we will survey important key results on the sensitivity of R and Q 1 . In Section 4 we give a refined perturbation analysis for showing in a simple way why the standard column pivoting strategy for A can be beneficial for certain aspects of the sensitivity of Q 1 . In Section 5 we analyze the perturbation in R, first by the straightforward matrix equation approach, then by the more detailed and tighter matrix-vector equation approach. We give numerical results and suggest practical condition estimators in Section 6, and summarize and comment on our findings in Section 7. PERTURBATION ANALYSES FOR THE QR FACTORIZATION 3 2. Notation and basics. To simplify the presentation, for any n \Theta n matrix X, we define the upper and lower triangular matrices so that up(X). For general X For symmetric X F \Gamma2 To illustrate a basic use of "up", we show that for any given n \Theta n nonsingular upper triangular R and any given n \Theta n symmetric M , the equation of the form (cf. (1.1)) always has a unique upper triangular solution U . Since UR \Gamma1 is upper triangular in we see immediately that so UR \Gamma1 and therefore U is uniquely defined. We will describe other uses later. Our perturbation bounds will be tighter if we bound separately the perturbations along the columns space of A and along its orthogonal complement. Thus we introduce the following notation. For real m \Theta n A, let P 1 be the orthogonal projector onto R(A), and P 2 be the orthogonal projector onto R(A) ? . For real m \Theta n \DeltaA define so 2 . When ffl ? 0 in (2.5) we also define so for the QR factorization We will use the following standard result. Lemma 2.1. For real m \Theta n A with rank n, real is the orthogonal projector onto R(A). 4 XIAO-WEN CHANG, CHRIS PAIGE AND G. W. STEWART Proof. Let A have the QR factorization which necessarily has full column rank if kQ T (A), the smallest singular value of A. But this inequality is just (2.8). 2 Corollary 2.2. If nonzero \DeltaA satisfies (2.8), then for ffl and G defined in (2.5) and (2.6), A + tG has full column rank and therefore a unique QR factorization for all jtj - ffl. 2 3. Previous norm-based results. In this section we summarize the strongest norm-based results by previous authors. We first give a derivation of what is essentially Sun's [13] and Stewart's [10] first order normwise perturbation result for R, since their techniques and results will be useful later. Theorem 3.1. [13]. Let A 2 R m\Thetan be of full column rank, with the QR factorization be a real m \Theta n matrix. has a unique QR factorization where Proof. Let G j \DeltaA=ffl (if the theorem is trivial). From Corollary 2.2 A+tG has the unique QR factorization for all jtj - ffl, where Notice that It is easy to verify that Q 1 (t) and R(t) are twice continuously differentiable for ffl from the algorithm for the QR factorization. Thus as in (1.1) we have which (see (2.4)) is a linear equation uniquely defining the elements of - in terms of the elements of Q T G. From upper triangular - R(0)R \Gamma1 in we see with (2.1) that PERTURBATION ANALYSES FOR THE QR FACTORIZATION 5 so with (2.3) and since from (2.5) to (2.7) kQ T The Taylor expansion for R(t) about so that which, combined with (3.8), gives (3.2). 2 This proof shows that the key point in deriving a first order perturbation bound for R is the use of (3.5) to give a good bound on the sensitivity k - we obtained the bounds directly from (3.7), this was a "matrix equation" analysis. We now show how a recent perturbation result for Q 1 given by Sun [15] can be obtained in the present setting, since the analysis can easily be extended to obtain a more refined result in a simple way. Note the hypotheses of the following theorem are those of Theorem 3.1, so we can use results from the latter theorem. Theorem 3.2. [15]. Let A 2 R m\Thetan be of full column rank, with the QR factorization be a real m \Theta n matrix. holds, then A + \DeltaA has a unique QR factorization where Proof. Let G j \DeltaA=ffl (if the theorem is trivial). From Corollary 2.2 A+tG has the unique QR factorization A(t) for all jtj - ffl. Differentiating these at It follows that so with any Q 2 such that Q 6 XIAO-WEN CHANG, CHRIS PAIGE AND G. W. STEWART Now using (2.1), we have with (3.7) in Theorem 3.1 that We see from this, (2.2), (3.11), and kGk from (2.7), that F and from the Taylor expansion for so that k\DeltaQ 1 4. Refined analysis for Q 1 . Note since both Q 1 and are orthonormal matrices in (3.1), and \DeltaQ 1 is small, \DeltaQ so the following analysis assumes n ? 1. The results of Sun [15] give about as good as possible overall bounds on the change \DeltaQ 1 in Q 1 . But by looking at how \DeltaQ 1 is distributed between and its orthogonal complement, and following the ideas in Theorem 3.2, we are able to obtain a result which is tight but, unlike the related tight result in [15], easy to follow. It makes clear exactly where any ill-conditioning lies. From (3.14) with square and orthogonal, and the key is to bound the first term on the right separately from the second. Note from (3.11) with (2.5) to (2.7) that where G can be chosen to give equality here for any given A. Hence is the true condition number for that part of \DeltaQ 1 in R(Q 2 Now for the part of \DeltaQ 1 in R(Q 1 ). We see from (3.12) that which is skew symmetric with clearly zero diagonal. Let R j and S j denote the leading blocks of R and S respectively, G j the matrix of the first j columns of G, and PERTURBATION ANALYSES FOR THE QR FACTORIZATION 7 where s j has j-1 elements, then from the upper triangular form of R in (4.2) ks ks F for any R n\Gamma1 equality is obtained by taking such that kR \GammaT It follows that the bound is tight in so the true condition number for that part of \DeltaQ 1 in R(Q 1 ) is not In some problems we are mainly interested in the change in Q 1 lying in R(Q 1 ), and this result shows its bound can be smaller than we previously thought. In particular if A has only one small singular value, and we use the standard column pivoting strategy in computing the QR factorization, then R n\Gamma1 will be quite well-conditioned compared with R, and we will have kR \Gamma1 5. Perturbation analyses for R. In Section 3 we saw the key to deriving first order perturbation bounds for R in the QR factorization of full column rank A is the equation (3.5), which has the general form of finding (bounding) X in terms of given R and F in the matrix equation upper triangular, R nonsingular: Sun [13] and Stewart [10] originally analyzed this using the matrix equation approach to give the result in Theorem 3.1. We will now analyze it in two new ways. The first, Stewart's [11, 3] refined matrix equation approach, gives a clear improvement on Theorem 3.1, while the second, Chang's [2, 3] matrix-vector equation approach, gives a further improvement still - provably tight bounds leading to the true condition number -R (A) for R in the QR factorization of A. Both approaches provide efficient condition estimators (see [2] for the matrix-vector equation approach), and nice results for the special case of permutation matrix giving the standard column pivoting, but we will only derive the matrix equation versions of these. The tighter but more complicated matrix-vector equation analysis for the case of pivoting is given in [2], and only the results will be quoted here. 5.1. Refined matrix equation analysis for R. Our proof of Theorem 3.1 used (3.5) to produce the matrix equation (3.7), and derived the bounds directly from this. We now look at this approach more closely, but at first using the general form (5.1) to keep our thinking clear. From this we see 8 XIAO-WEN CHANG, CHRIS PAIGE AND G. W. STEWART Let D n be the set of all n \Theta n real positive definite diagonal matrices. For any D = R. Note that for any matrix B we have up(BD). Hence if we define B R R \GammaT F T D) - R: With obvious notation, the upper triangular matrix . To bound this, we use the following result. Lemma 5.1. For n \Theta n B and D 2 D n , where 1-i!j-n Proof. Clearly But by the Cauchy-Schwarz theorem, We can now bound the solution X of (5.1) Since this is true for all D 2 D n , we know that for the upper triangular solution X of where i D is defined in (5.4). This gives the encouraging result (5. PERTURBATION ANALYSES FOR THE QR FACTORIZATION 9 Comparing (3.5) with (5.1), we see for the QR factorization, with (2.5) to (2.7), Hence and from (3.9) for a change in A we have where from (5.11) these are never worse than the bounds in Theorem 3.1. When we use the standard column pivoting strategy in permutation matrix, this analysis leads to a very nice result. Here the elements of R so r 2 nn . If D is the diagonal of R then i D - 1, and from (5.9) and But then it follows from [6, Theorem 8.13] that so since k - Thus when we use the standard pivoting strategy, we see the sensitivity of R is bounded for any n. Remark 5.1. Clearly -ME (A) is a potential candidate for the condition number of R in the QR factorization. From (5.9), -ME (A) depends solely on R, but it will only be the true condition number if for any nonsingular upper triangular R we can find an F in (5.1) giving equality in (5.8). From (5.7) this can only be true if every column of F T lies in the space of the right singular vectors corresponding to the maximum singular value of - R \GammaT . Such a restriction is in general too strong for (5.6) to be made an equality as well (see the lead up to (5.5)). However for a class of R this is possible. If R is diagonal, we can take and the first restriction on F disappears. Let i and j be such that i , so from G. W. STEWART So we see that, at least for diagonal R, the bounds are tight, and in this restricted case (A) is the true condition number. This refined matrix equation analysis shows to what extent the solution X of (5.1), and so the sensitivity of R in the QR factorization, is dependent on the row scaling in R. From the term D R \GammaT F T )D in (5.2), we saw multipliers occurred only with j ? i. As a result we obtained i D in our bounds rather than with equality if and only if the minimum element comes before the maximum on the diagonal. Thus we obtained full cancellation of D \Gamma1 with D in the first term on the right hand side of (5.2), and partial cancellation in the second. This gives some insight as to why R in the QR factorization is less sensitive than the earlier condition estimator indicated. If the ill-conditioning of R is mostly due to bad scaling of its rows, then correct choice of D in R can give - 2 ( - very near one. If at the same time i D is not large, then -(R; D) in (5.10) can be much smaller than pivoting always ensures that such a D exists, and in fact gives (5.14). However if we do not use such pivoting, then Remark 5.1 suggests that any relatively small earlier elements on the diagonal of R could correspond to poor conditioning of the factorization. We will return to -ME (A) and -(R; D) when we seek practical estimates of the true condition number that we derive in the next section. 5.2. Matrix-vector equation analysis for R. We can now obtain provably sharp, but less intuitive results by viewing the matrix equation (5.1) as a large matrix-vector equation. For any matrix C n\Thetan , denote by c (i) j the vector of the first i elements of c j . With this, we define ("u" denotes "upper") c (1)c (2)\Delta c (n) It is the vector formed by stacking the columns of the upper triangular part of C into one long vector. To analyze (3.5) we again consider the general form (5.1), repeated here for clarity, upper triangular, R nonsingular: which we saw via (2.4) has a unique upper triangular solution X. The upper and lower triangular parts of (5.1) contain identical information, and we now write the upper triangular part in matrix-vector, rather than matrix-matrix format. The first j elements of the jth column of (5.15) are given by R T f (j)Tf (j)T\Delta f (j)T PERTURBATION ANALYSES FOR THE QR FACTORIZATION 11 and by rewriting this, we can see how to solve for x (j) (R T r (1)T r (j \Gamma1)T (R T r (j)T r (j)T which, on dividing the last row of this by 2, gives 2 \Theta n(n+1) r 11 and ZR 2 R n(n+1) r 11 Since R is nonsingular, WR is also, and from (5.16) R ZR vec(F Remembering X is upper triangular, we see R ZR vec(F )k 2 where for any nonsingular upper triangular R, equality can be obtained by choosing vec(F ) to lie in the space spanned by the right singular vectors corresponding to the G. W. STEWART largest singular value of W \Gamma1 R ZR . It follows that (5.18) is tight, so from (5.8), derived from the matrix equation approach, and (5.11) Remark 5.2. Usually the first and second inequalities are strict. For exam- ple, let . Then we solving an optimization problem). But from Remark 5.1 the first inequality becomes an equality if R is diagonal, while the second also becomes an equality if R is an n \Theta n identity matrix with so the upper bound is tight. The structure of WR and ZR reveals that each column of WR is one of the columns of ZR , and so W \Gamma1 R ZR has an n(n + 1)=2 square identity submatrix, giving Remark 5.3. This lower bound is approximately tight for any n, as can be seen by taking for by taking and (5.10), we see from Remark 5.1 that These results, and the analysis in Section 4 for Q 1 , lead to our new first order normwise perturbation theorem. Theorem 5.2. Let be the QR factorization of A 2 R m\Thetan with full column rank, and let \DeltaA be a real m \Theta n matrix. holds, then there is a unique QR factorization such that where with WR and ZR as in (5.16), and -ME (A) as in (5.9) and (5.10), and where PERTURBATION ANALYSES FOR THE QR FACTORIZATION 13 Proof. From Corollary 2.2 A+ \DeltaA has the unique QR factorization (5.22). From (3.5), (5.15), (5.17) with G j \DeltaA=ffl and (2.5) we have R ZR vec(Q T so taking the 2-norm gives Combining this with and Thus, from the Taylor series (3.9) of R(t), (5.23) follows. The remaining results are restatements of (5.20), (5.19), (4.1) and (4.3). 2 Remark 5.4. From (5.24) we know the first order perturbation bound (5.23) is at least as sharp as (3.2), but it suggests it may be considerably sharper. In fact it can be sharper by an arbitrary factor. Consider the example in Remark 5.3, where taking and We see the first order perturbation bound (3.2) can severely overestimate the effect of a perturbation in A. Remark 5.5. If we take which is close to the upper bound This shows that relatively small early diagonal elements of R cause poor condition, and suggests if we do not use pivoting, then there is a significant chance that the condition of the problem will approach its upper bound, at least for randomly chosen matrices. When we use standard pivoting, we see from (5.24) and (5.14) but the following tighter result is shown in [2, Th. 2.2] Theorem 5.3. Let A 2 R m\Thetan be of full column rank, with the QR factorization when the standard column pivoting strategy is used. Then 14 XIAO-WEN CHANG, CHRIS PAIGE AND G. W. STEWART There is a parametrized family of matrices A('), ' 2 (0; -=2]; for which R ZR Theorem 5.3 shows that when the standard column pivoting strategy is used, -R (AP ) is bounded for fixed n no matter how large is. Many numerical experiments with this strategy suggest that -R (AP ) is usually close to its lower bound of one, but we give an extreme example in Section 6 where it is not. When we do not use pivoting, we have no such simple result for -R (A), and it is, as far as we can see, unreasonably expensive to compute or approximate -R (A) directly with the usual approach. Fortunately -ME (A) is apparently an excellent approximation to -R (A), and -ME (A) is quite easy to estimate. All we need to do is choose a suitable D in -(R; D) in (5.10). We consider how to do this in the next section. 6. Numerical experiments and condition estimators. In Section 5 we presented new first order perturbation bounds for the R factor of the QR factorization using two different approaches, defined R ZR k 2 as the true condition number for the R factor, and suggested -R (A) could be estimated in practice by -(R; D). Our new first order results are sharper than previous results for R, and at least as sharp for Q 1 , and we give some numerical tests to illustrate both this, and possible estimators for -R (A). We would like to choose D such that -(R; D) is a good approximation to the in (5.9), and show that this is a good estimate of the true condition number -R (A). Then a procedure for obtaining an O(n 2 ) condition estimator for R in the QR factorization (i.e. an estimator for -R (A)), is to choose such a D, use a standard condition estimator (see for example [5]) to estimate - 2 (D \Gamma1 R), and take -(R; D) in (5.10) as the appropriate estimate. By a well known result of van der Sluis [16], - 2 (D \Gamma1 R) will be nearly minimal when the rows of D \Gamma1 R are equilibrated. But this could lead to a large i D in (5.10). There are three obvious possibilities for D. The first one is choosing D to equilibrate R precisely. Specifically, take n. The second one is choosing D to equilibrate R as far as possible while keeping i D - 1. Specifically, take n. The third one is choosing Computations show that the third choice can sometimes cause unnecessarily large estimates, so we will not give any results for that choice. We specify the diagonal matrix D obtained by the first method and the second method by D 1 and D 2 respectively in the following. We give three sets of examples. The first set of matrices are n \Theta n Pascal matrices, (with elements a 15. The results are shown in Table 6.1 without pivoting, giving and in Table 6.2 with pivoting, giving ~ R. Note in Table 6.1 how can be far worse than the true condition number -R (A), which itself can be much greater than its lower bound of 1. In Table 6.2 pivoting is seen to give a significant improvement on -R (A), bringing very close to its lower bound, but of course still. Also we observe from Table 6.1 that both -(R; D 1 ) and -(R; D 2 ) are very good estimates for -R (A). The latter is a little better than the former. In Table 6.2 -( ~ (in fact are also good estimates for -R (AP ). PERTURBATION ANALYSES FOR THE QR FACTORIZATION 15 Table Results for Pascal matrices without pivoting, 7 6.6572e+02 4.4665e+03 4.2021e+03 2.1759e+05 2.1120e+06 8 2.4916e+03 1.8274e+04 1.6675e+04 2.7562e+06 2.9197e+07 9 9.4091e+03 7.4364e+04 6.6391e+04 3.6040e+07 4.1123e+08 2.0190e+06 1.9782e+07 1.6614e+07 1.2819e+12 1.8226e+13 14 7.7958e+06 7.9545e+07 6.6157e+07 1.8183e+13 2.6979e+14 Table Results for Pascal matrices with pivoting, ~ R 3 1.2892e+00 2.1648e+00 2.1648e+00 1.2541e+01 8.7658e+01 6 2.2296e+00 4.7281e+00 4.7281e+00 7.5426e+03 1.5668e+05 7 2.0543e+00 5.1140e+00 5.1140e+00 8.4696e+04 2.1120e+06 8 2.6264e+00 6.5487e+00 6.5487e+00 1.1777e+06 2.9197e+07 9 3.4892e+00 8.8383e+00 8.8383e+00 1.4573e+07 4.1123e+08 14 3.6106e+00 1.2386e+01 1.2386e+01 5.3554e+12 2.6979e+14 The second set of matrices are 10 \Theta 8 A j , which are all obtained from the same random 10 \Theta 8 matrix (produced by the MATLAB function randn), but with its jth column multiplied by 10 \Gamma8 to give A j . The results are shown in Table 6.3 without pivoting. All the results with pivoting are similar to that for 6.3, and so are not given here. For (A) are both close to their upper bound are significantly smaller than these results are what we expected, since the matrix R is ill-conditioned due to the fact that r jj is very small, but for the rows of R are already essentially equilibrated, and we do not expect -R (A) to be much G. W. STEWART better than (A). Also for the first seven cases the smallest singular value of the leading part R n\Gamma1 is close to that of R, so that -Q 1 (A) could not be much better than (A). For even though R is still ill-conditioned due to the fact that r 88 is very small, it is not at all equilibrated, and becomes well-conditioned by row scaling. Notice at the same time i D is close to 1, so -(R; D1), -(R; D 2 ), and therefore are much better than (A). In this case, the smallest singular value of R is significantly smaller than that of R n\Gamma1 . Thus -Q 1 (A), the condition number for the change in Q 1 lying in the range of Q 1 , is spectacularly better than (A). This is a contrived example, but serves to emphasize the benefits of pivoting for the condition of both Q 1 and R. Table pivoting 5 1.2467e+08 3.1208e+08 2.3992e+08 3.9067e+08 4.2323e+08 6 8.8237e+07 2.2252e+08 1.6584e+08 3.4710e+08 3.9061e+08 7 9.2648e+07 2.1010e+08 1.7127e+08 4.4303e+08 5.4719e+08 8 2.2580e+00 5.4735e+00 4.9152e+00 6.6109e+00 6.2096e+08 The third set of matrices are n \Theta n upper triangular These matrices were introduced by Kahan [7]. Of course I here, but the condition numbers depend on R only, and these are all we are interested in. The results for are shown in Table 6.5. Again we found D only list the results corresponding to D 1 . Table Results for Kahan matrices, In all these examples we see -(R; D 1 ) and -(R; D 2 ) gave excellent estimates for -R (A), with -(R; D 2 ) being marginally preferable. For the Kahan matrices, which correspond to correctly pivoted A, we see that in extreme cases, with large enough n, -R (A) can be large even with pivoting. This is about as bad a result as we can get PERTURBATION ANALYSES FOR THE QR FACTORIZATION 17 with pivoting (it gets a bit worse as ' ! 0 in R), since the Kahan matrices are the parameterized family mentioned in Theorem 5.3. Nevertheless -(R; still estimate -R (A) excellently. 7. Summary and conclusion. The first order perturbation analyses presented here show just what the sensitivity (condition) of each of Q 1 and R is in the QR factorization of full column rank A, and in so doing provide their true condition numbers (with respect to the measures used, and for sufficiently small \DeltaA), as well as efficient ways of approximating these. The key norm-based condition numbers we derived for A are: for that part of \DeltaQ 1 in R(A) ? , see (4.1), that part of \DeltaQ 1 in R(A), see (4.3), R ZR k 2 for R, see Theorem 5.2, ffl the estimate for -R (A), that is -ME (A) j inf D2Dn -(R; D), where -(R; D) j The condition numbers obey for while for R see (5.24). The numerical examples, and an analysis of the case (not given here), suggest that -(R; D), with D chosen to equilibrate - subject to i D - 1, gives an excellent approximation to -R (A) in the general case. In the special case of A with orthogonal columns, so R is diagonal, then Remark 5.1 showed by taking For general A when we use the standard column pivoting strategy in the QR factor- ization, we saw from (5.14) and [2] that As a result of these analyses we see both R and in a certain sense Q 1 can be less sensitive than was thought from previous analyses. The true condition numbers depend on any column pivoting of A, and show that the standard pivoting strategy often results in much less sensitive R, and sometimes leads to a much smaller possible change of Q 1 in the range of Q 1 , for a given size of perturbation in A. The matrix equation analysis of Section 5.1 also provides a nice analysis of an interesting and possibly more general matrix equation (5.1). By following the approach of Stewart [8, Th. 3.1], see also [12, Th. 2.11], it would be straightforward, but detailed and lengthy, to extend our first order results to provide strict perturbation bounds, as was done in [3]. We could also provide new component-wise bounds, but we chose not to do either of these here, in order to keep the material and the basic ideas as brief and approachable as possible. Our condition numbers and resulting bounds are asymptotically sharp, so there is less need for strict bounds. A new componentwise bound for R is given in [2]. G. W. STEWART Acknowledgement . We would like to thank Ji-guang Sun for his suggestions and encouragement, and for providing us with draft versions of his work. --R PhD Thesis A perturbation analysis for R in the QR factorization New perturbation analyses for the Cholesky fac- torization Matrix Computations A survey of condition number estimation for triangular matrices Accuracy and Stability of Numerical Algorithms and perturbation bounds for subspaces associated with certain eigenvalue problems Perturbation bounds for the QR factorization of a matrix On the perturbation of LU On the Perturbation of LU and Cholesky Factors Matrix perturbation theory Perturbation bounds for the Cholesky and QR factorization Componentwise perturbation bounds for some matrix decompositions On perturbation bounds for the QR factorization Condition numbers and equilibration of matrices A componentwise perturbation analysis of the QR decomposition --TR --CTR Younes Chahlaoui , Kyle A. Gallivan , Paul Van Dooren, An incremental method for computing dominant singular spaces, Computational information retrieval, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001

matrix equations;pivoting;condition estimation;QR factorization;perturbation analysis

263442

Optimal Local Weighted Averaging Methods in Contour Smoothing.

AbstractIn several applications where binary contours are used to represent and classify patterns, smoothing must be performed to attenuate noise and quantization error. This is often implemented with local weighted averaging of contour point coordinates, because of the simplicity, low-cost and effectiveness of such methods. Invoking the "optimality" of the Gaussian filter, many authors will use Gaussian-derived weights. But generally these filters are not optimal, and there has been little theoretical investigation of local weighted averaging methods per se. This paper focuses on the direct derivation of optimal local weighted averaging methods tailored towards specific computational goals such as the accurate estimation of contour point positions, tangent slopes, or deviation angles. A new and simple digitization noise model is proposed to derive the best set of weights for different window sizes, for each computational task. Estimates of the fraction of the noise actually removed by these optimum weights are also obtained. Finally, the applicability of these findings for arbitrary curvature is verified, by numerically investigating equivalent problems for digital circles of various radii.

Introduction There are numerous applications involving the processing of 2-D images, and 2-D views of 3-D images, where binary contours are used to represent and classify patterns of inter- est. Measurements are then made using the contour information (e.g. perimeter, area, moments, slopes, curvature, deviation angles etc. To obtain reliable estimates of these quantities, one must take into account the noisy nature of binary contours due to discrete sampling, binarization, and possibly the inherent fuzziness of the boundaries themselves 1 . In some cases, this can be done explicitly and exhaustively (see Worring & Smeulders [1] on curvature estimation). But more frequently it is done implicitly by smoothing. Following this operation, the measurements of interest can be obtained directly from the smoothed contour points, as in this paper, or from a curve fitted to these points. For a recent example of this last approach, see Tsai & Chen [2]. The smoothing of binary contours or curves for local feature extraction directly from 1 For example, the "borders" of strokes in handwriting. the discrete data is the focus of this article. More precisely, we will investigate optimum local weighted averaging methods for particular measurement purposes such as estimating point positions, derivatives (slopes of tangents), and deviation angles from point to point (see Fig. 2). In this article, the following definitions will be used. Let be the sequence of N points (4- or 8-connected) around the closed contour. Since the contour is cyclic, be the counter-clockwise elevation angle between v i and the horizontal x-axis. We have 4 where c i is the Freeman [3] chain code (see Fig. 1). For 4-connectivity, the values of c i are limited to even values. We also define d i , the differential chain code, as \Gamma'@ @ @ @I oe \Gamma\Psi @ @ @ Figure 1: Freeman chain code \Gamma\Psit Figure 2: Deviation angle at p i The deviation angle at point p i will be denoted OE i . It is the angle between the small vectors v i and v i+1 (See Fig. 2). Of course we have OE 4 . Finally, the local weighted averaging method investigated in sections II, III, and IV is defined as: j=\Gamman i is the contour point i after k smoothing steps (p (0) The window size is A. Variety of Approaches Used in Applications Due to limited computing power, early methods were quite simple and found justification in their "good results". Thus we find schemes removing/filling one-pixel wide protrusions/intrusions based on templates, or replacing certain pairs in the chain code sequence by other pairs or by singletons ([4]). However, from early on, local weighted averaging methods are the most frequently used. They are applied to differential chain codes ([5], [6], [7]), possibly with compensation for the anisotropy of the square grid ([8]); they are applied to cartesian coordinates ([9]), possibly with weights depending on neighboring pixel configuration ([10]) or varying with successive iterations ([11]); they are applied to deviation angles ([12]). With advances in computing power and insight into the smoothing problem, more complex methods were developed with more solid theoretical foundations. In this process, "Gaussian smoothing" has become very popular. One approach consists of applying local weighted averaging with Gaussian weights. Dill et al. [13] use normalized Gaussian- smoothed differential chain codes with fixed oe and window size. In Ansari & Huang [14], the weights and window size may vary from point to point based on e region of support. See also Pei Variable amounts of smoothing can be applied to the entire curve, taking the overall behaviour of the smoothed curve across scale as its complete description. Witkin [17] convolves a signal f(x) with Gaussian masks over a continuum of sizes: 2-oe F (x; oe) is called the scale-space image of f(x) and it is analyzed in terms of its inflection points. This concept of scale-space was also originally explored by Koenderink [18]. Asada & Brady [19] analyze the convolution of simple shape primitives with the first and second derivatives of G(x; oe) and then use the knowledge gained to extract these shape primitives from the contours of objects. Mokhtarian & Mackworth [20] compute November 20, 1997 the Gaussian-smoothed curvature using Y (t; oe) - where X(t; oe) and Y (t; oe) are the coordinates (x(t),y(t)) convolved with a Gaussian filter G(t; oe). The locus of points where -(t; is called the generalized scale-space image of the curve which they use for image matching purposes. Wuescher & Boyer [21] also use Gaussian-smoothed curvature but with a single oe to extract regions of constant curvature from contours. Multiscale shape representations are not necessarily associated with Gaussian smoothing. Saint-Marc et al. [22] propose scale-space representations based on adaptive smoothing, which is a local weighted averaging method where the smoothed signal S (k+1) (x) after smoothing steps is obtained as: \Gamma(x)X Several multiscale shape representations based on non-linear filters have also been used successfully. Maragos [23] investigated morphological opening/closing filters depending on a structuring element and a scale parameter; using successive applications of these operators and removing some redundancy, a collection of skeleton components (Reduced Skeleton Transform) can be generated which represents the original shape at various scales more compactly than multiscale filtered versions. See also Chen & Yan [24]. Recently Bangham et al. [25] used scale-space representations based on M - and N - sieves. For any 1D discrete signal f denoting as C r the set of all intervals of r consecutive integers, they define: Using the N-sieves of f are the sequence: f 1. The M-sieves of f are defined similarly, using M. Applying sieves horizontally and vertically to 2D images appears to preserve edges and reject impulsive noise better than Gaussian smoothing. B. Theoretical Foundations Regularization theory and the study of scale-space kernels are the two main areas which have provided insight into the special qualities of the Gaussian kernel for smoothing pur- poses. As will be seen, they do not warrant unqualified statements about the 'optimality' of Gaussian smoothing. Consider a one-dimensional function g(x), corrupted by noise j(x). The observed signal is then Assume that the information available is a sampling of this signal obtained for One approach to estimating g(x) is to find f(x) which minimizesn Z xn where - is the regularization parameter. The solution is a smoothing spline of order 2m (see [26], [27]). For equally spaced data and et al. [28] have shown that the cubic spline solution is very similar to a Gaussian. Canny's paper on edge detection[29] is also cited to support the optimality of Gaussian filtering. But the Gaussian is only an approximation to his theoretically obtained optimal filter. Babaud et al. [30] have considered the class of infinitely differentiable kernels g(x; y) vanishing at infinity faster than any inverse of polynomial, and one-dimensional signals f(x) that are continuous linear functionals on the space of these kernels. In this class, they have shown that only the Gaussian g(x; can guarantee that all first-order maxima (or minima) of the convolution will increase (or decrease) monotonically as y increases. Yuille & Poggio [31] extended the previous result by showing that, in any dimension, the Gaussian is the only linear filter that does not create generic zero crossings of the Laplacian as the scale increases. In the literature, such demonstrations are often designated as the scaling theorem. Mackworth & Mokhtarian [32] derived a similar result concerning their curvature scale-space representation. Wu & Xie [33] developed an elementary proof of the scaling theorem based on calculus, assuming signals represented by polynomials. Very recently, Anh et al. [34] provided another proof applicable to the broader class of band-limited signals and to a larger class of filtering kernels, by relaxing the smoothness constraint. This property of the Gaussian filter was established for continuous signals. Lindeberg [35] has studied the similar problem for discrete signals. He postulated that scale-space should be generated by convolution with a one-parameter family of kernels and that the number of local extrema in the convolved signal K f should not exceed the number of local extrema in the original signal. Imposing a semigroup requirement, he found the unique one-parameter family of scale-space kernels T (n; t) with a continuous scale parameter t; as t increases, it becomes less and less distinguishable from the discretized Gaussian. Recently, Pauwels et al. [36] demonstrated that imposing recursivity and scale-invariance on linear, isotropic, convolution filters narrows down the class of scale-space kernels to a one-parameter family of filters but is not restrictive enough to single out the Gaussian. The latter results for a parameter value of 2; for higher values, the kernel has zero crossings. Pauwels et al. also derive Lindeberg's results as special cases of theirs. Finally, Bangham et al. [25] showed that discrete 1D M-sieves and N-sieves do not introduce new edges or create new extrema as the scale increases. In addition, Bangham et al. [37] have proven that when differences are taken between sieving operations applied recursively with increasing scale on 1D discrete signals, a set of granule functions are obtained which map back to the original signal. Furthermore, the amplitudes of the granules are, to a certain extent, independent of one another. C. Practical Considerations For practical applications, regardless of the smoothing method one decides to use, some concrete questions must eventually be answered. For the regularization approach, what November 20, 1997 value should be used for -? For Gaussian smoothing, what value of oe and what finite window size? When scale-space representation is used, if we say that significant features are those which survive over "a wide range of scale", we must eventually put some actual figures on this 'wide' range. These decisions can be entirely data-driven or based on prior experience, knowledge of particular applications etc. In the end, they may play a significant role in both the performance of the selected method and its implementation cost. We now briefly present some of these aspects. In [38] [39], Shahraray & Anderson consider the regularization problem of equation 8, 4, and they argue that finding the best value of - is critical. For this purpose, they propose a technique based on minimizing the cross-validation mean square error (g [k] where g [k] n;- is the smoothing spline constructed using all samples except y k , and is then used to estimate y k . The method is said to provide a very good estimate of the best -, for equally-spaced periodic data assuming only a global minimum. Otherwise, a so-called generalized cross-validation function must be used. The presence of discontinuities to be preserved in the contours of interest brings more complexity into the optimal smoothing problem. One possible solution was already men- tioned: the adaptive smoothing of Saint-Marc et al. [22]. For one-dimensional regularization which preserves discontinuities, see Lee & Pavlidis [40]. For two-dimensional regularization which preserves discontinuities, see Chen & Chin [41]. For Gaussian smoothing which preserves discontinuities, see the methods of Ansari & Huang [14] and of Brady et al. [42]. Another problem is that repeated convolution of a closed curve with a kernel may not yield a closed contour or may cause shrinkage. For simple, convex, closed curves, Horn & Weldon [43] propose a new curve representation which guarantees closed curves. Mackworth Mokhtarian [32] offer a different solution involving the reparametrization of the Gaussian-smoothed curve by its normalized arc length parameter. Lowe [44] attenuates shrinking by adding a correction to the Gaussian-smoothed curve. Oliensis [45] applies FFT to the signal and resets amplitudes to 0 only for frequencies larger than some thresh- 9old. Recently, Wheeler & Ikeuchi [46] present a new low-pass filter based on the idea of iteratively adding smoothed residuals to the smoothed signals. Results are said to be comparable to Oliensis' and somewhat better than Lowe's. possible solution to the high computation and storage requirements of generating "continuous" scale-space. They show that an optimal L 1 approximation of the Gaussian can be obtained with a finite number of basis Gaussian filters: from which scale-space can be constructed efficiently. The above discussion exemplifies the potential complexity involved in implementing methods. Clearly, in practice, one should not lose track of the cost of these operations and how much smoothing is really required by the application of interest. It is not always necessary to attain the ultimate precision in every measurement. In many situations, simple and fast methods such as local weighted averaging with fixed weights and a small window size, will provide a very satisfactory solution in only 2 or 3 iterations (see [11], [5], [12]). Moreover, there is often little difference in the results obtained via different methods. Thus, Dill et al. [13] report similar results when a Gaussian filter and a triangular (Gallus-Neurath) filter of the same width are applied to differential chain codes; in Kasvand & Otsu [48], rectangular, triangular, and Gaussian kernels, with the same standard deviation, yield comparable outcomes (especially the latter two) for the smooth reconstruction of planar curves from piecewise linear approximations. D. Present Work Our interest in contour smoothing stems from practical work in handwriting recognition. In this and other applications, the discrimination of meaningful information from 'noise' is a complex problem which often plays a critical role in the overall success of the system. This filtering process can be handled across several stages (preprocessing, feature extrac- tion, even classification), with different methods. In the preprocessing stage of one of our recognition schemes [49], a triangular filter with first applied to contours of characters before deviation angles OE i were computed. Satisfactory results were obtained. Nevertheless, we were curious about the optimality of our choice of local weights. Initial review revealed that, in many practical applications, the smoothing operation is still performed by some local weighted averaging schemes 2 because they are simple, fast, and effective (see for example [14], [50], [51], [52], [21]). However, little theoretical investigation of these methods per se has been conducted. Some authors rapidly invoke the 'optimality' of Gaussian filtering and use Gaussian-derived weights. Their results may be satisfactory as the Gaussian may be a good approximation to the 'optimum' filter, but its discretization and truncation may cause it to further depart from 'optimum' behaviour. In this paper, we assume local weighted averaging with constant weights as a starting point and we investigate how these smoothing methods handle small random noise. To this end, we propose a simple model of a noisy horizontal border. The simplicity of the model allows a very pointed analysis of these smoothing methods. More precisely, for specific computational goals such as estimating contour point positions, derivatives (slopes of tangents), or deviation angles from the pixels of binary contours, we answer the following questions: what are the optimum fixed weights for a given window size? and what fraction of the noise is actually removed by these optimum weights? After deriving these results, we offer experimental evidence that their validity is not restricted to the limited case of noisy horizontal borders. This is done by considering digital circles. For each particular computational task, we find very close agreement between the optimum weights derived from our simple model and the ones derived numerically for circles over a wide range of radii. An important side-result concerns the great caution which should be exercised in speaking of 'optimal' smoothing. Even for our simple idealized model, we find that the smoothing coefficients which best restore the original noise-free pixel positions are not the same which best restore the original local slope, or the original local deviation angles; further- more, the best smoothing coefficients even depend on the specific difference method used to numerically estimate the slope. Hence, in choosing smoothing methods, researchers should probably first consider what it is they intend to measure after smoothing and in what manner. In relation to this, we point out the work of Worring & Smeulders [1]. They analyze 2 Once the Gaussian is discretized and truncated, it also simply amounts to a local weighted averaging method with particular weight values. noise-free digitized circular arcs and exhaustively characterize all centers and radii which yield a given digitization pattern; by averaging over all these, an optimum measure of radius or curvature can be obtained. If radius or curvature is the measurement of interest and if utmost precision is required (with the associated computing cost to be paid), then their approach is most suitable. Our work in contrast is not oriented towards measuring a single attribute. We focus on measurements such as position, slope and deviation angles because they are often of interest. But our model and approach can be used to investigate other quantities or other numerical estimates of the same quantities. The methods may be less accurate but they will be much less costly, and optimum in the category of local weighted averaging methods. The requirements of specific applications should dictate what is the best trade-off. The rest of the article is organized as follows. The next section briefly describes the methods investigated and provides a geometric interpretation for them. In section III, the simple model of a noisy horizontal border is used to derive optimal values of the smoothing parameters, in view of the above-mentioned computational goals. Finally, the applicability of our findings for varying curvature is explored experimentally in section IV. II. Local Weighted Averaging We begin our study of local weighted averaging, as defined by Eq. 1, with window size course, the smoothed contour points p (k) smoothing iterations, can be obtained directly from the original points p i as j=\Gamman 0 corresponding to a window size w 1, and the fi's are functions of the ff's and of k. The form of Eq. 1 is often computationnally more convenient. However, as long as k and n are finite, the study of local weighted averaging need only consider the case of a single iteration with finite width filters. When this is done, we will use the simpler notation p 0 i instead of p (1) We now impose a simple requirement to this large family of methods. Since our goal is to smooth the small 'wiggles' along boundaries of binary images, it seems reasonable to require that when p i and its neighbouring contour pixels are perfectly aligned, the November 20, 1997 smoothing operation should leave p i unchanged. In particular, consider the x-coordinates of consecutive horizontally-aligned pixels from p i\Gamman to p i+n . For Our requirement that x 0 j=\Gamman j=\Gamman For this to hold whatever the value of x i , we must have j=\Gamman Thus our requirement is equivalent to a normalization condition and a symmetry constraint on the ff's. A. Geometric Interpretation It is a simple matter to find a geometric interpretation for local weighted averaging. Using the above conditions, Eq. 1 can be rewritten as: Fig. 3. Geometric interpretation for For a single iteration of the simplest method 1), the last Eq. reduces to The points p are generally not aligned and the situation is illustrated in Fig. 3, where m is the middle of the base of the triangle. Eq. implies that the smoothed point i is always on the median of the triangle from point p i . Furthermore, the effect of the unique coefficient ff is clear since jp i p 0 As ff varies continuously from 0 to 0.5, p 0 i 'slides' from p i to m i1 . Similarly, in the more general situation, the vectors [ p (k\Gamma1) are the medians from p (k\Gamma1) i of the triangles \Deltap (k\Gamma1) i+j . Eq. 15 indicates that the smoothed point i is obtained by adding to p (k\Gamma1) i a weighted sum of the medians of these triangles, using 2ff j as weights. Thus, in a geometric sense, local weighted averaging as a contour smoothing method could be renamed median smoothing. III. Optimum Results from a Simple Digitization Noise Model This section addresses the question "If local weighted averaging is considered, what constant coefficients ff j should be used for smoothing binary contours in view of specific computational goals?". We develop an answer to this question, based on a simple model: an infinite horizontal border with random 1-pixel noise. Why use this model? Of course, we do not consider the horizontal line to be a very general object. Nor do we think that noise on any particular binary contour is a random phenomenom. We have noticed in our work that binary contours often bear small noise, commonly "1-pixel wiggles". Our goal is to perform an analytical study of the ability of local weighted averaging smoothing methods to remove such noise. Since the filters are meant to be used with arbitrary binary contours, it seems reasonable to consider that over a large set of images noise can be considered random. Furthermore, we do not make the very frequent implicit assumption that a smoothing filter can be optimal independently of the specific attributes one intends to measure or even the specific numerical estimation method used. For specific measurements and computation methods, we would like to find the best choice of smoothing coefficients for a given window size and an estimate of how much noise these coefficients remove; if the window size is increased 3 , what are then the best coefficients and how much more is gained compared to the smaller window size? These questions are very pointed and we have no workable expression for small random 3 assuming the feature structure scale allows this. noise on an arbitrary binary contour which would allow to derive answers analytically. Thus we choose to look at an ideal object for which we can easily model random 1-pixel noise and our study can be carried out. Similar approaches are often followed. For example, in studying optimal edge detectors, Canny [29] considers the ideal step edge. There is no implication that this is a common object to detect in practice; simply it makes the analytical investigation easier and can still allow to gain insight into the edge detection problem more generally. The practicality of our own findings concerning optimal local weighted averaging will be verified in section IV. We now give a definition for our simple model. The infinite horizontal border with random 1-pixel noise consists of all points Z, satisfying with probability with probability p An example of such a simple noisy boundary is shown in Fig. 4. y Fig. 4. Noisy Horizontal Border With this model, x It then follows from Eq. 15 that the smoothing operation will not change the x-coordinates and the local weighted averaging will only affect the y-coordinates. The best fitting straight line through this initial data is easy to obtain since it must be of the form y. It is obtained by minimizing the mean square distance with respect to ~ y. The best fitting line is simply We will consider this to be the Eq. of the ideal border which has been corrupted by the digitization process, yielding the situation of Eq. 18. We now examine the problem of applying "optimal" local weighted averaging to the data of our simple model. Our aim is to eliminate the 'wiggles' along the noisy horizontal border as much as possible. An alternate formulation is that we would want the border, after smoothing, to be "as straight as possible" and as close as possible to Several criteria can be used to assess the straightness of the border and optimize the smoothing process: ffl Minimize the mean square distance to the best fitting line after the data has been smoothed. ffl Minimize the mean square slope along the smoothed data points (ideally, the border is straight and its slope should be 0 everywhere). ffl Minimize the mean square deviation angle OE i (see Fig. 2) along the smoothed data (ideally, OE i should also be 0 everywhere). Each of the above criteria is sound and none can be said to be the best without considering the particular situation further. The first criterion is the most commonly used in the curve fitting literature. In this paper however, we want to derive optimal smoothing methods tailored for specific computational tasks; hence, we will consider each of the above criteria in turn. If our interest is simply to obtain numerical estimates of the slopes at contour points, the optimal ff j 's derived based on the second criterion should be preferred. And for estimating deviation angles OE i , the optimal coefficients derived from the third criterion would be better. For the first criterion, we will use d rms , the root mean square (r.m.s.) distance to the best fitting line, as our measure of noise before and after the smoothing step; for the second criterion, m rms , the r.m.s. slope along the border; with the third criterion, OE rms , the r.m.s. deviation angle along the border. For the original unsmoothed data, these noise measures can be computed using the probabilities of the possible configurations of 2 or 3 consecutive pixels. Thus for the original, unsmoothed data we have: r using Based on our simple model, we now derive the best smoothing parameters for each of the three criteria mentioned above. Once obtained, we will compute the corresponding noise measures for the smoothed data which we will denote by [w] d 0 rms rms ; and [w] OE 0 rms respectively. In this notation the 'prime' indicates a single smoothing step and w is the window size used. A. Best Parameters to Minimize d 0 rms The unsmoothed y-coordinate of our border points is a discrete random variable following a simple Bernouilli distribution for which the expected value is y and the variance is Obviously, the best fitting line is simply the expected value and d rms in Eq. 20 is the standard deviation of y i . After smoothing, the expected value of y 0 (finite) window size w. Denoting the expected value by E, we have from elementary probability theory: E(y 0 E( j=\Gamman j=\Gamman j=\Gamman Now we find the best choice of smoothing parameters and the corresponding noise measure rms . Denoting variance by oe 2 (), we have: j=\Gamman Each ff j y i+j is a discrete random variable (with two possible values) and its variance is . Thus j=\Gamman We must now minimize j=\Gamman subject to the constraint ff problem is typically solved using the Lagrange multipliers method (see [53], page 182). from which we obtain the simple result ff for each k. All coefficients are equal, hence of value 1=(2n+1) 4 . Substituting this value into Eq. 25, we obtain the corresponding 4 As expected, straight averaging reduces the variance of a collection of random variables faster than any other weighted average. 2 . Our findings can be summarized as follows: ffl For a single smoothing iteration with arbitrary window size w, for any value of p, the best choice of parameters to minimize the mean square distance is to set all ff j 's to 1=w, resulting in [w] d 0 s The fraction of the noise which is removed by the smoothing operation is sw Hence, for the noise is reduced by 42:3%; for by 55:3%; for 62:2%. Finally, contrary to what one might expect, we note that the optimum 5-point smoothing operation is not to apply the optimum 3-point operation twice. The latter is equivalent to a 5-point window with ff which gives d 0 This would remove approximately 51.6% of the noise. B. Best Parameters to Minimize m 0 rms In this section, we apply our second criterion for straightness and minimize the root mean square value of the slope after smoothing. We consider two different ways of computing the slope from contour points. B.1 Based on m 0 The simplest numerical estimate of the slope is given by the forward difference formula i . This gives j=\Gamman \Gamman y i\Gamman ff \Gamman Expanding the last squared summation and taking the mean, we obtain the following \Gamman ff n y i+1+n y i\Gamman \Gamman y 2 i\Gamman November 20, 1997 ff \Gamman \Gamman+1 For any s; t 2 Z; y s y Substituting these results, Eq. 27 can be further simplified by making use of Eq. 14. We obtain: j=\Gamman Using Eq. 14 again, some algebraic manipulation leads to an expression for involving only the n independent parameters ff Our task now is to minimize the mean squared slope. Differentiating this with respect to a system of n linear equations as shown below.B ff 6 ff Solutions are given in Table I for 1 - n - 6. The column before last gives the fraction of the noise which is removed by the optimum smoothing method. For comparison purposes, the last column provides the equivalent result when all weights are set equal to 1 w . Finally we note that for window size 5, the triangular filter using ff results in a noise reduction of 80.8%, slightly better than the equal-weights method. fraction of noise removed B.2 Based on m 0 A more accurate 5 estimate of the slope is given by m 0 Expanding it in terms of the original coordinates and following the same approach as in the preceding section, we arrive at: j=\Gamman When expressed in terms of the n independent ff's, this becomes: Minimizing with respect to ff k , for 1 - k - n, we obtain another set of n linear equations for which the solutions are listed in Table II for 1 - n - 6. Note that the distribution of coefficients from ff \Gamman to ff n is no longer unimodal. Fur- thermore, values of n. Finally we note that for window size w = 5, the 5 Provided the data resolution is fine enough. and fraction of noise removed triangular filter using ff results in a noise reduction of 66.7%, notably less than the equal-weights method. Parameters to Minimize Deviation Angles In this section, we examine the smoothing problem based on minimizing the deviation angles OE 0 . Here the problem is more complex and we will not obtain general expressions of the optimum smoothing parameters which are independant of the probability p involved in our model. We restrict our study to the cases 5. Our definitions of section I imply that OE 0 . From trigonometry, we have tan OE 0 Now tan ' 0 . Thus we can obtain tan OE 0 i from the smoothed coordinates and then OE 0 i from the value of the tangent. C.1 For For i at p 0 will depend on the original contour points in a 5-point neighbourhood around p i . For our model of Eq. 18, there are 2 possible configurations for such a neighbourhood, which must be examined for their corresponding OE 0 . Of course, these November 20, 1997 computations need not be performed manually; they can be carried out using a language for symbolic mathematical calculation. Adding together the contributions from the 32 possible configurations, weighted by the respective probabilities of these configurations, results in the following expression for For simplicity we have dropped the subscript on the unique parameter ff 1 . Numerical optimization was performed to find the value of ff which minimizes Eq. 34. No single value of ff will minimize tan 2 OE 0 i for all values of p. The results are shown in Fig. 5(a). The best value of ff is now a smooth function of p. However we note that the domain of variation is very little. We cannot compare the results obtained minimizing the mean squared tangent of OE 0 to the situation without smoothing, since tan 2 OE i is infinite. By taking the arc tangent function of Eq. 33, we can obtain the values of the angles OE 0 themselves and we can derive an expression for OE 0 2 in the same manner. Numerical optimization of this expression yields the results shown in Fig. 5(b). As can be seen, they are almost the same as those of Fig. 5(a). In a similar fashion, we can generate expressions for j tan OE 0 i j, for which the best smoothing parameters are shown in Fig. 5(c) and 5(d) respectively. Here there seems to be one predominant best parameter over a wide range of values for p. All the results shown in Fig. 5 were obtained numerically, for values of p ranging from to 0.995, in steps of 0.005. As expected, all these curves are symmetric about so we will limit our discussion to p ! 0:5. In Fig. 5(c), the best value of ff for 0:495), the best value is Between these two intervals, p increases almost linearly. In Fig. 5(d), the same values of ff are found: 0:25 is the best choice for 0:2857 is the best choice for p 2 (0:150; 0:495). In Fig. 5(a) and 5(b), the best value of ff is a smoothly varying function of p. But we notice that 0:2857 is an intermediate value of ff in the narrow range of best values. In fact, choosing all values of p, the value of tan 2 OE 0 i is always within 0.2% of the minimum possible. Despite the differences in the actual curves of Fig. 5, the corresponding ranges of best ff's are always quite narrow and very similar, independently of the exact criterion chosen. From now on, to maintain uniformity with the treatment of sections III-A and III-B, we will restrict ourselves to minimizing the mean squared angle. Eq. 22 provided a measure of the noise before smoothing: OE rms = -q . The fraction of this noise (1 \Gamma OE 0 rms =OE rms ) which is removed by a simple smoothing operation with computed for different values of ff. The results are displayed in Fig. 6. The solid line represents the best case and we see that approximately 73% of the r.m.s. noise is removed. The dashed line, representing the case where ff = 0:2857 is used for all values of p, is not distinguishable from the best case at this scale. The dash-dotted and the dotted lines represent the fraction of noise removed for In this last case, this fraction is a constant equal to 0.6655. C.2 For For possible configurations of a 7-point neighbourhood centered on p 0 must be considered to obtain the values of OE 0 after smoothing. In Eq. 34, there were 9 distinct terms involving p and ff. Now the computation of tan 2 OE 0 results in 35 distinct terms in p, ff 1 , and ff 2 . We will not reproduce this lengthy expression here. Fig. 7(a) presents the best choice of parameters to minimize OE 0 2 . We notice that there is very little variation in their values over the range of values of p. The optimum parameters are approximately ff These values are close to 2 9 and 1 9 , the values for the triangular 5-point filter. Fig. 7(b) shows the fraction of OE rms removed by the smoothing operation. The solid line represents the case where the optimum parameters are used for each value of p. In this situation, approximately 88:75% of the noise is removed. The dashed line represents the case ff 9 and ff 9 , for which 86% of the noise is removed approximately. We see that these results are close to the ideal situation. IV. Verifying Results for Varying Curvature In the preceding section, we have studied optimum local weighted averaging extensively, based on a model of a horizontal border with random 1-pixel noise. Particular solutions were derived based on error criteria chosen in light of specific computational tasks to be performed after the smoothing operation. But can these results be relied upon to handle digitization noise along arbitrary contours? Our results were obtained for a straight, horizontal border, i.e. a line of curvature 0. But for arbitrary contours, curvature may vary from point to point. Should optimum smoothing parameters vary with curvature and, if so, in what manner? For a given window size, can a fixed set of smoothing parameters be found which will give optimum (or near optimum) results across a wide range of curvature values? If so, how does this set of parameters compare with the one we have derived using our simple model? In this section, we try to answer these above questions by performing some experiments with digital circles. It should be clear that our interest is not with digital circles per se but rather, as explained above, with the variation of optimum smoothing parameters with curvature. The approach will be to examine, for digital circles of various radii, situations which are equivalent to the ones studied for the horizontal straight border in sections III-A, III-B, and III-C. Using numerical optimization, we will find the best choice of smoothing parameters for each situation, over a wide range of curvature values, and compare them with the values obtained previously. An example of a digital circle is shown in Fig. 8 for a radius R = 7. A. Minimizing Error on Distances to Center In this section, we consider the distances d i from the center to each pixel P i as approximations to the radius R. See Fig. 9(a). After smoothing, pixel P i is replaced by pixel P 0 which is at a distance d 0 i from the center of the circle. Our aim is to find the values of the smoothing parameters, for these parameters might vary depending on the radius of the circles. For reasons of symmetry, it is only necessary to consider one quadrant; with special attention to the main diagonal, we can restrict our attention to the first octant of each November 20, 1997 circle. Let N 1=8 be the number of pixels which are strictly within the first octant. The mean value of (R \Gamma d 0 obtained by adding twice the sum of (R \Gamma d 0 points, plus the value for the pixel at coordinates (R; 0), plus the value for the pixel on the main diagonal (if present). This sum is then divided by 2N 1=8 there is a pixel on the main diagonal). We now give a simple example, for 3. First we consider the situation before any smoothing is applied. For the point on the x-axis, the value of (R \Gamma d 0 ) 2 is always 0. For the next point (4; 1), (R \Gamma d 1 . For the next point (3; 2), (R . Finally, for the point on the main diagonal (3; 3), (R \Gamma d 3 . The contributions for points (4; 1) and (3; 2) are counted twice and added to the contributions for the diagonal point (3; 3). This sum is then divided by 6. Taking the square root of the result, we obtain (R After smoothing with the smallest window size, we have the following values for (R (R (R (R and we must minimize the expression 1i . Thus the best smoothing parameter for is found to be In the numerical computations it is possible to take advantage of the fact that, for small window sizes, the smoothing rarely affects the y-coordinates in the first octant; exceptions occur occasionally for the last pixel in the first octant (not on the diagonal) and for the pixel on the diagonal when the preceding pixel has the same x-coordinate. This last condition is found only for radii values of 1, 4, 11, 134, 373, 4552 etc. (see Kulpa [54]). The coordinates of the pixels for the first octant of the digital circles were generated using the simple procedure presented in Horn [55], with a small correction pointed out by Kulpa [56] (see also Doros [57]). The best smoothing parameters were obtained for integer values ranging from 2 pixels to 99 pixels, in steps of 1. The results are presented in November 20, 1997 Fig. 9(b) for 5. For comparison, the values derived in section III-A from our model of a noisy horizontal edge are shown with dashed lines. For we see that for radii values larger than 20 pixels the best ff \Sigma1 oscillates around3 , as derived from our model. Similarly, for 5, the best values of ff \Sigma1 and ff \Sigma2 are close to the predicted value of 0:2. For small radii values however, the optimum ff \Sigma2 -values are much lower than this value and the optimum ff \Sigma1 -values are correspondingly higher. This is easily understood since a 5-pixel neighbourhood covers a relatively large portion of the circumference in these cases (as much as one eighth of the total circumference for a radius of 6 pixels, one fourth for a radius of 3 pixels). In fact, for radii values of 2, 3, 4, 6, and 8 pixels, it is best to use ff using only the nearest neighbour. Fig. 9(d) compares the r.m.s. values of the errors on the radii without smoothing (solid line) to the best values possible after smoothing with window sizes of (dashed line) and (dotted line) respectively. For each value of the radius, we have also compared the noise reduction achieved using the optimum parameters to that achieved with the constant values 1and 1. The results are presented in Fig. 10(a) and 10(b), for respectively. For the 2 curves are indistinguishable for R ? pixels, and they are very close for R - 10. For smoothing with ff actually worse than no smoothing at all. But, for R ? 18, the best curve and that obtained with these fixed values are very close. Finally, for compares the mean noise reduction of 3 methods: the optimum method, corresponding to the variable parameters of Fig. 9(b) and Fig. 9(c); the fixed parameter method derived from our model of section III; and the best fixed parameter method obtained from numerical estimates. We see that the results are very close and that the method derived from our simple noise model compares very well with the numerically determined best fixed parameter method. B. Minimizing Error on Tangent Directions In this section, we compare the direction of the tangent to a circle at a given point to the numerical estimate of that direction, obtained for digital circles. The situation is November 20, 1997 Window Method ff \Sigma1 ff \Sigma2 Mean noise reduction optimum variable 0.484317 best fixed 0.3329 0.482673 optimum variable variable 0.6184 best fixed 0.2148 0.1703 0.6117 III Mean noise reduction for illustrated in Fig. 11(a). Since the slope of the tangent is infinite at pixel (R; 0) of the digital circle, we will consider instead the angle which the tangent line makes with the x-axis. The radius from the center of the circle to pixel P i makes an angle ' i with the positive x-axis. Now consider the point where this radius intersects the continuous circle. Theoretically, the angle between the tangent to the circle at that point and the x-axis is - On the other hand, the numerical estimate of this angle is given by - is the angle between the horizontal axis and the perpendicular bisector of the segment from P i\Gamma1 to Fig. 11(a)). The difference between these angles, (' is the error on the elevation of the tangent to the circle at the point of interest. Our goal is to minimize the r.m.s. value of (' 0 primes refer to the quantities after smoothing. The values of ' i and ' i are readily computed in terms of the original coordinates of the digital circle as follows: The values of ' 0 are obtained similarly, in terms of the coordinates after smoothing. Once again, the r.m.s. error for the entire circle can be computed by considering only the first octant; and the best smoothing parameters were obtained for integer radii values ranging from 4 pixels to 99 pixels, in steps of 1. The results are presented respectively in Fig. 11(b) for 5. In section III-B.2, for the optimum value derived for ff 5, the optimum values derived for ff \Sigma1 and ff \Sigma2 were 1and 3respectively. These values are shown with horizontal dashed lines in Fig. 11(b) and 11(c). The best smoothing parameters vary with the values of R. However, when we compute their means for 4 - R - 99, the results obtained are very close to the predicted values. Thus, for (compared to 0.4); for (compared to 0.1429) and ff (compared to 0.2143). Fig. 11(d) compares the r.m.s. values of the errors on the elevation of the tangents to a circle without smoothing (solid line) to the best values possible after smoothing with window sizes (dashed line) and (dotted line). The noise reduction produced by smoothing is equal to 1:0 \Gamma (' 0 . For each value of the radius, we have compared the noise reduction achieved using the optimum parameters to that achieved with the constant values ff 5 , for 5. The results are presented in Fig. 12(a) and 12(b) respectively. As can be seen, the constant values predicted by our simple model yield noise reduction results which are very close to optimum. Finally, for 4 - R - 99 and window sizes the mean noise reduction of 3 methods as explained previously. The best smoothing parameters derived from our simple noise model and the numerically determined best fixed parameters are almost the same and their performance is nearly optimal. C. Minimizing Error on Deviation Angles In this section, we compare the deviation angles along the circumference of a circle to the numerical estimates obtained for digital circles. The situation is illustrated in Fig. 13(a). Consider 3 consecutive pixels P on the circumference of a digital circle. The deviation angle at P i is denoted by OE i . Now the line segments from the center of the circle to these 3 pixels (partly represented by dashed lines in the figure) have elevation angles of ' respectively. The intersection of these line segments with the circle are the true circle points Q elevations. Connecting these November 20, 1997 Window Method ff \Sigma1 ff \Sigma2 Mean noise reduction optimum variable 0.58084 best fixed 0.4026 0.57478 optimum variable variable 0.7658 14 0.7572 best fixed 0.1445 0.2088 0.7576 IV Mean noise reduction for 4 - R - 99 points by line segments defines a deviation angle ffi i at Q i , for which OE i is a numerical estimate. The difference between ffi i and OE i is the error on the deviation angle at the point of interest. Our goal in this section is to find the optimum parameters which will minimize the r.m.s. value of this error, after smoothing. In terms of the pixel coordinates deviation angle OE i is equal to To compute ffi i , we first obtain the elevation angles as ' then the coordinates of the circle points Q i as Finally, ffi i is computed as in Eq. 37, using (~x; ~ y) instead of (x; y). The best smoothing parameters were obtained for integer radii ranging from 4 pixels to pixels, in steps of 1. The results are presented in Fig. 13(b) for 5. In section III-C, for the optimum value derived for ff \Sigma1 was 0.2857; for 5, the optimum values derived for ff \Sigma1 and ff \Sigma2 were 0.2381 and 0.1189 respectively. These values are shown with dashed lines in Fig. 13(b) and 13(c). Again, the best smoothing parameters vary with the values of R. However, when we compute their means for 4 - R - 99, the November 20, 1997 results obtained are very close to the predicted values. Thus, for Fig. 13(d) compares the r.m.s. values of the errors on the deviation angles without smoothing (solid line) to the best values possible after smoothing with window sizes (dashed line) and (dotted line). The noise reduction produced by smoothing is equal to 1:0 \Gamma (ffi 0 . For each value of the radius, we have compared the noise reduction achieved using the optimum parameters to that achieved with the constant values ff 5. The results are presented in Fig. 14(a) and 14(b) respectively. As can be seen, the constant values predicted by our simple model yield noise reduction results which are very close to optimum. For 4 - R - 99 and window sizes compares the mean noise reduction of 3 methods as explained previously. The best smoothing parameters derived from our simple noise model and the numerically determined best fixed parameters are even closer than in the 2 previous cases and their performance is very nearly optimal. Window Method ff \Sigma1 ff \Sigma2 Mean noise reduction optimum variable 0.770903 best fixed 0.2865 0.765732 optimum variable variable 0.910833 best fixed 0.2386 0.1190 0.906917 Mean noise reduction for 4 - R - 99 Finally, for one prefers to use ff 1(computationnally convenient for deviation angle measurements), the mean error reduction level is 0.8832. This is very good but not quite as effective as the optimum methods previously discussed. A comparison of the error reduction with the best possible solution, for every value of R, appears in Fig. 15. V. Conclusion This paper has presented a different avenue to solve the problem of optimum smoothing of 2-D binary contours. Several approaches were reviewed with particular emphasis on their theoretical merits and implementation difficulties. It was argued that most methods are eventually implemented as a local weighted average with particular weight values. Hence we adopted this scheme as the starting point of our investigation into optimum methods. Furthermore, there are many applications where smoothing is performed to improve the precision of specific measurements to be computed from the contour points. In such cases, the smoothing parameters should be chosen based on the nature of the computations intended, instead of relying on a single, general 'optimality' criterion. Thus our work was focused on optimum local weighted averaging methods tailored for specific computational goals. In the present article, we have considered three such goals: obtaining reliable estimates of point positions, of slopes, and of deviation angles along the contours. To study the problem, a simple model was defined to represent 1-pixel random noise along a straight horizontal border. Based on this simple model, an in-depth analytical investigation of the problem was carried out, from which precise answers were derived for the 3 chosen criteria. Despite its simplicity, this model captures well the kind of perturbations which digitization noise causes in the numerical estimation of various quantities along 2-D binary contours even with arbitrary curvature. This was indeed verified, for window sizes of and by finding the best smoothing parameters, using equivalent criteria, for digital circles over a wide range of radii. In this general case, the best smoothing parameters were found to vary according to the length of the radius. Thus, in order to take full advantage of these optimum filters, it would be necessary to compute local estimates of the radius of curvature for groups of consecutive pixels along the contour, and then apply the best parameters found for these radii. This would significantly reduce the efficiency of the smoothing operation. However, it is not really necessary to go to that extent since the performance of these varying-weight optimum filters can be very nearly approached by methods with a fixed set of parameters. The latter were derived by numerical computation, for a wide range of radii. And it turned out that their values were very close to those predicted using our simple digitization noise model. These numerical computations with varying radius of curvature validate our proposed model and confer added confidence to the results obtained from it. Researchers requiring simple and effective local weighted averaging filters before making numerical estimates of specific quantities can thus rely on this model to derive optimum methods tailored to their particular needs. VI. Acknowledgements The authors would like to thank Dr. Louisa Lam for helpful comments and suggestions made on an earlier draft of this paper. This work was supported by the National Networks of Centres of Excellence research program of Canada, as well as research grants awarded by the Natural Sciences and Engineering Research Council of Canada and by an FCAR team research grant awarded by the Ministry of Education of Quebec. --R "Digitized circular arcs: Characterization and parameter estimation," "Curve fitting approach for tangent angle and curvature measurements," "On the encoding of arbitrary geometric configurations," "Extraction of invariant picture sub-structures by computer," "Analysis of the digitized boundaries of planar objects," "On the digital computer classification of geometric line patterns," "Improved computer chromosome analysis incorporating preprocessing and boundary analysis," "Computer recognition of partial views of curved objects," "The medial axis of a coarse binary image using boundary smoothing," "Performances of polar coding for visual localisation of planar objects," "Curve smoothing for improved feature extraction from digitized pictures," "Measurements of the lengths of digitized curved lines," "Multiple resolution skeletons," "Non-parametric dominant point detection," "On the detection of dominant points on digital curves," "Fitting digital curve using circular arcs," "Scale-space filtering," "The structure of images," "The curvature primal sketch," "Scale-based description and recognition of planar curves and two-dimensional shapes," "Robust contour decomposition using a constant curvature criterion," "Adaptive smoothing: A general tool for early vision," "Pattern spectrum and multiscale shape representation," "A multiscale approach based upon morphological filtering," "Scale-space from nonlinear filters," "Smoothing by spline functions," "Spline functions and the problem of graduation," "A regularized solution to edge detection," "A computational approach to edge detection," "Uniqueness of the gaussian kernel for scale-space filtering," "Scaling theorems for zero crossings," "The renormalized curvature scale space and the evolution properties of planar curves," "Scaling theorems for zero-crossings," "Scaling theorems for zero crossings of bandlimited signals," "Scale-space for discrete signals," "An extended class of scale-invariant and recursive scale space filters," "Multiscale nonlinear decomposition: The sieve decomposition theorem," "Optimal estimation of contour properties by cross-validated regularization," "Optimal smoothing of digitized contours," "One-dimensional regularization with discontinuities," "Partial smoothing splines for noisy boundaries with corners," "Describing "Filtering closed curves," "Organization of smooth image curves at multiple scales," "Local reproducible smoothing without shrinkage," "Iterative smoothed residuals: A low-pass filter for smoothing with controlled shrinkage," "Optimal L1 approximation of the gaussian kernel with application to scale-space construction," "Regularization of digitized plane curves for shape analysis and recognition," "Refining curvature feature extraction to improve handwriting recognition," "Multistage digital filtering utilizing several criteria," "Computer recognition of unconstrained hand-written numerals," "Speed, accuracy, flexibility trade-offs in on-line character recognition," John Wiley "On the properties of discrete circles, rings, and disks," "Circle generators for display devices," "A note on the paper by "Algorithms for generation of discrete circles, rings, and disks," --TR --CTR Ke Chen, Adaptive Smoothing via Contextual and Local Discontinuities, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.10, p.1552-1567, October 2005 Helena Cristina da Gama Leito , Jorge Stolfi, A Multiscale Method for the Reassembly of Two-Dimensional Fragmented Objects, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.9, p.1239-1251, September 2002

optimal local weighted averaging;contour smoothing;gaussian smoothing;digitization noise modeling

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Bias in Robust Estimation Caused by Discontinuities and Multiple Structures.

AbstractWhen fitting models to data containing multiple structures, such as when fitting surface patches to data taken from a neighborhood that includes a range discontinuity, robust estimators must tolerate both gross outliers and pseudo outliers. Pseudo outliers are outliers to the structure of interest, but inliers to a different structure. They differ from gross outliers because of their coherence. Such data occurs frequently in computer vision problems, including motion estimation, model fitting, and range data analysis. The focus in this paper is the problem of fitting surfaces near discontinuities in range data.To characterize the performance of least median of the squares, least trimmed squares, M-estimators, Hough transforms, RANSAC, and MINPRAN on this type of data, the "pseudo outlier bias" metric is developed using techniques from the robust statistics literature, and it is used to study the error in robust fits caused by distributions modeling various types of discontinuities. The results show each robust estimator to be biased at small, but substantial, discontinuities. They also show the circumstances under which different estimators are most effective. Most importantly, the results imply present estimators should be used with care, and new estimators should be developed.

Introduction Robust estimation techniques have been used with increasing frequency in computer vision applications because they have proven effective in tolerating the gross errors (outliers) characteristic of both sensors and low-level vision algorithms. Most often, robust estimators are used when fitting model parameters - e.g. the coefficients of either a polynomial surface, an affine motion model, a pose estimate, or a fundamental matrix - to a data set. For these applications, robust estimators work reliably when the data contain measurements from a single structure, such as a single surface, plus gross errors. Sometimes, however, the data are more complicated than this, presenting a challenge to robust estimators not anticipated in the robust statistics literature. This complication occurs when the data are measurements from multiple structures while still being corrupted by gross outliers. These structures may be different surfaces in depth measurements or multiple moving objects in motion estimation. Here, the difficulty arises because robust estimators are designed to extract a single fit. Thus, to estimate accurate parameters modeling one of the structures - which one is not important - they must treat the points from all other structures as outliers. After successfully estimating the fit parameters of one structure, the robust estimator may be re-applied, if desired, to estimate subsequent fits after removing the first fit's inliers from the data. An example using synthetic range data illustrates the potential problems caused by multiple structures. Figure 1 shows (non-robust) linear least-squares fits to data from a single surface and to data from a pair of surfaces forming a step discontinuity. In the single surface example, the least-squares fit is skewed slightly by the gross outliers, but the points from the surface are still generally closer to the fit than the outliers. Thus, the fit estimated by a robust version of least-squares will not be significantly corrupted by these outliers. In the multiple surface example, the least-squares fit is skewed so much that it crosses (or "bridges") the point sets from both surfaces, placing the fit in close proximity to both point sets. Since robust estimators use fit proximity to distinguish inliers and outliers and downgrade the influence of outliers, this raises two concerns about the accuracy of robust fits. First, an estimator that iteratively refines an initial least squares fit will have a local, and potentially a b Figure 1: Examples demonstrating the effects of (a) gross outliers and (b) both gross outliers and data from multiple structures on linear least-squares fits. global, minimum fit that is not far from the initial, skewed fit. This is because points from both surfaces will have both small and large residuals, making it difficult for the estimator to "pull away" from one of the surfaces. Second, and more important, for the robust estimate be the correct fit, thereby treating the points from one surface as inliers and points from the other as outliers, the estimator's objective function must be lower for the smaller inlier set of the correct fit than the larger inlier set of the bridging fit. By varying both the proximity of the two surfaces and the relative sizes of their point sets, all robust estimators studied here can be made to "fail" on this data, producing fits that are heavily skewed. Motivated by the foregoing discussion, the goal of this paper is to study how effectively robust estimators can estimate fit parameters given a mixture of data from multiple struc- tures. Stating this "pseudo outliers problem" abstractly, to obtain an accurate fit a robust technique must tolerate two different types of outliers: gross outliers and pseudo outliers. Gross outliers are bad measurements, which may arise from specularities, boundary effects, physical imperfections in sensors, or errors in low-level vision computations such as edge detection or matching algorithms. Pseudo outliers are measurements from one or more additional structures. (Without losing generality, inliers and pseudo outliers are distinguished by assuming the inliers are points from the structure contributing the most points and pseudo outliers are points from the other structures.) The coherence of pseudo outliers distinguishes them from gross outliers. Because data from multiple structures are common in vision ap- plications, robust estimators' performance on this type of data must be understood to use them effectively. Where they prove ineffective, new and perhaps more complicated robust techniques will be needed. 1 To study the pseudo outliers problem, this paper develops a measure of "pseudo outlier bias" using tools from the robust statistics literature [10, pages 81-95] [12, page 11]. Pseudo outlier bias will measure the distance between a robust estimator's fit to a ``target'' distribution and its fit to an outlier corrupted distribution. The target distribution will model the distribution of points drawn from a single structure without outliers, and the outlier corrupted mixture distribution [27] will combine distributions modeling the different structures and a gross outlier distribution. The optimal fit is found by applying the functional form of an estimator to these distributions, rather than by applying the estimator's standard form to particular sets of points generated from these distributions. This gives a theoretical mea- sure, avoids the need for extensive simulations, and, most importantly, shows the inherent limitations of robust estimators by studying their objective functions independent of their search techniques. The bias of a number of estimators - M-estimators [12, Chapter 7], least median of squares (LMS) [16, 21], least trimmed squares (LTS) [21], Hough transforms [13], RANSAC [7], and MINPRAN [26] - will be studied as the target and mixture distributions vary. The application for studying the pseudo outliers problem is fitting surfaces to range data taken from the neighborhood of a surface discontinuity. While this is a simple application for studying the pseudo outliers problem, the problem certainly arises in other applications as well - essentially any application where the data could contain multiple structures - and the results obtained here should be used as qualitative predictions of potential difficulties in these applications. In the context of the range data application, three idealized discontinuity models are used to develop mixture distributions: step edges, crease edges and parallel surfaces. Step edges model depth discontinuities, where points from the upper surface of the step are pseudo outliers to the lower surface. Crease edges model surface orientation discontinuities, where points from one side of the crease are pseudo outliers to the other. versions of these techniques actually exist for fitting surfaces to range data. Their effectiveness, however, depends in part on the accuracy of an initial set of robust fits. Finally, parallel surfaces model transparent or semi-transparent surfaces, where a background surface appears through breaks in the foreground surface, and data from the background are pseudo outliers to the foreground. A final introductory comment is important to assist in reading this paper. The paper defines the notion of "pseudo outlier bias" using techniques common in mathematical statistics but not in computer vision, most importantly, the "functional form" of a robust estimator. The intuitive meaning of functional forms and their use in pseudo outlier bias are discussed at the start of Section 4, which then proceeds with the main derivations. Readers uninterested in the mathematical details should be able to skip Sections 4.2 through 4.6 and still follow the analysis results. Robust Estimators This section defines the robust estimators studied. These definitions are converted to functional forms suitable for analysis in Section 4. Because the goal of the paper is to expose inherent limitations of robust estimators, the focus in defining the estimators is their objective functions rather than their optimization techniques. Special cases of iterative optimization techniques where local minima are potentially problematic will be discussed where appropriate. The data are (~x is an image coordinate vector - the independent vari- able(s) - and z i is a range value - the dependent variable. Each fit is a function z = '(~x), often restricted to the class of linear or quadratic polynomials. The notation - '(~x) indicates the fit that minimizes an estimator's objective function, with - ' called the "estimate". Each esti- mator's objective function evaluates hypothesized fits, '(~x), via the residuals, r 2.1 M-Estimators A regression M-estimate [12, Chapter 7] is ae(r i;' =-oe); (1) oe is an estimate of the true scale (noise) term, oe, and ae(u) is a robust "loss" function which grows subquadratically for large juj to reduce the effect of outliers. (Often, as discussed ' and - oe are estimated jointly.) M-estimators are categorized into three types [11] by the behavior of one estimator of each type is studied. Monotone M-estimators (Figure 2a), such as Huber's [12, Chapter 7], have non-decreasing, bounded /(u) functions. Hard redescenders (Figure 2b), such as Hampel's [9] [10, page 150], force hence c is a rejection point, beyond which a residual has no influence. Soft redescenders (Figure 2c), such as the maximum likelihood estimator of Student's t-distribution [5], do not have a finite rejection point, but force /(u) ! 0 as juj !1. The three robust loss functions are shown in Figure 2 and in order they are (2) ae h and ae s The ae functions' constants are usually set to optimize asymptotic efficiency relative to a given target distribution [11] (e.g. Gaussian residuals). M-estimators typically minimize using iterative techniques [11] [12, Chapter 7]. The objective functions of hard and soft redescending M-estimators are non-convex and may have multiple local minima. In general, - oe must be estimated from the data. Hard-redescending M-estimators often use the median absolute deviation (MAD) [11] computed from the residuals to an initial fit, Monotone Hard Soft ae(u) (a) (b) (c) Figure 2: ae(u) and /(u) functions for three M-estimators. for consistency at the normal distribution and at Student's t-distribution (when Other M-estimators jointly estimate - oe and - ' as ';oe ae(r In particular, Huber [12, Chapter 7] uses =oe) is from equation 2 and a is a tuning parameter; Mirza and Boyer [5] use ae s (r =oe) is from equation 4. When fitting surfaces to range data, a different option for obtaining - oe is often used [3]. If oe depends only on the properties of the sensor then - oe may be estimated once and fixed for all data sets. Theoretically, when - oe is fixed, the M-estimators described by equation 1 are no longer true M-estimators since they are not scale equivariant [10, page 259]. To reflect this, when - oe is fixed a priori, they are called "fixed-scale M-estimators." Both standard M-estimators and fixed-scale M-estimators are studied here. 2.2 Fixed-Band Techniques: Hough Transforms and RANSAC Hough transforms [13], RANSAC [4, 7], and Roth's primitive extractor [20] are examples of "fixed-band" techniques [20]. For these techniques, - ' is the fit maximizing the number of points within ' \Sigma r b , where r b is an inlier bound which generally depends on - oe (i.e. r some constant c). Equivalently, viewing fixed-band techniques as minimizing the number of outliers, they become a special case of fixed-scale M-estimators with a simple, discontinuous loss function ae f Fixed-band techniques search for - using either random sampling or voting techniques. 2.3 LMS and LTS Least median of squares (LMS), introduced by Rousseeuw [21], finds the fit minimizing the median of squared residuals. (See [16] for a review.) Specifically, the LMS estimate is fmedian i Most implementations of LMS use random sampling techniques to find an approximate minimum. Related to LMS and also introduced by Rousseeuw [21] is the least trimmed squares estimator (LTS). The LTS estimate is where the (r 2 are the (non-decreasing) ordered squared residuals of fit '. Usually implementations also use random sampling. 2.4 MINPRAN MINPRAN searches for the fit minimizing the probability that a fit and a collection of inliers to the fit could be due to gross outliers [24, 26]. It is derived by assuming that relative to any hypothesized fit '(x) the residuals of gross outliers are uniformly distributed 2 in the range \SigmaZ 0 . Based on this assumption, the probability that a particular gross outlier could be within '(~x i Furthermore, if all n points are gross outliers, the probability k or more of them could be within '(~x) \Sigma r is Given n data points containing an unknown number of gross outliers, MINPRAN evaluates hypothesized fits '(~x) by finding the inlier bound, r, and the associated number of points (inliers), k r;' , within \Sigmar of '(~x), minimizing the probability that the inliers could actually be gross outliers. Thus MINPRAN's objective function in evaluating a particular fit is min r and MINPRAN's estimate is [min r MINPRAN is implemented using random sampling techniques (see [26]). Modeling Discontinuities The important first step in developing the pseudo outlier bias analysis technique is to model the data taken from near a discontinuity as a probability distribution. Attention here is restricted to discontinuities in one-dimensional structures, since this will be sufficient to demonstrate the limitations of robust estimators. 3.1 Outlier Distributions To set the context for developing the distributions modeling discontinuities, consider the one- dimensional, outlier corrupted distributions used in the statistics literature to study robust location estimators [10, page 97] [12, page 11]: 2 MINPRAN has been generalized to any known outlier distribution [26]. z x x d Figure 3: Example data set for points near a step discontinuity. Here, F 1 is an inlier distribution (also called a "target distribution"), such as a unit variance Gaussian, and G is an outlier distribution, such as a large variance Gaussian or an uniform distribution over a large interval. The parameter " is the outlier proportion. A set A of N points sampled from this distribution will contain on average "N outliers. Robust location estimators are analyzed using distribution F rather than using a series of point sets sampled from F . 3.2 Mixture Distributions Modeling Discontinuities The present paper analyzes robust regression estimators by examining their behavior on distributions modeling discontinuities. These mixture distributions [27] will be of the form will be inlier, pseudo outlier and gross outlier distributions, respectively, and control the proportion of points drawn from the three distributions. To and to set " s and " of data points taken from the vicinity of a discontinuity. For example, S might be the points in Figure 3 whose x coordinate falls in the interval [x modeled as a two-dimensional distribution of points (x; z) with x values in an interval [x losing generality, more points are from the left side of the discontinuity location than the right. (Using a two-dimensional distribution could be counterintuitive since the x values, which may be thought of as image positions at which depth measurements are made, are usually fixed.) Here, x is treated as uniform in the interval [x modeling the uniform spacing of image positions. 3 The depth measurement for an inlier is z = fi 1 (x)+e, where e is independent noise controlled by the Gaussian density g(e; oe 2 ) with mean 0 and variance oe 2 . fi 1 (x) models the ideal curve from which the inliers are measured. The pseudo outlier distribution, H 2 , can be defined similarly, with x values uniform in [x d for both distributions H 1 and H 2 , the densities of x and z can be combined to give the joint density 0; otherwise. bound the uniform distribution on the x interval. For the distribution of gross outliers in S, again x values are uniformly distributed, but this time over the entire interval [x z values are governed by density g o (z), which will be uniform over a large range. This gives the joint density for a gross outlier: 0; otherwise. The mixture proportions " s and " in (14) are easily specified. " just the fraction of gross outliers. " s is the "relative fraction" of inliers, i.e. the fraction points that are not gross outliers and that are from the inlier side of the discontinuity. Assuming the density of x values does not change across the discontinuity, " s is determined by x d : Equivalently, inliers and pseudo outliers, Notice that the "actual fraction" of inliers is " Depending on which estimator is being analyzed, either the relative or the actual fraction or both will be important. 3 For any point set sampled from this distribution, the x values will not be uniformly spaced, in general, but their expected values are. This expected behavior is captured when using the distribution itself in the analysis rather than points sets sampled from the distribution. Using these mixture proportions, the above densities can be combined into a single, mixed, two-dimensional density: Observe that the "target density" is just h 1 (x; z) and the "target distribution" is H 1 (x; z). The mixture distribution H(x; z) and the target distribution H 1 (x; z) can be calculated from h(x; z) and h 1 (x; z) respectively. Using mixture density h(x; z), data can be generated to form step edges and crease edges. The appropriate model is determined by the two curve functions fi 1 and fi 2 . For example, a step edge of height \Deltaz is modeled by setting fi 1 c. A crease edge is modeled when fi 1 and fi 2 are linear functions and lines with overlapping x domains can be created by using fi 1 and fi 2 from step edges, but setting x proportion of points from the lower line. In this case, the mixture proportions are divorced from the location of the discontinuity, which has no meaning. Thus, all three desired discontinuities can be modeled. 4 Functional Forms and Mixture Models To analyze estimators on distributions H, each estimator must be rewritten as a functional, a mapping from the space of probability distributions to the space of possible estimates. This section derives functional forms of the robust estimators defined in Section 2. It starts, in Section 4.1 by giving intuitive insight. Then, Section 4.2 introduces functional forms and empirical distributions on a technical level, using univariate least-squares location estimates as an example. Next, Section 4.3 derives several important distributions needed in the functionals. The remaining sections derive the required functionals. Readers uninterested in the technical details should read only Section 4.1 and then skip ahead to Section 5. 4.1 Intuition To illustrate what it means for a functional T to be applied to a distribution H, consider least-squares regression. When applied to a set containing points objective function is i;' , which is proportional to the second moment of the residuals conditioned on ', and the least squares estimate is the fit - minimizing this conditional second moment. A similar second moment, conditioned on ', may be calculated for distribution H(x; z), and the fit - ' minimizing this conditional second moment may be found. This is the least-squares regression functional. The functional form of an M-estimator, by analogy, returns the fit minimizing a robust version of the second moment of the conditional residual distribution calculated from H. Intuitions about the functional forms of other estimators are similar. The estimate T (H) can be used to represent or characterize the estimator's performance on point sets sampled from H. Although the robust fit to any particular point set may differ from T (H), if T (H) is skewed by the pseudo and gross outliers, then the fit to the point set will likely be skewed as well. Indeed, when an estimator's minimization technique is an iterative search, the skew may be worse than that of T (H) because it may stop at a local minimum. 4.2 One-Dimensional Location Estimators To introduce functional forms on a more technical level, this section examines the least-squares location estimate for univariate data. For a finite sample fx g, the location estimate is 'n which is the sample mean or expected value. The functional form of this is the location estimate of the distribution F from which the x i 's are drawn: Z Z Z the population mean or expected value. The functional form of the location estimate is derived from the sample location estimate by writing the latter in terms of the "empirical distribution" of the data, denoted by F n , and then replacing F n with F , the actual distribution. The empirical density of fx is where ffi(\Delta) is the Dirac delta function, and the empirical distribution is where u(\Delta) is the unit step function. When the x i 's are independent and identically dis- converges to F as n ! 1. The least squares location estimate is written in terms of the empirical density by using the sifting property of the delta function [8, page 56]: argmin 'n 'n Z Z Z Replacing f n with the population density yields the functional form of the location estimate as desired (20). 4.3 Residual Distributions and Empirical Distributions Before deriving functional forms for the robust regression estimators, the mixture distribution H(x; z) must be rewritten in terms of the distribution of residuals relative to a hypothesized fit, '. This is because the estimators' objective functions depend directly on residuals r and only indirectly on points (x; z). In addition, several empirical versions of this "residual distribution" are needed. Two different residual distributions are required: one for signed residuals and one for their absolute values. Let the distribution and density of signed residuals be F s (rj'; H) and (including H in the notation to make explicit the dependence on the mixture distribution). These are easily seen to be (Figure 4a) F s Z '(x)+r a b Figure 4: The cumulative distribution of residual r relative to fit '(x) is the integral of the point densities, h 1 and h 2 , from the curves and from the gross outlier density, h over the region bounded above by '(x) bounded on the sides by (a) unbounded below for signed residuals or (b) bounded below by '(x) \Gamma r for absolute residuals. Both figures show the region of integration for functions fi i and x boundaries modeling a step edge. and Let the distribution and density of absolute residuals be F a (rj'; H) and f a (rj'; H), where r - 0. These are (Figure 4b) F a Z '(x)+r and f a Appendix A evaluates these integrals. Replacing h with h 1 in the above equations yields the residual distributions and densities for the target (inlier) distribution. several empirical distributions are needed below. First, given n points sampled from h(x; z), the empirical density of the data is simply should not be confused with h i from equation 15). Next, the empirical density of the signed residuals follows from h n (x; z) using the sifting property of the ffi function [8, page 56]: \Gamma1n =n Finally, the empirical distribution of the absolute residuals is F a Z r \Gammar 4.4 M-Estimators and Fixed-Band Techniques The functionals for the robust regression estimators can now be derived, starting with that of fixed-scale M-estimators. The first step is to write equation 1 in a slightly modified form, which does not change the estimate: 'n ae(r i;' =-oe); Next, writing this in terms of the empirical distribution produces argmin 'n ae(r i;' 'n 'n ZZ ZZ Replacing the empirical density h n (x; z) with the mixture density h(x; z) yields T ae ZZ The change of variables simplifies things further, T ae Z ae(r=-oe) Z Z This is the fixed-scale M-estimator functional. Substituting equations 2, 3 and 4 gives respectively for the M-estimators studied here. For the M-estimators that jointly estimate - ' and - oe (see equations 7 and 8), the functional is obtained by replacing ae(r=-oe) with ae(r; oe) in equation 27, producing T ae;s Z Finally, recalling that fixed-band techniques are special cases of fixed-scale M-estimators, their functional is obtained by substituting equation 9 into equation 27, yielding T b Observe that [1 \Gamma F a (r b j'; H)] is the expected fraction of outliers. 4.5 LMS and LTS Deriving the functional equivalent to LMS requires first deriving the cumulative distribution of the squared residuals and then writing the median in terms of the inverse of this distribution. Defining the empirical distribution of squared residuals is F n;y since it is simply the percentage of points whose absolute residuals relative to fit ' are less than y. Now, n;y (1=2j'; h); (30) In other words, the median is the inverse of the cumulative, evaluated at 1=2. 4 This is the standard functional form of the median [10, page 89]. Substituting equation 4 When LMS is implemented using random sampling where p points are chosen to instantiate a fit, the median residual is taken from among the remaining points. To reflect this, the 1=2 in equation could be replaced by (n replacing the empirical distribution F n;y with F y produces the LMS y Turning now to LTS, normalizing its objective function and writing it in terms of the empirical density of residuals yieldsn Z rm n;y (1=2j'; H n ) is the empirical median square residual. The functional form of LTS then is easily written as T T \GammaF 4.6 MINPRAN MINPRAN's functional is derived by first re-writing MINPRAN's objective function, replacing the binomial distribution with the incomplete beta function [19, page 229]: min r where \Gamma(v)\Gamma(w) Z pt and \Gamma(\Delta) is the gamma function. This is done because I(v; w; p) only requires v; w 2 the binomial distribution requires integer values for k r;' and n. Now, since F a is the empirical distribution of the absolute residuals (see equation 26), k objective function can be re-written equivalently as min r I(n \Delta F a Replacing F a n by F a and substituting equation 13 gives the functional min r Observe that n, the number of points, is still required here, but TM (H) is considered a functional [10, page 40]. 5 Pseudo Outlier Bias Now that the functional forms of the robust estimators have been derived, the pseudo outlier bias metric can be defined. Given a particular mixture distribution H(x; z), target distribution These fits are assumed to minimize the estimator's objective functional globally. Then, pseudo outlier bias is defined as the normalized L 2 distance between the fits: oe 1=2 As is easily shown, this metric is invariant to translation and independent scaling of both x and z. (For fixed-scale M-estimators, - oe, which is provided a priori, must be scaled as well. For MINPRAN, the outlier distribution must be scaled appropriately.) When the set of the possible curves '(x) includes fi 1 (x), it can be shown that for each of the functionals derived in Section 4, T In other words, the estimator's objective function is minimized by fi 1 . 5 When T the pseudo outlier bias metric becomes oe 1=2 Intuitively, pseudo outlier bias measures the L 2 norm distance between the two estimates, T (H) and T (H 1 ), normalized by the length of the x interval over which H(x; z) is non-zero and by the standard deviation of the noise in the z values. Since T (H 1 for the cases studied here, a metric value of 0 implies that T is not at all corrupted by the presence of either gross or pseudo outliers, and a metric value of 1 implies that on average over the x domain T (H) is one standard deviation away from fi 1 . 5 In the analysis results given in Section 6, the set of curves will be linear functions of the form will also be linear. These curves are continuous and have infinite extent in x, unlike the densities modeling data drawn from them. z d s a a z d Step Creaseb (x) zx Parallel Figure 5: Parameters controlling the curve models for step edges, crease edges, and parallel lines. In each case, fi 1 (x) is the desired correct fit and points from fi 2 (x) are pseudo outliers. \Deltaz=oe is the scaled discontinuity magnitude, and " s controls the percentage of points from 6 Bias Caused by Surface Discontinuities Pseudo outlier bias (or "bias" for short) can now be used to analyze robust estimators' accuracy in fitting surfaces to data from three different types of discontinuities: step edges, crease edges, and parallel lines with overlapping x domains. To do this, Section 6.1 parameterizes the mixture density, outlines the technique to find T (H), and discusses the relationship between results presented here and results for higher dimensions. Then, analysis results for specific estimators are presented: fixed-scale M-estimators and fixed-band techniques (Sec- tion 6.2) which require a prior estimate of - oe, standard M-estimators (Section 6.3) which estimate - oe, and LMS, LTS and MINPRAN (Section 6.4) which are independent of - oe. In each case, the bias is examined as both the discontinuity magnitude and mixture of inliers, pseudo outliers and gross outliers vary. -5 -T Figure Surface plot of the objective functional of T ae h (H), i.e. R the hard redescending, fixed-scale M-estimator on a step edge with " when fits have the form b. (The plot shows the negation of the objective functional, so local minima in the functional appear as local maxima in the plot.) There are three local optimal: one at the second at and the third at a heavily biased fit, 1:91. The biased fit is the global optimum. 6.1 Discontinuity Models and Search Figure 5 shows the models of step edges, crease edges, and parallel lines. The translation and scale invariance of both the estimators and pseudo outlier bias, along with several realistic assumptions, allow these discontinuities to be described with just a few parameters. (Refer back to Section 3 for the exact parameter definitions.) For all models, and the x interval is [0; 1]. For step edges, fi 1 retaining the oe parameter to make clear the scale invariance - and x these values, " To move from step to crease edges, only the curves fi 1 (x) and fi 2 (x) must be changed. Referring to Figure 5b, these functions are no explicit role because it is not scale invariant. For parallel lines (Figure 5c), fi 1 (x) and fi 2 (x) are the same as for step edges, x and the parameter x d plays no role. Finally, the outlier distribution g o (z) is uniform for z within \Sigmaz 0 =2 of \Deltaz=2 and 0 otherwise. The foregoing shows that the parameters " s , " \Deltaz=oe, and z 0 completely specify a two surface discontinuity model, the resulting mixture density, h(x; z), and therefore, the distri- bution, H(x; z). Hence, after specifying the class of functions (linear, here) for hypothesized fits, a given robust estimator's pseudo outlier bias can be calculated as a function of these parameters. This calculation requires an iterative, numerical search to minimize T (H), and may require several starting points to avoid local minima. (See Figure 6 for an example plot of T ae h 's objective functional.) Thus, for a particular type of discontinuity and for a particular robust estimator, the parameters may be varied to study their effect on the estimator's pseudo outlier bias, thereby characterizing how accurately the estimator can fit surfaces near discontinuities. As a final observation, although the results are presented for one-dimensional image domains, they have immediate extension to two dimensions. For example, a two-dimensional analog of the step edge presented here is fi 1 (x; It is straightforward to show that this model results in exactly the same pseudo outlier bias as a one-dimensional step model having the same mixture parameters and gross outlier distribution. Similar results are obtained for natural extensions of the crease edge and parallel lines models. Thus, one-dimensional discontinuities are sufficient to establish limitations in the effectiveness of robust estimators. 6.2 Fixed-Scale M-Estimators and Fixed-Band Techniques The first analysis results are for fixed-band techniques and fixed-scale M-estimators. These techniques represent an ideal case where the noise parameter - oe = oe is known and fixed in advance. Figure 7 shows the bias of fixed-band techniques (T F ) and three fixed-scale M-estimators ) as a function \Deltaz=oe when " 0:8. The bias of the least-squares estimator, calculated by substituting is included for comparison. The ae function tuning parameters values are directly from the literature page 167], and . Interestingly, the proportion of gross outliers, " has no effect on the results. This is because the fraction of the outlier distribution within r of a fit is the same for all fits ' and for all r except when '(x) \Sigma r is extreme enough to cross outside the bounds of the gross outlier distribution. The sharp drops in bias shown in Figure 7 (a) and (b) for fixed-band techniques and the hard redescending M-estimator (and to some extent for the soft redescending M-estimator in (b)) correspond to - shifting from the local minimum associated with a heavily biased fit to the local minimum near fi 1 (x), the optimum fit to the target distribution. Plotting the step height at which this drop occurs as a function of " s gives a good summary of these estimators' bias on step edges. Figure 8 does this, referring to this height as the "small bias height" and quantifying it as the step height at which the bias drops below 1.0. The plots in Figures 7 and 8(a) show that fixed-band techniques and fixed-scale M-estimators are biased nearly as much as least-squares for significant step edge and crease edge discontinuity magnitudes. The estimators fare much better on parallel lines (Figure 7(e) and (f)); apparently, asymmetric positioning of pseudo outliers causes the most bias. To give an intuitive feel for the significance of the bias, Figure 9 shows step edge data generated using model parameters for which the robust estimators are strongly biased. Overall, the hard redescending, fixed-scale M-estimator is the least biased of the techniques studied thus far. Compared to other fixed-scale M-estimators, its finite rejection point - the point at which outliers no longer influence the fit - makes it less biased by pseudo outliers than monotone and soft redescending fixed-scale M-estimators. On the other hand, it is less biased than fixed-band techniques because it retains the statistical efficiency of least-squares for small residuals. The hard redescending, fixed-scale M-estimator can be made less biased by reducing the values of its tuning parameters, as shown in Figure 8(b), effectively narrowing ae h and reducing its finite rejection point. (The parameter set a = 1:0, 2:0 comes from [2]; the set a = 1:0, chosen as an intermediate set of values.) Using small parameter values has two disadvantages, however: the optimum statistical efficiency of the standard parameters is lost, giving less accurate fits to the target distribution, and some good data may be rejected as outliers. Despite these disadvantages, lower tuning parameters should be used since avoiding heavily biased fits is the most important objective. Finally, in practice, the non-convex objective functions of hard and soft redescending Bias Step Height Least-squares Fixed-band Bias Step Height Least-squares Monotone Fixed-band Hard a b Crease Bias Crease Height Least-squares Monotone Fixed-band Hard0.20.611.41.8 Bias Crease Height Least-squares Monotone Fixed-band Hard c d Parallel Bias Relative Height Least-squares Monotone Fixed-band Bias Relative Height Least-squares Monotone Fixed-band Hard Figure 7: Bias of fixed-band techniques, fixed-scale M-estimators and least-squares on step edges, (a) and (b), crease edges, (c) and (d), and parallel lines, (e) and (f), as a function of height when " 0:8. The horizontal axis is the relative discontinuity magnitude (height), \Deltaz=oe, and the vertical axis is the bias (see equation 35). Plots not shown in (a) are essentially equivalent to the least-squares plots. Small Bias Cut-off Height Fraction of Points on Lower Half of Step Fixed-band Small Bias Cut-off Height Fraction of Points on Lower Half of Step 1.31, 2.04, 4.0 1.0, 2.0, 3.0 1.0, 1.0, 2.0 a b Figure 8: Small bias cut-off heights as a function of " s , the relative fraction of points on the lower half of the step. Plots in (a) show the heights for fixed-band techniques and two fixed-scale M-estimators. Plots in (b) show the heights for different tuning parameters of the hard redescending fixed-scale M-estimator. Heights not plotted for small " s are above When height is not plotted for large " s , bias is never greater than 1.0. fixed-scale M-estimators can lead to more biased results than indicated here. Iterative search techniques, especially when started from a non-robust fit, may stop at a local minimum corresponding to a biased fit when the fit to the target distribution is the global minimum of the objective function. Therefore, to avoid local minima, fixed-scale M-estimators should use either a random sampling search technique or a Hough transform. 6.3 M-Estimators Next, consider standard M-estimators, which estimate - oe from the data. To calculate T (H) for the monotone and soft redescending M-estimators, simply calculate - any mixture distribution using equation 7 or 8 as the objective functional. For the hard redescending M-estimator, which estimates - oe from an initial fit, the optimum fit to the mixture distribution is found in three stages: first find the optimum LMS fit, then calculate the median absolute deviation (MAD) [10, page 107] to this fit, scaling it to estimate - oe, and finally calculate - oe fixed. Two different scale factors for estimating - oe are considered: the first, 1:4826, ensures consistency at the normal distribution; the second, 1:14601, ensures consistency at Student's t-distribution (with 1:5). Using the latter allows accurate comparison between the hard and soft redescending M-estimators since the Figure 9: Example step edge data generated when " where each the objective function of each robust estimator (except LTS) is minimized by a biased fit. The example fit shown is - '(x) for the hard redescending, fixed-scale M-estimator. latter is the maximum likelihood estimate for Student's t distribution [5]. Figure shows bias plots for the soft redescending M-estimator and for the hard re- descending M-estimator using the two different scale factors (plot "Hard-N" for the normal distribution and plot "Hard-t" for the t-distribution). Results for the monotone M-estimator are not shown since its bias matches that of least-squares almost exactly. Overall, the results are substantially worse than for fixed-scale M-estimators, especially for " This is a direct result of - oe being a substantial over-estimate of oe: for example, when " oe=oe - 2:4 for all estimates. (See [22] for analysis of bias in estimating - oe.) These over-estimates allow a large portion of the residual distribution to fall in the region where ae is quadratic, causing the estimator to act more like least-squares. Because of this, M-estimators are heavily biased by discontinuities when they must estimate - oe from the data. 6.4 LMS, LTS and MINPRAN The last estimators examined are LMS, LTS, and MINPRAN, methods which neither require oe a priori nor need to estimate it while finding - '(x). Figure shows bias plots for these estimators on step edges, crease edges and parallel lines, using " Figure shows small bias cut-off heights on step edges for LMS, LTS and MINPRAN, and it demonstrates the effects of changes in the mixture proportions on LMS and LTS. LMS and LTS work as well as any technique studied as long as the actual of fraction Bias Step Height Least-squares Hard-N Bias Step Height Least-squares Hard-N Hard-t a b Crease Bias Crease Height Least-squares Hard-N Hard-t0.20.611.41.8 Bias Crease Height Least-squares Hard-N Hard-t c d Parallel Bias Relative Height Least-squares Hard-N Bias Relative Height Least-squares Hard-N Hard-t Figure 10: Bias of M-estimators and least-squares on step edges, (a) and (b), crease edges, Bias Step Height Least-squares MINPRAN LMS Bias Step Height Least-squares MINPRAN LMS a b Crease Bias Crease Height Least-squares MINPRAN LMS Bias Crease Height Least-squares MINPRAN LMS c d Parallel Bias Relative Height Least-squares MINPRAN LMS Bias Relative Height Least-squares MINPRAN LMS Figure 11: Bias of MINPRAN, LMS, LTS and least-squares on step edges, (a) and (b), crease Small Bias Cut-off Height Relative Fraction from Lower Half of Step MINPRAN LMS Small Bias Cut-off Height Relative Fraction from Lower Half of Step0.1a b261014 Small Bias Cut-off Height Actual Fraction from Lower Half of Step0.1048120.5 0.55 0.6 0.65 0.7 0.75 Small Bias Cut-off Height Actual Fraction from Lower Half of Step0.1c d Figure 12: Small bias cut-off heights. Plot (a) shows these for LMS, LTS, MINPRAN, and the modified MINPRAN optimization criteria (MINPRAN2) as a function of " s , the relative fraction of inliers. Plot (b) shows these for LTS as a function of " s for different gross outlier percentages " Plots (c) and (d) show these for LTS and LMS respectively as a function of , the actual fraction of inliers. Heights not plotted for small " s or are above When height is not plotted for large " s or never greater than oe. inliers - data from fi 1 (x) - is above 0.5. Since this fraction is the bias of LMS and LTS, unlike that of M-estimators, depends heavily on both " sampling implementations of LMS and LTS, where p points instantiate a hypothesized fit and the objective function is evaluated on the remaining points, the bias curves in Figure 11 and the steep drop in cut-off heights in Figure 12 will shift to the right, but only marginally since usually n AE p.) Figures 12b and c demonstrate this dependence in two ways for LTS. Figure shows small bias cutoffs as a function of " s , the relative fraction of inliers - points on the lower half of the step. The bias cutoffs are lower for lower " simply because fewer gross outliers imply more actual inliers when " s remains fixed. Figure 12c shows small bias cutoffs as a function of the actual fraction of inliers. In this context, varying " while fixed changes the fraction of gross outliers versus pseudo outliers. As the plot shows, the coherent structure of the pseudo outliers causes more bias than the random structure of gross outliers. This same effect is shown for LMS in Figure 12d. Finally, the magnitude of z 0 , which controls the gross outlier distribution, has little effect on the bias results, except in the unrealistic case where it approaches the discontinuity magnitude. LTS is less biased than LMS, especially when the actual fraction of inliers is only slightly above 0.5. This can be seen most easily by comparing the low bias cutoff plots in Figure 12c and d. Like the advantage of hard redescending M-estimators over fixed-band techniques (Section 6.2), this occurs because LTS is more statistically efficient than LMS [21] - its objective function depends on the smallest 50% of the residuals rather than just on the median residual. It is important to note that although LMS's efficiency can be improved by application of a one-step M-estimator starting from the LMS estimate, this will not improve substantially a heavily biased fit, since a local minimum of the M-estimator objective function will be near this fit. With a minor modification to its optimization criteria, MINPRAN can be made much less sensitive to pseudo outliers, improving dramatically on the poor performance shown in Figures 11 and 12. The idea is to find two disjoint fits (no shared inliers), - ' a and - inlier bounds - r a and - r b and inlier counts k - 'a ;-r a and k - , minimizing F(-r a +-r b ; k - ' 1 ;-r a [23, 26]. If - ' is the single fit minimizing the criterion function, with inlier bound - r and inlier count k -r , then the two fits - ' a and - ' b are chosen instead of the single fit - 'a ;-r a Thus, the modified optimization criteria tests whether one or two inlier distributions are more likely in the data [27]. Figure 12 shows the step edge small bias cut-off heights for this new objective function, denoted by MINPRAN2. These are substantially lower than those of the other techniques, including LTS. Further, these results, unlike those of MINPRAN, are only marginally affected by the parameters " . Unfortunately, the search for - ' a and ' b is computationally expensive, and so the present implementation of MINPRAN2 uses a simple search heuristic that yields [23, 26] more biased results than the optimum shown here. It is, however, as effective as the fixed-scale, hard redescending M-estimator and, unlike LMS and LTS, it does not fail dramatically when there are too few inliers. 6.5 Discussion and Recommendations Overall, the results show that all the robust estimators studied estimate biased fits at small but substantial discontinuity magnitudes. This bias, which relative to the bias of least-squares is greater for crease and step edges and less for parallel lines, occurs even if - oe or the distribution of gross outliers or both are known a priori. Further, it must be emphasized that this bias is not an artifact of the search process: the functional form of each estimator returns the fit corresponding to the global minimum of the estimator's objective function. The reason for the bias can be seen by examining the cumulative distribution functions (cdfs) of absolute residuals. Figure 13 plots this cdf, F a (rj'; H), when ' is the target fit is the least-squares fit to H, for H modeling crease and step discontinuities. For the cdf of the biased fit is almost always greater that of the target fit, meaning that in a discrete set of samples, the biased fit, which crosses through both point sets, will on average yield smaller magnitude residuals than the target fit, which is close to only the target point set. (The situation is somewhat better when \Deltaz=oe = 9:0.) Therefore, robust estimators, such as the ones studied, whose objective functions are based solely on residuals, are unlikely to estimate unbiased fits at small magnitude discontinuities. CDF Absolute Residual Biased Fit Target Fit0.20.61 CDF Absolute Residual Biased Fit Target Fit a b Crease CDF Absolute Residual Biased Fit Target Fit0.20.61 CDF Absolute Residual Biased Fit Target Fit c d Figure 13: Each figure plots the cumulative distribution functions (cdf) of absolute residuals for the target fit and for a biased (least-squares) fit: (a) and (b) are relative to a step discontinuity, and (c) and (d) are relative to a crease discontinuity. For all plots, the mixture fractions are fixed at " robust estimators are substantially biased at both step and crease discontinuities. While none of the estimators works as well as desired, the following recommendations for choosing among them are based on the results presented above: oe is known a priori , one should use a hard redescending M-estimator objective function such as Hampel's with reduced tuning parameter values and either a random-sampling search technique or a weighted Hough transform. To ensure all inliers are found and to obtain greater statistical efficiency, an one-step M-estimator with larger tuning parameters should be run from the initial optimum fit. This technique is preferable to LTS and LMS because it is less sensitive to the number of gross outliers. oe is not known a priori, but the distribution of gross outliers is known, one should use the modified MINPRAN algorithm, MINPRAN2 [23, 26]. ffl When neither - oe nor the distribution of gross outliers is known, LTS should be used, although its performance degrades quickly when there are too few inliers. LTS is preferable to LMS because of its statistical efficiency. 7 Summary and Conclusions This paper has developed the pseudo outlier bias metric using techniques from mathematical statistics to study the fitting accuracy of robust estimators on data taken from multiple structures - surface discontinuities, in particular. Pseudo outlier bias measures the distance between a robust estimator's optimum fit to a target distribution and its optimum fit to an outlier corrupted mixture distribution. Here, the target distribution models the points from a single surface and the mixture distribution models points from multiple surfaces plus gross outliers. Each estimator's optimum fit is found by applying its functional form to one of these model distributions. Thus, like other analysis tools from the robust statistics literature, pseudo outlier bias depends on point distributions rather than on particular point sets drawn from these distributions. While this has some limitations - the actual fitting error for particular points sets may be more or less than the pseudo outlier bias and it ignores problems that may arise from multiple local minima in an objective function - it represents a simple, efficient, and elegant method of analyzing robust estimators. Pseudo outlier bias was used to analyze the performance of M-estimators, fixed-band techniques (Hough transforms and RANSAC), least median of squares (LMS), least trimmed squares (LTS) and MINPRAN in fitting surfaces to three different discontinuity models: step edges, crease edges and parallel lines. For each of these discontinuities, two surfaces generate data, with the larger set of surface data forming the inliers and the smaller set forming the pseudo outliers. By characterizing these discontinuity models using a small number of parameters, formulating the models as mixture distributions, and studying the bias of the robust estimators as the parameters varied, it was shown that each robust estimator is biased for substantial discontinuity magnitudes. This effect, which relative to that of least-squares is strongest for step edges and crease edges, persists even when the noise in the data or the gross outlier distribution or both are known in advance. It is disappointing because in vision data - not just in range data - multiple structures (pseudo outliers) are more prevalent than gross outliers. In spite of the disappointment, however, specific recommendations, which depend on what is known about the data, were made for choosing between current techniques. 6 These negative results indicate that care should be used when robustly estimating surface parameters in range data, either to obtain local low-order surface approximations or to initialize fits for surface growing algorithms [3, 5, 6, 15]. (Similar problems may occur for the "layers" techniques that have been applied to motion analysis [1, 6, 28].) Robust estimates will be accurate for large scale depth discontinuties and sharp corners, but will be skewed at small magnitude discontinuites, such as near the boundary of a slightly raised or depressed area of a surface. Obtaining accurate estimates near these discontinuities will require new and perhaps more sophisticated robust estimators. Acknowledgements The author would like to acknowledge the financial support of the National Science Foundation under grants IRI-9217195 and IRI-9408700, the assistance of James Miller in various aspects of this work, and the insight offered by the anonymous reviewers which led to sub- 6 See [14, 17] for new, related techniques. stantial improvements in the presentation. Appendix A: Evaluating F s This appendix shows how to evaluate the conditional cumulative distribution and conditional density of signed residuals, F s 22). The distribution and density of the absolute residuals are obtained easily from these. Expanding the expression in equation 21 for F s (rj'; H), using equation h, gives F s Here, Z '(x)+r is the cumulative distribution of the gross outliers, and for Z '(x)+r To simplify evaluating F s variables and then change the order of integration. Starting with the change of variables, make the substitutions (intuitively, v is the fit residual at x), define Then, the integral becomes Z OE(x)+r Since the integrand is now independent of x, rewriting the integral to integrate over strips parallel to the x axis will produce a single integral. Consider a strip bounded by v and v+ \Deltav Figure 14). The integral over this strip is approximately - i g(v)w(v)\Deltav, where w(v) is the width of the integration region at v. In the limit as \Deltav ! 0, this becomes exact and the integral over the entire region becomes is the maximum of OE(x) r f x Figure 14: Calculating F s requires integrating the point density for curve over strips of width \Deltav parallel to the x axis. The density g(v; oe 2 ) is constant over these strips. Evaluating w(v) depends on OE(x). This paper studies linear fits and linear curve models, so OE(x) is linear. In this case, let and Figure 14). Then, using G to denote the cdf of the gaussian, A similar result is obtained when m ! 0, and when To compute the density f s (rj'; H), start from the mixture density in equation 22 and integrate each component density separately. This is straightforward when the density g o is uniform and, as above, '(x) and fi i (x) are linear. --R Layered representation of motion video using robust maximum likelihood estimation of mixture models and MDL encoding. Robust window operators. Segmentation through variable-order surface fitting A Ransac-based approach to model fitting and its application to finding cylinders in range data The Robust Sequential Estimator: A general approach and its application to surface organization in range data. Cooperative robust estimation using layers of support. Random Sample Consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Linear Systems The change-of-variance curve and optimal redescending M-estimators Robust Statistics: The Approach Based on Influence Functions. Robust regression using iteratively reweighted least- squares Robust Statistics. A survey of the Hough transform. Robust adaptive segmentation of range images. Segmentation of range images as the search for geometric parametric models. Robust regression methods for computer vision: A review. MUSE: Robust surface fitting using unbiased scale esti- mates Performance evaluation of a class of M-estimators for surface parameter estimation in noisy range data Numerical Recipes in C: The Art of Scientific Computing. Extracting geometric primitives. Least median of squares regression. Alternatives to the median absolute deviation. A new robust operator for computer vision: Application to range images. A new robust operator for computer vision: Theoretical analysis. Expected performance of robust estimators near discontinuities. MINPRAN: A new robust estimator for computer vision. Statistical Analysis of Finite Mixture Distri- butions Layered representation for motion analysis. --TR --CTR Cesare Alippi, Randomized Algorithms: A System-Level, Poly-Time Analysis of Robust Computation, IEEE Transactions on Computers, v.51 n.7, p.740-749, July 2002 Alireza Bab-Hadiashar , David Suter, Robust segmentation of visual data using ranked unbiased scale estimate, Robotica, v.17 n.6, p.649-660, November 1999 Ulrich Hillenbrand, Consistent parameter clustering: Definition and analysis, Pattern Recognition Letters, v.28 n.9, p.1112-1122, July, 2007 Klaus Kster , Michael Spann, MIR: An Approach to Robust Clustering-Application to Range Image Segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.5, p.430-444, May 2000 Thomas Kmpke , Matthias Strobel, Polygonal Model Fitting, Journal of Intelligent and Robotic Systems, v.30 n.3, p.279-310, March 2001 Christine H. Mller , Tim Garlipp, Simple consistent cluster methods based on redescending M-estimators with an application to edge identification in images, Journal of Multivariate Analysis, v.92 n.2, p.359-385, February 2005 Hanzi Wang , David Suter, Robust Adaptive-Scale Parametric Model Estimation for Computer Vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.26 n.11, p.1459-1474, November 2004 Hanzi Wang , David Suter, MDPE: A Very Robust Estimator for Model Fitting and Range Image Segmentation, International Journal of Computer Vision, v.59 n.2, p.139-166, September 2004 Philip H. S. Torr , Colin Davidson, IMPSAC: Synthesis of Importance Sampling and Random Sample Consensus, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.3, p.354-364, March T. Vieville , D. Lingrand , F. Gaspard, Implementing a Multi-Model Estimation Method, International Journal of Computer Vision, v.44 n.1, p.41-64, August 2001 P. H. S. Torr, Bayesian Model Estimation and Selection for Epipolar Geometry and Generic Manifold Fitting, International Journal of Computer Vision, v.50 n.1, p.35-61, October 2002 Myron Z. Brown , Darius Burschka , Gregory D. Hager, Advances in Computational Stereo, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.8, p.993-1008, August

parameter estimation;multiple structures;outliers;discontinuities;robust estimation

263444

Affine Structure from Line Correspondences With Uncalibrated Affine Cameras.

AbstractThis paper presents a linear algorithm for recovering 3D affine shape and motion from line correspondences with uncalibrated affine cameras. The algorithm requires a minimum of seven line correspondences over three views. The key idea is the introduction of a one-dimensional projective camera. This converts 3D affine reconstruction of "line directions" into 2D projective reconstruction of "points." In addition, a line-based factorization method is also proposed to handle redundant views. Experimental results both on simulated and real image sequences validate the robustness and the accuracy of the algorithm.

Introduction Using line segments instead of points as features has attracted the attention of many researchers [1], [2], [3], [4], [5], [6], [7], [8], [9] for various tasks such as pose estima- tion, stereo and structure from motion. In this paper, we are interested in structure from motion using line correspondences across mutiple images. Line-based algorithms are generally more difficult than point-based ones for the following two reasons. The parameter space of lines is non linear, though lines themselves are linear subspaces, and a line-to-line correspondence contains less information than a point-to-point one as it provides only one component of the image plane displacement instead of two for a point correspondence. A minimum of three views is essential for line correspondences, whereas two views suffice for point ones. In the case of calibrated perspective cameras, the main results on structure from line correspondences were established in [4], [10], [5]: With at least six line correspondences over three views, nonlinear algorithms are possible. With at least thirteen lines over three views, a linear algorithm is possible. The basic idea of the thirteen-line linear algorithm is similar to the "eight-point" one [11] in that it is based on the introduction of a redundant set of intermediate parameters. This significant over-parametrization of the problem leads to the instability of the algorithm reported in [4]. The thirteen-line algorithm was extended to uncalibrated camera case in [12], [9]. The situation here might be expected to be better, as more free parameters are introduced. However, the 27 tensor components that are introduced as intermediate parameters are still subject to 8 complicated algebraic constraints. The algorithm Long QUAN is with CNRS-GRAVIR-INRIA, ZIRST 655, avenue de l'Europe, 38330 Montbonnot, France. E-mail: [email protected] Takeo KANADE is with The Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. E-mail: [email protected] can hardly be stable. A subsequent nonlinear optimization step is almost unavoidable to refine the solution [5], [4], In parallel, there has been a lot of work [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [14], [16], [23], [17], [24], [25] on structure from motion with simplified camera models varing from orthographic projections via weak and paraperspective to affine cameras, almost exclusively for point features. These simplified camera models provide a good approximation to perpsective projection when the width and depth of the object are small compared to the viewing distance. More importantly, they expose the ambiguities that arise when perspective effects diminish. In such cases, it is not only easier to use these simplified models but also advisable to do so, as by explicitly eliminating the ambiguities from the algorithm, one avoids computing parameters that are inherently ill-conditioned. Another important advantage of working with uncalibrated affine cameras is that the reconstruction is affine, rather than projective as with uncalibrated projective cameras. Motivated on the one hand by the lack of satisfactory line-based algorithms for projective cameras and on the other by the fact that the affine camera is a good model for many practical cases, we investigate the properties of projection of lines by affine cameras and propose a linear algorithm for affine structure from line correspondences. The key idea is the introduction of a one-dimensional projective camera. This converts the 3D affine reconstruction of "line direc- tions" into 2D projective reconstruction of "points". The linear algorithm requires a minimum of seven lines over three images. We also prove that seven lines over three images is the strict minimum data needed for affine structure from uncalibrated affine cameras and that there are always two possible solutions. This result extends the previous results of Koenderink and Van Doorn [14] for affine structure with a minimum of two views and five points. To deal with redundant views, we also present a line-based factorisation algorithm which extends the previous point-based factorisation methods [18], [21], [22]. A preliminary version of this work was presented in [26]. The paper is organized as follows. In Section II, the affine camera model is briefly reviewed. Then, we investigate the properties of projection of lines with the affine camera and introduce the one-dimensional projective camera in Section III. Section IV is focused on the study of the un-calibrated one-dimensional camera, and in this section we present also a linear algorithm for 2D projective reconstruction which is equivalent to the 3D affine reconstruction of IEEE-PAMI, VOL. *, NO. *, 199* line directions. Later, the linear estimation of the translational component of the uncalibrated affine camera is given in Section V and the affine shape recovery is described in Section VI. To handle redundant views, a line-based factorisation method is proposed in Section IX. The passage to metric structure from the affine structure using known camera parameters will be described in Section XI. Finally in Section XIII, discussions and some concluding remarks are given. Throughout the paper, tensors and matrices are denoted in upper case boldface, vectors in lower case boldface and scalars in either plain letters or lower case Greek. II. Review of the affine camera model For a projective (pin-hole) camera, the projection of a point of P 3 to a point be described by a 3 \Theta 4 homogeneous projection matrix For a restricted class of camera models, by setting the third row of the perspective camera P 3\Theta4 to (0; 0; 0; ), we obtain the affine camera initially introduced by Mundy and Zisserman [27], The affine camera A 3\Theta4 encompasses the uncalibrated versions of the orthographic, weak perspective and paraperspective projection models. These reduced camera models provide a good approximation to the perspective projection model when the depth of the object is small compared to the viewing distance. For more detailed relations and applications, one can refer to [20], [22], [28], [29], [13]. For points in the affine spaces IR 3 and IR 2 , they are naturally embedded into by the mappings w a 7! We have thus . If we further use relative coordinates of the points with respect to a given reference point (for instance, the centroid of the set of points), the vector t 0 is cancelled and we obtain the following linear mapping between space points and image points: \Deltaw This is the basic equation of the affine camera for points. III. The affine camera for lines Now consider a line in IR 3 through a point x 0 , with direction The affine camera A 3\Theta4 projects this to an image line A 3\Theta4 passing through the image point with direction This equation describes a linear mapping between direction vectors of 3D lines and those of 2D lines, and reflects a key property of the affine camera: lines parallel in 3D remain parallel in the image. It can be derived even more directly using projective geometry by considering that the line direction d x is the point at infinity the projective line in P 3 and the line direction dw is the point at infinity of the projective line in Equation (4) immediately follows as the affine camera preserves the points at infinity by its very definition. Comparing Equation (4) with Equation (1)-a projection from P 3 to P 2 , we see that Equation (4) is nothing but a projective projection from P 2 to P 1 if we consider the 3D and 2D "line directions" as 2D and 1D projective "points". This key observation allows us to establish the following. The affine reconstruction of line directions with a two-dimensional affine camera is equivalent to the projective reconstruction of points with a one-dimensional projective camera. One of the major remaining efforts will be concerned with projective reconstruction from the points in P 1 . There have been many recent works [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [10], [42], [43] on projective reconstruction and the geometry of multi-views of two dimensional uncalibrated projective cameras. Particularly, the tensorial formalism developed by Triggs [36] is very interesting and powerful. We now extend this study to the case of the one-dimensional camera. It turns out that there are some nice properties which were absent in the 2D case. IV. Uncalibrated one-dimensional camera A. Trilinear tensor of the three views First, rewrite Equation (4) in the following form: in which we use of dw and d x to stress that we are dealing with "points" QUAN: AFFINE STRUCTURE FROM LINE CORRESPONDENCES 3 in the projective spaces P 2 and P 1 rather than "line direc- tions" in the vector spaces IR 3 and IR 2 . We now examine the matching constraints between multiple views of the same point. Since two viewing lines in the projective plane always intersect in a point, no constraint is possible for less than three views. There is one constraint only for the case of 3 views. Let the three views of the same point x be given as follows: These can be rewritten in matrix form as@ M u x which is the basic reconstruction equation for a one-dimensional camera. The vector be zero, so fi fi fi fi fi fi The expansion of this determinant produces a trilinear constraint of three viewsX or in short homogeneous tensor whose components T ijk are 3 \Theta 3 minors of the following 6 \Theta 3 joint projection matrix:@ M The components of the tensor can be made explicit as 2: (11) where the bracket [ij 0 k 00 ] denotes the 3 \Theta 3 minor of i- row vector of the above joint projection matrix and bar " " in i, j and k denotes the dualization It can easily be seen that any constraint obtained by adding further views reduces to a trilinearity. This proves the uniqueness of the trilinear constraint. Moreover, the homogeneous tensor T 2\Theta2\Theta2 has d.o.f., so it is a minimal parametrization of three views since three views have exactly 3 \Theta (2 \Theta 3 \Gamma d.o.f. up to a projective transformation in P 2 . Each point correspondence over three views gives one linear constraint on the tensor components T ijk . We can establish the following. The tensor components T ijk can be estimated linearly with at least 7 points in P 1 . At this point, we have obtained a remarkable result that for a one-dimensional projective camera, the trilinear tensor encapsulates exactly the information needed for projective reconstruction in P 2 . Namely, it is the unique matching constraint, it minimally parametrizes the three views and it can be estimated linearly. Contrast this to the 2D projective camera case in which the multilinear constraints are algebraically redundant and the linear estimation is only an approximation based on over-parametrization. B. Retrieving normal forms for projection matrices The geometry of the three views is most conveniently, and completely represented by the projection matrices associated with each view. In the previous section, the trilinear tensor was expressed in terms of the projection matrices. Now we seek a map from the trilinear tensor representation back to the projection matrix representation of the three views. Without loss of generality, we can always take the following normal forms for the three projection matrices Actually, the set of projection matrices fM;M parametrized this way has more than the minimum of 7. Further constraints can be imposed. We can observe that any projective transformation in P 2 of the I 2\Theta2 0 for an arbitrary 2-vector v leaves M invariant and transforms ~ A c As c cannot be a zero vector, it can be normalized such that c T c = 1. If we further choose an arbitrary vector v to be \GammaA T c, then ~ A. It can now be easily verified that ~ This amounts to saying that ~ A in 4 IEEE-PAMI, VOL. *, NO. *, 199* ~ M 0 can be taken to be a rank 1 matrix up to a projective transformation, i.e. ~ a 1 aea 1 a for a non-zero scalar ae. The 2-vector c is then (\Gammaa 2 ; a 1 ) T . Hence M 0 can be represented as a by two parameters, the ratio a 1 : a 2 and ae. Therefore, a minimal 7 parameter representation for the set of projection matrices has been obtained. With the projection matrices given by (13), the trilinear tensor (T ijk ) defined by (11) becomes represents the dulization (12). If we consider the tensor (T ijk ) as an 8-vector the eight homogeneous equations of (15) can be rearranged into 7 non-homogeneous ones by taking the ratios t l : t 8 for 7. By separating the entries of M 0 from those of G 7\Theta6@ d e where the matrix G 7\Theta6 is given Since the parameter vector (d; cannot be zero, the 7 \Theta 6 matrix in Equation (16) has at most rank 5. Thus all of its 6 \Theta 6 minors must vanish. There are such minors which are algebraically independent, and each of them gives a quadratic polynomial in a 1 , a 2 and ae as follows: ae By eliminating ae, we obtain a homogeneous quadratic equation in a 1 and a where This quadratic equation may be easily solved for a 1 =a 2 . Then ae is given by the following linear equation for each of two solutions of a 1 =a 2 Thus, we obtain two possible solutions for the projection Finally, the 6-vector (d; for the projection matrix M 00 is linearly solved from Equation (16) (for instance, using SVD) in terms of M 0 . C. 2D projective reconstruction-3D affine line direction reconstruction With the complete determination of the projection matrices of the three views, the projective reconstruction of "points" in P 2 , which is equivalent to the affine reconstruction of "line directions" in IR 3 , can be performed. From the projection equation each point of a view homogeneous linear equation in the unknown point x in P 2 2 are the first and second row vector of the matrix M. With one point correspondence in three views we have the following homogeneous linear equation system,@ A where designates a constant entry. This equation system can be easily solved for x, either considered as a point in or as an affine line direction in IR 3 . V. Uncalibrated translations To recover the full affine structure of the lines, we still need to find the vectors t 3\Theta1 of the affine cameras defined in (2). These represent the image translations and magnification components of the camera. Recall that line correspondences from two views-now a 2D view instead of 1D view-do not impose any constraints on camera motion: The minimum number of views required is three. If the interpretation plane of an image line for a given view is defined as the plane going through the line and the projection center, the well-known geometric interpretation of the constraint available for each line correspondence across three views (cf. [3], [5]) is that the interpretation planes from different views must intersect in a common line in space. QUAN: AFFINE STRUCTURE FROM LINE CORRESPONDENCES 5 If the equation of a line in the image is given by l T then substituting produces the equation of the interpretation plane of l in space: l T A 3\Theta4 The plane is therefore given by the 4-vector p which can also be expressed as the normal vector of the plane. An image line of direction nw can be written as its interpretation plane being The 2 \Theta 3 submatrices M 2\Theta3 representing uncalibrated camera orientations have already been obtained from the two-dimensional projective reconstruction. Now we proceed to recover the uncalibrated translations. For each interpretation plane (n x ; p) T of each image line, its direction component is completely determined by the previously computed fM;M as Only its fourth component remains undetermined. This depends linearly on t. Notice that as the direction vector can still be arbitrarily and individually rescaled, the interpretation plane should be properly written as Hence the ratio = is significant, and this justifies the homogenization of the vector t. So far we have made explicit the equation of the interpretation planes of lines in terms of the image line and the projection matrix, the geometric constraint of line correspondences on the camera motion gives a 3 \Theta 4 matrix whose rows are the three interpretation planes@ which has rank at most two. Hence all of its 3 \Theta 3 minors vanish. Only two of the total of four minors are algebraically independent, as they are connected by the quadratic identities [44]. The vanishing of any two such minors provides the two constraints on camera motion for a given line correspondence of three views. The minor formed by the first three columns contains only known quantities. It provides the constraint on the directions. It is easy to show that it is equivalent to the tensor by using suitable one-dimensional projective transformations. By taking any two of the first three columns, say the first two, and the last column, we obtain the following vanishing determinant: fi fi fi fi fi fi l T t l 0T t 0 l 00T t 00 where the " " designates a constant entry. Expanding this minor by cofactors in the last column gives a homogeneous linear equation in t, t 0 and \Theta \Theta \Theta where the "\Theta" designates a constant 3-vector in a row. Collecting all these vanishing minors together, we obtainB @ \Theta \Theta \Theta \Theta \Theta \ThetaC A n\Theta9@ for n line correspondences in three views. At this stage, since the origin of the coordinate frame in space is not yet fixed, we may take up to a scaling factor, say t so the final homogeneous linear equations to solve for \Theta \Theta \Theta \ThetaC A n\Theta7@ t 0 This system of homogeneous linear equations can be nicely solved by Svd factorisation. The least squares solution for subject to jj(t is the right singular vector corresponding to the smallest singular value. VI. Affine shape The projection matrices of the three views are now completely determined up to a common scaling factor. From now on, it is a relatively easy task to compute the affine shape. Two methods to obtain the shape will be described, one based on the projective representation of lines and another on the minimal representation of lines, inspired by [5]. A. Method 1: projective representation A projective line in space can be defined either by a pencil of planes (a pencil of planes is defined by two projective planes) or by any two of its points. The matrix WP =@ 6 IEEE-PAMI, VOL. *, NO. *, 199* should have rank 2, so its kernel must also have dimension 2. The range of WP defines the pencil of planes and the null space defines the projective line in space. Once again, using Svd to factorize WP gives us everything we want. Let be the Svd of WP with ordered singular values. Two points of the line might be taken to be v 3 and v 4 , so the line is given by One advantage of this method is that, using subset selection [45], near singular views can be detected and discarded. B. Method 2: Minimal representation As a space line has 4 d.o.f., it can be minimally represented by four parameters. One such possibility is suggested by [5] which uses a 4-vector l such that the line is defined as the intersection of two planes (1; 0; \Gammaa; \Gammax and (0; 1; \Gammab; \Gammay 0 ) T with equations: Geometrically this minimal representation gives a 3D line with direction (a; b; 1) T and passing through the point This representation excludes, therefore, the lines of direction (a; b; parallel to the xy plane. Two other representations are needed, each excluding either the directions (0; b; c) T or (a; 0; c) T . These 3 representations together completely describe any line in space. In our case, we have no problem in automatically selecting one of the three representations, as the directions of lines have been obtained in the first step of factorisation, allowing us to switch to one of the three representations. There remain only two unknown parameters x 0 and y 0 for each line. To get a solution for x 0 and y 0 , as the two planes defining the line belong to the pencil of planes defined by WP , we can still stack these two planes on the top of WP to get the matrix Since this matrix still has rank 2, all its 3 \Theta 3 minors vanish. Each minor involving x 0 and y 0 gives a linear equation in x 0 and y 0 . With n views, a linear equation system is obtained An\Theta2 This can be nicely solved using least squares for each line. VII. Affine-structure-from-lines theorem Summarizing the results obtained above, we have established the following. For the recovery of affine shape and affine motion from line correspondences with an uncalibrated affine camera, the minimum number of views needed is three and the minimum number of lines required is seven for a linear solu- tion. There are always two solutions for the recovered affine structure. This result can be compared with that of Koenderink and Doorn [14] for affine structure with a minimum of two views and five points. We should also note the difference with the well-known results established for both calibrated and uncalibrated projective cameras [3], [4], [5], [39]: A minimum of 13 lines in three views is required to have a linear solution. It is important to note that with the affine camera and the method presented in this paper, the number of line correspondences for achieving a linear solution is reduced from 13 to 7, which is of great practical importance. VIII. Outline of the 7-line \Theta 3-view algorithm The linear algorithm to recover 3D affine shape/motion from at least 7 line correspondences over three views with uncalibrated affine cameras may be outlined as follows: 1. If an image line segment is represented by its end-points the direction vector of the line this as the homogeneous coordinates of a point in P 1 . 2. Compute the tensor components (T ijk ) defined by Equation linearly with at least 7 lines in 3 views. 3. Retrieve the projection matrices fM;M of the one-dimensional camera from the estimated tensor using Equations (17), (18) and (16). There are always two solutions. 4. Perform 2D projective reconstruction using equation which recovers the directions of the affine lines in space and the uncalibrated rotations of the camera motion. 5. Solve the uncalibrated translation vector (t; t using Equation (20) by linear least squares. 6. Compute the final affine lines in space using Equation (21) or (22). IX. Line-based factorisation method from an image stream The linear affine reconstruction algorithm described above deals with redundant lines, but is limited to three views. In this section we discuss redundant views, extending the algorithm from the minimum of three to any number N ? 3 of views. In the past few years, a family of algorithms for structure from motion using highly redundant image sequences called factorisation methods have been extensively studied QUAN: AFFINE STRUCTURE FROM LINE CORRESPONDENCES 7 [18], [19], [20], [21], [22] for point correspondences for affine cameras. Algorithms of this family directly decompose the feature points of the image stream into object shape and camera motion. More recently, a factorisation based algorithm has been proposed by Triggs and Sturm [36], [37] for 3D projective reconstruction. We will accomodate our line-based algorithm to this projective factorisation schema to handle redundant views. A. 2D projective reconstruction by rescaling According to [36], [37], 3D projective reconstruction is equivalent to the rescaling of the 2D image points. We have already proven that recovering the directions of affine lines in space is equivalent to 2D projective reconstruction from one-dimensional projective images. Therefore, a re-construction of the line directions in 3D can be obtained by rescaling the direction vectors, viewed as points of P 1 . For each 1D image point in three views (cf. Equation (6)), the scale factors , 0 and 00 -taken individually-are ar- bitrary. However, taken as a whole (; the projective structure of the points x in P 2 . One way to recover the scale factors (; is to use the basic reconstruction equation (7) directly or alternatively to observe the following matrix identity:@ M u The rank of the left matrix is therefore at most 3. All 4 \Theta 4 minors vanish, and three them are algebraically independent, for instance, M u Each of them can be expanded by cofactors in the last column to obtain a linear homogeneous equation in ; Therefore can be solved linearly using@ A@ where designate a known constant entry in the matrix. For each triplet of views, the image points can be consistently rescaled according to Equation (23). For the case of n ? 3 views, we can take appropriate triplets among n views such that each view is contained in at least two triplets. Then, the rescaling equations of all triplets of views for any given point can be chained together over n views to give a consistent (; B. Direction factorisation-step 1 Suppose we are given m line correspondences in n views. The view number is indexed by a superscript and the line number by a subscript. We can now create the 2n \Theta m measurement matrix WD of all lines in all views by stacking all the direction vectors d (j) properly rescaled by (j) of m lines in n views as follows: wm (n) Since the following matrix equation holds for the measurement the rank of WD is at most of three. The factorisation method can then be applied to WD . Let be the Svd factorisation (cf. [45], [46]) of WD . The 3 \Theta 3 diagonal matrix \Sigma D3 is obtained by keeping the first three singular values (assuming that singular values are ordered) of \Sigma and UD3 (VD3 ) are the first 3 columns (rows) of U (V). Then, the product UD3 \Sigma D3V T D3 gives the best rank 3 approximation to WD . One possible solution for " D may be taken to be For any nonsingular 3 \Theta 3 matrix A 3\Theta3 -either considered as a projective transformation in P 2 or as an affine transformation in MA 3\Theta3 and " D are also valid solutions, as we have This means that the recovered direction matrix " D and the rotation matrix " are only defined up to an affine transformation C. Translation factorisation-Step 2 We can stack all of the interpretation planes from different views of a given line to form the following n \Theta 4 measurement matrix of planes: l T t l 0T t 0 8 IEEE-PAMI, VOL. *, NO. *, 199* This matrix WP geometrically represents a pencil of planes, so it still has rank at most 2. For any three rows of WP , taking any minor involving the t (i) , we obtain fi fi fi fi fi fi fi l (i) T l (j) T t (j) l 0: Expanding this minor by cofactors in the last column gives a homogeneous linear equation in t (i) , t (j) and t \Theta \Theta \Theta \Delta@ where each "\Theta" designates a constant 3-vector in a row. Collecting all these minors together, we \Theta \Theta \Theta 0 0 We may take up to a scaling factor, say so the final homogeneous linear equations to solve for are \Theta \Theta 0 0 Once again, this system of equations can be nicely solved by Svd factorisation of W T . The least squares solution for subject to jj(t the singular vector corresponding to the smallest singular value of W T . Note that the efficiency of the computation can be further improved if the block diagonal structure of W T is exploited. D. Shape factorisation-Step 3 The shape reconstruction method developed for three views extends directly to more than 3 views. Given n views, for each line across n views, we just augment the matrix W p from a 3 \Theta 4 to n \Theta 4 matrix, then apply exactly the same method. X. Outline of the line-based factorisation algorithm The line-based factorisation algorithm can be outlined as follows: 1. For triplets of views, compute the tensor (T ijk ) associated with each triplet, then rescale the directions of lines of the triplet using Equation (23). 2. Chain together all the rescaling factors (; for each line across the sequence. 3. Factorise the rescaled measurement matrix of direction to get the uncalibrated rotations and the directions of the affine lines 4. Factorise the measurement matrix using the constraints on the motion to get the uncalibrated translation vector 5. Factorise the measurement matrix of the interpretation planes for each line correspondence over all views to get two points of the line XI. Euclidean structure from the calibrated affine camera So far we have worked with an uncalibrated affine camera, the recovered shape and motion are defined up to an affine transformation in space. If the cameras are calibrated, then the affine structure can be converted into a Euclidean one up to an unknown global scale factor. Following the decomposition of the submatrix M 2\Theta3 of the affine camera A 3\Theta4 as introduced in [22], the metric information from the calibrated affine camera is completely contained in the affine intrinsic parameters KK T . Each view with the associated uncalibrated rotation is subject to for the unknown affine transformation X which upgrades the affine structure to a Euclidiean one. A linear solution may be expected as soon as we have three views if we QUAN: AFFINE STRUCTURE FROM LINE CORRESPONDENCES 9 solve for the entries of XX T . However it may happen that the linear estimate of XX T is not positive-definite due to noise. An alternative non-linear solution using Cholesky parametrization that ensures the positive-definiteness can be found in [22]. Once we obtain the appropriate " carry the rotations of the camera and the directions of lines. The remaining steps are the same as the uncalibrated affine camera case. If we take the weak perspective as a particular affine camera model, with only the aspect ratio of the camera, Euclidean structure is obtained this way. XII. Experimental results A. Simulation setup We first use simulated images to validate the theoretical development of the algorithm. To preserve realism, the simulation is set up as follows. First, a real camera is calibrated by placing a known object of about 50 cm 3 in front of the camera. The camera is moved around the object through different positions. A calibration procedure gives the projection matrices at different positions, and these projection matrices are rounded to affine projection matrices. Three different positions which cover roughly 45 o of the field of view are selected. A set of 3D line segments within a cube of generated synthetically and projected onto the different image planes by the affine projection matrices. All simulated images are of size 512 \Theta 512. Both 3D and 2D line segments are represented by their endpoints. The noise-free line segments are then perturbed as follows. To take advantage of the relatively higher accuracy of line position obtained by the line fitting process in practice, each 2D line segment is first re-sampled into a list of evenly spaced points of the line segment. The position of each point is perturbed by varying levels of noise of uniform distribution. The final perturbed line is obtained by a least squares fit to the perturbed point data. Reconstruction is performed with 21 line segments and two different re-sample rates. The average residual error is defined to be the average distance of the midpoint of the image line segment to the reprojected line in the image plane from the 3D reconstructed line. In Table I, the average residual errors of reconstruction are given with various noise levels. The number of points used to fit the line is the length of the line segment in pixels, this re-sample rate corresponds roughly to the digitization process. Table II shows the results with the number of points used to fit the line equal to only one fourth the length of the line segment. We can notice that the degradation with the increasing noise level is very graceful and the reconstruction results remain acceptable with up to \Sigma5:5 pixel noise. These good results show that the reconstruction algorithm is numerically stable. While comparing Table I and II, it shows that higher re-sample rate gives better results, this confirms the importance of the line fitting procedure-the key advantage of line features over point features. Another influential factor for the stability of the algorithm is the number of lines used. Table III confirms that the more lines used, the better the results obtained. In this test, the pixel error is set to \Sigma1:5. Lines # 8 13 17 21 Average residual error 1.9 1.6 0.59 0.26 III Average residual errors of reconstruction with \Sigma 1.5 pixel noise and various number of lines. B. The experiment with real images A Fujinon/Photometrics CCD camera is used to aquire a sequence of images of a box of size 12 \Theta 12 \Theta 12:65cm. The image resolution is 576 \Theta 384. Three of the frames in the sequence used by the experiments are shown in Figure 1. A Canny-like edge detector is first applied to each image. The contour points are then linked and fitted to line segments by least squares. The line correspondences across three views are selected by hand. There are a total of 46 lines selected, as shown in Figure 2. Fig. 2. Line segments selected across the image sequence. The reconstruction algorithm generates infinite 3D lines, each defined by two arbitrary points on it. 3D line segments are obtained as follows. We reproject 3D lines into one image plane. In the image plane selected, the corresponding original image line segments are orthogonally projected onto the reprojected lines to obtain the reprojected line segments. Finally by back-projecting the reprojected line segments to space, we obtain the 3D line segments, each defined by its two endpoints. Excellent reconstruction results are obtained. An average residual error of one tenth of a pixel is achieved. Figure 3 shows two views of the reconstructed 3D line segments. We notice that the affine structure of the box is almost perfectly recovered. Table IV shows the influence of the number of line segments Average residual error 0.045 0.061 0.10 0.15 0.20 0.25 I Average residual errors with various noise levels for the reconstruction with 21 lines over three views. The number of points to fit the line is the length of the line segment in pixels. Average residual error 0.077 0.26 0.31 0.44 0.65 1.1 II Average residual errors of reconstruction with various noise levels. The number of points to fit the line segment is one fourth the length of the line segment. Fig. 1. Three original images of the box used for the experiments. used by the algorithm. The reconstruction results degrade gracefully with decreasing number of lines. Average residual error 1.3 0.88 0.28 0.12 IV Table of residual errors of reconstruction with different number of line segments. Table V shows the influence of the distribution of line segments in space. For instance, one degenerate case for structure from motion is that when all line segments in space lie on the same plane. Actually, in our images, line segments lie on three different planes-pentagon face, star shape face and rectangle face-of the box. We also performed experiments with line segments lying on only two planes. Table V shows the results with various different two-plane configu- rations. Compared with the three-plane configuration, the reconstruction algorithm does almost equally well. To illustrate the effect of using affine camera model as an approximation to the perspective camera, we used a bigger cube of size 30 \Theta 30 \Theta 30cm, which is two and a half times the size of the first smaller cube. The affine approximation to the perspective camera is becoming less accurate than it was with the smaller cube. A sequence of images of this cube is aquired in almost the same conditions as for the smaller cube. The perspective effect of the big cube is slightly more pronounced as shown in Figure 4. The configuration of line segments is preserved. A total of 39 line segments of three views is used to perform the recon- struction. Figure 5 illustrates two reprojected views of the reconstructed 3D line segments. Compared with Figure 3, the reconstruction is slightly degraded: in the top view of Figure 5, we notice that one segment falls a little apart from the pentagon face of the cube. Globally, the degradation is quite graceful as the average residual error is only 0.3 pixels, compared with 0.12 pixels for the smaller cube. The affine structures obtained can be converted to Euclidean ones (up to a global scaling factor) as soon as we know the aspect ratio [22], which is actually 1 for the camera used. Figure 6 shows the rectified affine shape illustrated in Figure 3. The two sides of the box are accurately orthogonal to each other. XIII. Discussion A linear line-based structure from motion algorithm for uncalibrated affine cameras has been presented. The algorithm requires a minimum of seven line correspondences over three views. It has also been proven that seven lines over three views are the strict minimum data needed to recover affine structure with uncalibrated affine cameras. In other words, in contrast to projective cameras, the linear algorithm is not based on the over-parametrization. This gives the algorithm intrinsic stability. The previous results of Koenderink and Van Doorn [14] on affine structure from motion using point correspondences are therefore extended to line correspondences. To handle the case of redundant views, a factorisation method was also developed. The experimental results based on real and simulated image sequences demonstrate the accuracy and the stability of the algorithms. As the algorithms presented in this paper are developed within the same framework as suggested in [22] for points, it is straightforward to integrate both points and lines into the same framework. QUAN: AFFINE STRUCTURE FROM LINE CORRESPONDENCES 11 Line configuration star+rect.+pent. star+rect. pent.+rect. star+pent. Average residual error 0.12 0.078 0.14 0.28 Table of residual errors of reconstruction with different data. Fig. 3. Reconstructed 3D line segments: a general view and a top view. Fig. 4. One original image of the big cube image sequence. Fig. 5. Two views of the reconstructed line segments for the big box: a general view and a top view. Fig. 6. A side view of the Euclidean shape obtained by using the known aspect ratio of the camera. Acknowledgement This work was supported by CNRS and the French Min- ist'ere de l'Education which is gratefully acknowledged. I would like to thank D. Morris, N. Chiba and B. Triggs for their help during the development of this work. --R "Deter- mination of the attitude of 3D objects from sigle perspective view" Stereovision and Sensor Fusion "Estimation of rigid body motion using straight line correspondences" "A linear algorithm for motion estimation using straight line correspondences" "Motion and structure from point and line matches" 3D Dynamique Scene Analysis "Motion and structure from line correspondences: Closed-form solution, uniqueness, and op- timization" "Optimal estimation of object pose from a single perspective view" "Motion of points and lines in the uncalibrated case" "A unified theory of structure from motion" "A computer program for reconstructing a scene from two projections" "Projective reconstruction from line correspon- dences" The Interpretation of Visual Motion "Affine structure from mo- tion" "Affine shape representation from motion through reference points" "Finding point correspondences and determining motion of a rigid object from two weak perspective views" "Recursive affine structure and motion from image sequences" "Shape and motion from image streams under orthography: A factorization method" "Linear and incremental acquisition of invariant shape models from image sequences" "3D motion recovery via affine epipolar geometry" "A paraperspective factorization method for shape and motion recovery" "Self-calibration of an affine camera from multiple views" "Object pose: the link between weak perspective, para perspective, and full perspec- tive" PhD thesis "Recognition by linear combinations of models" "A factorization method for affine structure from line correspondences" Geometric Invariance in Computer Vision "Obtaining surface orientation from texels under perspective projection" "Perspective approximations" "What can be seen in three dimensions with an un-calibrated stereo rig?" "Stereo from uncalibrated cameras" Matrice fondamentale et autocalibration en vision par ordinateur "Canonic representations for the geometries of multiple projective views" "Relative 3D reconstruction using multiple uncalibrated images" "Invariants of six points and projective reconstruction from three uncalibrated images" "Matching constraints and the joint image" "A factorization based algorithm for multi-image projective structure and motion" "On the geometry and algebra of the point and line correspondences between n images" "Lines and points in three views - an integrated approach" "Algebraic functions for recognition" "Structure from motion using line correspondences" "Dual computation of projective shape and camera positions from multiple images" "Ac- tive visual navigation using non-metric structure" Algorithms in Invariant Theory Matrix Computation Numerical Recipes in C --TR --CTR Nassir Navab , Yakup Genc , Mirko Appel, Lines in One Orthographic and Two Perspective Views, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.7, p.912-917, July Jong-Seung Park, Interactive 3D reconstruction from multiple images: a primitive-based approach, Pattern Recognition Letters, v.26 n.16, p.2558-2571, December 2005 Fredrik Kahl , Anders Heyden, Affine Structure and Motion from Points, Lines and Conics, International Journal of Computer Vision, v.33 n.3, p.163-180, Sept. 1999 Long Quan, Two-Way Ambiguity in 2D Projective Reconstruction from Three Uncalibrated 1D Images, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.23 n.2, p.212-216, February 2001 Yichen Wei , Eyal Ofek , Long Quan , Heung-Yeung Shum, Modeling hair from multiple views, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005 Adrien Bartoli , Peter Sturm, Structure-from-motion using lines: representation, triangulation, and bundle adjustment, Computer Vision and Image Understanding, v.100 n.3, p.416-441, December 2005 Magnus Oskarsson , Kalle strm , Niels Chr. Overgaard, The Minimal Structure and Motion Problems with Missing Data for 1D Retina Vision, Journal of Mathematical Imaging and Vision, v.26 n.3, p.327-343, December 2006 Kalle strm , Magnus Oskarsson, Solutions and Ambiguities of the Structure and Motion Problem for 1DRetinal Vision, Journal of Mathematical Imaging and Vision, v.12 n.2, p.121-135, April 2000 Olivier Faugeras , Long Quan , Peter Strum, Self-Calibration of a 1D Projective Camera and Its Application to the Self-Calibration of a 2D Projective Camera, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.10, p.1179-1185, October 2000 Kalle strm , Fredrik Kahl, Ambiguous Configurations for the 1D Structure and Motion Problem, Journal of Mathematical Imaging and Vision, v.18 n.2, p.191-203, March Loong-Fah Cheong , Chin-Hwee Peh, Depth distortion under calibration uncertainty, Computer Vision and Image Understanding, v.93 n.3, p.221-244, March 2004 Ben Tordoff , David Murray, Reactive Control of Zoom while Fixating Using Perspective and Affine Cameras, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.26 n.1, p.98-112, January 2004 Hayman , Torfi Thrhallsson , David Murray, Tracking While Zooming Using Affine Transfer and Multifocal Tensors, International Journal of Computer Vision, v.51 n.1, p.37-62, January

structure from motion;one-dimensional camera;uncalibrated image;factorization method;affine structure;affine camera;line correspondence

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A Sequential Factorization Method for Recovering Shape and Motion From Image Streams.

AbstractWe present a sequential factorization method for recovering the three-dimensional shape of an object and the motion of the camera from a sequence of images, using tracked features. The factorization method originally proposed by Tomasi and Kanade produces robust and accurate results incorporating the singular value decomposition. However, it is still difficult to apply the method to real-time applications, since it is based on a batch-type operation and the cost of the singular value decomposition is large. We develop the factorization method into a sequential method by regarding the feature positions as a vector time series. The new method produces estimates of shape and motion at each frame. The singular value decomposition is replaced with an updating computation of only three dominant eigenvectors, which can be performed in O(P2) time, while the complete singular value decomposition requires O(FP2) operations for an FP matrix. Also, the method is able to handle infinite sequences, since it does not store any increasingly large matrices. Experiments using synthetic and real images illustrate that the method has nearly the same accuracy and robustness as the original method.

Introduction Recovering both the 3D shape of an object and the motion of the camera simultaneously from a stream of images is an important task and has wide applicability in many tasks such as navigation and robot manipulation. Tomasi and Kanade[1] first developed a factorization method to recover shape and motion under an orthographic projection model, and obtained robust and accurate results. Poelman and Kanade[2] have extended the factorization method to scaled-orthographic projection and paraperspective projection. This method closely approximates perspective projection in most practical situations so that it can deal with image sequences which contain perspective distortions. O O FP 2 Although the factorization method is a useful technique, its applicability is so far limited to off-line computations for the following reasons. First, the method is based on a batch-type computation; that is, it recovers shape and motion after all the input images are given. Second, the singular value decompo- sition, which is the most important procedure in the method, requires operations for features in frames. Finally, it needs to store a large measurement matrix whose size increases with the number of frames. These drawbacks make it difficult to apply the factorization method to real-time applications. This report presents a sequential factorization method that considers the input to the system as a vector time series of feature positions. The method produces estimates of shape and motion at each input frame. A covariance-like matrix is stored instead of feature positions, and its size remains constant as the number of frames increases. The singular value decomposition is replaced with a computation, updating only three dominant eigenvectors, which can be performed in time. Consequently, the method becomes recursive. We first briefly review the factorization method by Tomasi and Kanade. We then present our sequential factorization method in Section 3. The algorithm's performance is tested using synthetic data and real images in Section 4. 2. Theory of the Factorization Method: Review 2.1 Formalization The input to the factorization method is a measurement matrix , representing image positions of tracked features over multiple frames. Assuming that there are features over frames, and letting be the image position of feature at frame , is a matrix such that O FP 2 O Proceedings of 1994 ARPA Image Understanding Workshop, November, 1994, Monterey CA Vol II pp. 1177-1188 . (1) Each column of contains all the observations for a single point, while each row contains all the observed x-coordinates or y-coordinates for a single frame. Suppose that the camera orientation at frame is represented by orthonormal vectors , , and , where corresponds to the x-axis of the image plane and to the y-axis. The vectors and are collected over frames into a motion matrix such that . (2) Let be the location of feature in a fixed world coordinate system, whose origin is set at the center-of-mass of all the feature points. These vectors are then collected into a shape matrix such that . (3) Note that . (4) With this notation, the following equation holds by assuming an orthographic projection. Tomasi and Kanade[1] pointed out the simple fact that the rank of is at most 3 since it is the product of the motion matrix and the shape matrix . Based on this rank theory, they developed a factorization method that robustly recovers the matrices and from . 2.2 Subspace Computation The actual procedure of the factorization method consists of two steps. First, the measurement matrix is factorized into two matrices of rank 3 using the singular value decomposi- tion. Assume, without loss of generality, that . By computing the singular value decomposition of , we can obtain orthogonal matrices and such that In reality, the rank of W is not exactly 3, but approximately 3. is made from the first three columns of the left singular matrix of . Likewise, consists of the first three singular values and is made from the first three rows of the right singular matrix. By setting and (7) we can factorize into where the product is the best possible rank three approximation to . It is well known that the left singular vectors span the column space of while the right singular vectors span its row space. The span of , namely motion space, determines the motion, and the span of , namely shape space, determines the shape. The rank theory claims that the dimension of each subspace is at most three, and the first step of the factorization method finds those subspaces in the high dimensional input spaces. Both spaces are said to be dual in the sense that one of them can be computed from the other. This observation helps us to further develop the sequential factorization method. 2.3 Metric Transformation The decomposition of equation (8) is not completely unique: it is unique only up to an affine transformation. The second step of the method is necessary to find a non-singular matrix , which transforms and into the true solutions and as follows. Observing that rows and of must satisfy the normalization constraints, and , (11) we obtain the system of overdetermined equations such that x 11 . x 1P x F1 . x FP y 11 . y 1P y F1 . y FP f MS U U U where is a symmetric matrix and, and are the rows of . By denoting , , and the system (12) can be rewritten as where , , and are defined by and The simplest solution of the system is given by the pseudo-inverse method such that . (18) The vector determines the symmetric matrix , whose eigendecomposition gives . As a result, the motion and the shape are derived according to equations (9) and (10). The matrix is an affine transform which transforms into in the motion space, while the matrix transforms into in the shape space. Obtaining this transform is the main purpose of the second step of the factorization method, which we call metric transformation. 3. A Sequential Factorization Method 3.1 Overview In the original factorization method, there was one measurement matrix containing tracked feature positions through-out the image sequence. After all the input images are given and the feature positions are collected into the matrix , the motion and shape are then computed. In real-time applica- tions, however, it is not feasible to use this batch-type scheme. It is more desirable to obtain an estimate at each moment sequentially. The input to the system must be viewed as a vector time series. At frame , two vectors containing feature positions such that and (19) are given. Immediately after receiving these vectors, the system must compute the estimates of the camera coordinates , and the shape at that frame. At the next frame, new samples and arrive and new camera coordinates and are to be computed as well as an updated shape estimate . The key to developing such a sequential method is to observe that the shape does not change over time. The shape space is stationary, and thus, it should be possible to derive from without performing expensive computations. More specifically, we store the feature vectors and in a covariance-type matrix defined recursively by . (20) As shown later, the rank of is at most three and its three dominant eigenvectors span the shape space. Once is obtained, the camera coordinates at frame can be computed simply by multiplying the feature vectors and the eigenvectors as follows. This framework makes it possible to estimate camera coordinates immediately after receiving feature vectors at each frame. All information obtained up to the frame is accumulated in and used to produce the estimates at that frame. In equation (20), the size of is fixed to , which only depends on the number of feature points. Therefore, the algorithm does not need to store any matrices whose sizes increase over time. The computational effort in the original factorization method is dominated by the cost of the singular value decomposition. In the framework above, we need to compute eigenvectors of . Note that, however, we only need the first three dominant eigenvectors. Fortunately, several methods exist to compute only the dominant eigenvectors with much less computation necessary to compute all the eigenvectors. Before describing A l 2 l 4 l 5 l 3 l 5 l 6 Gl c - l R 6 - c R 3F G l 1 l 6 F l G T G l L A M f f the details of our algorithm, we briefly review these techniques in the following section. 3.2 Iterative Eigenvector Computation Among the existing methods which can compute dominant eigenvectors of a square matrix, we introduce two methods, the power method and orthogonal iteration[3]. The power method is the simplest, which computes the most dominant eigenvector, i.e., an eigenvector associated with the largest eigenvalue. It provides the starting point for most other tech- niques, and is easy to understand. The method of orthogonal iteration, which we adopt in our method, is able to compute several dominant eigenvectors. 3.2.1 Power Method Assume that we want to compute the most dominant eigen-vectors of an matrix . Given a unit 2-norm , the power method iteratively computes a sequence of vectors : for The second step of the iteration is simply a normalization that prevents from becoming very large or very small. The vectors generated by the iteration converge to the most dominant eigenvector of . To examine the convergence property of the power method, suppose that is diagonaliz- able. That is, with an orthogonal matrix , and . If and , then it follows that where is a constant. Since , equation shows that the vectors point more and more accurately toward the direction of the dominant eigenvector , and the convergence factor is the ratio . 3.2.2 Orthogonal Iteration A straightforward generalization of the power method can be used to compute several dominant eigenvectors of a symmetric matrix. Assume that we want to compute dominant eigenvectors of a symmetric matrix , where . Starting with an matrix with orthonormal columns, the method of orthogonal iteration generates a sequence of matrices : for The second step of the above iteration is the QR factorization of , where is an orthogonal matrix and is an upper triangular matrix. The QR factorization can be achieved by the Gram-Schmidt process. This step is viewed as a normalization process that is similar to the normalization used in the power method. Suppose that is the eigendecomposition of with an orthogonal matrix , and . It has been shown in [3] that the subspace generated by the iteration converges to at a rate proportional to , i.e., where and is a constant. The function dist represents the subspace distance defined by The method offers an attractive alternative to the singular value decomposition in situations where is a large matrix and a few of its largest eigenvalues are needed. In our case, these conditions clearly hold. In addition, the rank theory of the factorization method[1] guarantees that the ratio is very small, and as a result, we should achieve fast convergence for computing the first three eigenvectors. 3.3 Sequential Factorization Algorithm As in the original method, the sequential factorization method consists of two steps, sequential shape space computation and sequential metric transformation. 3.3.1 A Sequential Shape Space Computation In the sequential factorization method, we consider the feature vectors, and , as a vector time series. Let us denote the measurement matrix in the original factorization method at frame by . Then, it grows in the following manner: - BX diag l 1 . l n l j l 1 l 1 l 2 . l n range span x 1 . x p { } l p 1 dist range Q k l p dist range Q k l 4 l 3 , , . (26) Now let us define a matrix time series by . (27) From the definition, it follows that . (28) Since the rank of is at most three, the rank of is also at most three. If is the singular value decomposition of , where and are orthogonal matrices, and , then . (30) This means the eigenvectors of are equivalent to the right singular vectors of . Hence, it is possible to obtain the shape space by computing the eigenvectors of . To compute , we combine orthogonal iteration with updating by equation (27). Given a matrix with orthonormal columns and a null matrix , the following algorithm generates a sequence of matrices : [Algorithm (1)] for (1) (2) The index corresponds to the frame number and each iteration is performed frame by frame. The matrix generated by the algorithm is expected to converge to the eigenvectors of . While the original orthogonal iteration works with a fixed matrix, the above algorithm works with the matrix , which varies from iteration to iteration incorporating new fea- tures. In other words, the sequential factorization method folds the update of into the orthogonal iteration. If the randomly changes over time, no convergence is expected to appear. However, it can be shown that , for all . (31) Therefore, is stationary and converges to as in the orthogonal iteration. Even when noise exists, if the noise is uncorrelated or the noise space is orthogonal to the signal space , then is still equal to and the convergence can be shown. The following convergence rate of the algorithm is deduced from the convergence rate of the orthogonal iteration. 3.3.2 Stationary Basis for the Shape Space Algorithm (1) presented in the previous section produces the matrix , which converges to the matrix that spans the shape space. The true shape and motion are determined from the shape space by a metric transformation. It is not straight-forward at this point, however, to apply the metric transformation sequentially. The problem is that, even though is stationary, the matrix itself changes as the number of frames increases. This is due to the nature of singular vectors. They are the basis for the row and column sub-spaces of a matrix, and the singular value decomposition chooses them in a special way. They are more than just orthonormal. As a result, they rotate in the 3D subspace . Recall that the matrix obtained in metric transformation (9) is a transform from (or ) to in the subspace . Since changes at each frame, also changes. Consequently, the matrix also changes frame by frame. For clarity, let us denote an matrix at frame as . The fact that changes at each frame makes it difficult to estimate iteratively and efficiently. Thus we need to add an additional process to obtain stationary basis for the shape space to update matrix . Let us define a projection matrix onto the by where is the output from Algorithm (1). Needless to say, the rank of is at most three. Since (= ) is stationary, the projection matrix must be stationary. It is thus possible to obtain the stationary basis for the shape space by replacing with the eigenvectors of An iterative method similar to Algorithm (1) can be used to reduce the amount of computation. Given a matrix with orthonormal columns, the iterative method below generates a matrix , which provides the stationary basis for the shape space. U Z f range range range range range range dist range Q f f range range range A f A f A f A f A f range range [Algorithm (2)] for 3.3.3 Sequential Metric Transformation In the previous section, we derived the shape space in terms of . Once is obtained, it is possible to compute camera coordinates and as These coordinates are used to solve the overdetermined equations (12) and the true camera coordinates are recovered in the same way as in the original method. Doing so, however, requires storing all the coordinates and , the number of which may be very large. Instead, we use the following sequential algorithm. [Algorithm (3)] for Let and be the matrices and at frame , where and are defined in Section 2.3 From the definition, it follows that . (36) Assigning equations (35) and (36) to equation (18), we have which gives the symmetric matrix . The eigendecomposition of yields the affine transform and, as a result, the camera coordinates and the shape are obtained as follows: Algorithm (3) followed by equations (37), (38), and (39) completes the sequential method. The size of matrices and are fixed to and , and the method does not store any matrices that grow, even in the sequential metric transformation. 4. Experiments 4.1 Synthetic Data In this section we compare the accuracy of our sequential factorization method with that of the original factorization method. Since both methods are essentially based on the rank theory, we do not expect any difference in the results. Our purpose here is to confirm that the sequential method has the same accuracy of shape and motion recovery as the original method. 4.1.1 Data Generation The object in this experiment consists of 100 random feature points. The sequences are created using a perspective projection of those points. The image coordinates of each point are perturbed by adding Gaussian noise, which we assume to simulate tracking error and image noise. The standard deviation of the Gaussian noise is set to two pixels of a pixel image. The distance of the object center from the camera is fixed to ten times the object size. The focal length is chosen so that the projection of the object covers the whole image. The camera is rotated as shown in Figure 1, while the object is translated to keep its image at the image center. Quantization errors are not added since we assume that we are able to track features with a subpixel resolution. When discussing the accuracy of the sequential method, one needs to consider its dynamic property regarding the 3D recovery. The accuracy of the recovery at a particular frame by the sequential method depends on the total amount of motion up to that time, since the recovery is made only from the information obtained up to that time. At the beginning of an image sequence, for example, the motion is generally small, so high accuracy can not be expected. The accuracy generally improves as the motion becomes larger. The original method does not have this dynamic property, since it is based on a batch-type scheme and uses all the information throughout the sequence. In order to compare both methods under the same conditions, we perform the following computations beforehand. First, we form a submatrix , which only contains the feature positions up to frame . The original factorization is applied to the submatrix, then the results are kept as solutions at frame . They are the best estimates given by the original method. Repeating this process for each frame, we derive the best esti- mates, with which our results are compared. 4.1.2 Accuracy of the Sequential Shape Space Computation We first discuss the convergence property of the sequential shape space computation. The sequential factorization method starts with Algorithm (1) in Section 3.3.1, iteratively generating the matrix which is an estimate for the true shape space . Let us represent the estimation error with respect to the c l f D f- E f A f A f f f true shape space by Recall that the function dist provides a notion of difference between two spaces. On the other hand, the original method produces the best estimate for the shape space by computing the right singular vectors of the submatrix , and its error with respect to the true shape space is also represented by Comparing both errors, Figure 2 shows that they are almost identical. That is, the errors given by the sequential method are almost equal to those given by the original method. At the beginning of the sequence, the amount of motion is small and both errors are relatively large. The ratio of the 4th to 3rd singular values, shown in Figure 3, also indicates that it is difficult to achieve good accuracy at the beginning. Both errors, however, quickly become smaller as the camera motion becomes larger. After about the 20th frame, constant errors of are observed in this experiment. The solutions given by the two methods are so close that the graphs are completely overlapped. Thus, we also plot their difference defined by in Figure 4. Although is relatively large at the beginning, it quickly becomes very small. In fact, after about the 30th frame, is less than , while and are both dist span Q f dist span Q f Frame103050Rotation (deg.) Roll Pitch Yaw Figure 1: True camera motion The camera roll, pitch, and yaw are varied as shown in this figure. The sequence consists of 150 frames. Frame Subspace distance Sequential factorization Original factorization Figure 2: Shape space errors Shape space estimation errors by the sequential method (solid line) and the original method (dashed line) with respect to the true shape space. The errors are defined by subspace distance and plotted logarithmically. Frame Figure 3: Singular value ratio The ratio of the 4th to 3rd singular values, that is . Frame Difference Figure 4: Difference of shape space errors The difference of the estimates by the sequential and original methods, versus the frame number. The difference is plotted logarithmically. 4.1.3 Accuracy of the Motion and Shape Recovery The three plots of Figure 5 show errors in roll, pitch, and yaw in the recovered motion: the solid lines correspond to the sequential method, the dotted lines to the original method. The difference in motion errors between the original and sequential methods is quite small. Both results are unstable for a short period at the beginning of the sequence. After that, they show two kinds of errors: random and structural. Random errors are due to Gaussian noise added to the feature positions. Structural errors are due to perspective distortion, and relate to the motion patterns. The structural errors show a negative peak at about the 60th frame and are almost constant between the 90th and 120th frames. Note the pattern corresponds to the motion pattern shown in Figure 1. Of course, these intrinsic errors cannot be eliminated in the sequential method. The point to observe is that the differences between the two solutions are sufficiently smaller than the intrinsic errors. Shape errors which are compared in Figure 6 also indicate the same results. Again, the differences between the two methods are quite small compared to the intrinsic errors which the original method possesses. Note that no Gaussian noise appears in the shape errors since they are averaged over all the feature points. We conclude from these results that the sequential method is nearly as accurate as the original method except that some extra frames are required to converge. 4.2 Real Images Experiments were performed on two sets of real images. The first set is an image sequence of a satellite rotating in space. Another experiment uses a long video recording (764 images) of a house taken with a hand-held camera. These experiments demonstrate the applicability of the sequential factorization method in real situations. In both experiments, features are selected and tracked using the method presented by Tomasi and Kanade[1]. 4.2.1 Satellite Images Figure 7 shows an image of the satellite with selected features indicated by small squares. The image sequence was digitized from a video recording[4] actually taken by a space shuttle astronaut. The feature tracker automatically selected and tracked features throughout the sequence of 101 images. Of these, five features on the astronaut maneuvering around the satellite were manually eliminated because they had a different motion. Thus, the remaining 27 features were pro- Frame -0.40.4Yaw error (deg.) Sequential Original Frame -0.40.4Pitch error (deg.) Sequential Original Frame -0.40.4Roll error (deg.) Sequential Original Figure 5: Motion errors Errors of recovered camera roll (top), pitch (middle), and yaw (bottom). The errors given by the sequential method are plotted with solid lines, while the errors given by the original method are plotted with dotted lines. Frame Shape error Sequential Original Figure This figure compares the shape errors given by the two method. The errors given by the sequential method are plotted with solid lines, while the errors given by the original method are plotted with dotted lines. The errors are computed as the root-mean-square errors of the recovered shape with respect to the true shape, at each frame. cessed. Figure 8 shows the recovered motion in terms of roll, pitch, and yaw. The side view of the recovered shape is displayed in Figure 9, where the features on the solar panel are marked with opaque squares and others with filled squares. No ground-truth is available for the shape or the motion in this experiment. Yet, it appears that the solutions are satisfac- tory, since the features on the solar panel almost lie in a single line in the side view. 4.2.2 House Images Figure shows the first image of the sequence used in the second experiment. Using a hand-held camera, one of the authors took this sequence while walking. It consists of 764 images which correspond to about 25 seconds. The feature tracker detected and tracked 62 features. The recovered motion and shape are shown in Figures 11 and 12. It is clearly seen that the shape is qualitatively correct. It is also reason-able to observe that only the camera yaw is increasing, because the camera is moving parallel to the ground. In addi- tion, note that the computed roll motion reveals the pace of the recorder's steps, which is about 1 step per second. Further evaluation of accuracy in these experiments is diffi- cult. However, this qualitative analysis of the results with real images, and quantitative analysis of the results with synthetic data essentially shows that the sequential method works as well with real images as the original batch method. 4.3 Computational Time Finally, we compare the processing time of the sequential method with the original method. The computational complexity of the original method is dominated by the cost of the singular value decomposition, which needs computations for a measurement matrix with [5]. Note that corresponds to the number of frames and to the number of features. On the other hand, the complexity of the sequential method is . Computing the solution for frame F, therefore, takes only using the sequential method, while the original method would require operations. Figure 13 shows the actual processing time of the sequential 54P O O FP 2 Figure 7: An image of a satellite The first frame of the satellite image sequence. The superimposed squares indicate the selected features. Frame261014 Rotation (deg.) Roll Pitch Yaw Figure 8: Recovered motion of satellite Recovered camera roll (solid line), pitch (dashed line), and (dotted line) for the satellite image sequence. Figure 9: Side view of the recovered shape A side view of the recovered shape of the satellite. The features on the solar panel are shown with opaque squares and others with filled squares. Notice that the features on the solar panel correctly lie in a single plane. Figure 10: An image of a house The first frame of the house image sequence. The super-imposed squares indicate the selected features. Frame515Rotation (deg.) Roll Pitch Yaw Figure Recovered motion of house Recovered camera roll (solid line), pitch (dashed line), and (dotted line) for the house image sequence. Figure 12: Top view of the recovered shape A view of the recovered shape of the house from above. The features on the two side walls are correctly recovered. Figure 13: Processing time The processing time of the sequential method on a Sun4/ compared with that of the original method (dotted line), as a function of the number of features which is varied from 10 to 500. The number of frames is fixed at 120. Number of features Time (ms) Sequential Original method on a Sun4/10 compared together with that of the original method. The number of features varied from 10 to 500, while the number of frames was fixed at 120. The processing time for selecting and tracking features was not included. The singular value decomposition of the original method is based on a routine found in [6]. The results sufficiently agree with our analysis above. In addition, when the number of features is less than 40, the sequential method can be run in less than 1/30 of a second, which enables video-rate processing on a 5. Conclusions We have presented the sequential factorization method, which provides estimates of shape and motion at each frame from a sequence of images. The method produces as accurate and robust results as the original method, while significantly reducing the computational complexity. The reduction in complexity is important for applying the factorization method to real-time applications. Furthermore, the method does not require storing any growing matrices so that its implementation in VLSI or DSP is feasible. Faster convergence in the shape space computation could be achieved using more sophisticated algorithms such as the orthogonal iteration with Ritz acceleration[3] instead of the basic orthogonal iteration. Also, it is possible to use scaled orthographic projection or paraperspective projection[2] to improve the accuracy of the sequential factorization method. Acknowledgments The authors wish to thank Conrad J. Poelman and Richard Madison for their helpful comments. --R "Shape and Motion from Image Streams Under Orthog- raphy: A Factorization Method," A paraperspective factorization method for shape and motion recovery Matrix computa- tions Satellite rescue in space: highlights of shuttle flights 41C Tracking a few extreme singular values and vectors in signal processing Numerical recipes in C: the art of scientific computing --TR --CTR Xiaolong Xu , Koichi Harada, Sequential projective reconstruction with factorization, Machine Graphics & Vision International Journal, v.12 n.4, p.477-487, April Pui-Kuen Ho , Ronald Chung, Stereo-Motion with Stereo and Motion in Complement, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.2, p.215-220, February 2000 Yiannis Xirouhakis , Anastasios Delopoulos, Least Squares Estimation of 3D Shape and Motion of Rigid Objects from Their Orthographic Projections, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.4, p.393-399, April 2000 Pedro M. Q. Aguiar , Jos M. F. Moura, Rank 1 Weighted Factorization for 3D Structure Recovery: Algorithms and Performance Analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.9, p.1134-1049, September Pei Chen , David Suter, An Analysis of Linear Subspace Approaches for Computer Vision and Pattern Recognition, International Journal of Computer Vision, v.68 n.1, p.83-106, June 2006 Lin Chai , William A. Hoff , Tyrone Vincent, Three-dimensional motion and structure estimation using inertial sensors and computer vision for augmented reality, Presence: Teleoperators and Virtual Environments, v.11 n.5, p.474-492, October 2002

image understanding;feature tracking;shape from motion;3D object reconstruction;real-time vision;singular value decomposition

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On the Sequential Determination of Model Misfit.

AbstractMany strategies in computer vision assume the existence of general purpose models that can be used to characterize a scene or environment at various levels of abstraction. The usual assumptions are that a selected model is competent to describe a particular attribute and that the parameters of this model can be estimated by interpreting the input data in an appropriate manner (e.g., location of lines and edges, segmentation into parts or regions, etc.). This paper considers the problem of how to determine when those assumptions break down. The traditional approach is to use statistical misfit measures based on an assumed sensor noise model. The problem is that correct operation often depends critically on the correctness of the noise model. Instead, we show how this can be accomplished with a minimum of a priori knowledge and within the framework of an active approach which builds a description of environment structure and noise over several viewpoints.

Introduction (a) (b) (c) a) A shaded range image scanned from above a wooden mannequin lying face down. Superellipsoids fitted to segmented data from (a). The dark dots are the range data points. Note that the mannequin's right arm has failed to segment and only a single model has been fitted where two would have been preferable c) Detail of the superellipsoid fitted to the mannequin's right arm in (b). The dark lines on the surface show the position of the range scans. Although the model doesn't match our perceptual notions of what the arm should look like, it does fit the data well. Figure 1. Superellipsoid models fitted to a range data from a Wooden Mannequin Many strategies in computer vision assume the existence of general purpose models that can be used to characterize a scene or environment at various levels of abstrac- tion. They span the range from local characterizations of orientation and curvature [3, 24], to intermediate level representations involving splines and parametric surfaces [1,7,8,24], to still more global representations for solid shape [5,14,19]. The usual assumptions are that a selected model is competent to describe a particular 2 On the Sequential Determination of Model Misfit attribute, and that the parameters of this model can be estimated by appropriate interpretation of input data. But many of these estimation problems are ill-conditioned inverse problems that cannot be solved without additional constraints derived from knowledge about the environment [15]. This leads to a classical chicken and egg problem where model selection and parameter estimation must be dealt with concurrently, a problem difficult to solve given a single static view of the world. In this paper we describe an active strategy that permits solution of both prob- lems, i.e. model parameter estimation and model validation. The context is a system for computing an articulated, 3-D geometric model of an object's shape from a sequence of views obtained by a mobile sensor (laser rangefinder) that is free to select its viewpoint [23]. Shape is characterized by general purpose models consisting of conjunctions of volumetric primitives [5]. An active approach is used where the current state of the model, determined from a bottom-up analysis, is used to predict the locations of surfaces not visible in the current view. Gaze is directed to surfaces where the prediction is least certain (maximum variance), and from there additional measurements are made and used to update the model parameters. The validity of the model is tested against its ability to correctly predict the locations of hidden sur- faces. Initially both the applicability of the model and estimates of its parameters are uncertain, but as the process unfolds with each successive planning cycle (calculation of new gaze point, measurement, updating of model parameters), such assessments become increasingly clear. The emphasis of this paper is the model validation problem. Knowing when a particular model fails can provide at least two significant pieces of information. First, it can indicate when assumptions about the scene are wrong and trigger the search for other models that provide a plausible alternative, that is, it can initiate a model selection process. Second, it can indicate when the processes leading to the determination of model parameters have gone awry. This can be used to initiate a backtracking procedure to re-interpret the data, particularly if the validation procedure is also able to indicate the location of where the model breaks down. Such would be the case if a model is known to be valid but insufficient data are available from which to correctly apply the model or estimate its parameters. The example shown in Figure 1 is a case in point and part of the motivation for this research. Figure 1a shows a laser rangefinder image of a wooden mannequin rendered as a shaded image. Based on analysis of surface features, the image is partitioned into regions corresponding to the different parts of the mannequin [6]. A further abstraction is computed by fitting superquadric primitives to each region with the result shown in Figure 1b [5]. At first glance the result appears to capture each of 2. Estimating Parameters and Planning Gaze 3 the parts of the mannequin. However, on closer examination (Figure 1c), it can be seen that the partitioning algorithm has missed the cues separating the arm at the right elbow. Superquadrics are appropriate shape descriptors provided that parts are convex, but as Figure 1c shows, do not fit the data well otherwise. contextual knowledge, it is difficult to detect such an error given a single view of the object because there is little basis from which to reject the resulting fit. One would have to know the loci of the occluded surfaces in order assess the model's true fit to the data. However such knowledge is often not possible, e.g. inaccessible viewpoints; or is expensive to obtain, e.g. time required to acquire measurements. The compromise advocated in this paper is a sequential process that incrementally builds its descriptions by optimizing measurement to maximize the certainty of each model, then tests them by verifying their consistency from view to view. The remainder of the paper is as follows. Section 2 begins with a brief overview of optimization strategy used to plan gaze and estimate model parameters. It provides the necessary background for Section 3 which describes the model validation process, and presents the results of experiments which demonstrate the resulting algorithms at different noise levels and for different noise models. Section 4 shows how the situation shown in Figure 1c can be identified using the gaze planning strategy and model validation procedures. Finally, we conclude with some observations and briefly outline remaining work. 2. Estimating Parameters and Planning Gaze In earlier work we have considered the problem of how to best direct the gaze of a laser range scanner in order to improve estimates of model parameters and knowledge of object surface position over a sequence of views [20, 21, 23]. The laser scanner is capable (after appropriate transformations) of providing the 3-D coordinates of points sampled from surfaces in the scene. In this scenario it is assumed that the scene is well represented by a conjunction of parametric volumetric primitives, and that data is collected by moving the scanner around on the end-effector of a robot arm (Figure 2). Using methods described in [5] the data are partitioned into sets corresponding to the parts of each visible object. It is assumed that each data set corresponds to a sample of the surface of a single model 1 . Given one of these data sets fx i we wish to infer those parameters "a that best estimate the true parameters a of the model in the scene from which the data was collected. In general an exact solution cannot be found because the 1 In this paper superellipsoids are used to represent parts, but the approach generalizes to other parametric models. 4 On the Sequential Determination of Model Misfit Figure 2. Mobile scanner setup. A laser rangefinder with a 1m 3 field of view is mounted on the end-effector of an inverted Puma 560 robot. scanner measurements are subject to both systematic and random errors, but a good estimate can be obtained by finding the parameters that minimize the squared sum (1) of distances D(x a) of the data points from the surface of the model. Except for very simple models and distance metrics one usually must resort to iterative techniques, e.g. the Levenburg-Marquardt method [12, 16], to perform the minimization. Provided the estimated parameters fall within the region of parameter space around the true parameters where D is reasonably approximated by its first order linear terms, the classic statistical theory of linear models can describe the parameter errors [16]. This theory tells us that when the errors described by the distance metric are randomly sampled from a normal distribution then the error in the estimated parameters a can be described by a p-variate normal distribution dispersed 2. Estimating Parameters and Planning Gaze 5 in the different parameter directions by an amount determined from the matrix of covariances C. Furthermore the quadratic form ffia T C \Gamma1 ffia that defines the distribution is itself randomly sampled from a distribution that obeys a chi-square law with degrees of freedom. In that case we can find the point of the chi-square distribution and use it to define the ellipsoid of confidence (2) that is an ellipsoidal region of parameter space around the estimated parameters and in which there is a probability of fl that the true parameters lie. Because of the noise in the model, and because the data are often incompletely sampled, e.g. only one side of the model is visible from a single viewpoint, the parameters will often be under constrained and exhibit large estimation errors. These errors can be reduced by collecting more data, but there are liabilities in terms of cost and accessibility; e.g. the time taken to plan and move the scanner, memory and cpu resources consumed to process additional data, and limits on accessible viewpoints. Ideally we would like to minimize the amount of data collected and the complexity of the movements necessary to place the scanner in the correct position. To do so requires the formulation of a precise relationship between the parameters that govern the data acquisition process and those related to the model being fit. This task is somewhat difficult because the scanner collects data in the 3-D space of the scene, thus making it difficult to predict the effect that newly collected data points will have on the parameter errors in the p-dimensional space of model parameters. The approach that we have taken to solve this problem is to think of the estimated model as a predictor of surfaces in the scene, and to quantify this error in terms of an interval around each point on the predicted surface. We call this the surface prediction error interval and have shown [20] that an "error bar" protruding from a point x s on the estimated model's surface is given by the quantity s @D @a @a where (@D=@a) is the gradient of the distance metric evaluated for the point x s on the surface of model "a, and \Delta 2 fl is a confidence interval chosen from a chi-square distribution as for the ellipsoid of confidence in (2). The practical use of this representation for optimizing data collection via gaze planning can be explained with the aid of Figure 3. The figure shows the surface prediction error interval corresponding to the model fit to the arm shown earlier in Figure 1c. In Figure 3a, the interval is coded such that darker shading represents 6 On the Sequential Determination of Model Misfit higher uncertainty in surface positions as predicted by the model. Even though the data leading to the model are acquired from a single viewpoint, the resulting prediction extends beyond the visible surfaces and can thus serve as a basis for planning the next gaze direction. An intuitive strategy for doing so would be to direct the scanner to the viewpoint corresponding to the highest uncertainty of prediction. Theoretically we can show that updating model parameters with additional data obtained from this view will minimize the determinant of the parameter covariances [22, 23]. Figure 3a shows a parameterization of the uncertainty surface in the coordinates of a view sphere centered on the the model (uncertainty map), and as can be seen the uncertainty is lowest at the current scanner position, but rises rapidly to a maximum on the opposite side of the view sphere. The optimum strategy here is to move the scanner to the other side of the model, to sample additional data there, and to update the model parameters. However the general problem of gaze planning is much more complex than implied by our example. First, the prediction afforded by the surface error prediction interval is local, so it is unlikely that a complete set of constraining views can be determined on the basis of the model computed from a single viewpoint. In fact the additional data will completely alter the uncertainty map so it must be recomputed after each iteration. Second, the prediction does not take accessibility constraints into account, e.g. certain views may either be unreachable by the scanner, or occluded by surfaces not visible from the current viewpoint. So, as in our example, it is often the case that "the other side" of the model cannot be reached. In spite of these difficulties, we have found that using uncertainty to plan incremental displacements of gaze angle relative to the current viewpoint can result in a successful strategy [21, 23]. We apply a hill climbing algorithm to the changing uncertainty map and use the resulting path to guide the trajectory of the mobile scanner. This can result in a near optimum a data collection strategy with respect to the rate of convergence of model parameters. Also, lack of accessibility is often not that great a problem. For example when representing convex surfaces with superel- lipsoid primitives we have observed that well constrained parameter estimates can be obtained by taking data with the scanner displaced approximately of the initial gaze position. This is because the model can interpolate across large "holes" in the data set. However the success of the exploration strategy hinges on the central assumption that the model fits the data. If this is not the case the parameter covariances, and therefore the surface prediction error, do not accurately reflect the constraints 2. Estimating Parameters and Planning Gaze 7 Latitude a b The figure shows predicted surface uncertainty as an uncertainty map (a) where U is plotted as a function of view sphere latitude and longitude, and as an uncertainty surface (b) where the surface of the model is shaded such that darker shading corresponds to higher values of U . The lines on the top of the uncertainty surface show the data collected when the scanner was positioned at the north pole of the view sphere (latitude=90 ffi ), and to which the model was fitted. As can be seen U is low where data exists, but increases as the model attempts to extrapolate away from the data. The maximum uncertainty lies under the sharp ends of the model, and is marked by the tall peaks on the uncertainty map. The scanner is initially located at the right edge of the uncertainty map. When it moves to the next view position it will follow the local uncertainty gradient, and will therefore move up the center of the broad ridge extending out between the two peaks, i.e. towards the south pole along a longitude of approximately 220 ffi . This corresponds to a path that samples the side of the model facing the viewer in (b). Figure 3. Two different representations for the surface prediction error interval 8 On the Sequential Determination of Model Misfit placed on the model by the data. Thus to ensure a meaningful sensor trajectory it is necessary to test the validity of the model at each iteration. 3. The Detection of Misfits Implicit in our "bottom-up" approach to vision is the notion of "increasing speci- ficity" as processing moves from the lower to the higher layers. By doing things this way we can build computationally intensive lower layers that operate very generally, yet still provide usable data to higher layers designed for specific tasks. However specialized algorithms are usually tuned to a set of assumptions more restrictive than can be truthfully applied to input data processed by the lower layers. Consequently it is necessary to check the validity of the data before proceeding. Such a necessity becomes apparent when we fit volumetric models to segmented range data. The segmentation algorithm we use [5] deliberately avoids detailed assumptions about the exact shape of the primitives (e.g. that they be symmetric) and requires only that they be convex. To this segmented data we fit models designed to represent the kinds of shapes expected in the world. In our case, because they can economically portray a wide range of symmetrical shapes, we use superellipsoids. The problem is that not all convex shapes are superellipsoids, so while the segmentation algorithm may have correctly processed its input data, there is no guarantee that a valid superellipsoid model can be made to fit it. The most straightforward means of evaluating the validity of the data is simply to fit and see. If the model fits well then all of the data should lie on or close to its surface. If not there will be significant residual errors, the model can be declared a misfit, and the flow of processing altered to take remedial action. Because the data are subject to random fluctuations it is not possible to conclude that there has been misfit (or that there has not) with complete certainty. We show how to deal with this problem using methods found in the statistical field of decision theory [4, 13]. In the theory that follows we develop three lack-of-fit statistics, each one useful in different situations. The first of these (/L 1 ) requires an accurate model of the data noise, and knowledge of the parameters of that model, in particular the value of the noise level. When the noise level is not known but the noise model is, then the second lack-of-fit statistic (/L 2 ) can be used. It requires repeat measurements of the data in order to provide an independent estimate of data noise, and therefore incurs additional time and processing costs (e.g. it takes about 12 seconds to scan a 256 \Theta 256 image with the McGill-NRC scanner). In situations where a rapid response is required, and where the noise not known, we propose an incremental lack-of-fit statistic (/L 3 ) which "learns" the local noise level as the scanner moves through the 3. The Detection of Misfits 9 scene. Our experimental results suggest that the measure is able to detect model misfit even if the real noise is not well modeled by the theory. 3.1. Theory. In the discussion that follows we will assume that we have at our disposal a sequence of n s data sets S 0 ns of 3-D coordinates S obtained by moving a laser range scanner along some trajectory through the scene. The S j are not necessarily the original data scans, but are subsets picked out by a segmentation algorithm as having come from the same convex surface. There is also a finite chance that the segmentation algorithm has incorrectly partitioned the data. 3.1.1. Known sensor noise. We will first consider the case for which the data noise meets the conditions assumed by the fitting procedure, i.e. that the data is normally distributed in a direction radially about the surface of the true model with zero mean and known variance oe 2 . For each step j in the sequence of views we find the model parameters "a j that minimize the least squared error of the combined data sets S T We do this by iteratively minimizing the following functional, is the implicit equation of the surface of a superellipsoid [18, 20]. Despite the nonlinearity of the model we will assume that a global minimum error has been found and that the errors are small enough so we can linearize the model and apply the well-known result from linear least squares theory - that an unbiased estimate " j of the true variance oe 2 can be found from the squared sum of the residuals (which are measured by the D 4 metric), where N j is the total number of data points and p is the number of parameters used to fit the model (p = 11 for superellipsoids). Unexpectedly large values of " indicate that the residual errors are not solely due to noise, and therefore give us grounds for believing that the model fits the data badly. A simple strategy to detect misfit is to find those cases for which where k v is a threshold used to decide whether models should be accepted or rejected. On the Sequential Determination of Model Misfit Because of random data noise it is impossible to find a value of k v that correctly classifies the models in all situations, and we must learn to live with two types of detection errors. The first of these, the Type I error, occurs when a model fits well but chance variations increase the value of " enough that the model is erroneously rejected. The other, the Type II error, is the alternative; that a model fits the data badly but random variations result in a reduction of " large enough to cause the model to be erroneously accepted. In general there is a tradeoff - larger values of decrease the chance of Type I errors but increase the possibility of Type II errors. It is possible to evaluate the Type I error. When a model fits the data and the residuals are distributed normally, the statistic is known to be sampled from a chi-squared distribution with degrees of freedom. Thus the probability of a Type I error is when the model fits Graphically it is the area under the chi-squared probability distribution to the right of (n \Gamma p) k v . However it usually makes more sense to work the other way around; that is from the probability distribution find the value of k v which gives a tolerable Type I error. The level is often expressed in terms of a confidence level fl, or the probability of correctly classifying the good models as good. Knowing that (7) follows a chi-squared law we can find the point 2 fl;n\Gammap on the distribution for which the probability of a Type I error is reject models as misfits at the fl level of confidence when In contrast to the Type I error, it is very difficult to find the expected levels of Type II error. The reason for this is that the Type II error is the probability that given that the model does not fit the data. The number of different data configurations that can lead to this situation is so huge, and the interaction of the fitting algorithm to them so unpredictable, that it is impractical to find the probability distribution of " oe 2 j that takes into account all the ways in which a model can be misfitted. 3. The Detection of Misfits 11 3.1.2. Unknown sensor noise level. When the true level of data noise is unknown we can use an estimate of it, provided that estimate is independent of the model fitting process. One way to do this is to exploit repeated measurements. Suppose at some stage during an experiment the laser beam has hit locations on surfaces of the scene, and at each location we have made m i measurements. An estimate of the variance, often called the pure estimate " oe 2 R , is R where is the mean value of the measured surface coordinates at location i. If a model fits the data well " computed for the first data sets should be approximately equal . A lack-of-fit statistic that uses the weighted difference of the two estimates relative to the pure estimate is [2] can be shown to be sampled from an F ratio distribution with numerator and m R denominator degrees of freedom. Models can be rejected at the fl level of confidence when / 3.1.3. Consecutive Estimates of Variance. When repeated measurements cannot be taken we propose that misfit can be detected by comparing consecutive estimates of variance. If the model "a j \Gamma1 fitted to the first j data sets S 0 the estimated variance " should be a valid estimate of data noise. If on the next iteration the variance " oe 2 found after adding S j is significantly greater than " have grounds for believing that the model cannot account for the additional data and that it is therefore unacceptable. It is difficult however to evaluate the Type I errors, and therefore to design a test at the appropriate level of confidence. One might think that because " oe 2 are sampled from chi-squared distributions an F distribution would correctly account for their ratio. Unfortunately, this relationship is true only if the chi-squared distributions are independent. Because they share coordinates from the first j data sets, such is obviously not the case. To avoid any confusion, the index j is added to variable subscripts to indicate the sequential order of data samples and their statistics, e.g. " oe 2 is the sample variance computed over the first data sets. 12 On the Sequential Determination of Model Misfit The approach we have taken is to minimize the dependency by using only the residuals of the newly added points to estimate the data noise. First we compute " oe 2 in the usual way (5) from all of the data in the first j data sets S T The other estimate of variance " oe 2 computed using only the data in S j , that is where in this case n j is the number of data in S j , but the model "a j is the least squares fit to all of the data S T . Because the residuals are distributed normally then " oe 2 are sampled from a chi-squared distributions with of freedom respectively. Therefore when the two variance estimates are independent the incremental lack-of-fit statistic is sampled from an F ratio distribution with n j numerator, and degrees of freedom. Models can be rejected as misfits at the fl level of confidence when However the / should be used with caution because the estimate of " oe 2 In effect some of the data variability is used to compute the model parameters, and this loss results in an estimate of variance lower than it should be. We compensate for the loss in (5) by dividing by that is p points have been used up fitting the p model parameters. When we take only a subset of the data as in (12) it is hard to arrive at an appropriate compensatory figure; mainly because it is difficult to evaluate the relative influence exerted by the subset on the fit. The lower bias of " will be compensated to some degree by a narrower confidence interval in an F distribution with a higher number of degrees of freedom so the overall effect is probably minor and will in any case decrease as the number of data points increases. With the / L 3 metric, " calculated over a more localized region of the surface. Given that surface features causing misfit are most likely to be in the newly scanned region then the mean squared residual error here will be higher than if it were computed over the entire region so far scanned. The result is an apparent increase in the metric's sensitivity to misfit error. However this sensitivity is offset by a higher confidence threshold due to a lower number of degrees of freedom in the chi-squared distribution of " calculated from fewer data points. 3. The Detection of Misfits 13 An implicit assumption when using the / L 3 statistic is that "a j \Gamma1 is a valid fit, and that " oe 2 is a valid estimate of the data noise level. By induction it must also be true that the initial estimate "a 0 be a valid fit, so in practice it is up to us to select the appropriate initial conditions which make sure that this is the case. Generally this can be done without great difficulty, and with only a rough a-priori knowledge of the scene being explored. For example by knowing the minimum size of objects in the scene one could limit the initial scan to a small region, and validly fit it to the surface of almost any large model (even a planar patch). 3.2. Simulation Experiments. In the experiments that follow we used a scene synthesized from two superellipsoid models, a sphere and a cylinder both of 50mm radius, but joined so as to blend smoothly and form an squat cylindric shape with a spherically domed top (Figure 4). This shape was chosen because the overall convexity of the surface ensures that it will not be partitioned by the segmentation algorithms. Data collected from the top of the scene can be initially modeled with a superellipsoid, but as the scanner moves from the top of the scene to a view of the bottom the misfit increases, at first slowly as more of the cylindrical edge is exposed, then abruptly when the flat bottom surface comes into view. Range data is sampled from the scene using a computer simulation of the McGill- NRC range scanner we have in our laboratory [17]. The camera is always directed so that its line of sight is towards the origin of the scene coordinate system, but is allowed to move around on the surface of a view sphere of radius %, also centered on the scene origin. Camera position is specified by a latitude and longitude (#; ') set up with respect to the scene coordinate frame such that the positive Z axis intersects the view sphere at its north pole so that the X-Z plane cuts the view sphere around the meridian of zero longitude. Our scanner uses two mirrors to sweep the laser beam over a field of view 36:9 ffi by 29:2 ffi along the camera's X and Y axes respectively. In both cases the mirror angles are controlled by an index between 0 and 256 that divides the field of view into equi-angular increments. The sampling is specified by two triples of numbers fi min where the X mirror is moved from index i min to i max in steps of i inc , and likewise for the Y mirror and the j indices. If a mirror is not moved, for example when only a single scan line is taken, then the redundant maximum and incremental values will be dropped. We call the array of data collected by scanning the X and Y mirrors a range image. An advantage of a simulated range scanner is that it is very easy to implement and investigate the effect of different noise models. For these experiments we will use a radial noise model in which normally distributed noise (i.e. Gaussian noise) is added so as to displace the data point from the surface in a direction radial to 14 On the Sequential Determination of Model Misfit20 -2525a) We use a scene composed of two superel- lipsoid models, a sphere and a cylinder, joined to make a smooth transition. Although the compound model is convex it cannot be described by a single superellip- soid surface. The scene above is as seen from a view sphere latitude of about \Gamma20 ffi . b) A typical sequence of data. The dots mark the data points, and the lines show the direction of the scan lines, which are doubled to obtain repeat measurements. A radial noise model used. Figure 4. The 3-D Scene and Data used in these experiments the model's center. This noise model matches the assumptions upon which the least squares minimization is based, and therefore those of the tests that detect misfit. Unless otherwise stated all of the following experiments will be performed with sets of data collected in the following way. Initially the scanner is moved to on a view sphere of radius and the scene is twice sampled coarsely (f0; 256; 64g \Theta f96; 160; 16g) to give 50 points (a repeated 5 \Theta 5 range image). The field of view is such that the scanner sees only the spherical surface. After an initial scan, additional data from a sequence of views is taken by moving the camera along the meridian of 0 ffi longitude in 10 ffi increments of latitude until it reaches the south pole. At each position a single line of data (f0; 256; 32g \Theta f128g) is twice scanned to give a set usually containing 10 points (a repeated 5 \Theta 1 image). The result is a sequence of 19 data sets ordered according to the latitude, so S 0 is the initial set collected from the north pole, and S is collected from the south pole. In every 3. The Detection of Misfits 15 data set there are repeat samples of each point, so we can evaluate all 3 lack-of-fit statistics under exactly the same conditions. Figure 4 shows a 3-D rendition of a typical sequence of scans with added radial noise The first set of experiments was designed to evaluate the performance of the three lack-of-fit measures for radially distributed noise, i.e noise in agreement with the assumptions upon which the lack-of-fit statistics are based. A large number of trials were performed at two different noise levels: one about twice that typically observed in our laboratory and the other observed when sampling from the limits of the scanner's range 2000). In each trial a sequence of data was obtained by moving the scanner in 10 ffi steps as described above. At each step the three lack-of-fit statistics were evaluated, and accumulated into the corresponding histogram at that step. On completion we obtained a sequence of histograms showing the progression of each lack-of-fit statistic as the scanner discovered the model surface while moving from the top to the bottom of the scene. The results are shown in Figure 5. We obtain the theoretically expected results for viewsphere latitudes from 90 ffi down to 0 ffi . Here the scanner is just sampling the surface of the sphere, so a valid superellipsoid fit can be obtained. The histograms indicate that for the / statistics approximately 1% of the trials exceed the 99% confidence level, and that the histogram value is close to the expected value of 1%. The misfit level is somewhat lower than expected for the / statistic, and the histogram peak is also displaced downward. As mentioned in the theoretical discussion, an effect like this could be due to overestimation of the degrees of freedom when calculating " Only 5 data points were used in these computations, so an additional degree of freedom would cause a significant decrease in the value of / L 3 . For latitudes below 0 ffi there is a gradual rise in the rate of misfits, until by \Gamma40 ffi almost all of the trials are classified as such. This behaviour also matches that expected, with the slow increase marking the transition region where it becomes increasingly difficult to describe the surface shape as superellipsoid, and the abrupt change indicating gross violations of the assumed symmetry. The / statistic is not as sensitive as the other two in detecting misfit, and again we would expect that behaviour. Because it compares variance estimates at adjacent latitudes, the / L 3 statistic is really detecting incremental increase in misfit, and can therefore be fooled when the misfit is increased slowly. Another way of looking at this is to think of / L 3 as adapting to "learn" the noise. Thus we see that unlike / the / L 3 statistic does not reject fits at latitudes # ! \Gamma60 ffi because it has adapted itself to the very high levels of " j found at On the Sequential Determination of Model Misfit a) 1.% 0.9% 0.9% 1.% 0.9% 0.7% 0.7% 0.9% 1.3% 95.1% 100% 100% 100% 100% 100% 100% 1.3% 0.9% 1.% 1.% 0.9% 0.7% 0.5% 1.% 91.9% 100.% 100% 100% 100% 100% 100% 0.1% 0.5% 77.2% 84.4% 100% 100% 100% 80.3% 1.3% 1.2% 0.7% 0.5% 0.9% 0.7% 0.7% 0.9% 0.7% 0.7% 2.4% 14.6% 41.5% 98.9% 99.8% 99.9% 100% 1.2% 1.2% 1.5% 1.1% 1.1% 1.% 0.9% 0.7% 0.9% 0.9% 1.1% 3.8% 13.6% 97.4% 99.% 99.5% 99.8% 0.1% 0.1% 0.1% 2.6% 7.5% 93.5% 80.9% 34.2% Histograms are rendered radially at their corresponding view sphere latitude. Each bin is coloured a level of grey in proportion to the number of values falling within it. The number of trials used to compute the histograms is shown in parentheses above each figure. Each histogram has been computed by dividing the theoretical one-sided 99% confidence interval (shown underneath the histograms) into 11 bins. The first 10 bins split the 95% interval up into equal parts, while the remaining one shows the other 4%. Values exceeding 99% confidence threshold are all accumulated into the outer bin and the percentage falling here is indicated beside it. When the model fits we would expect this figure to be 1%. The dotted circle marks a lack-of-fit statistic value of 1:0. It should coincide with the histogram maximum. Figure 5. Comparison of / L 1 , / model with noise levels of (a) 1mm and (b) 4mm. 3. The Detection of Misfits 17 At the higher noise level three statistics are close to the limits of their ability to discriminate, and only the gross misfit is detected. In fact the noise level is so high that it is starting to obscure the viewer's perception of the corner of the cylinder in the profile data. Noise of this level would not be encountered on our scanner except when measuring surfaces at the limits of its range. 3.3. Real experiments. The simulations confirm the correctness of theory but to what extent is this true when using real scanners for which the theoretical noise models are only an approximation? To test this we have used the apparatus shown in Figure 6 to perform the same experiments but with real data. The apparatus consists of the McGill-NRC scanner mounted in a fixed position with a view of an object clamped to the rotational axis of a small stage. Different parts of object can be scanned by using stepper motors to rotate the stage about two orthogonal axes. Before the experiment begins a calibration procedure is run to determine the orientation and position of the two rotational axes. Once known, the angles of rotation can be used to map scanner range coordinates into a scene frame attached to the rotating object. Figure 6. Rotary stage used in the real experiments showing the compound model comprised of a smoothly joined cylinder and block On the Sequential Determination of Model Misfit A side effect of the calibration procedure is that it provides us with an estimate of the sensor noise oe. The axes are found by measuring, at several different rotations, the orientation of an inclined plane attached to the stage. For each orientation we can estimate the sensor noise from the residual errors left after fitting a plane to the scanned data. Figure 7 shows that oe varies with orientation, that it depends mainly on the angle the plane makes with the scan direction, and that it is minimum when the surface is normal to the scanner's line of sight. It is well known that oe also varys with the distance to the surface, that it depends on surface properties, and that it can change with time. In general these factors make it very difficult to choose a constant value of oe demanded by the misfit statistics, but for these experiments we have taken the average minimum value over several calibration runs Our choice is motivated by the fact that most of the data are taken from surfaces normal to the scan beam, and that the distance to the surface is approximately that of the calibration plane. Angle Y Angle0.180.22Std Dev Angle Figure 7. Sensor noise as a function of surface orientation. The figure shows oe as a function of the angle between surface to the scanner's X and Y axes. Most of the variation is due to surface slope in the direction of the scan line (the X direction). The results of the first experiment are shown in Figure 8. The procedure used was essentially the same as that described in Section 3.2, though we used the smoothly 3. The Detection of Misfits 19 2.6% 2.2% 1.7% 1.7% 1.7% 1.9% 1.7% 1.7% 1.9% 2.% 2.4% 4.3% 100% 100% 100% 100% 100% 100% 26.8% 28.1% 26.6% 30.4% 36.7% 43.9% 50.2% 54.7% 99.4% 100% 100% 100% 100% 100% 100% 1.1% 0.9% 0.9% 2.4% 2.1% 2.4% 48.9% 99.8% 100% 100% 3.4% 2.1% Figure 8. Comparison of / tests for real data obtained using McGill-NRC rangefinder. joined cylinder and block shown in Figure 6 because it was easier to fabricate than the spherically capped cylinder. The sampling was also changed to take into account the different configuration, and to prevent inclusion of points not on the object's surface. The results were accumulated from 536 trials. It took approximately 2 minutes for each trial and around 20 hours to collect the complete data set. In general the results indicate that the / L1 statistic overestimates the amount of misfit slightly, that the / L2 statistic is in gross error, but that the / L3 statistic still behaves very much as predicted by the theory. The qualitative behaviour of the / L1 lack-of-fit statistic matches that in the simu- lations, except the percentage of trials exceeding the 99% confidence level is about twice that expected (1.7%-2.8% or 9-13 trials). The cause of this discrepancy is indicated in Figure 9 where we show a histogram of the residual errors left after fitting a superellipsoid to a patch of range data scanned from the cylindrical part of the surface Figure 6). When compared to the normal distribution with standard deviation oe computed from (5) we observe that the residuals depart from the assumption of normality: there is asymmetry, and the tails of the histogram are somewhat thicker than expected when compared with the width of the peak. One has the impression that the distribution is composed of two or more normal distributions with different 20 On the Sequential Determination of Model Misfit variances and offset means, which is the kind of effect expected due to the variation of oe with surface orientation and distance. In addition we observed an overall upward drift in the residual errors over the 20 hour duration of the experiment, indicating that the actual sensor noise worsened during this period. The net result is that the values of " oe obtained from the residual errors are greater than the assumed sensor noise, so the values of the / L1 statistic are higher than expected. ex4/cylblk.rids N=2048 Mean=-0.016 StdDev=0.3250100150 Figure 9. Histogram of residual errors left after fitting a superellip- soid to cylindrical data. The solid line shows the normal distribution with the same mean as the residual errors and with a standard deviation oe computed using equation (5). The dotted line indicates the normal distribution assumed for a sensor noise level of The / L2 statistic performs very badly, with the number of trials exceeding the 99% confidence level at around 30 times that expected for a good fit. The reason for the poor performance is that the errors in the repeat data sets are not independent as demanded by the theory. In Figure 10, where we show 8 successive scans of the same patch of surface, it can be seen that there is a noticeable amount of coherency from scan to scan. For example there are similar patterns of variation in scans 4, for the first 15 mirror positions, and in scans 6, 7, & 8 for the last 12. As a result the noise " oe R estimated by looking at the differences between successive scans will be significantly less than the variation along a scan, and since the latter is effectively the residual variation left after fitting it will look like misfit to the / L3 lack-of-fit statistic. The reason for the repeatability in the "noise" from scan to scan is not exactly known, but we have seen it in other laser range scanners as well. One possibility is that it is caused by speckle interference induced when the laser beam passes through the scanner's optics. However even if this kind of noise was 3. The Detection of Misfits 21 not present in the sensor, exactly the same problem would arise if the surface was roughly textured or patterned. We must conclude that the / L2 statistic will only be useful in very specific circumstances. 1mm Figure 10. Repeated Scans. The figure shows 8 sequential scans taken approximately 1 second apart from exactly the same place on the cylindrical part of the surface used in the experiments. The scans have been offset from each other and a plotted horizontally as a function of the scanner's X mirror index. The vertical scale of each scan is indicated by the 1 mm bar on the left. In comparison the / L3 statistic still behaves as expected, even though the scanner noise characteristics depart from the underlying theoretical assumptions. In fact the results seem to match the theory better than those obtained in the simulations (5) where we observed a lower than expected number of trials exceeding the 99% confidence interval. A possible reason for this is that in the real experiments over twice as many points (12 vs 5) were taken in each scan line so the / L3 statistic will be less sensitive to underestimation of degrees of freedom. A curious point, and one which highlights a limitation of the / L3 statistic, is the dip in misfit at a latitude of . If this feature is statistically significant (1% represents only 5 trials in this experiment) the interpretation is that the data obtained at this latitude fits the current model better than the data from all the higher latitudes. That can happen when the measured surface is not exactly superellipsoidal (e.g. because of small errors in the stage calibration). The fitted surface would position itself to minimize the residual error so some parts of it would be inside the measured surface and some parts outside. If the last scan happens to fall near the place the fitted and measured surfaces cross, then the residual errors for it will be lower than 22 On the Sequential Determination of Model Misfit average resulting in a low value of / L3. We cannot expect the / L3 statistic to detect slow departures from the valid class of models, but we can expect it to function well when changes are abrupt (e.g. segmentation errors). 4. An Example Figure 1 illustrates a scenario which typifies the misfit problem - the jointed right arm of the mannequin has been described using a single model, rather than two as one would expect. In this particular situation segmentation should take place along the local concave creases marking the join between the upper and lower arm. However the discrete sampling of the scanner has "skipped" over the fine detail of the elbow joint. A crease is detected around the elbow, but it is not continuous enough to completely sever the arm data into two surface patches. It can be argued that a more detailed analysis could handle this situation, e.g. [9-11, 24], but there will always be times when it is just not possible to segment smoothly joined, articulated objects at such a low level. Consider the out-stretched human arm - how is the boundary that separates it into the upper and lower arms precisely delineated? Instead we have to rely on more global models of the surface to provide additional clues as to when data should be partitioned. Model misfit is one such clue, and where it occurs may, under the right conditions, indicate good places to re-partition. However for the arm of the mannequin there is no clue that anything is wrong - the surface and the scanner have conspired to produce unsegmented data that can be fit very well by a superellipsoid model. Only by collecting more data can the structure of the mannequin be correctly inferred and resolved, which brings us back to the gaze planning strategy described in Section 2. Recall that the strategy operates by directing the scanner to that position on the surface of the current model that exhibits highest uncertainty, or in the case of incremental planning, to a position along the direction of the uncertainty gradient [23]. According to the theory we expect that when data collected at the new sensor position are added to the model " oe will not increase by any significant degree. This can be confirmed by applying an appropriate lack of fit statistic (Table 1). In the event that misfit is detected, further data acquisition can be inhibited until the problem is resolved, e.g. by re-applying the segmentation algorithm to the composite data set. Another object which can cause the gaze planning strategy to fail is the small owl shown in Figure 11. The problem here is that the crease separating the head of the owl from its body does not completely encircle the neck (Figure 12 top). If the initial data is taken from the back of the owl (Figure 12 bottom) a single model is fit to both the head and the body but the strategy will cause the scanner to move towards 4. An Example 2310-55 a b Figure 11. The owl. a) View of the owl mounted in the rotary stage. b) A typical sequence of scans collected from a band encircling the region around the owl's neck. the front of the owl where two models are more appropriate. We investigated the behaviour of the / L3 statistic in this situation by mounting the owl in the stage so data could be collected from the smooth portion of the back. The initial model fit was cylindrical, though both the / L1 and / L2 statistics rejected it outright. The initial misfit is unsurprising given that the soapstone surface is roughly textured, and that the back is slightly concave. A sequence of single line scans was collected by rolling the owl's body over until it faced the scanner - the direction predicted as being the quickest way to improve knowledge of the model surface according to the gaze planning techniques discussed in Section 2. A typical set of scans is shown in Figure 11b and the / L3 lack-of-fit histograms in Figure 13. The scale of the histogram has been expanded (the confidence interval is 99.99999%) to reveal the pattern of change even when the misfit is large. Initially the value of the statistic stays below the 99% confidence level but rises rapidly as soon as data are scanned from part of the owl's wing at a latitude of 40 ffi . After this the statistic starts to adapt to the variation exhibited by the wing parts until by 0 ffi the misfit levels have almost dropped back to normal. The abrupt jump at occurs when the scanner encounters the crease around the owl's neck, but the On the Sequential Determination of Model Misfit Figure 12. Two views of the owl. Depending on whether it is viewed from the front (top view) or from the back (bottom view), the owl can be represented by either two models or a single model respectively. statistic adapts to this change as well, falling to near normal levels by the time the face is fully in view. As can be seen by examining the trace of histogram peaks in Figure 13, the L3 statistic provides a stable indication of misfit errors associated with the surface boundaries that would normally be determined by segmentation. In practice we have found close agreement between misfit indications based on the / L3 statistic and empirically determined modeling errors observed in our laboratory system. The assumptions regarding the use of the / L3 and the other lack-of-fit statistics are summarized below in Table 1. 5. Discussion and Conclusions 25 Assumption / L1 / L2 / Sensor noise is normally distributed 3 yes yes yes Sensor noise level oe known yes no no Sensor noise level is constant 4 yes yes weakly Residual errors due only to sensor noise yes yes no Residual errors spatially independent 5 yes yes yes Residual errors temporally independent 6 yes yes no Repeat measurements available no yes no no no yes Table 1: Assumptions used in the different lack-of-fit statistic 5. Discussion and Conclusions The results we obtain match those our intuition leads us to expect. Perhaps this is better illustrated by considering the analogy of an archaeologist who has discovered a object shaped as above but with only the top of the joint protruding from the sand. So great is the antiquity of this object that the original surface detail has eroded, and the discoverer can only guess at its true nature. Initially it appears to be the top of a container of unusual design, perhaps a burial casket, but only further excavation will tell. From the exposed shape the object looks significantly longer than it is wide, and it will therefore be more economical to begin digging down the objects side. This is done and as the excavation proceeds the initial expectations are confirmed - the object still appears to be a casket. However at some depth further digging suddenly reveals a concavity in the objects surface so pronounced that the archaeologist is forced to drop the casket hypothesis and consider others. 3 In practice the assumption of normality can be weakened. The factor of real importance is that the cumulative lack-of-fit distribution is accurate at the chosen confidence level, because we can then make accurate predictions about the expected rate of misfit due to random chance. 4 Constancy of the noise can also be weakened in practice, particularly with the / L3 statistic. 5 By spatially independent we mean that errors at different surface locations do not depend on each other. It is not strictly necessary that this be the case, for example the errors could be Markovian provided the scale of interaction is much smaller than the spatial extent of the measurements. 6 By temporally independent we mean that the errors from exactly the same surface location at different times are independent. For example, the residual errors resulting from a rough surface are not temporally independent. 26 On the Sequential Determination of Model Misfit Thus it is with the arm of the mannequin and the back of the owl. Initially the laser scanner exposes only a part of the surface so our knowledge of the global shape is extremely uncertain. To resolve this uncertainty we must explore, and to guide our exploration we need an initial hypothesis - that the shape is superellipsoid. However we must always be on guard lest that hypothesis fail. This is the role of the test for misfit - to tell us to reconsider, either by choosing a different hypothesis or by re-examining the data. We are particularly interested in the latter scenario because it is common in an active vision context. Very often we have strong prior knowledge about the appropriate model to use for a given task, but fail because the data used to fit the model is wrong, e.g. segmentation errors. Can we gain any insight into the nature and location of such errors from the exploration procedure? This would be of obvious advantage to a backtracking procedure. In general the answer appears to be no. While we can determine the exact point at which the model fails, we still cannot ascertain whether this is due to the data already collected or to the data newly acquired. In the case of failures due to partitioning errors, our only alternative thus far is to go back and re-sample the data at higher precision such that the segmentation algorithm [6, 10,11] has a better chance of detecting the missing boundary. In this paper we have outlined a framework for this process of what we call autonomous exploration. We have shown that by using the current estimate of a model to predict the locations of surfaces in yet to be explored regions of a scene, we can both improve estimates of model parameters as well as validate its ability to describe the scene. Knowing when we are wrong is not sufficient. In an unstructured environment an autonomous system must act to correct that wrong either by selecting a more appropriate model or by re-interpreting the data in light of cues provided by the failure of the model. These topics are currently under investigation in our laboratory. --R Segmentation through variable-order surface fitting Model discrimination for nonlinear regression models. Shading flows and scenel bundles: A new approach to shape form shading. Probability and statistics for the engineering Darboux frames SNAKES: active contour models. Describing complicated objects by implicit polyno- mials Toward a computational theory of shape: An overview. Finding the parts of objects in range images. Partitioning range images using curvature and scale. Introduction to the Theory of Statistics. Closed form solutions for physically based shape modelling and recognition. Computational vision and regularization theory. Numeric Recipes in C - The Art of Scientific Computing Laser range finder based on synchronized scanners. Recovery of parametric models from range images: The case for superquadrics with global deformations. Dynamic 3D models with local and global deformations: Deformable superquadrics. From uncertainty to visual exploration. Uncertain views. Autonomous exploration: Driven by uncertainty. Autonomous exploration: Driven by uncertainty. The Organization of Curve Detection: Coarse Tangent Fields and Fine Spline Coverings. --TR --CTR Francesco G. Callari , Frank P. Ferrie, Active Object Recognition: Looking for Differences, International Journal of Computer Vision, v.43 n.3, p.189-204, July/August, 2001 Francesco G. Callari , Frank P. Ferrie, Active Object Recognition: Looking for Differences, International Journal of Computer Vision, v.43 n.3, p.189-204, July/August, 2001 Tal Arbel , Frank P. Ferrie, On the Sequential Accumulation of Evidence, International Journal of Computer Vision, v.43 n.3, p.205-230, July/August, 2001

lack-of-fit statistics;active vision;autonomous exploration;misfit

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Approximating Bayesian Belief Networks by Arc Removal.

AbstractI propose a general framework for approximating Bayesian belief networks through model simplification by arc removal. Given an upper bound on the absolute error allowed on the prior and posterior probability distributions of the approximated network, a subset of arcs is removed, thereby speeding up probabilistic inference.

Introduction Today, more and more applications based on the Bayesian belief network 1 formalism are emerging for reasoning and decision making in problem domains with inherent uncertainty. Current applications range from medical diagnosis and prognosis [1], computer vision [10], to information retrieval [2]. As applications grow larger, the belief networks involved increase in size. And as the topology of the network becomes more dense, the run-time complexity of probabilistic inference increases dramatically, reaching a state where real-time decision making eventually becomes prohibitive; exact inference in general with Bayesian belief networks has been proven to be NP-hard [3]. For many applications, computing exact probabilities from a belief network is liable to be unrealistic due to inaccuracies in the probabilistic assessments for the network. Therefore, in general, approximate methods suffice. Furthermore, the employment of approximate methods alleviates probabilistic inference on a network at least to some extend. Approximate methods provide probability estimates either by employing simulation methods for approximate first introduced by Henrion [7], or through methods based on model simplification, examples are annihilating small probabilities [8] and removal of weak dependencies [13]. With the former approach, stochastic simulation methods [4] provide for approximate inference based on generating multisets of configurations of all the variables from a belief network. From this multiset, (conditional) probabilities of interest are estimated from the occurrence frequencies. These probability estimates tend to approximate the true probabilities Part of this work has been done at Utrecht University, Dept. of Computer Science, The Netherlands. 1 In this paper we adopt the term Bayesian belief network or belief network for short. Belief networks are also known as probabilistic networks, causal networks, and recursive models. if the generated multiset is sufficiently large. Unfortunately, the computational complexity of approximate methods is still known to be NP-hard [5] if a certain accuracy of the probability estimates is demanded for. Hence, just like exact methods, simulation methods have an exponential worst-case computational complexity. As has been demonstrated by Kjaerulff [13], forcing additional conditional independence assumptions portrayed by a belief network provides a promising direction towards belief net-work approximation in view of model simplification. However, Kjaerulff's method is specifically tailored to the Bayesian belief universe approach to probabilistic inference [9] and model simplification is not applied to a network directly but to the belief universes obtained from a belief network. The method identifies weak dependencies in a belief universe of a network and removes these by removing specific links from the network thereby enforcing additional conditional independencies portrayed by the network. As a result, a speedup in probabilistic inference is obtained at a cost of a bounded error in inference. In this paper we propose a general framework for belief network approximation by arc removal. The proposed approximation method adopts a similar approach as Kjaerulff's method [13] with respect to the means for quantifying the strength of arcs in a network in terms of the Kullback-Leibler information divergence statistic. In general, the Kullback-Leibler information divergence statistic [14] provides a means for measuring the divergence between a probability distribution and an approximation of the distribution, see e.g. [22]. However, there are important differences to be noted between the approaches. Firstly, the type of independence statements enforced in our approach renders the direct dependence relationship portrayed by an arc superfluous, in contrast to Kjaerulff's method where other links may be rendered superfluous as well. As a consequence, we apply more localized the changes to the network which allows a large set of arcs to be removed simultaneously. Secondly, as has been mentioned above, Kjaerulff's method operates only with the Bayesian belief universe approach to probabilistic inference using the clique-tree propagation algorithm of Lauritzen and Spiegelhalter [16]. In contrast, the framework we propose operates on a network directly and therefore applies to any type of method for probabilistic inference. Finally, given an upper bound on the posterior error in probabilistic inference allowed, a (possibly large) set of arcs is removed simultaneously from a belief network requiring only one pre-evaluation of the network in contrast to Kjaerulff's method in which conditional independence assumptions are added to the network one at a time. The rest of this paper is organized as follows. Section 2 provides some preliminaries from the Bayesian belief network formalism and introduces some notions from information theory. In Section 3, we present a method for removing arcs from a belief network and analyze the consequences of the removals on the represented joint probability distribution. In Section 4, some practical approximation schemes are discussed, aimed at reducing the computational complexity of inference on a belief network. To conclude, in Section 5 the advantages and disadvantages of the presented method are compared to other existing methods for approximating networks. Preliminaries In this section we briefly review the basic concepts of the Bayesian belief network formalism and some notions from information theory. In the sequel, we assume that the reader is well acquainted with probability theory and with the basic notions from graph theory. 2.1 Bayesian Belief Networks Bayesian belief networks allow for the explicit representation of dependencies as well as independencies using a graphical representation of a joint probability distribution. In general, undirected and directed graphs are powerful means for representing independency models, see e.g. [21, 22]. Associated with belief networks are algorithms for probabilistic inference on a network by propagating evidence, providing a means for reasoning with the uncertain knowledge represented by the network. A belief network consists of a qualitative and a quantitative representation of a joint probability distribution. The qualitative part takes the form of an acyclic digraph G in which each vertex represents a discrete statistical variable for stating the truth of a proposition within a problem domain. In the sequel, the notions of vertex and variable are used interchangeably. Each arc in the digraph, which we denote as called the tail of the arc, and vertex called the head of the arc, represents a direct causal influence between the vertices discerned. Then, vertex V i is called an immediate predecessor of vertex V j and vertex V j is called an immediate descendant of vertex V i . Furthermore, associated with the digraph of a belief network is a numerical assessment of the strengths of the causal influences, constituting the quantitative part of the network. In the sequel, for ease of exposition, we assume binary statistical variables taking values in the domain ftrue; falseg. However, the generalization to variables taking values in any finite domain is straightforward. Each variable V i represents a proposition where is denoted as v i and false is denoted as :v i . For a set of variables V , the conjunction is called the configuration scheme of V ; a configuration c V of V is a conjunction of value assignments to the variables in V . In the sequel, we use the concept of configuration scheme to denote that a specific property holds for all possible configurations of a set of variables. Definition 2.1 A Bayesian belief network is a tuple is an acyclic digraph with (G)g is a set of real-valued functions \Theta fC G called assessment functions, such that for each configuration of the set G (V i ) of immediate predecessors of vertex V i we have that A probabilistic meaning is assigned to the topology of the digraph of a belief network by means of the d-separation criterion [18]. The criterion allows for the detection of dependency relationships between the vertices of the network's digraph by traversing undirected paths, called chains, comprised by the directed links in the digraph. Chains can be blocked by a set of vertices as is stated more formally in the following definition. Definition 2.2 Let be an acyclic digraph. Let be a chain in G. Then is blocked by a set of vertices contains three consecutive vertices which one of the following three conditions is fulfilled: are on the chain and are on the chain and are on the chain and oe the set of vertices composed of X 2 and all its descendants. Note that a chain is blocked by ; if and only if contains . In this case, vertex X 2 is called a head-to-head vertex with respect to [6]. Definition 2.3 Let be an acyclic digraph and let X, Y , Z ' V (G) be disjoint subsets of vertices from G. The set Y is said to d-separate the sets X and Z in G, denoted G , if for each every chain from V i to V j in G is blocked by Y . The d-separation criterion provides for the detection of probabilistic independence relations from the digraph of a belief network, as is stated more formally in the following definition. Definition 2.4 Let be an acyclic digraph. Let Pr be a joint probability distribution on V (G). Digraph G is an I-map for Pr if hX j Z j Y i d G implies X?? Pr Y j Z for all disjoint subsets X, Y , Z ' V (G), i.e. X is conditionally independent of Z given Y in Pr. By the chain-rule representation of a joint probability distribution from probability theory, the initial probability assessment functions of a belief network provide all the information necessary for uniquely defining a joint probability distribution on the set of variables discerned that respects the independence relations portrayed by the digraph [11, 18]. Theorem 2.5 Let (G; \Gamma) be a belief network as defined in Definition 2.1. Then, Y defines a joint probability distribution Pr on V (G) such that G is an I-map for Pr. A belief network therefore uniquely represents a joint probability distribution. For computing (conditional) probabilities from a network, several efficient algorithms have been developed from which Pearl's polytree algorithm with cutset conditioning [18, 19] and the method of clique-tree propagation by Lauritzen and Spiegelhalter [16] (and combinations [20]) are the most widely used algorithms for exact probabilistic inference. Simulation methods provide for approximate probabilistic inference, see [4] for an overview. 2.2 Information Theory The Kullback-Leibler information divergence [14] has several important applications in sta- tistics. One of which is for measuring how well one joint probability distribution can be approximated by another with a simpler dependence structure, see e.g. [22]. In the sequel, we will make extensive use of the Kullback-Leibler information divergence. Before defining the Kullback-Leibler information divergence more formally, the concept of continuity is introduced [14]. Definition 2.6 Let V be a set of statistical variables and let Pr and Pr 0 be joint probability distributions on V . Then Pr is absolutely continuous with respect to Pr 0 over a subset of variables denoted as Pr Pr 0 k X, if Pr(c X configurations c X of X. We will write Pr Pr 0 for Pr Pr 0 k V for short. Note that the continuity relation is a reflexive and transitive relation on probability distributions. Furthermore, the continuity relation satisfies ffl if Pr Pr 0 k X, then Pr Pr 0 k Y for all subsets of variables X, Y ' V with Y ' X; subsets of variables X, Y ' V and each configuration c Y of Y with That is, if a joint probability distribution Pr is absolutely continuous with respect to a distribution Pr 0 over some set of variables X, then Pr is also absolutely continuous with respect to Pr 0 over any subset of X. In addition, any posterior distribution configuration c Y of Y is also absolutely continuous with respect to the posterior distribution Definition 2.7 Let V be a set of statistical variables and let X ' V . Let Pr and Pr 0 be joint probability distributions on V . The Kullback-Leibler information divergence or cross entropy of Pr with respect to Pr 0 over X, denoted as I(Pr; Pr 0 ; X), is defined as In the sequel, we will write short. Note that the information divergence is not symmetric in Pr and Pr 0 and is finite if and only if Pr is absolutely continuous with respect to Pr 0 . Furthermore, the information divergence I satisfies subsets of variables X ' V , especially only if Pr(CX subsets of variables subsets of variables X, Y ' V if and Y are independent in both Pr and Pr 0 . In principle, the base of the logarithm for the Kullback-Leibler information divergence is immaterial, providing only a unit of measure; in the sequel, we use the natural logarithm. With this assumption the following property holds. Proposition 2.8 Let V be a set of statistical variables and let Pr and Pr 0 be joint probability distributions on V . Furthermore, let I be the Kullback-Leibler information divergence as defined in Definition 2.7. Then, for all X ' V . Hence, the Kullback-Leibler information divergence provides for an upper bound on the absolute divergence jPr(c X configurations c X of X, a property of the Kullback-Leibler information divergence known as the information inequality [15]. r s CC appoximation CTP appoximation Figure 1: Reducing the complexity of cutset conditioning (CC) and clique-tree propagation (CTP) by removing arc 3 Approximating a Belief Network by Removing Arcs In this section we propose a method for removing arcs from a belief network and we investigate the consequences of the removal on the computational resources and the error introduced. For ease of exposition, a method for removing a single arc from a belief network is introduced first. Then, based on this method and the observations made, a method for multiple simultaneous arc removals is presented. 3.1 Reducing the Complexity of a Belief Network by Removing Arcs The computational complexity of exact probabilistic inference on a belief network depends to a large extend on the connectivity of the digraph of the network. Removing an arc from the digraph of the network may substantially reduce the complexity of probabilistic inference on the network. For Pearl's polytree algorithm with the method of cutset conditioning [18, 19], undirected cycles, called loops [18], can be broken resulting in smaller loop cutsets to be used. The size of the cutset determines the computational complexity of inference on the network to a large extend. For the method of clique-tree propagation [16], a belief network is first transformed into a decomposable graph. Here, the computational complexity of inference depends to a large extend on the size of the largest clique in the decomposable graph. Removal of an appropriate arc or edge results in splitting cliques into several smaller cliques, see e.g. the method of Kjaerulff [13], yielding a reduction in computational complexity of inference on the decomposable graph. In Figure 1 we have depicted the effect of removing an arc from the digraph of a belief network for the method of cutset conditioning and for the method of clique-tree propagation. For cutset conditioning, a vertex in the cutset (e.g. the vertex drawn in shading) is required to break the loop. Since removal of arc breaks the loop, a smaller cutset may be necessary. For clique-tree propagation, the decomposable graph obtained from the example belief network has three cliques, each with 4 vertices. Removal of arc results in a decomposable graph with four smaller cliques, one with 2 and three with 3 vertices. For approximate methods, the computational complexity of for example forward simulation [4] depends to some extend on the distance from a root vertex to a leaf vertex. Therefore, the removal of arcs may also yield a reduction in the complexity of approximate inference. However, it is more difficult to analyze and measure the amount of reduction in complexity in general in comparison to exact methods and in the sequel we will discuss arc removal in view of exact methods for probabilistic inference. 3.2 Removing an Arc from a Belief Network Although several methods for removing an arc from a belief network can be devised, the method for removal of an arc as defined in the following definition is the most natural choice. This will be made clear when we analyze the effects of the removal. Definition 3.1 (G; \Gamma) be a belief network and let Pr be the joint probability distribution defined by B. Let V r be an arc in G. We define the tuple B is the acyclic digraph with V (G Vr6!Vs (G)g is the set of functions with Note that network B resulting after removal of an arc V r from the digraph G of a belief network B, again constitutes a belief network. In this network the assessment functions for the head vertex of the arc are changed only. In the sequel, we will refer to B Vr6!Vs as the approximated belief network after removal of arc and the operation of computing B Vr6!Vs will be referred to as approximating the network. Removal of an arc from a belief network may result in a change of the represented joint probability distribution. However, the represented dependency structure of the distribution portrayed by the graphical part of the network may be retained by introducing a virtual arc between the two vertices for which a physical arc is removed. A virtual arc may serve for the detection of dependencies and independencies in the original probability distribution using the d-separation criterion. A virtual arc, however, is not used in probabilistic inference, still allowing for a faster, approximate computation of prior and posterior probabilities from the simplified network. 3.3 The Error Introduced by Removing an Arc Removing an arc from a belief network yields a (slightly) simplified network that is faster in inference but exhibits errors in the marginal and conditional probability distributions. In this section we will analyze the errors introduced in the prior and posterior distributions upon belief network approximation by removal of an arc. These effects can be summarized as introducing both a change in the qualitative (ignoring any virtual arcs) as well as a change in the quantitative representation of a joint probability distribution. The Qualitative Error in Prior and Posterior Distributions The change in the qualitative belief network representation of the probabilistic dependency structure by removing an arc from a belief network is described by the following lemma. Lemma 3.2 Let G be an acyclic digraph and let V r be an arc in G. Let be the digraph G with arc removed, that is, g. Then, we have that hfV r g j G Vr 6!Vs Proof. To prove that hfV r G Vr 6!Vs holds, we show that every chain from vertex V r to vertex V s in G Vr6!Vs is blocked by the set G Vr 6!Vs (V s ). For such a chain from V r to V s two cases can be distinguished: ffl comprises an arc chain is blocked by G Vr 6!Vs (V s ); ffl comprises an arc is acyclic, must contain a head-to-head vertex V k , i.e. a vertex with two converging arcs on . Since oe G Vr 6!Vs blocked by G Vr 6!Vs (V s ).The property states that after removing arc digraph G of a belief network, the simplified graphical representation now yields that variable V r is conditionally independent of variable V s given G Vr 6!Vs (V s ) being the set of immediate predecessors of V s in the digraph G with arc The Quantitative Error in the Prior Distribution The change in the qualitative dependency structure portrayed by the network has its quantitative counterpart as the two are inherently linked together in the belief network formalism. To analyze the error of the approximated prior probability distribution, similar to [13, 22] we use the Kullback-Leibler information divergence for a quantitative comparison in terms of the divergence between the joint probability distribution defined by a belief network and the approximated joint probability distribution obtained after removing an arc from the network. To facilitate the investigation, we will give an expression for the approximated joint probability distribution in terms of the original distribution. First, we will introduce some additional notions related to arcs in a digraph that are useful for describing the properties that These notions are build on the observation that the set of immediate predecessors G Vr 6!Vs (V s ) d-separates tail vertex V r from head vertex V s in the digraph G with arc removed. Definition 3.3 Let be an acyclic digraph and let V r be an arc in G. We define the arc block of V r denoted as fi G as the set of vertices g. Furthermore, we define the arc environment of V r in G, denoted as j G as the set of vertices The joint probability distribution defined by the approximated belief network can be factorized in terms of the joint probability distribution defined by the original network. Lemma 3.4 Let (G; \Gamma) be a belief network and let Pr be the joint probability distribution defined by B. Let V r be an arc in G and let B be the approximated belief network after removal of V r defined in Definition 3.1. Then the joint probability distribution Pr Vr6!Vs defined by B Vr6!Vs satisfies Pr Vr6!Vs (C V (G) where is the arc environment of V r as defined in Definition 3.3. Proof. From Theorem 2.5, the joint probability distribution Pr Vr6!Vs defined by network Pr Vr6!Vs (C V (G) Y where Exploiting Definition 3.1 leads to Y Now, since fl Vs (V s j C G (Vs ) )Clearly, this property links the graphical implications of removing an arc from a belief net-work with the numerical probabilistic consequences of the removal; variable V r is rendered conditionally independent of variable V s given G Vr 6!Vs (V s ) after removal of an arc V r Now, one of the most important consequences to be investigated is the amount of absolute divergence between the prior probability distribution and the approximated distribu- tion. From the information inequality we have for all subsets X ' V , where Pr and Pr Vr6!Vs are joint probability distributions on the set of variables V defined by a belief network and the network with arc removed respectively. However, we recall that this bound is finite only if Pr is absolutely continuous with respect to Pr Vr6!Vs . We prove this property in the following lemma. Lemma (G; \Gamma) be a belief network. Let V r be an arc in G and let be the approximated belief network after removal of V r defined in Definition 3.1. Then the joint probability distribution Pr defined by B is absolutely continuous with respect to the joint probability distribution Pr Vr6!Vs defined by B Vr6!Vs over Proof. To prove that Pr is absolutely continuous with respect to Pr Vr6!Vs over V (G), we prove that Pr(c V (G) implies that Pr Vr6!Vs configurations c V (G) of (G). First observe that from the chain rule of probability theory we have that where is the arc environment of arc s in G as defined by Definition 3.3. Now consider a configuration c V (G) of V (G) with configuration we have that Pr(c j G (Vr !Vs . Furthermore, Pr(c Vr " c Vs j c G (Vs )nfVr implies that These observations lead to Hence, if Pr(c we conclude that Pr Pr Vr6!Vs . 2 From this property of absolute continuity, the Kullback-Leibler information divergence provides a proper upper bound on the error introduced in the joint probability distribution by removal of an arc from the network. However, the bound can be rather coarse as it can be expected that removing an arc may not always affect the prior probabilities of some specific marginal distributions defined by the network. This observation is formalized by the following lemma which states that the divergence in the prior marginal distributions is always zero for sets of vertices that are not descendants of the head vertex of an arc that is removed. In fact, this property is a direct result from the chain-rule representation of the joint probability distribution by a belief network. Lemma 3.6 Let (G; \Gamma) be a belief network and let Pr be the joint probability distribution defined by B. Let V r be an arc in G and let B be the approximated belief network after removal of V r defined in Definition 3.1. Then the joint probability distribution Pr Vr6!Vs defined by B Vr6!Vs satisfies Pr Vr6!Vs (C Y for all Y ' V (G) n oe denotes the set comprised by V s and all its descendants Proof. First, we will prove that Y G (V s ). By applying Theorem 2.5 and by marginalizing Pr we obtain Y Y Y for all configurations c X of X with the assumption that the configurations that occur within the sum adhere to c V . Now since G (V k we find by rearranging terms Y Y Y for all configurations c X of X. Hence, we have Y By a similar exposition for network B Vr6!Vs , we have Pr Vr6!Vs (CX Y where Now observe that from Definition 3.1 and we obtain Pr Vr6!Vs (CX by principle of marginalization we conclude that Pr Vr6!Vs (C Y This property provides the key observation for the applicability of multiple arc removals as will be described in Section 3.4. The Quantitative Error in Posterior Distributions Belief networks are generally used for reasoning with uncertainty by processing evidence. That is, the probability of some hypothesis is computed from the network given some evidence. In the belief network framework, this amounts to computing the revised probabilities from the posterior probability distribution given the evidence. We will investigate the implications on posterior distributions after removal of an arc. We begin our investigation by exploring some general properties of the Kullback-Leibler information divergence. Lemma 3.7 Let V be a set of statistical variables and let X, Y ' V be subsets of V . Let Pr and Pr 0 be joint probability distributions on V . Then the Kullback-Leibler information divergence I satisfies Proof. We distinguish two cases: the case that Pr Pr and the case that ffl Assume that Pr Pr 0 k X[Y . This assumption implies that the information divergence 2.7 we therefore have that c X[Y Here, we used the fact that if for some configuration c 0 Y of the set of variables Y the probability distribution Y ) is undefined, that is, if Pr(c 0 any configuration c 0 of X the probability Pr(c 0 log(Pr(c 0 definition. Therefore, we let the first sum in the last equality above range over all configurations c Y of Y for which Pr(c Y ) ? 0. Now by rearranging terms we find log Note that I(Pr; Pr ffl Assume that Pr 6 Pr This implies that I(Pr; Pr show that I(Pr; Pr observe that from the assumption there exists a configuration c 0 Y of X [ Y such that Pr(c 0 two cases are distinguished: the case that Pr 0 (c 0 and the case that Pr 0 (c 0 - Assume that Pr 0 (c 0 implies that Pr(c 0 yields that Pr 6 Pr 0 k Y . By Definition 2.7 I(Pr; Pr using the fact that the divergence I is non-negative to - Assume that Pr 0 (c 0 for the configurations c 0 X and Y . Hence, and by Definition 2.7 this implies that non-negative, we conclude that I(Pr; Pr property of the Kullback-Leibler information divergence leads to the following lemma stating an upper bound on the absolute divergence of the posterior probability distribution defined by a belief network given some evidence and the (approximated) posterior probability distribution defined by another (approximated) network. Lemma 3.8 Let V be a set of statistical variables and let Pr and Pr 0 be joint probability distributions on V such that Pr Pr 0 . Let I be the Kullback-Leibler information divergence. Then, for all subsets of variables X, Y ' V and all configurations c Y of Y with Furthermore, this upper bound on the absolute divergence is finite. Proof. Consider two subsets X, Y ' V and a configuration c Y of Y with this configuration, Pr Pr 0 implies that Pr 0 hence, the posterior distributions are well-defined. Furthermore, since Pr Pr 0 also implies that it follows from Proposition 2.8 that we have the finite upper bound Furthermore, Lemma 3.7 yields that Y )?0 When we consider the divergence isolation, we have since for any configuration c 0 Y of Y with Pr(c 0 divergence Y is finite and non-negative. From these observations we finally find the finite upper bound Y )Now, from this property of the information divergence, the absolute divergence between the posterior distribution given evidence c Y for a subset of variables Y of a belief network B and the approximated network B Vr6!Vs after removal of an arc V r ! V s is bounded by where Pr is the joint probability distribution defined by B and Pr Vr6!Vs is the joint probability distribution defined by B Vr6!Vs . This bound is finite since Pr is absolutely continuous with respect to Pr Vr6!Vs . Furthermore, from this bound we find that in the worst case, i.e. the error in probabilistic inference on an approximated belief net-work is inversely proportional to the square root of the probability of the evidence; the more unlikely the evidence, the larger the error may be. 3.4 Multiple Arc Removals In this section we generalize the method of single arc removal from belief networks to a method of multiple simultaneous arc removals, thereby still guaranteeing a finite upper bound on the error introduced in the prior and posterior distributions. We recall from Definition 3.1 that removing an arc yields an appropriate change of the assessment functions only for the head vertex of the arc to be removed. Therefore, this operation can be applied in parallel for all arcs not sharing the same head vertex. To formalize this requirement, we introduce the notion of a linear subset of arcs of a digraph. be an acyclic digraph with the set of vertices indexed in ascending topological order. The relation OE G ' A(G) \Theta A(G) on the set of arcs of G is defined as V r pairs of arcs G. Furthermore, let A ' A(G) be a subset of arcs in G. Then we say that A is linear with respect to G if the order OE G is a total order on A, that is, either for each pair of distinct arcs Note that a linear subset of arcs from a digraph contains no pair of arcs that have a head vertex in common. Now, we formally define the simultaneous removal of a linear set of arcs from a belief network. (G; \Gamma) be a belief network. Let A ' A(G) be a linear subset of arcs in G. We define the multiply approximated belief network, denoted as the network resulting after the simultaneous removal of all arcs A from B by Definition 3.1. That is, we obtain network the digraph with V (GA (G)g the set of functions To analyze the error introduced in the prior as well as in the posterior distribution after removal of a linear set of arcs from a belief network, we once more exploit the information inequality. For obtaining a proper upper bound, the essential requirement is that the joint probability distribution defined by the original network is absolutely continuous with respect to the distribution defined by the multiply approximated network. To prove this, we will exploit the ordering relation on the arcs of a digraph as defined above. This ordering relation induces a total order on the arcs of a linear subset of arcs in a digraph and we show that a consecutive removal of arcs from a belief network in arc linear order yields a multiply approximated network. Then, by transitivity of the continuity relation, this directly implies that the joint probability distribution defined by the original network is absolutely continuous with respect to the distribution defined by the multiply approximated network. Lemma 3.11 Let (G; \Gamma) be a belief network and let Pr be the joint probability distribution defined by B. Let 1, be a linear subset of arcs in G ordered with respect to OE G as defined in Definition 3.9, i.e. for all pairs of arcs V r we have that i ! j. Now, be the multiply approximated belief network after removal of all arcs A as defined in Definition 3.10. Then, Vrn 6!Vsn where each (approximated) network on the right-hand side is approximated by removal of an defined in Definition 3.1. Proof. The proof is by induction on the cardinality of A. Base case For holds as the hypothesis for induction. Now, consider arc A. Then, by principle of in- duction, to prove that now have to prove that . Obviously, the digraphs obtained after removal of this arc are identical, i.e we have This leaves us with a proof for the probability assessment functions. First, observe that the simultaneous removal of all arcs A from network B yields network BA with probability assessment functions where we have that fl 0 observe that the removal of arc V rn!Vsn from network B AnfVrn !Vsn g yields probability assessment functions which we find that fl 00 it remains to prove that fl 0 Vsn , or equivalently, that Pr(V sn j C G (Vsn )nfVrn observe that from the ordering relation OE G we find that all arcs A n fV rn ! V sn g that are removed from B are 'below' arc V rn ! V sn in the digraph G of B, i.e. by assuming an ascending topological order of the vertices this implies that s i ? s n g. Hence, ( G (V sn )[fV sn g)" oe and by the induction hypothesis, we can apply Lemma 3.6 for each arc in to find that Pr(V sn "C G (Vsn )nfVrn thermore, this yields that Pr(V sn j C G (Vsn )nfVrn Vsn and we conclude that As a result of this property of multiple arc removals, the Kullback-Leibler information divergence of the joint probability distribution defined by a belief network with respect to the distribution defined by the multiply approximated network is finite. Furthermore, arc linearity implies the following additive property of the Kullback-Leibler information divergence. Lemma 3.12 Let (G; \Gamma) be a belief network and let Pr be the joint probability distribution defined by B. Let A ' A(G) be a linear subset of arcs in G and let be the multiply approximated belief network after removal of all arcs A as defined in Definition 3.10. Let PrA be the joint probability distribution defined by BA . Then the Kullback-Leibler information divergence I satisfies Proof. First, we prove that Pr PrA . Assume that the arcs in the linear set A are ordered according to the relation OE G as defined in Definition 3.9, i.e. for all pairs of arcs V r i we have that i ! j. From Lemma 3.5 we find that Pr Pr Vr 1 (Pr Vr 1 , . , is transitive, we conclude that Pr PrA by application of Lemma 3.11. Now, with this observation we find is linear, we have for each arc new probability assessment function fl 0 for each This leads to log Vs that linearity of a set of arcs to be removed is a sufficient condition for the property stated above, yet not a necessary one. From these observations, we have that the information inequality provides a finite upper bound on the error introduced in the prior and posterior distributions of an approximated belief network after simultaneous removal of a linear set of arcs. This bound is obtained by summing the information divergences between the joint probability distribution defined by the network and the approximated distribution after removal of each arc individually from the set of arcs. Example 1 Consider the belief network (G; \Gamma) where G is the digraph depicted in Figure 2. Figure 2: Information divergence for each arc in the digraph of an example belief network. A Table 1: Information inequality and absolute divergence of an approximated example belief network. The set \Gamma consists of the probability assessment functions fl For each arc V r digraph G, the information divergence I(Pr; Pr Vr6!Vs ) between the joint probability distribution Pr defined by B and the joint probability distribution Pr Vr6!Vs defined by the approximated network B Vr6!Vs after removal of V r computed and depicted next to each arc in Figure 2. Note that despite the presence of arc are conditionally independent given variable V 7 from the fact that fl V9 (V 9 this graphically portrayed dependence can be rendered redundant and arc can be removed without introducing an error in the probability distribution since I(Pr; Pr V86!V9 as shown in Figure 2. Table 1 gives the upper bound provided by the information inequality and the absolute divergence of the approximated joint probability distributions after removal of various linear subsets of arcs A from the network's digraph. The table is compressed by leaving out all linear sets containing arc for the set fV 8 because the second and third column are both unchanged after leaving out this arc. Note that any subset of arcs containing both arcs 7 is not linear. From this example, it can be concluded that the upper bound provided by the information inequality exceeds the absolute divergence by a factor of 2 to 3. Furthermore, note that some arcs have more weight in the value of the absolute divergence. For example, the absolute divergence for all sets containing arc Approximation Schemes In this section we will present static and dynamic approximation schemes for belief networks. These schemes are based on the observations made in the previous section. 4.1 A Static Approximation Scheme Clearly, arcs that significantly reduce the computational complexity of inference on a belief network upon removal are most desirable to remove. However, the error introduced upon removal may not be too large. For each arc, the error introduced upon removal of the arc is expressed in terms of the Kullback-Leibler information divergence. Efficiently Computating the Information Divergence for each Arc Unfortunately, straightforward computation of the Kullback-Leibler information divergence is computationally far too expensive as it requires summing over all configurations of the entire set of variables, an operation in the order of O(2 jV (G)j ). However, the following property of the Kullback-Leibler information divergence can be exploited to compute the information divergence locally. Lemma 4.1 Let V be a set of statistical variables and let X, Y , Z ' V be mutually disjoint subsets of V . Let Pr and Pr 0 be joint probability distributions on V such that Pr 0 (C the Kullback-Leibler information divergence I satisfies Proof. By exploiting the factorization of Pr 0 in terms of Pr we find that Pr Pr 0 . Using Definition 2.7 we derive log Pr(c X[Y [Z ) Now, since yields that c X[Y[Z c X[Y[Z efficiently computing the Kullback-Leibler information divergence I(Pr; Pr Vr6!Vs ) for each of a linear subset of arcs A of the digraph of a belief network, it suffices to sum over all configurations of the arc block fi G only, which amounts to computing the quantity c G (Vs )[fVsg log which is derived by application of the chain rule from probability theory. Hence, the computation of the information divergence I(Pr; Pr Vr6!Vs ) only requires the probabilities Pr(C G (Vs ) ), to be computed from the original belief net- work. In fact, the latter two sets of probabilities can simply be computed from the former set of probabilities using marginalization: and these conditional probabilities are further used to compute Furthermore, once the probabilities Pr(C G (Vs ) are known, the divergence I(Pr; Pr Vr6!Vs ) for that share the same head vertex V s can be computed simultaneously since these computations only require the probabilities Pr(C G (Vs ) ). Selecting a Set of Arcs for Removal For selecting an optimal set of linear arcs for removal one should carefully weight the advantage of the reduction in computational complexity in inference on a belief network and the disadvantage of the error introduced in the represented joint probability distribution after removal of the arcs. Given a linear subset of arcs A from the digraph of a belief network B, we define the function expressing the exact reduction in computational complexity of inference on network B as where K is a cost function expressing the computational complexity of inference on a net- work. Furthermore, we define the exact divergence function d given arcs A on the probability distribution Pr defined by network B as the absolute divergence Note that function K depends on the algorithms used for probabilistic inference. For example, if the clique-tree propagation algorithm of Lauritzen and Spiegelhalter is employed, K(B) expresses the sum of the number of configurations of the sets of variables of the cliques of the decomposable graph rendered upon moralization and subsequent triangulation of the digraph. Then, K(BA ) expresses this complexity in terms of the approximated network B after removal of arcs A. Here, we assume an optimal triangulation of the moral graphs of B and BA , since a bad triangulation of the moral graph of BA may even yield a negative value for c(B; A). If Pearl's polytree algorithm with cutset conditioning is employed, K(B) equals the number of configurations of the set of variables of the loop cutset of the digraph. Now, an optimal selection method weights the advantage expressed by c(B; disadvantage expressed by removal of a set of arcs A from network B. Unfortunately, an optimal selection scheme will first of all depend heavily on the algorithms used for probabilistic inference and, secondly, will depend on the purpose of the network within a specific application. Furthermore, it is rather expensive from a computational point of view to evaluate the exact measures c and d for all possible linear subsets of arcs. In general, the employment of heuristic measures for the selection of a near optimal set of arcs for removal will suffice. To avoid costly evaluations for all possible subsets of arcs, the heuristic measures should be based on combining the local advantages (or disadvantages) of removing each arc individually. Such heuristic functions ~ c and " d for respectively c and d, expressing the impact on the computational complexity and error introduced by removing an arc may be defined with various degrees of sophistication. In fact, the Kullback-Leibler information divergence measures how well one joint probability distribution can be approximated by another exhibiting a simpler dependence structure [22, 13]. Hence, instead of computing the absolute divergence, the information inequality can be used: where is the information divergence associated with each arc as described in the previous section. Note that " d now combines the divergence of removing each arc separately and independently. For defining a heuristic function ~ c valuing the reduction in computational complexity of inference with exact methods for probabilistic inference upon removal of a set of arcs from a belief network, the following scheme can be employed. The complexity of methods for exacts inference depends to a large extend on the connectivity of the digraph of a belief network. With each arc in the digraph G, a set of loops (undirected cycles), denoted as loopset loopset of an arc consists of all loops in the digraph containing the arc; a loopset of an arc provides local information on the role of the arc in the connectivity of the digraph. This set can be found by a depth-first search for all chains from in the graph, backtracking for all possibilities and storing the set of vertices found along each chain in the form of bit-vector. Now, we define the heuristic function ~ c as ~ c(B; loopset i.e. ~ c expresses the number of distinct loops that are broken by removal of a set of arcs from the digraph plus a fraction ff 2 (0; 1] of the the total number of arcs rendered superfluous. The optimal value for ff depends on the algorithm used for exact probabilistic inference. Now, a combined measure reflecting the trade-off between the advantage ~ c and disadvantage d of arc removal may have the form as suggested by Kjaerulff [13] where is chosen such that ~ c(B; A) is comparable to " Function w expresses the desirability of removing a set of arcs from a belief network. Now suppose that a maximum absolute error " ? 0 is allowed in probabilistic inference on a multiply approximated belief network and further suppose that the probability of the evidence to be processed is never smaller than some constant . Observe that from Lemma 3.8 a set of arcs A can be safely removed from the network if 1I(Pr; PrA )=" 2 . Hence, an optimal set of arcs can be found for removal if we solve the following optimization problem: maximize w(B;A) for A ' A(G) subject to " and A is linear. Note that the constraint ensures that the error in the prior and posterior probability distribution never exceeds ". This optimization problem can be solved by employing a simulated annealing technique [12], or by using an evolutionary algorithm [17], to find a linear set of arcs for removal that is nearly optimal. A 'real' optimal solution is not appropriate to search for, since only heuristic functions are involved in the search process. Example 2 Consider once more the belief network from Example 1. Suppose that the probability of evidence to be processed by the approximated belief network does not exceed and further suppose that the maximum absolute error allowed for the (conditional) probabilities to be inferred from the approximated network is First, three loops in G can be identified: loop 1 constitutes vertices g. Thus, the loopset of arc and the loopset of arc c. The following table is obtained for " A ~ c(B; The linear set is the most desirable set of arcs for removal (w(B; 4:9547). Note that after removal, the graph GA is singly connected and, therefore, the network is at least twice as fast for probabilistic inference compared to the original network using either Pearl's polytree algorithm with cutset conditioning or the method of clique-tree propagation. in probability Probability of evidence Observed Upper bound Figure 3: Posterior error in probabilities inferred from an approximated example belief network. Actually, the probability of evidence that can be processed with the approximated net-work such that the error in inferred probabilities is bounded by " requires that Pr(c Y In Figure 3 we show the observed maximum absolute error and upper bound obtained for all evidence c Y , Y ' V (G), with Pr(c Y ) 0:205. 3 Efficiently Computing an Approximation of a Belief Network Removal of a linear set of arcs from a belief network requires the computation of new set of probability assessment functions that reflect the introduced qualitative conditional independence with a quantitative conditional independence. We recall from Definition 3.1 that we have that the new probability assessment functions removal of an arc V r selected for removal only if the Kullback-Leibler information divergence I(Pr; Pr Vr6!Vs ) is sufficiently small in order that the error introduced by approximating the network after removal of bounded. The probabilities Pr(V are in fact already computed by the computation of the information divergence I(Pr; Pr Vr6!Vs ) for all arcs in the digraph of a belief network. When these probabilities are stored temporarily, it suffices to assign these probabilities to the new probability assessment functions of the head vertex of each arc that is selected for removal. 4.2 A Dynamic Approximation Scheme In this section we will consider belief networks with singly connected digraphs as a special case for approximation. A singly connected digraph exhibits no loops, that is, at most one chain exists between any two vertices in the digraph. For these networks, arcs can be removed dynamically while evidence is being processed in contrast to a static removal of arcs as a preprocessing phase before inference as described in the previous section. Therefore, the computational complexity of processing evidence can be reduced depending on the evidence itself and no estimate for a lower bound for the probability of the evidence has to be provided in advance. A detailed description and analysis of the method is beyond the scope of this paper. However, a practical outline of the scheme will be presented which is based on Pearl's polytree algorithm. First, we will show that all variables in the network retain their prior probabilities upon removal of an arc. Lemma 4.2 Let (G; \Gamma) be a belief network with a singly connected digraph G. Let Pr be the joint probability distribution defined by B. Furthermore, let be an arc in G and let B be the approximated belief network after removal of defined in Definition 3.1. Let Pr Vr6!Vs be the joint probability distribution defined by B Vr6!Vs . Then, Pr Vr6!Vs Proof. Assume that the vertices of the singly connected digraph are indexed in ascending topological order, i.e. for each pair of vertices directed path from V i to in G we have that i ! j. The proof is by induction on the index i of variable V i . Base case s: from Lemma 3.6 we have that Pr Vr6!Vs For i s, we apply the chain rule and the principle of marginalization to obtain Pr Vr6!Vs c G Vr 6!Vs c G Vr 6!Vs where singly connected, all variables are mutually independent by the d-separation criterion. Hence, we have that Pr Vr6!Vs (c G Vr 6!Vs ). By the assumption that the vertices in G are ordered in ascending topological order, for each by the induction hypothesis assume that for each by applying the principle of induction, we find Pr Vr6!Vs c G Vr 6!Vs Y c G Vr 6!Vs Y consider an in a singly connected digraph. In a singly connected digraph no other chain exists from V r to V s except for the chain constituting the arc V r Therefore, G Vr 6!Vs holds on the singly connected digraph G Vr6!Vs for any subset of variables Y ' V (G). From this observation, we have that the independence relationship between the variables V r and V s given G Vr 6!Vs (V s ) remain unchanged after evidence is given for any subset of variables. Informally speaking, this means that after evidence is processed in a belief network, we can compute the Kullback-Leibler information divergence between the posterior probability distribution defined by a belief network and the posterior distribution of the approximated network after removal of an arc locally. Then, by a similar exposition for the properties of the Kullback-Leibler information divergence applied on general belief networks for multiple arc removals as presented in the previous sections, it can be shown that for belief network consisting of a singly connected digraph, where Pr is the joint probability distribution defined by the network and PrA is the joint probability distribution defined by the multiple approximated network after removal of all arcs A. We note that the computation of the divergence is as expensive on the computational resources as the computation of the causal and diagnostic messages for vertex V s in Pearl's polytree algorithm assuming that logarithms require one time unit. Furthermore, in fact, by using Pearl's polytree algorithm, arcs do not have to be physically removed, the blocking of causal and diagnostic messages for updating the probability distribution will suffice. With this observation, we envisage an approximate wave-front version of the polytree algorithm where the sending of messages is blocked between two connected vertices in the graph if the probabilistic dependency relationship between the vertices is very weak. That is, we block all messages for which the information divergence per blocked arc is small such that the total sum of the information divergences over all blocked arcs does not exceed some predetermined constant for the maximum absolute error allowed in probabilistic 5 Discussion and Related Work We have presented a scheme for approximating Bayesian belief networks based on model simplification through arc removal. In this section we will compare the proposed method with other methods for belief network approximation. Existing belief network approximation methods are annihilating small probabilities from belief universes [8], and removal of weak dependencies from belief universes [13]. Both methods have proven to be very successful in reducing the complexity of inference on a belief network on real-life applications using the Bayesian belief universe approach [9]. The method of annihilating small probabilities by Jensen and Andersen reduces the computational effort of probabilistic inference when the method of clique-tree propagation is used for probabilistic inference. The basic idea of the method is to eliminate configurations with small probabilities from belief universes, accepting a small error in the probabilities inferred from the network. To this end, the k smallest probability configurations are selected for each belief universe where k is chosen such that the sum of the probabilities of the selected configurations in the universe is less than some predetermined constant ". The constant " determines the maximum error of the approximated prior probabilities. The belief universes are then compressed to take advantage of the zeros introduced. Jensen and Andersen further point out that if the range of probabilities of evidence is known in advance, the method can be applied to approximate a belief network such that the error of the approximated posterior probabilities computed from the network are bounded by some predetermined constant. Similar to the method of annihilating small probabilities, the method of removal of weak dependencies by Kjaerulff reduces the computational effort of probabilistic inference when the method of clique-tree propagation is used. Kjaerulff's approximation method and the method of annihilation are complementary techniques [13]. The basic idea of the method is to remove edges from the chordal graph constructed from the digraph of a belief network that model weak dependencies. The weaker the dependencies, the smaller the error introduced in the represented joint probability distribution approximated upon removal of an edge. The method operates on the junction tree of a belief network only. Given a constant ", a set of edges can be removed sequentially such that the error introduced in the prior distribution is smaller than ". Removal of an edge results in the decomposition of the clique containing the edge into two or more smaller cliques which results in a simplification of the junction tree thereby reducing the computational complexity of inference on the network. In comparing the methods for approximating belief networks, we first of all find that the method of annihilating small probabilities from belief universes introduces an error that is inversely proportional to the probability of the evidence [8] while the methods based on removing arcs introduces an error that is inversely proportional to the square root of the probability of the evidence. Furthermore, since the original joint probability distribution is absolutely continuous with respect to the approximated probability distribution, the processing of evidence in an approximated belief network by our method is safe in the sense that no undefined conditional probabilities will arise for evidence with a nonzero probability in the original distribution; the evidence that can be processed in an approximated belief network is a superset of the evidence that can be processed in the original network. Once more, this is in contrast to the method of annihilating small probabilities from belief universes. On the other hand, however, the advantage of annihilating small probabilities is that the method operates on the quantitative part of a belief network only whereas arc removal methods change the qualitative representation as well. This can be remedied by introducing virtual arcs to replace removed arcs. Virtual arcs are not used in probabilistic inference. The method presented in this paper has some similarities to Kjaerulff's method of removal of weak dependencies from belief universes [13]. Both methods aim at reducing inference on a belief network by removing arcs or edges. However, the independency statements we enforce are of the in contrast to V r ??V s j C n fV r by Kjaerulff's method where C ' V (G) denotes the clique containing the edge removed by Kjaerulff's method. Furthermore, Kjaerulff's method of removal is based on the clique-tree propagation algorithm only and restricts the removal to one edge from a clique at a time in order that the error introduced is bounded by some predetermined constant. In contrast, our method allows a larger set of arcs (edges) to be removed in parallel, still guaranteeing that the introduced error to be bounded by some predetermined constant regardless of the algorithms for probabilistic inference used. To summarize the conclusions, the scheme we propose for approximating belief networks operates directly on the digraph of a belief network, has a relatively low computational com- plexity, provides a bound on the posterior error in the presence of evidence, and is independent of the algorithms used for probabilistic inference. Acknowledgements The author would like to acknowledge valuable discussions with Linda van der Gaag of Utrecht University, The Netherlands. --R Index Expression Belief Networks for Information Disclosure The Computational Complexity of Probabilistic Inference using Bayesian Belief Networks A Tutorial to Stochastic Simulation Algorithms for Belief Networks Approximating Probabilistic Inference in Bayesian Belief Networks is NP-hard Propagating uncertainty in Bayesian networks by probabilistic logic sam- pling Approximations in Bayesian Belief Universes for Knowledge-based Systems Bayesian updating in causal probabilistic networks by local computations Use of Causal Probabilistic Networks as High LEvel Models in Computer Vision Journal of the Australian Mathematical Society A Reduction of Computational Complexity in Bayesian Networks through Removal of Weak Dependencies Information Theory and Statistics A lower bound for discriminating information in terms of variation Local computations with probabilities on graphical structures and their application to expert systems Genetic Algorithms Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference Probabilistic inference in multiply connected belief networks using loop cutsets A Combination of Exact Algorithms for Inference on Bayesian Belief Networks On Substantive Research Hypothesis Graphical Models in Applied Multivariate Statistics --TR --CTR Helge Langseth , Olav Bangs, Parameter Learning in Object-Oriented Bayesian Networks, Annals of Mathematics and Artificial Intelligence, v.32 n.1-4, p.221-243, August 2001 Marek J. Druzdzel , Linda C. van der Gaag, Building Probabilistic Networks: 'Where Do the Numbers Come From?' Guest Editors' Introduction, IEEE Transactions on Knowledge and Data Engineering, v.12 n.4, p.481-486, July 2000 Helge Langseth , Thomas D. Nielsen, Fusion of domain knowledge with data for structural learning in object oriented domains, The Journal of Machine Learning Research, 4, 12/1/2003 Magnus Ekdahl , Timo Koski, Bounds for the Loss in Probability of Correct Classification Under Model Based Approximation, The Journal of Machine Learning Research, 7, p.2449-2480, 12/1/2006 Rina Dechter , Irina Rish, Mini-buckets: A general scheme for bounded inference, Journal of the ACM (JACM), v.50 n.2, p.107-153, March Russell Greiner , Christian Darken , N. Iwan Santoso, Efficient reasoning, ACM Computing Surveys (CSUR), v.33 n.1, p.1-30, March 2001

belief network approximation;approximate probabilistic inference;information theory;model simplification;bayesian belief networks

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Closure properties of constraints.

Many combinatorial search problems can be expressed as constraint satisfaction problems and this class of problems is known to be NP-complete in general. In this paper, we investigate the subclasses that arise from restricting the possible constraint types. We first show that any set of constraints that does not give rise to an NP-complete class of problems must satisfy a certain type of algebraic closure condition. We then investigate all the different possible forms of this algebraic closure property, and establish which of these are sufficient to ensure tractability. As examples, we show that all known classes of tractable constraints over finite domains can be characterized by such an algebraic closure property. Finally, we describe a simple computational procedure that can be used to determine the closure properties of a given set of constraints. This procedure involves solving a particular constraint satisfaction problem, which we call an indicator problem.

Introduction Solving a constraint satisfaction problem is known to be NP-complete [20]. However, many of the problems which arise in practice have special properties which allow them to be solved efficiently. The question of identifying restrictions to the general problem which are sufficient to ensure tractability is important from both a practical and a theoretical viewpoint, and has been extensively studied. Such restrictions may either involve the structure of the constraints, in other words, which variables may be constrained by which other variables, or they may involve the Earlier versions of parts of this paper were presented at the International Conferences on Constraint Programming in 1995 and 1996. (See references [14] and [15].) nature of the constraints, in other words, which combinations of values are allowed for variables which are mutually constrained. Examples of the first approach can be found in [7, 9, 12, 22, 23] and examples of the second approach can be found in [4, 14, 16, 8, 17, 22, 28, 29]. In this paper we take the second approach, and investigate those classes of constraints which only give rise to tractable problems whatever way they are combined. A number of distinct classes of constraints with this property have previously been identified and shown to be maximal [4, 14, 16]. In this paper we establish that any class of constraints which does not give rise to NP-complete problems must satisfy a certain algebraic closure condition, and hence this algebraic property is a necessary condition for a class of constraints to be tractable (assuming that P is not equal to NP). We also show that many forms of this algebraic closure property are sufficient to ensure tractability. As an example of the wide applicability of these results, we show that all known examples of tractable constraint classes over finite domains can be characterized by an algebraic condition of this kind, even though some of them were originally defined in very different ways. Finally, we describe a simple computational procedure to determine the algebraic closure properties of a given set of constraints. The test involves calculating the solutions to a fixed constraint satisfaction problem involving constraints from the given set. The work described in this paper represents a generalization of earlier results concerning tractable subproblems of the Generalized Satisfiability problem. Schaefer [26] identified all possible tractable classes of constraints for this problem, which corresponds to the special case of the constraint satisfaction problem in which the variables are Boolean. The tractable classes described in [26] are special cases of the tractable classes of general constraints described below, and they are given as examples. A number of tractable constraint classes have also been identified by Feder and Vardi in [8]. They define a notion of 'width' for constraint satisfaction problems in terms of the logic programming language Datalog, and show that problems with bounded width are solvable in polynomial time. It is stated in [8] that the problem of determining whether a fixed collection of constraints gives rise to problems of bounded width is not known to be decidable. However, it is shown that a more restricted property, called 'bounded strict width', is decidable, and in fact corresponds to an algebraic closure property of the form described here. Other tractable constraint classes are also shown to be characterised by a closure property of this type. This paper builds on the work of Feder and Vardi by examining the general question of the link between algebraic closure properties and tractability of constraints, and establishing necessary and sufficient conditions for these closure properties. A different approach to identifying tractable constraints is taken in [28], where it is shown that a property of constraints, referred to as 'row-convexity', together with path-consistency, is sufficient to ensure tractability in binary constraint satisfaction prob- lems. It should be noted, however, that because of the additional requirement for path- consistency, row-convex constraints do not constitute a tractable class in the sense defined in this paper. In fact, the class of problems which contain only row-convex constraints is NP-complete [4]. The paper is organised as follows. In Section 2 we give the basic definitions, and in Section 3 we define what we mean by an algebraic closure property for a set of relations and examine the possible forms of such a closure property. In Section 4 we identify which of these forms are necessary conditions for tractability, and in Section 5 we identify which of them are sufficient for tractability. In Section 6 we describe a computational method to determine the closure properties satisfied by a set of relations. Finally, we summarise the results presented and draw some conclusions. 2.1 The constraint satisfaction problem Notation 2.1 For any set D, and any natural number n, we denote the set of all n-tuples of elements of D by D n . For any tuple t 2 D n , and any i in the range 1 to n, we denote the value in the ith coordinate position of t by t[i]. The tuple t may then be written in the form ht[1]; A subset of D n is called an n-ary relation over D. We now define the (finite) constraint satisfaction problem which has been widely studied in the Artificial Intelligence community [18, 20, 22]. Definition 2.2 An instance of a constraint satisfaction problem consists of ffl a finite set of variables, ffl a finite domain of values, D; ffl a set of constraints fC g. Each constraint C i is a pair (S i list of variables of length m i , called the constraint scope, and R i is an m i -ary relation over D, called the constraint relation. (The tuples of R i indicate the allowed combinations of simultaneous values for the variables in S i .) The length of the tuples in the constraint relation of a given constraint will be called the arity of that constraint. In particular, unary constraints specify the allowed values for a single variable, and binary constraints specify the allowed combinations of values for a pair of variables. A solution to a constraint satisfaction problem is a function from the variables to the domain such that the image of each constraint scope is an element of the corresponding constraint relation. Deciding whether or not a given problem instance has a solution is NP-complete in general [20] even when the constraints are restricted to binary constraints. In this paper we shall consider how restricting the allowed constraint relations to some fixed subset of all the possible relations affects the complexity of this decision problem. We therefore make the following definition. Definition 2.3 For any set of relations, \Gamma, CSP(\Gamma) is defined to be the decision problem with INSTANCE: An instance, P , of a constraint satisfaction problem, in which all constraint relations are elements of \Gamma. Does P have a solution? If there exists an algorithm which solves every problem instance in CSP(\Gamma) in polynomial time, then we shall say that CSP(\Gamma) is a tractable problem, and \Gamma is a tractable set of relations. Example 2.4 The binary disequality relation over a set D, denoted 6= D , is defined as Note that CSP(f6=D g) corresponds precisely to the Graph jDj-Colorability problem [11]. This problem is tractable when jDj - 2 and NP-complete when jDj - 3.Example 2.5 Consider the ternary relation ffi over the set which is defined by The problem CSP(fffig) corresponds precisely to the Not-All-Equal Satisfiability problem [11], which is NP-complete [26]. 2 Example 2.6 We now describe three relations which will be used as examples of constraint relations throughout the paper. Each of these relations is a set of tuples of elements from the domain as defined below: The problem CSP(fR 1 consists of all constraint satisfaction problem instances in which the constraint relations are all chosen from the set fR 1 g. The complexity of CSP(\Gamma) for arbitrary subsets \Gamma of fR 1 will be determined using the techniques developed later in this paper (see Example 6.6). 2 2.2 Operations on relations In Section 4 we shall examine conditions on a set of relations \Gamma which allow known NP-complete problems to be reduced to CSP(\Gamma). The reductions will be described using standard operations from relational algebra [1], which are described in this section. Definition 2.7 We define the following operations on relations. ffl Let R be an n-ary relation over a domain D and let S be an m-ary relation over D. The Cartesian product R \Theta S is defined to be the (n +m)-ary relation R \Theta ffl Let R be an n-ary relation over a domain D. Let 1 - n. The equality selection oe i=j (R) is defined to be the n-ary relation oe ffl Let R be an n-ary relation over a domain D. Let i be a subsequence of n. The projection - (R) is defined to be the m-ary relation It is well-known that the combined effect of two constraints in a constraint satisfaction problem can be obtained by performing a relational join operation [1] on the two constraints [12]. The next result is a simple consequence of the definition of the relational join operation. Lemma 2.8 Any relational join of relations R and S can be calculated by performing a sequence of Cartesian product, equality selection, and projection operations on R and S. In view of this result, it will be convenient to use the following notation. Notation 2.9 The set of all relations which can be obtained from a given set of relations, \Gamma, using some sequence of Cartesian product, equality selection, and projection operations will be denoted Note contains exactly those relations which can be obtained as 'derived' relations in a constraint satisfaction problem instance with constraint relations chosen from \Gamma [2]. 3 Closure operations We shall establish below that significant information about CSP(\Gamma) can be determined from algebraic properties of the set of relations \Gamma. In order to describe these algebraic properties we need to consider arbitrary operations on D, in other words, arbitrary functions from D k to D, for arbitrary values of k. For the results below we shall be particularly interested in certain special kinds of operations. We therefore make the following definition: Definition 3.1 Let\Omega be a k-ary operation from D k to D. If\Omega is such that, for all d 2 then\Omega is said to be idempotent. ffl If there exists an index i 2 kg such that for all hd 1 is a non-constant unary operation on D, then\Omega is called essentially unary. (Note that f is required to be non-constant, so constant operations are not essentially unary.) If f is the identity operation, then\Omega is called a projection. ffl If k - 3 and there exists an index i 2 kg such that for all d 1 with but\Omega is not a projection, then\Omega is called a semiprojection [25, 27]. then\Omega is called a majority operation. are binary operations on D such that hD; +; \Gammai is an Abelian group [27], is called an affine operation. Any operation on D can be extended to an operation on tuples over D by applying the operation in each coordinate position separately (i.e., pointwise). Hence, any operation defined on the domain of a relation can be used to define an operation on the tuples in that relation, as follows: Definition 3.2 D be a k-ary operation on D and let R be an n-ary relation over D. For any collection of k tuples, t necessarily all distinct) the defined as follows: Finally, we define\Omega (R) to be the n-ary relation Using this definition, we now define the following closure property of relations. Definition 3.3 Let\Omega be a k-ary operation on D, and let R be an n-ary relation over D. The relation R is closed under\Omega if\Omega (R) ' R. Example 3.4 Let 4 be the ternary majority operation defined as follows: x otherwise. The relation R 2 defined in Example 2.6 is closed under 4, since applying the 4 operation to any 3 elements of R 2 yields an element of R 2 . For example, The relation R 3 defined in Example 2.6 is not closed under 4, since applying the 4 operation to the last 3 elements of R 3 yields a tuple which is not an element of R 3 For any set of relations \Gamma, and any operation\Omega , if every R 2 \Gamma is closed under\Omega , then we shall say that \Gamma is closed under\Omega . The next lemma indicates that the property of being closed under some operation is preserved by all possible projection, equality selection, and product operations on relations, as defined in Section 2.2. Lemma 3.5 For any set of relations \Gamma, and any closed under\Omega , closed under\Omega . Proof: Follows immediately from the definitions. Notation 3.6 For any set of relations \Gamma with domain D, the set of all operations on D under which \Gamma is closed will be denoted \Gamma . The set of closure operations, \Gamma , can be used to obtain a great deal of information about the problem CSP(\Gamma), as we shall demonstrate in the next two Sections. As a first example of this, we shall show that the operations in \Gamma can be used obtain a reduction from one problem to another. Proposition 3.7 For any set of finite relations \Gamma, and there is a polynomial-time reduction from CSP(\Gamma) to CSP(\Omega (\Gamma)), \Gammag. Proof: Let P be any problem instance in CSP(\Gamma) and consider the instance P 0 obtained by replacing each constraint relation R i of P by the relation\Omega (R i ). It is clear that P 0 can be obtained from P in polynomial-time. It follows from Definition 3.3 that P 0 has a solution if and only if P has a solution. It follows from this result that if \Gamma contains a non-injective unary operation, then CSP(\Gamma) can be reduced to a problem over a smaller domain. One way to view this is that the presence of a non-injective unary operation in \Gamma indicates that constraints with relations chosen from \Gamma allow a form of global 'substitutability', similar to the notion defined by Freuder in [10]. does not contain any non-injective unary operations, then we shall say that \Gamma is reduced. The next theorem uses a general result from universal algebra [25, 27] to show that the possible choices for \Gamma are quite limited. Theorem 3.8 For any reduced set of relations \Gamma, on a finite set, either contains essentially unary operations only, or 2. \Gamma contains an operation which is (a) a constant operation; or (b) a majority operation; or (c) an idempotent binary operation (which is not a projection); or (d) an affine operation; or (e) a semiprojection. Proof: The set of operations \Gamma contains all projections and is closed under composition, hence it constitutes a 'clone' [3, 27]. It was shown in [25] that any non-trivial clone on a finite set must contain a minimal clone, and that any minimal clone contains either 1. a non-identity unary operation; or 2. a constant operation; or 3. a majority operation; or 4. an idempotent binary operation (which is not a projection); or 5. an affine operation; or 6. a semiprojection. Furthermore, if \Gamma is reduced, and \Gamma contains any operations which are not essentially unary, then it is straightforward to show, by considering such an operation of the smallest possible arity, that \Gamma contains an operation in one of the last five of these classes [27, 19]. In the next two Sections we shall examine each of these possibilities in turn, in order to establish what can be said about the complexity of CSP(\Gamma) in the various cases. 4 A necessary condition for tractability In this Section we will show that any set of relations which is only closed under essentially unary operations will give rise to a class of constraint satisfaction problems which is NP-complete Theorem 4.1 For any finite set of relations, \Gamma, over a finite set D, if \Gamma contains essentially unary operations only then CSP(\Gamma) is NP-complete. Proof: When jDj - 2, then we may assume without loss of generality that D ' f0; 1g, where 0 corresponds to the Boolean value false and 1 corresponds to the Boolean value true. It follows that the problem CSP(\Gamma) corresponds to the Generalised Satisfiability problem over the set of Boolean relations \Gamma, as defined in [26] (see also [11]). It was established in [26] that this problem is NP-complete unless one of the following conditions holds: 1. Every relation in \Gamma contains the tuple (0; 2. Every relation in \Gamma contains the tuple (1; 3. Every relation in \Gamma is definable by a formula in conjunctive normal form in which each conjunct has at most one negated variable; 4. Every relation in \Gamma is definable by a formula in conjunctive normal form in which each conjunct has at most one unnegated variable; 5. Every relation in \Gamma is definable by a formula in conjunctive normal form in which each conjunct contains at most 2 literals; Every relation in \Gamma is the set of solutions of a system of linear equations over the finite field GF(2). It is straightforward to show that in each of these cases \Gamma is closed under some operation which is not essentially unary (see [13] for details). Hence the result holds when 2. For larger values of jDj we proceed by induction. Assume that jDj - 3 and the result holds for all smaller values of jDj. Let M be an m by n matrix over D in which the columns consist of all possible m-tuples over D (in some order). Let R 0 be the relation consisting of all the tuples occuring as rows of M . The only condition we place on the choice of order for the columns of M is that - 1;2 (R 0 is the binary disequality relation over D, as defined in Example 2.4. We now construct a relation - R 0 which is the 'closest approximation' to R 0 that we can obtain from the relations in \Gamma and the domain D using the Cartesian product, equality selection and projection operations: Since this is a finite intersection, and intersection is a special case of join, we have from Lemma 2.8 that - In other words, the relation - R 0 can be obtained as a derived constraint relation in some problem belonging to CSP(\Gamma). There are now two cases to consider: 1. If there exists some tuple t then we will construct, using an appropriate operation under which \Gamma is closed. Define the is the unique column of M corresponding to the m-tuple hd We will show that \Gamma is closed under\Omega . Choose any R 2 \Gamma, and let p be the arity of R. We are required to show that R is closed under\Omega . Consider any sequence t of tuples of R (not necessarily be the m-tuple ht 1 [i]; t each pair of indices, i; j, such that c apply the equality selection oe i=j to R, to obtain a new relation R 0 . Now choose a maximal set of indices, I g, such that the corresponding c i are all distinct, and construct the relation R Finally, permute the coordinate positions of R 00 (by a sequence of Cartesian product, equality selection, and projection operations), such that R 00 ' R 0 (this is always possible, by the construction of R 0 and R 00 ). Since R 00 we know that t 0 is a tuple of R 00 , by the definition of - R 0 . Hence the appropriate projection of t 0 is an element of R, and R is closed under\Omega . If\Omega is not essentially unary, then we have the result. Otherwise, let f : D ! D be the corresponding unary operation, and set By the choice of t 0 , f cannot be injective, so jf(D)j ! jDj. By the inductive hypothesis, we know that either CSP(f (\Gamma)) is NP-complete (in which case CSP(\Gamma) must also be NP-complete) or else f (\Gamma) is closed under some operation \Phi which is not essentially unary (in which case \Gamma is closed under the operation \Phif , which is also not essentially unary). Hence, the result follows by induction in this case. 2. Alternatively, if - R 0 contains no tuple t such that . But this implies that CSP(f6=D g) is reducible to CSP(\Gamma), since every occurence of the constraint relation 6= D can be replaced with an equivalent collection of constraints with relations chosen from \Gamma. However, it was pointed out in Example 2.4 that CSP(f6=D g) corresponds to the Graph jDj-Colorability problem [11], which is NP-complete when jDj - 3. Hence, this implies that CSP(\Gamma) is NP-complete, and the result holds in this case also. Combining Theorem 4.1 with Theorem 3.8 gives the following necessary condition for tractability. Corollary 4.2 Assuming that P is not equal to NP, any tractable set of reduced relations must be closed under either a constant operation, or a majority operation, or an idempotent binary operation, or an affine operation, or a semiprojection. Note that the arity of a semiprojection is at most jDj, so for any finite set D there are only finitely many operations matching the given criteria, which means that there is a finite procedure to check whether this necessary condition is satisfied (see Corollary 6.5, below). 5 Sufficient conditions for tractability We have shown in the previous section that when \Gamma is a tractable set of relations, then \Gamma must contain an operation from a limited range of types. We now consider each of these possibilities in turn, to determine whether or not they are sufficient to ensure tractability. 5.1 Constant operations Closure under a constant operation is easily shown to be a sufficient condition for tractability. Proposition 5.1 For any set of relations \Gamma, if \Gamma is closed under a constant operation, then CSP(\Gamma) is solvable in polynomial time. Proof: If every relation in \Gamma is closed under some constant operation\Omega , with constant value d, then every non-empty relation in \Gamma must contain the tuple hd; in this case, the decision problem for any constraint satisfaction problem instance P in CSP(\Gamma) is clearly trivial to solve, since P either contains an empty constraint, in which case it does not have a solution, or else P allows the solution in which every variable is assigned the value d. The class of sets of relations closed under some constant operation is a rather trivial tractable class. It is referred to in [14] as Class 0. Example 5.2 Let ? denote the unary operation on the domain which returns the constant value 1. The constraint R 3 defined in Example 2.6 is closed under ?, since applying the ? operation to any element of R 3 yields the tuple h1; 1i, which is an element of R 3 . The constraint R 2 defined in Example 2.6 is not closed under ?, since applying the ? operation to any element of R 2 yields the tuple h1; 1; 1i, which is not an element of R 2 . In fact, R 2 is clearly not closed under any constant operation. 2 Example 5.3 When there are only two possible constant operations on D. The first two tractable subproblems of the Generalised Satisfiability problem identified by Schaefer in [26] correspond to the tractable classes of relations characterised by closure under these two constant operations. 2 5.2 Majority operations We will now show that closure under a majority operation is a sufficient condition for tractability. We first establish that when a relation R is closed under a majority operation, any constraint involving R can be decomposed into binary constraints. Proposition 5.4 Let R be a relation of arity n which is closed under a majority opera- tion, and let C be any constraint constraining the variables in S with relation R. For any problem P with constraint C, the problem P 0 which is obtained by replacing C by the set of constraints ng has exactly the same solutions as P. Proof: It is clear that any solution to P is a solution to P 0 , since P 0 is obtained by taking binary projections of a constraint from P. Now let oe be any solution to We shall prove, by induction on n, that t 2 R, thereby establishing that oe is a solution to P. For 3 the result holds trivially, so assume that n - 3, and that the result holds for all smaller values. Let I = ng be the set of indices of positions in S and choose Proposition 3.5 and the inductive hypothesis, applied to - Infi j g (R), there is some t j 2 R which agrees with t at all positions except i j , for 3. Since R is closed under a majority operation, applying this operation to t Example 5.5 Recall the relation R 2 defined in Example 2.6. It was shown in Example 3.4 that R 2 is closed under the operation 4. Since this operation is a majority operation, we know by Proposition 5.4 that any constraint with relation R 2 can be decomposed into a collection of binary constraints with the following relations: It is, of course, not always the case that a constraint can be replaced by a collection of binary constraints on the same variables. In many cases the binary projections of the constraint relation allow extra solutions, as the following example demonstrates. Example 5.6 Recall the relation ffi on domain defined in Example 2.5. The binary projections of ffi are as follows: The join of these binary projections contains the tuples h0; 0; 0i and h1; 1; 1i, which are not elements of ffi. It clearly follows that ffi cannot be replaced by any set of binary constraints on the same variables. 2 Theorem 5.7 Let \Gamma be any set of relations over a finite domain, D. If \Gamma is closed under a majority operation, then CSP(\Gamma) is solvable in polynomial time. Proof: For any problem instance P in CSP(\Gamma) we can impose strong (jDj consistency [5] in polynomial time to obtain a new instance P 0 with the same solutions. All of the constraints in P 0 are elements so they are closed under a majority operation, by Proposition 3.5. Hence, all of the constraints of P 0 are decomposable into binary constraints by Proposition 5.4. Hence, by Corollary 3.2 of [5], P 0 is solvable in polynomial time. Example 5.8 When there is only one possible majority operation on D, (which is equal to the 4 operation defined in Example 3.4). It is easily shown that all possible binary Boolean relations are closed under 4. Hence, it follows from Proposition 5.4 that the Boolean relations of arbitrary arity which are closed under this majority operation are precisely the relations which are definable by a formula in conjunctive normal form in which each conjunct contains at most 2 literals. Hence, if a set of Boolean relations \Gamma is closed under a majority operation, then CSP(\Gamma) is equivalent to the 2-Satisfiability problem (2-Sat) [24], which is well-known to be a tractable subproblem of the Satisfiability problem [26]. 2 Recall the class of tractable constraints identified independently in [4] and [17], and referred to as 0/1/all constraints or implicational constraints. (This class of tractable constraints is referred to as Class I in [14].) It was shown in [14] that these constraints are in fact precisely the relations closed under the majority operation 4 defined in Example 3.4. This result is rather unexpected, in view of the fact that 0/1/all constraints were originally defined purely in terms of their syntactic structure [4]. However, we remark here that the class of tractable sets of relations defined by closure under some majority operation is a true generalization of the class containing all sets of 0/1/all constraints. In other words, there exist tractable sets of relations which are closed under some majority operation but are not closed under the 4 operation, as the following example demonstrates. Example 5.9 Let - be the ternary majority operation on which returns the median value of its three arguments (in the standard ordering of D). Recall the relation R 3 defined in Example 2.6. It is easy to show that R 3 is closed under -, since applying the - operation to any 3 elements of R 3 yields an element of R 3 . For example, Hence, by Theorem 5.7, CSP(fR 3 g) is tractable. However, it was shown in Example 3.4 that R 3 is not closed under 4, and hence R 3 is not a 0/1/all constraint. 2 5.3 Binary operations We first show that closure under an arbitrary idempotent binary operation is not in general sufficient to ensure tractability. Lemma 5.10 There exists a set of relations \Gamma closed under an idempotent binary operation (which is not a projection) such that CSP(\Gamma) is NP-complete. Proof: Consider the binary operation 2 on the set which is defined by the following table: This operation is idempotent but it is not a projection (in fact, it is an example of a form of binary operation known as a 'rectangular band' [21].) Now consider the functions which return the first and second bit in the binary expression for the numerical value of each element of D. Using these functions, we define ternary relations R 1 and R 2 over D, as follows: Finally, we define R It is easily shown that R is closed under 2, since applying the operation 2 to any 2 elements of R yields an element of R. However, it can also be shown that the Not-All-Equal Satisfiability problem [26], which is known to be NP-complete, is reducible in polynomial time to CSP(fRg). Hence, CSP(fRg) is NP-complete, and the result follows. We now describe some additional conditions which may be imposed on binary operations. It will be shown below that closure under any binary operation satisfying these additional conditions is a sufficient condition for tractability. D be an idempotent binary operation on the set D such that, for all d Then u is said to be an ACI operation. We will make use of the following result about ACI operations, which is well-known from elementary algebra [3, 21]. Lemma 5.12 Let u be an ACI operation on the set D. The binary relation R on D defined by is a partial order on D in which any two elements have a least upper bound given by u(d It follows from Lemma 5.12 that any (finite) non-empty set D 0 ' D which is u-closed contains a least upper bound with respect to the partial order R. This upper bound will be denoted u(D 0 ). Using Lemma 5.12, we now show that relations which are closed under some arbitrary ACI operation form a tractable class. Theorem 5.13 For any set of relations \Gamma over a finite domain D, if \Gamma is closed under some ACI operation, then CSP(\Gamma) is solvable in polynomial time. be a set of relations closed under the ACI operation u, and let P be any problem instance in CSP(\Gamma). First enforce pairwise consistency to obtain a new instance with the same set of solutions which is pairwise consistent. Such a P 0 can be obtained by forming the join of every pair of constraints in P, replacing these constraints with the (possibly smaller) constraints obtained by projecting down to the original scopes, and then repeating this process until there are no further changes in the constraints. The time complexity of this procedure is polynomial in the size of P, and the resulting P 0 is a member of CSP(\Gamma all the constraint relations in P 0 are closed under u, by Proposition 3.5. Now let D(v) denote the set of values allowed for variable v by the constraints of P 0 . Since D(v) equals the projection of some u-closed constraint onto v, it must be u-closed, by Proposition 3.5. There are two cases to consider: 1. If any of the sets D(v) is empty then P 0 has no solutions, so the decision problem is trivial. 2. On the other hand, if all of these sets are non-empty, then we claim that assigning the value u(D(v)) to each variable v gives a solution to P 0 , so the decision problem is again trivial. To establish this claim, consider any constraint with relation R of arity n, and scope S. For each there must be some tuple t i 2 R such that t i by the definition of D(S[i]). Now consider the tuple We know that t 2 R, since R is closed under u. Furthermore, for each i, because u(D(S[i])) is an upper bound of D(S[i]), so for all d 2 D(S[i]). Hence the constraint C allows the assignment of u(D(v)) to each variable v in S. Since C was arbitrary, we have shown that this assignment is a solution to P 0 , and hence a solution to P. Example 5.14 When there are only two idempotent binary operations on D (which are not projections), corresponding to the logical AND operation and the logical OR operation. These two operations are both ACI operations, and they correspond to the two possible orderings of D. It is well-known [6, 16, 24] that a Boolean relation is closed under AND if and only if it can be defined by a Horn sentence (that is, a conjunction of clauses each of which contains at most one unnegated literal). Hence, if a set of Boolean relations \Gamma is closed under AND, then CSP(\Gamma) is equivalent to the Horn clause satisfiability problem, Hornsat [24], which is a tractable subproblem of the Satisfiability problem [26]. Similarly, a Boolean relation is closed under OR if and only if it can be defined by a conjunction of clauses each of which contains at most one negated literal, and this class or relations also gives rise to a tractable subproblem of the Satisfiability problem [26].Example 5.15 Let D be a finite subset of the natural numbers. The operation MAX : which returns the larger of any pair of numbers is an ACI operation. The following types of arithmetic constraints (amongst many others) are closed under this operation: where upper-case letters represent variables and lower-case letters represent positive con- stants. Hence, by Theorem 5.13 it is possible to determine efficiently whether any collection of constraints of these types has a solution. These constraints include (and extend) the 'basic' arithmetic constraints allowed by the well-known constraint programming language, CHIP [29]. 2 The class of tractable constraints first identified in [16], and referred to as max-closed constraints, are in fact relations closed under an u operation with the additional property that the partial order R, defined in Lemma 5.12, is a total ordering of D. Hence, a set of constraints is max-closed if and only if the constraint relations are closed under some specialized ACI operation of this kind (see, for example, Example 5.15). This class of tractable constraints is referred to as Class II in [14]. However, we remark here that the class of tractable sets of relations defined by closure under some ACI operation is a true generalization of the class containing all sets of max- closed constraints. In other words, there exist tractable relations which are closed under some ACI operation but are not closed under the maximum operation associated with any (total) ordering of the domain. (An example of such a relation is the relation R 1 defined in Example 2.6, see Example 6.4 below.) 5.4 Affine operations We will now show that closure under an affine operation is a sufficient condition for tractability. This result was established in [14] using elementary methods, for the special case when the domain D contains a prime number, p, of elements. It was shown in [14] that, in this special case, constraints which are closed under an affine operation correspond precisely to constraints which may be expressed as conjunctions of linear equations modulo p. (This class of tractable constraints is referred to as Class III in [14].) We now generalise this result to arbitrary finite domain sizes by making use of a result stated by Feder and Vardi in [8]. Theorem 5.16 ([8]) For any finite group G, and any set \Gamma of cosets of subgroups of direct products of G, CSP(\Gamma) is solvable in polynomial time. Corollary 5.17 For any set of relations \Gamma, if \Gamma is closed under an affine operation, then CSP(\Gamma) is solvable in polynomial time. Proof: By Definitions 3.1 and 3.2, any relation R which is closed under an affine operation is a subset of a direct product of some Abelian group, with the property that R. However, this is equivalent to saying that R is a coset of a subgroup of this direct product group [21], so we may apply Theorem 5.16 to \Gamma to obtain the result. Example 5.18 Let r be the affine operation on which is defined by addition and subtraction are both modulo 3. The relation R 2 defined in Example 2.6 is closed under r, since applying the r operation to any 3 elements of R 2 yields an element of R 2 . For example, Since jDj is prime, the results of [14] indicate that R 2 must be the set of solutions to some system of linear equations over the integers modulo 3. In fact, we have Example 5.19 Let G be the Abelian group hD; +; \Gammai, where 3g and the operation is defined by the following table: Now let r be the affine operation on which is defined by r(d addition and subtraction are as defined in G. Any relation R over D which is a coset of a subgroup of a direct product of G will be closed under r, and hence CSP(fRg) will be tractable by Corolllary 5.17. One example of such a relation is the following: It is easily seen that in this case R is not the set of solutions to any system of linear equations over a field. 2 Example 5.20 When there is only one possible Abelian group structure over D and hence only one possible affine operation on D. If a set of Boolean relations \Gamma is closed under this affine operation, then CSP(\Gamma) is equivalent to the problem of solving a set of simultaneous linear equations over the integers modulo 2. This corresponds to the final tractable subproblem of the generalised Satisfiability problem identified by Schaefer in [26]. 2 Semiprojections We now show that closure under a semiprojection operation is not in general a sufficient condition for tractability. In fact we shall establish a much stronger result, which shows that even being closed under all semiprojections is not sufficient to ensure tractability. Lemma 5.21 For any finite set D, with jDj - 3, there exists a set of relations \Gamma over D, such that \Gamma is closed under all semiprojections on D, and CSP(\Gamma) is NP-complete. Proof: Let D be a finite set with jDj - 3 and let d 1 ; d 2 be elements of D. Consider the ig. This relation is closed under all semiprojections on D, since any 3 elements of R contain at most two distinct values in each coordinate position. However, if we identify d 1 with the Boolean value true and d 2 with the Boolean value false, then it is easy to see that CSP(fRg) is isomorphic to the Not-All-Equal Satisfiability problem [11], which is NP-complete [26] (see Example 2.5). It is currently unknown whether there are tractable sets of relations closed under some combination of semiprojections, unary operations and binary operations which are not included in any of the tractable classes listed above. However, when the situation is very simple, as the next example shows. Example 5.22 When there are no semiprojections on D, so there are no subproblems of the Satisfiability problem which are characterised by a closure operation of this form. 2 6 Calculating closure operations For any set of relations \Gamma, over a set D, the operations under which \Gamma is closed are simply mappings from D k to D, for some k, which satisfy certain constraints, as described in Definition 3.3. In this Section we show that it is possible to identify these operations by solving a single constraint satisfaction problem in CSP(\Gamma). In fact, we shall show that these closure operations are precisely the solutions to a constraint satisfaction problem of the following form. Definition 6.1 Let \Gamma be a set of relations over a finite domain D. For any natural number m ? 0, the indicator problem for \Gamma of order m is defined to be the constraint satisfaction problem IP (\Gamma; m) with ffl Set of variables D m ; ffl Domain of values D; ffl Set of constraints fC g, such that for each R 2 \Gamma, and for each sequence of tuples from R, there is a constraint C is the arity of R and Example 6.2 Consider the relation R 1 over defined in Example 2.6. The indicator problem for fR 1 g of order 1, IP(fR 1 g; 1), has 3 variables and 4 con- straints. The set of variables is and the set of constraints is The indicator problem for fR 1 g of order 2, IP(fR 1 g; 2), has 9 variables and 16 con- straints. The set of variables is and the set of constraints is Further illustrative examples of indicator problems are given in [15]. 2 Solutions to the indicator problem for \Gamma of order m are functions from D m to D, or in other words, m-ary operations on D. We now show that they are precisely the m-ary operations under which \Gamma is closed. Theorem 6.3 For any set of relations \Gamma over domain D, the set of solutions to IP (\Gamma; m) is equal to the set of m-ary operations under which \Gamma is closed. Proof: By Definition 3.3, we know that \Gamma is closed under the m-ary operation\Omega if and only if\Omega satisfies the for each possible choice of R (not necessarily all distinct). But this is equivalent to saying that \Omega satisfies all the constraints in IP (\Gamma; m), so the result follows. Example 6.4 Consider the relation R 1 over defined in Example 2.6. The indicator problem for fR 1 g of order 1, defined in Example 6.2, has 2 solutions, which may be expressed in tabular form as follows: Variables Solution Solution One of these solutions is a constant operation, so CSP(fR 1 g) is tractable, by Proposition 5.1. In fact, any problem in CSP(fR 1 g) has the solution which assigns the value 0 to each variable, so the complexity of CSP(fR 1 g) is trivial. The indicator problem for fR 1 g of order 2, defined in Example 6.2, has 4 solutions, which may be expressed in tabular form as follows: Variables Solution Solution Solution Solution The first of these solutions is a constant operation, and the second and third are essentially unary operations. However, the fourth solution shown in the table is more interesting. It is easily checked that this operation is an associative, commutative, idempotent (ACI) binary operation, so we have a second proof that CSP(fR 1 g) is tractable, by Theorem 5.13. Furthermore, this result shows that R 1 can be combined with any other relations (of any arity) which are also closed under this ACI operation to obtain larger tractable problem classes. 2 Corollary 6.5 For any set of relations \Gamma over a domain D, with jDj - 3, if all solutions to IP (\Gamma; jDj) are essentially unary, then CSP(\Gamma) is NP-complete. Proof: Follows from Theorem 3.8, Theorem 4.1, and Theorem 6.3. Example 6.6 Recall the relations R 1 , R 2 and R 3 defined in Example 2.6. It has been shown in Examples 6.4, 5.18, and 5.2 that a set containing any one of these relations on its own is tractable. For any set \Gamma containing more than one of these relations, it can be shown, using Corollary 6.5, that CSP(\Gamma) is NP-complete. 2 In the special case when we obtain an even stronger result. Corollary 6.7 For any set of relations \Gamma over a domain D, with solutions to IP (\Gamma; are essentially unary then CSP(\Gamma) is NP-complete, otherwise it is polynomial. Proof: It has been shown in Examples 5.3, 5.8, 5.14, 5.20, and 5.22 that when possible closure operations of the restricted types specified in Corollary 4.2 are sufficient to ensure tractability. This result demonstrates that solving the indicator problem of order 3 provides a simple and complete test for tractability of any set of relations over a domain with 2 elements. This answers a question posed by Schaefer in 1978 [26] concerning the existence of an efficient test for tractability in the Generalised Satisfiability problem. Note that carrying out the test requires finding the solutions to a constraint satisfaction problem with just 8 Boolean variables. 7 Conclusion In this paper we have shown how the algebraic properties of relations can be used to distinguish between sets of relations which give rise to tractable constraint satisfaction problems and those which give rise to NP-complete problems. Furthermore, we have proposed a method for determining the operations under which a set of relations is closed by solving a particular form of constraint satisfaction problem, which we have called an indicator problem. For problems where the domain contains just two elements these results provide a necessary and sufficient condition for tractability (assuming that P is not equal to NP), and an efficient test to distinguish the tractable sets of relations. For problems with larger domains we have described algebraic closure properties which are a necessary condition for tractability. We have also shown that in many cases these closure properties are sufficient to ensure tractability. In particular, we have shown that closure under any constant operation, any majority operation, any ACI operation, or any affine operation, is a sufficient condition for tractability. It can be shown using the results of [13] that for any operation of one of these types, the set, \Gamma, containing all relations which are closed under that operation is a maximal set of tractable relations. In other words, the addition of any other relation which is not closed under the same operation changes CSP(\Gamma) from a tractable problem into an NP-complete problem. Hence, the tractable classes defined in this way are as large as possible. We are now investigating the application of these results to particular problem types, such as temporal problems involving subsets of the interval algebra. We are also attempting to determine how the presence of particular algebraic closure properties in the constraints can be used to derive appropriate efficient algorithms for tractable problems. Acknowledgments This research was partially supported by EPSRC research grant GR/L09936 and by the 'British-Flemish Academic Research Collaboration Programme' of the Belgian National Fund for Scientific Research and the British Council. We are also grateful to Martin Cooper for many helpful discussions and for suggesting the proof of Theorem 5.7. --R "A relational model of data for large shared databanks" "Derivation of constraints and database rela- tions" "Characterizing tractable constraints" "From local to global consistency" "Structure identification in relational data" "Network-based heuristics for constraint-satisfaction prob- lems" "Monotone monadic SNP and constraint satisfaction" "A sufficient condition for backtrack-bounded search" "Eliminating interchangeable values in constraint satisfaction prob- lems" Computers and intractability: a guide to the theory of NP-completeness "Decomposing constraint satisfaction problems using database techniques" "An algebraic characterization of tractable constraints" "A unifying framework for tractable con- straints" "A test for tractability" "Tractable constraints on ordered domains" "Fast parallel constraint satisfaction" "On binary constraint problems" "Classifying essentially minimal clones" "Consistency in networks of relations" "Networks of constraints: fundamental properties and applications to picture processing" "Constraint relaxation may be perfect" Computational Complexity "Minimal clones I: the five types" "The complexity of satisfiability problems" Clones in "On the Minimality and Decomposability of Row-Convex Constraint Networks" "A generic arc-consistency algorithm and its specializations" --TR A sufficient condition for backtrack-bounded search Network-based heuristics for constraint-satisfaction problems Constraint relaxation may be perfect From local to global consistency Structure identification in relational data A generic arc-consistency algorithm and its specializations Fast parallel constraint satisfaction Monotone monadic SNP and constraint satisfaction On binary constraint problems Decomposing constraint satisfaction problems using database techniques Characterising tractable constraints Tractable constraints on ordered domains A relational model of data for large shared data banks Computers and Intractability A Unifying Framework for Tractable Constraints The complexity of satisfiability problems --CTR Andrei Bulatov , Andrei Krokhin , Peter Jeavons, The complexity of maximal constraint languages, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.667-674, July 2001, Hersonissos, Greece Yuanlin Zhang , Roland H. C. Yap, Consistency and set intersection, Eighteenth national conference on Artificial intelligence, p.971-972, July 28-August 01, 2002, Edmonton, Alberta, Canada Richard E. Stearns , Harry B. Hunt, III, Exploiting structure in quantified formulas, Journal of Algorithms, v.43 n.2, p.220-263, May 2002 Hubie Chen, The expressive rate of constraints, Annals of Mathematics and Artificial Intelligence, v.44 n.4, p.341-352, August 2005 D. A. Cohen, Tractable Decision for a Constraint Language Implies Tractable Search, Constraints, v.9 n.3, p.219-229, July 2004 Peter Jeavons , David Cohen , Justin Pearson, Constraints and universal algebra, Annals of Mathematics and Artificial Intelligence, v.24 n.1-4, p.51-67, 1998 Lane A. Hemaspaandra, SIGACT news complexity theory column 34, ACM SIGACT News, v.32 n.4, December 2001 Lefteris M. Kirousis , Phokion G. Kolaitis, The complexity of minimal satisfiability problems, Information and Computation, v.187 n.1, p.20-39, November 25, Andrei Bulatov , Martin Grohe, The complexity of partition functions, Theoretical Computer Science, v.348 n.2, p.148-186, 8 December 2005 Lane A. Hemaspaandra, SIGACT news complexity theory column 43, ACM SIGACT News, v.35 n.1, March 2004 Andrei A. Bulatov, H-Coloring dichotomy revisited, Theoretical Computer Science, v.349 n.1, p.31-39, 12 December 2005 Victor Dalmau , Peter Jeavons, Learnability of quantified formulas, Theoretical Computer Science, v.306 n.1-3, p.485-511, 5 September David Cohen , Peter Jeavons , Richard Gault, New Tractable Classes From Old, Constraints, v.8 n.3, p.263-282, July David Cohen , Peter Jeavons , Richard Gault, New tractable constraint classes from old, Exploring artificial intelligence in the new millennium, Morgan Kaufmann Publishers Inc., San Francisco, CA, David Cohen , Peter Jeavons , Peter Jonsson , Manolis Koubarakis, Building tractable disjunctive constraints, Journal of the ACM (JACM), v.47 n.5, p.826-853, Sept. 2000 Hubie Chen, Periodic Constraint Satisfaction Problems: Tractable Subclasses, Constraints, v.10 n.2, p.97-113, April 2005 Richard Gault , Peter Jeavons, Implementing a Test for Tractability, Constraints, v.9 n.2, p.139-160, April 2004 Harry B. Hunt, III , Madhav V. Marathe , Richard E. Stearns, Strongly-local reductions and the complexity/efficient approximability of algebra and optimization on abstract algebraic structures, Proceedings of the 2001 international symposium on Symbolic and algebraic computation, p.183-191, July 2001, London, Ontario, Canada Vctor Dalmau, A new tractable class of constraint satisfaction problems, Annals of Mathematics and Artificial Intelligence, v.44 n.1-2, p.61-85, May 2005 Peter Jonsson , Andrei Krokhin, Recognizing frozen variables in constraint satisfaction problems, Theoretical Computer Science, v.329 n.1-3, p.93-113, 13 December 2004 Martin Grohe , Dniel Marx, Constraint solving via fractional edge covers, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.289-298, January 22-26, 2006, Miami, Florida Georg Gottlob , Nicola Leone , Francesco Scarcello, Hypertree decompositions and tractable queries, Proceedings of the eighteenth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, p.21-32, May 31-June 03, 1999, Philadelphia, Pennsylvania, United States David A. Cohen , Martin C. Cooper , Peter G. Jeavons , Andrei A. Krokhin, The complexity of soft constraint satisfaction, Artificial Intelligence, v.170 n.11, p.983-1016, August 2006 Andrei A. Bulatov, A dichotomy theorem for constraint satisfaction problems on a 3-element set, Journal of the ACM (JACM), v.53 n.1, p.66-120, January 2006 Georg Gottlob , Francesco Scarcello , Martha Sideri, Fixed-parameter complexity in AI and nonmonotonic reasoning, Artificial Intelligence, v.138 n.1-2, p.55-86, June 2002 Andrei A. Bulatov , Vctor Dalmau, Towards a dichotomy theorem for the counting constraint satisfaction problem, Information and Computation, v.205 n.5, p.651-678, May, 2007 Moshe Y. Vardi, Constraint satisfaction and database theory: a tutorial, Proceedings of the nineteenth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, p.76-85, May 15-18, 2000, Dallas, Texas, United States Hubie Chen, A rendezvous of logic, complexity, and algebra, ACM SIGACT News, v.37 n.4, December 2006 Georg Gottlob , Nicola Leone , Francesco Scarcello, The complexity of acyclic conjunctive queries, Journal of the ACM (JACM), v.48 n.3, p.431-498, May 2001

complexity;NP-completeness;indicator problem;constraint satisfaction problem

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Constraint tightness and looseness versus local and global consistency.

Constraint networks are a simple representation and reasoning framework with diverse applications. In this paper, we identify two new complementary properties on the restrictiveness of the constraints in a networkconstraint tightness and constraint loosenessand we show their usefulness for estimating the level of local consistency needed to ensure global consistency, and for estimating the level of local consistency present in a network. In particular, we present a sufficient condition, based on constraint tightness and the level of local consistency, that guarantees that a solution can be found in a backtrack-free manner. The condition can be useful in applications where a knowledge base will be queried over and over and the preprocessing costs can be amortized over many queries. We also present a sufficient condition for local consistency, based on constraint looseness, that is straightforward and inexpensive to determine. The condition can be used to estimate the level of local consistency of a network. This in turn can be used in deciding whether it would be useful to preprocess the network before a backtracking search, and in deciding which local consistency conditions, if any, still need to be enforced if we want to ensure that a solution can be found in a backtrack-free manner. Two definitions of local consistency are employed in characterizing the conditions: the traditional variable-based notion and a recently introduced definition of local consistency called relational consistency.

Introduction Constraint networks are a simple representation and reasoning framework. A problem is represented as a set of variables, a domain of values for each variable, and a set of constraints between the variables. A central reasoning task is then to find an instantiation of the variables that satisfies the constraints. In spite of the simplicity of the framework, many interesting problems can be formulated as constraint networks, including graph coloring, scene labeling, natural language parsing, and temporal reasoning. In general, what makes constraint networks hard to solve is that they can contain many local inconsistencies. A local inconsistency is an instantiation of some of the variables that satisfies the relevant constraints that cannot be extended to an additional variable and so cannot be part of any global solution. If we are using a backtracking search to find a solution, such an inconsistency can lead to a dead end in the search. This insight has led to the definition of conditions that characterize the level of local consistency of a network [12, 18, 21] and to the development of algorithms for enforcing local consistency conditions by removing local inconsistencies (e.g., [2, 7, 10, 18, 21]). Local consistency has proven to be an important concept in the theory and practice of constraint networks for primarily two reasons. First, a common method for finding solutions to a constraint network is to first preprocess the network by enforcing local consistency conditions, and then perform a back-tracking search. The preprocessing step can reduce the number of dead ends reached by the backtracking algorithm in the search for a solution. With a similar aim, local consistency techniques can be interleaved with backtracking search. The effectiveness of using local consistency techniques in these two ways has been studied empirically (e.g., [4, 6, 13, 14, 23]). Second, much previous work has identified conditions for when a certain level of local consistency is sufficient to guarantee that a network is globally consistent or that a solution can be found in a backtrack-free manner (e.g., [5, 7, 11, 12, 21, 29]). In this paper, we identify two new complementary properties on the restrictiveness of the constraints in a network-constraint tightness and constraint looseness-and we show their usefulness for estimating the level of local consistency needed to ensure global consistency, and for estimating the level of local consistency present in a network. In particular, we present the following results. We show that, in any constraint network where the constraints have arity r or less and tightness of m or less, if the network is strongly ((m+ 1)(r consistent, then it is globally consistent. Informally, a constraint network is strongly k-consistent if any instantiation of any k \Gamma 1 or fewer variables that satisfies the constraints can be extended consistently to any additional variable. Also informally, given an r-ary constraint and an instantiation of r \Gamma 1 of the variables that participate in the constraint, the parameter m is an upper bound AAAI-94 [27]. on the number of instantiations of the rth variable that satisfy the constraint. In general, such sufficient conditions, bounding the level of local consistency that guarantees global consistency, are important in applications where constraint networks are used for knowledge base maintenance and there will be many queries against the knowledge base. Here, the cost of preprocessing will be amortized over the many queries. They are also of interest for their explanatory power, as they can be used for characterizing the difficulty of problems formulated as constraint networks. We also show that, in any constraint network where the domains are of size d or less, and the constraints have looseness of m or greater, the network is strongly (dd=(d \Gamma m)e)-consistent 2 . Informally, given an r-ary constraint and an instantiation of r \Gamma 1 of the variables that participate in the constraint, the parameter m is a lower bound on the number of instantiations of the rth variable that satisfy the constraint. The bound is straightforward and inexpensive to determine. In contrast, all but low-order local consistency is expensive to verify or enforce as the optimal algorithms to enforce k-consistency are O(n k d k ), for a network with n variables and domains of size at most d [2, 24]. The condition based on constraint looseness is useful in two ways. First, it can be used in deciding which low-order local consistency techniques will not change the network and thus are not useful for processing a given constraint network. For example, we use our results to show that the n-queens problem, a widely used test-bed for comparing backtracking algorithms, has a high level of inherent local consistency. As a consequence, it is generally fruitless to preprocess such a network. Second, it can be used in deciding which local consistency conditions, if any, still need to be enforced if we want to ensure that a solution can be found in a backtrack-free manner. Two definitions of local consistency are employed in characterizing the condi- tions: the traditional variable-based notion and a recently introduced definition of local consistency called relational consistency [9, 29]. Definitions and Preliminaries A constraint network R is a set of n variables fx a domain D i of possible values for each variable x and a set of t constraint relations t. Each constraint relation R S i , t, is a subset of a Cartesian product of the form We say that constrains the variables fx j g. The arity of a constraint relation is the number of variables that it constrains. The set fS is called the scheme of R. We assume that S i Because we assume that variables have names, the order of the variables constrained by a relation is not important (see [26, pp. 43-45]). We use subsets of the integers ng and subsets of the variables fx 2 dxe, the ceiling of x, is the smallest integer greater than or equal to x. ng be a subset of the variables in a constraint network. An instantiation of the variables in Y is an element of . Given an instantiation -a of the variables in Y , an extension of -a to a variable x is the tuple (-a, a i ), where a i is in the domain of x i . We now introduce a needed operation on relations adopted from the relational calculus (see [26] for details). S be a relation, let S be the set of variables constrained by R S , and let S 0 ' S be a subset of the variables. The projection of RS onto the set of variables S 0 , denoted \Pi S 0 (RS ), is the relation which constrains the variables in S 0 and contains all tuples u 2 D j such that there exists an instantiation - a of the variables in S \Gamma S 0 such that the tuple (u; -a) 2 R S . ng be a subset of the variables in a constraint network. An instantiation -a of the variables in Y satisfies or is consistent with a relation R S i , the tuple \Pi S i (f-ag) 2 R S i . An instantiation -a of the variables in Y is consistent with a network R if and only if for all S i in the scheme of R such that a satisfies R S i . A solution to a constraint network is an instantiation of all n of the variables that is consistent with the network. constraint relation R of arity k is called m-tight if, for any variable x i constrained by R and any instantiation -a of the remaining constrained by R, either there are at most m extensions of -a to x i that satisfy R, or there are exactly jD i j such extensions. Definition 3 (m-loose) A constraint relation R of arity k is called m-loose if, for any variable x i constrained by R and any instantiation -a of the remaining constrained by R, there are at least m extensions of - a to x i that satisfy R. The tightness and looseness properties are complementary properties on the constraints in a network. Tightness is an upper bound on the number of extensions and looseness is a lower bound. With the exception of the universal relation, a constraint relation that is m 1 -loose and m 2 -tight has case where there are exactly jD i j extensions to variable x i is handled specially in the definition of tightness). Every constraint relation is 0-loose and a constraint relation is d-loose if and only if it is the universal relation, where the domains of the variables constrained by the relation are of size at most d. Every constraint relation is (d \Gamma 1)-tight and a constraint relation is 0-tight if and only if it is either the null relation or the universal relation. In what follows, we are interested in the least upper bound and greatest lower bound on the number of extensions of the relations of a constraint network. It is easy to check that: Given a constraint network with t constraint relations each with arity at most r and each with at most e tuples in the relation, determining the least m such that all t of the relations are m-tight requires O(tre) time. The same bound applies to determining the greatest m such that the relations are m-loose. Example 1. We illustrate the definitions using the following network R with variables fx 1 domains (b,b,b), (b,c,a), (c,a,b), (c,b,a), (c,c,c)g, g. The projection of is given by, (R (c,c)g. The instantiation - (a,c,b) of the variables in is consistent with R since \Pi S2 . The instantiation of the variables in Y is not consistent with R since \Pi S2 (a,a,a,b) be an instantiation of all of the variables fx 1 g. The tuple - a is consistent with R and is therefore a solution of the network. The set of all solutions of the network is given by, f(a,a,a,b), (a,a,c,b), (a,b,c,a), (b,a,c,b), (b,c,a,c), (c,a,b,b), (c,b,a,c), (c,c,c,a)g. It can be verified that all of the constraints are 2-tight and 1-loose. As a partial verification of the binary constraint R S3 , consider the extensions to variable x 3 given instantiations of the variable x 4 . For the instantiation - a = (a) of x 4 there is 1 extension to x 3 ; for - a = (b) there are 3 extensions (but so the definition of 2-tightness is still satisfied); and for - a = (c) there is 1 extension. Local consistency has proven to be an important concept in the theory and practice of constraint networks. We now review previous definitions of local consistency, which we characterize as variable-based and relation-based. 2.1 Variable-based local consistency Mackworth [18] defines three properties of networks that characterize local con- sistency: node, arc, and path consistency. Freuder [10] generalizes this to k- consistency, which can be defined as follows: Definition 4 (k-consistency, global consistency) A constraint network R is k-consistent if and only if given 1. any k \Gamma 1 distinct variables, and 2. any instantiation - a of the variables that is consistent with R, there exists an extension of -a to any kth variable such that the k-tuple is consistent with R. A network is strongly k-consistent if and only if it is j-consistent for all j - k. A strongly n-consistent network is called globally consistent, where n is the number of variables in the network. Node, arc, and path consistency correspond to one-, two-, and three-consis- tency, respectively. Globally consistent networks have the property that a solution can be found without backtracking [11]. (a) (b) Figure 1: (a) not 3-consistent; (b) not 4-consistent Example 2. We illustrate the definition of k-consistency using the well-known n-queens problem. The problem is to find all ways to place n-queens on an n \Theta n chess board, one queen per column, so that each pair of queens does not attack each other. One possible constraint network formulation of the problem is as follows: there is a variable for each column of the chess board, x the domains of the variables are the possible row positions, D and the binary constraints are that two queens should not attack each other. Consider the constraint network for the 4-queens problem. It can be seen that the network is 2-consistent since, given that we have placed a single queen on the board, we can always place a second queen such that the queens do not attack each other. However, the network is not 3-consistent. For example, given the consistent placement of two queens shown in Figure 1a, there is no way to place a queen in the third column that is consistent with the previously placed queens. Similarly the network is not 4-consistent (see Figure 1b). 2.2 Relation-based local consistency In [29], we extended the notions of arc and path consistency to non-binary re- lations. The new local consistency conditions were called relational arc- and path-consistency. In [9], we generalized relational arc- and path-consistency to relational m-consistency. In the definition of relational m-consistency, the relations rather than the variables are the primitive entities. As we shall see in subsequent sections, this allows expressing the relationships between the restrictiveness of the constraints and local consistency in a way that avoids an explicit reference to the arity of the constraints. The definition below is slightly weaker than that given in [9]. Definition 5 (relational m-consistency) A constraint network R is relationally m-consistent if and only if given 1. any m distinct relations R 2. any x 2 3. any instantiation - a of the variables in ( that is consistent with R, there exists an extension of - a to x such that the extension is consistent with the relations. A network is strongly relationally m-consistent if it is relationally j-consistent for every j - m. Example 3. Consider the constraint network with variables fx 1 domains (b,a,c)g. g. The constraints are not relationally 1-consistent. For example, the instantiation (a,b,b) of the variables in is consistent with the network (trivially so, since S 1 6' Y and does not have an extension to x 5 that satisfies R S1 . Similarly, the constraints are not relationally 2-consistent. For example, the instantiation (c,b,a,a) of the variables in fx 1 is consistent with the network (again, trivially so), but it does not have an extension to x 5 that satisfies both R S1 and RS2 . If we add the constraints R fx2g fag and R fbg, the set of solutions of the network does not change, and it can be verified that the network is both relationally 1- and 2-consistent. When the constraints are all binary, relational m-consistency is identical to variable-based (m the conditions are different. While enforcing variable-based m-consistency can be done in polynomial time, it is unlikely that relational m-consistency can be achieved tractably since even solves the NP-complete problem of propositional satisfiability (see Example 6). A more direct argument suggesting an increase in time and space complexity is the fact that an algorithm may need to record relations of arbitrary arity. As with variable-based local-consistency, we can improve the efficiency of enforcing relational consistency by enforcing it only along a certain direction or linear ordering of the variables. Algorithms for enforcing relational consistency and directional relational consistency are given in [9, 28]. 3 Constraint Tightness vs Global Consistency In this section, we present relationships between the tightness of the constraints and the level of local consistency sufficient to ensure that a network is globally consistent. Much work has been done on identifying relationships between properties of constraint networks and the level of local consistency sufficient to ensure global consistency. This work falls into two classes: identifying topological properties of the underlying graph of the network (e.g., [7, 8, 11, 12]) and identifying properties of the constraints (e.g., [3, 15, 16, 21, 29]). Dechter [5] identifies the following relationship between the size of the domains of the variables, the arity of the constraints, and the level of local consistency sufficient to ensure the network is globally consistent. Theorem 1 (Dechter [5]) If a constraint network with domains that are of size at most d and relations that are of arity at most r is strongly consistent, then it is globally consistent. For some networks, Dechter's theorem is tight in that the level of local consistency specified by the theorem is really required (graph coloring problems formulated as constraint networks are an example). For other networks, Dechter's theorem overestimates. Our results should be viewed as an improvement on Dechter's theorem. By taking into account the tightness of the constraints, our results always specify a level of strong consistency that is less than or equal to the level of strong consistency required by Dechter's theorem. The following lemma is needed in the proof of the main result. l be l relations that constrain a variable x, let d be the size of the domain of variable x, and let - a be an instantiation of all of the variables except for x that are constrained by the l relations (i.e., - a is an instantiation of the variables in (S 1 [ 1. each relation is m-tight, for some 2. for every subset of fewer relations from fR l g, there exists at least one extension of - a to x that satisfies each of the relations in the subset, then there exists at least one extension of - a to x that satisfies all l relations. Proof. Let a 1 ; a a d be the d elements in the domain of x. We say that a relation allows an element a i if the extension (-a; a i ) of - a to x satisfies the relation. Assume to the contrary that an extension of - a to x that satisfies all of the l relations does not exist. Then, for each element a i in the domain of x there must exist at least one relation that does not allow a i . Let c i denote one of the relations that does not allow a i . By construction, the set is a set of relations for which there does not exist an extension of -a to x that satisfies each of the relations in the set (every candidate a i is ruled out since c i does not allow a i ). For every possible value of m, this leads to a contradiction. Case 1 The contradiction is immediate as is a set of relations of size m+ 1 for which there does not exist an extension to x that satisfies every relation in the set. This contradicts condition (2). Case The nominal size of the set 2. We claim, however, that there is a repetition in c and that the true size of the set is m+1. Assume to the contrary that c i 6= c j for i 6= j. Recall c i is a relation that does not allow a i , g. This is a set of m+ 1 relations so by condition (2) there must exist an a i that every relation in the set allows. The only possibility is a d . Now consider fc g. Again, this is a set of m relations so there must exist an a i that every relation in the set allows. This time the only possibility is a d\Gamma1 . Continuing in this manner, we can show that relation c 1 must allow a d ; a must allow exactly m+1 extensions. This contradicts condition (1). Therefore, it must be the case that c j. Thus, the set c is of size m and this contradicts condition (2). Case 3 The remaining cases are similar. In each case we argue that (i) there are repetitions in the set (ii) the true size of the set c is a contradiction is derived by appealing to condition (2). Thus, there exists at least one extension of - a to x that satisfies all of the relations.We first state the result using variable-based local consistency and then state the result using relation-based local consistency. Theorem 2 If a constraint network with relations that are m-tight and of arity at most r is strongly ((m+1)(r \Gamma 1)+1)-consistent, then it is globally consistent. Proof. Let 1. We show that any network with relations that are m-tight and of arity at most r that is strongly k-consistent is consistent for any i - 1. Without loss of generality, let be a set of k variables, let - a be an instantiation of the variables in X that is consistent with the constraint network, and let x k+i be an additional variable. Using Lemma 1, we show that there exists an extension of - a to x k+i that is consistent with the constraint network. Let R l be l relations which are all and only the relations which constrain only x k+i and a subset of variables from X. To be consistent with the constraint network, the extension of - a to x k+i must satisfy each of the l relations. Now, condition (1) of Lemma 1 is satisfied since each of the l relations is m-tight. It remains to show that condition (2) is satisfied. By definition, the requirement of strong ((m 1)-consistency ensures that any instantiation of any (m+ 1)(r \Gamma 1) or fewer variables that is consistent with the network, has an extension to x k+i such that the extension is also consistent with the network. Note, however, that since each of the l relations is of arity at most r and constrains x k+i , each relation can constrain at most r \Gamma 1 variables that are not constrained by any of the other relations. Therefore, the requirement of strong ((m 1)-consistency also ensures that for any subset of m+1 or fewer relations from fR l g, there exists an extension of - a to x k+i that satisfies each of the relations in the subset. Thus, condition (2) of Lemma 1 is satisfied. Therefore, from Lemma 1 it follows that there is an extension of - a to x k+i that is consistent with the constraint network. 2 Theorem 2 always specifies a level of strong consistency that is less than or equal to the level of strong consistency required by Dechter's theorem (Theo- rem 1). The level of required consistency is equal only when As well, the theorem can sometimes be usefully applied if theorem cannot. As the following example illustrates, both r, the arity of the constraints, and can change if the level of consistency required by the theorem is not present and must be enforced. The parameter r can only increase; m can decrease, as shown below, but also increase. The parameter m will increase if all of the following hold: (i) there previously was no constraint between a set of variables, (ii) enforcing a certain level of consistency results in a new constraint being recorded between those variables and, (iii) the new constraint has a larger m value than the previous constraints. Example 4. Nadel [22] introduces a variant of the n-queens problem called confused n-queens. The problem is to find all ways to place n-queens on an n \Theta n chess board, one queen per column, so that each pair of queens does attack each other. One possible constraint network formulation of the problem is as follows: there is a variable for each column of the chess board, x the domains of the variables are the possible row positions, D and the binary constraints are that two queens should attack each other. The constraint relation between two variables x i and x The problem is worth considering, as Nadel [22] uses confused n-queens in an empirical comparison of backtracking algorithms for solving constraint networks. Thus it is important to analyze the difficulty of the problems to set the empirical results in context. As well, the problem is interesting in that it provides an example where Theorem 2 can be applied but Dechter's theorem can not (since d - 1). Independently of n, the constraint relations are all 3-tight. Hence, the theorem guarantees that if the network for the confused n-queens problem is strongly 5-consistent, the network is globally consistent. First, suppose that n is even and we attempt to either verify or achieve this level of strong consistency by applying successively stronger local consistency algorithms. Kondrak [17] has shown that the following analysis holds for all n, even. 1. Applying an arc consistency algorithm results in no changes as the network is already arc consistent. 2. Applying a path consistency algorithm does tighten the constraints between the variables. Once the network is made path consistent, the constraint relations are all 2-tight. Now the theorem guarantees that if the network is strongly 4-consistent, it is globally consistent. 3. Applying a 4-consistency algorithm results in no changes as the network is already 4-consistent. Thus, the network is strongly 4-consistent and therefore also globally consistent. Second, suppose that n is odd. This time, after applying path consistency, the networks are still 3-tight and it can be verified that the networks are not 4-consistent. Enforcing 4-consistency requires 3-ary constraints. Adding the necessary 3-ary constraints does not change the value of m; the networks are still 3-tight. Hence, by Theorem 2, if the networks are strongly 9-consistent, the networks are globally consistent. Kondrak [17] has shown that recording 3-ary constraints is sufficient to guarantee that the networks are strongly 9-consistent for all n, Hence, independently of n, the networks are globally consistent once strong 4-consistency is enforced. Recall that Nadel [22] uses confused n-queens problems to empirically compare backtracking algorithms for finding all solutions to constraint networks. Nadel states that these problems provide a "non-trivial test-bed" [22, p.190]. We believe the above analysis indicates that these problems are quite easy and that any empirical results on these problems should be interpreted in this light. Easy problems potentially make even naive algorithms for solving constraint networks look promising. To avoid this potential pitfall, backtracking algorithms should be tested on problems that range from easy to hard. In general, hard problems are those that require a high level of local consistency to ensure global consistency. Note also that these problems are trivially satisfiable. Example 5. The graph k-colorability problem can be viewed as a problem on constraint networks: there is a variable for each node in the graph, the domains of the variables are the k possible colors, and the binary constraints are that two adjacent nodes must be assigned different colors. Graph k-colorability provides examples of networks where both Theorems 1 and 2 give the same bound on the sufficient level of local consistency since the constraints are tight. We now show how the concept of relation-based local consistency can be used to alternatively describe Theorem 2. Theorem 3 If a constraint network with relations that are m-tight is strongly relationally (m 1)-consistent, then it is globally consistent. Proof. We show that any network with relations that are m-tight that is strongly relationally (m + 1)-consistent is k-consistent for any k - 1 and is therefore globally consistent. Without loss of generality, let be a set of let -a be an instantiation of the variables in X that is consistent with the constraint network, and let x k be an additional variable. Using Lemma 1, we show that there exists an extension of -a to x k that is consistent with the constraint network. Let R l be l relations which are all and only the relations which constrain only x k and a subset of variables from X. To be consistent with the constraint network, the extension of -a to x k must satisfy each of the l rela- tions. Now, condition (1) of Lemma 1 is satisfied since each of the l relations is m-tight. Further, condition (2) of Lemma 1 is satisfied since, by definition, the requirement of strong relational (m+ 1)-consistency ensures that for any subset of fewer relations, there exists an extension of -a to x k that satisfies each of the relations in the subset. Therefore, from Lemma 1 it follows that there is an extension of - a to x k that is consistent with the constraint network.As an immediate corollary of Theorem 3, if we know that the result of applying an algorithm for enforcing strong relational (m 1)-consistency will be that all of the relations will be m-tight, we can guarantee a priori that the algorithm will return an equivalent, globally consistent network. Example 6. Consider networks where the domains of the variables are of size two. For example, the satisfiability of propositional CNFs provide an example of networks with domains of size two. Relations which constrain variables with domains of size two are 1-tight and any additional relations that are added to the network as a result of enforcing strong relational 2-consistency will also be 1-tight. Thus, the consistency of such networks can be decided by an algorithm that enforces strong relational 2-consistency. A different derivation of the same result is already given by [5, 29]. A backtracking algorithm constructs and extends partial solutions by instantiating the variables in some linear order. Global consistency implies that for any ordering of the variables the solutions to the constraint network can be constructed in a backtrack-free manner; that is, a backtracking search will not encounter any dead-ends in the search. Dechter and Pearl [7] observe that it is often sufficient to be backtrack-free along a particular ordering of the variables and that local consistency can be enforced with respect to that ordering only. Frequently, if the property of interest (in our case tightness and looseness) is satisfied along that ordering we can conclude global consistency restricted to that ordering as well. Enforcing relational consistency with respect to an ordering of the variables can be done by a general elimination algorithm called Directional-Relational-Consistency, presented in [9]. Such an algorithm has the potential of being more effective in practice and in the worst-case as it requires weaker conditions. Directional versions of the tightness and looseness properties and of the results presented in this paper are easily formulated using the ideas presented in [7, 9]. The results of this section can be used as follows. Mackworth [19] shows that constraint networks can be viewed as a restricted knowledge representation and reasoning framework. In this context, solutions of the constraint network correspond to models of the knowledge base. Our results which bound the level of local consistency needed to ensure global consistency, can be useful in applications where constraint networks are used as a knowledge base and there will be many queries against the knowledge base. Preprocessing the constraint network so that it is globally consistent means that queries can be answered in a backtrack-free manner. An equivalent globally consistent representation of a constraint network is highly desirable since it compiles the answers to many queries and it can be shown that there do exist constraint networks and queries against the network for which there will be an exponential speed-up in the worst case. As an exam- ple, consider a constraint network with no solutions. The equivalent globally consistent network would contain only null relations and an algorithm answering a query against this constraint network would quickly return "yes." Of course, of more interest are examples where the knowledge base is consistent. Queries which involve determining if a value for a variable is feasible-can occur in a model of the network-can be answered from the globally consistent representation by looking only at the domain of the variable. Queries which involve determining if the values for a pair of variables is feasible-can both occur in a single model of the network-can be answered by looking only at the binary relations which constrain the two variables. It is clear that a general algorithm to answer a query against the original network, such as backtracking search, can take an exponential amount of time to answer the above queries. In general, a globally consistent representation of a network will be useful whenever it is more compact than the set of all solutions to the network. With the globally consistent representation we can answer any query on a subset of the variables Y ' ng by restricting our attention to the smaller network which consists of only the variables in Y and only the relations which constrain the variables in Y . The global consistency property ensures that a solution for all of the variables can also be created in a backtrack-free manner. However, how our results will work in practice is an interesting empirical question which remains open. The results of this section are also interesting for their explanatory power, as they can be used for characterizing the difficulty of problems formulated as constraint networks (see the discussion at the end of the next section). 4 Constraint Looseness vs Local Consistency In this section, we present a sufficient condition, based on the looseness of the constraints and on the size of the domains of the variables, that gives a lower bound on the inherent level of local consistency of a constraint network. It is known that some classes of constraint networks already possess a certain level of local consistency and therefore algorithms that enforce this level of local consistency will have no effect on these networks. For example, Nadel [22] observes that an arc consistency algorithm never changes a constraint network formulation of the n-queens problem, for n ? 3. Dechter [5] observes that constraint networks that arise from the graph k-coloring problem are inherently strongly k-consistent. Our results characterize what it is about the structure of the constraints in these networks that makes these statements true. The following lemma is needed in the proof of the main result. l be l relations that constrain a variable x, let d be the size of the domain of variable x, and let - a be an instantiation of all of the variables except for x that are constrained by the l relations (i.e., - a is an instantiation of the variables in (S 1 [ 1. each relation is m-loose, for some 2. l - l d d\Gammam there exists at least one extension of -a to x that satisfies all l relations. Proof. Let a 1 ; a a d be the d elements in the domain of x. We say that a relation allows an element a i if the extension (-a; a i ) of - a to x satisfies the relation. Now, the key to the proof is that, because each of the l relations is m-loose, at least m elements from the domain of x are allowed by each relation. Thus, each relation does not allow at most d \Gamma m elements, and together the l relations do not allow at most l(d \Gamma m) elements from the domain of x. Thus, if it cannot be the case that every element in the domain of x is not allowed by some relation. Thus, if l - d there exists at least one extension of - a to x that satisfies all l relations. 2 We first state the result using variable-based local consistency and then state the result using relation-based local consistency. Let binomial(k; r) be the binomial coefficients, the number of possible choices of r different elements from a collection of k objects. If k ! r, then binomial(k; Theorem 4 A constraint network with domains that are of size at most d and relations that are m-loose and of arity at least r, r - 2, is strongly k-consistent, where k is the minimum value such that the following inequality holds, Proof. Without loss of generality, let be a set of variables, let - a be an instantiation of the variables in X that is consistent with the constraint network, and let x k be an additional variable. To show that the network is k-consistent, we must show that there exists an extension of -a to x k that is consistent with the constraint network. Let R l be l relations which are all and only the relations which constrain only x k and a subset of variables from X. To be consistent with the constraint network, the extension of - a to x k must satisfy each of the l relations. From Lemma 2, such an extension exists if l - dd=(d \Gamma m)e \Gamma 1. Now, the level of strong k-consistency is the minimum number of distinct variables that can be constrained by the l relations. In other words, k is the minimum number of variables that can occur in l . We know that each of the relations constrains the variable x k . Thus, is the minimum number of variables in (S fxg). The minimum value of c occurs when all of the relations have arity r and thus each (S l, is a set of r \Gamma 1 variables. Further, we know that each of the l relations constrains a different subset of variables; i.e., if i 6= j, then S i l. The binomial coefficients binomial(c; r \Gamma 1) tell us the number of distinct subsets of cardinality r \Gamma 1 which are contained in a set of size c. Thus, us the minimum number of variables c that are needed in order to specify the remaining r \Gamma 1 variables in each of the l relations subject to the condition that each relation must constrain a different subset of variables. 2 Constraint networks with relations that are all binary are an important special case of Theorem 4. Corollary 1 A constraint network with domains that are of size at most d and relations that are binary and m-loose is strongly d d\Gammam -consistent. Proof. All constraint relations are of arity 2. Hence, the minimum value of k such the inequality in Theorem 4 holds is when Theorem 4 always specifies a level of local consistency that is less than or equal to the actual level of inherent local consistency of a constraint network. That is, the theorem provides a lower bound. However, given only the looseness of the constraints and the size of the domains, Theorem 4 gives as strong an estimation of the inherent level of local consistency as possible as examples can be given for all m ! d where the theorem is exact. Graph coloring problems provide an example where the theorem is exact for n-queens problems provide an example where the theorem underestimates the true level of local consistency. Example 7. Consider again the well-known n-queens problem discussed in Example 2. The problem is of historical interest but also of theoretical interest due to its importance as a test problem in empirical evaluations of backtracking algorithms and heuristic repair schemes for finding solutions to constraint networks (e.g., [13, 14, 20, 22]). For n-queens networks, each of the domains is of size n and each of the constraints is binary and (n \Gamma 3)-loose. Hence, Theorem 4 predicts that n-queens networks are inherently strongly (dn=3e)- consistent. Thus, an n-queens constraint network is inherently arc-consistent for inherently path consistent for n - 7, and so on, and we can predict where it is fruitless to apply a low-order consistency algorithm in an attempt to simplify the network (see Table 1). The actual level of inherent consistency is bn=2c for n - 7. Thus, for the n-queens problem, the theorem underestimates the true level of local consistency. Table 1: Predicted (dn=3e) and actual (bn=2c, for n - 7) level of strong local consistency for n-queens networks pred. actual Example 8. Graph k-colorability provides an example where Theorem 4 is exact in its estimation of the inherent level of local consistency (see Example 5 for the constraint network formulation of graph coloring). As Dechter [5] states, graph coloring networks are inherently strongly k-consistent but are not guaranteed to be strongly 1)-consistent. Each of the domains is of size k and each of the constraints is binary and 1)-loose. Hence, Theorem 4 predicts that graph k-colorability networks are inherently strongly k-consistent. Example 9. Consider a formula in 3-CNF which can be viewed as a constraint network where each variable has the domain ftrue, falseg and each clause corresponds to a constraint defined by its models. The domains are of size two and all constraints are of arity 3 and are 1-loose. The minimum value of k such that the inequality in Theorem 4 holds is when 3. Hence, the networks are strongly 3-consistent. We now show how the concept of relation-based local consistency can be used to alternatively describe Theorem 4. Theorem 5 A constraint network with domains that are of size at most d and relations that are m-loose is strongly relationally d d\Gammam -consistent. Proof. Follows immediately from Lemma 2. 2 The results of this section can be used in two ways. First, they can be used to estimate whether it would be useful to preprocess a constraint network using a local consistency algorithm, before performing a backtracking search (see, for example, [6] for an empirical study of the effectiveness of such preprocessing). Second, they can be used in conjunction with previous work which has identified conditions for when a certain level of local consistency is sufficient to ensure a solution can be found in a backtrack-free manner (see, for example, the brief review of previous work at the start of Section 3 together with the new results presented there). Sometimes the level of inherent strong k-consistency guaranteed by Theorem 4 is sufficient, in conjunction with these previously derived conditions, to guarantee that the network is globally consistent and therefore a solution can be found in a backtrack-free manner without preprocessing. Oth- erwise, the estimate provided by the theorem gives a starting point for applying local consistency algorithms. The results of this section are also interesting for their explanatory power. We conclude this section with some discussion on what Theorem 2 and Theorem 4 contribute to our intuitions about hard classes of problems (in the spirit of, for example, [1, 30]). Hard constraint networks are instances which give rise to search spaces with many dead ends. The hardest networks are those where many dead ends occur deep in the search tree. Dead ends, of course, correspond to partial solutions that cannot be extended to full solutions. Networks where the constraints are that are close to d, the size of the domains of the variables, are good candidates to be hard problems. The reasons are two-fold. First, networks that have high looseness values have a high level of inherent strong consistency and strong k-consistency means that all partial solutions are of at least size k. Second, networks that have high tightness values require a high level of preprocessing to be backtrack-free. Computational experiments we performed on random problems with binary constraints provide evidence that networks with constraints with high looseness values can be hard. Random problems were generated with and is the probability that there is a binary constraint between two variables, and q=100 is the probability that a pair in the Cartesian product of the domains is in the constraint. The time to find one solution was measured. In the experiments we discovered that, given that the number of variables and the domain size were fixed, the hardest problems were found when the constraints were as loose as possible without degenerating into the trivial constraint where all tuples are allowed. In other words, we found that the hardest region of loose constraints is harder than the hardest region of tight constraints. That networks with loose constraints would turn out to be the hardest of these random problems is somewhat counter-intuitive, as individually the constraints are easy to satisfy. These experimental results run counter to Tsang's [25, p.50] intuition that a single solution of a loosely constrained problem "can easily be found by simple backtracking, hence such problems are easy," and that tightly constrained problems are "harder compared with loose problems." As well, these hard loosely-constrained problems are not amenable to preprocessing by low-order local consistency algorithms, since, as Theorem 4 states, they possess a high level of inherent local consistency. This runs counter to Williams and Hogg's [30, p.476] speculation that preprocessing will have the most dramatic effect in the region where the problems are the hardest. Conclusions We identified two new complementary properties on the restrictiveness of the constraints in a network: constraint tightness and constraint looseness. Constraint tightness was used, in conjunction with the level of local consistency, in a sufficient condition that guarantees that a solution to a network can be found in a backtrack-free manner. The condition can be useful in applications where a knowledge base will be queried over and over and the preprocessing costs can be amortized over many queries. Constraint looseness was used in a sufficient condition for local consistency. The condition is inexpensive to determine and can be used to estimate the level of strong local consistency of a network. This in turn can be used in deciding whether it would be useful to preprocess the network before a backtracking search, and in deciding which local consistency conditions, if any, still need to be enforced if we want to ensure that a solution can be found in a backtrack-free manner. We also showed how constraint tightness and constraint looseness are of interest for their explanatory power, as they can be used for characterizing the difficulty of problems formulated as constraint networks and for explaining why some problems that are "easy" locally, are difficult globally. We showed that when the constraints have low tightness values, networks may require less pre-processing in order to guarantee that a solution can be found in a backtrack-free manner and that when the constraints have high looseness values, networks may require much more search effort in order to find a solution. As an example, the confused n-queens problem, which has constraints with low tightness values, was shown to be easy to solve as it is backtrack-free after enforcing only low-order local consistency conditions. As another example, many instances of crossword puzzles are also relatively easy, as the constraints on the words that fit each slot in the puzzle have low tightness values (since not many words have the same length and differ only in the last letter of the word). On the other hand, graph coloring and scheduling problems involving resource constraints can be quite hard, as the constraints are inequality constraints and thus have high looseness values. Acknowledgements The authors wish to thank Peter Ladkin and an anonymous referee for their careful reading of a previous version of the paper and their helpful comments. --R Where the really hard problems are. An optimal k-consistency algorithm Characterising tractable constraints. Enhancement schemes for constraint processing: Backjump- ing From local to global consistency. Experimental evaluation of preprocessing techniques in constraint satisfaction problems. Tree clustering for constraint networks. Local and global relational consistency. Synthesizing constraint expressions. A sufficient condition for backtrack-free search A sufficient condition for backtrack-bounded search Experimental case studies of backtrack vs. waltz-type vs Increasing tree search efficiency for constraint satisfaction problems. A test for tractability. Fast parallel constraint satisfaction. Personal Communication. Consistency in networks of relations. The logic of constraint satisfaction. Solving large-scale constraint satisfaction and scheduling problems using a heuristic repair method Networks of constraints: Fundamental properties and applications to picture processing. Constraint satisfaction algorithms. Hybrid algorithms for the constraint satisfaction problem. On the complexity of achieving k-consistency Foundations of Constraint Satisfaction. Principles of Database and Knowledge-Base Systems On the inherent level of local consistency in constraint net- works Constraint tightness versus global consis- tency On the minimality and global consistency of row-convex constraint networks Using deep structure to locate hard problems. --TR A sufficient condition for backtrack-bounded search Principles of database and knowledge-base systems, Vol. I Network-based heuristics for constraint-satisfaction problems Tree clustering for constraint networks (research note) An optimal <italic>k</>-consistency algorithm Enhancement schemes for constraint processing: backjumping, learning, and cutset decomposition Constraint satisfaction algorithms From local to global consistency The logic of constraint satisfaction Fast parallel constraint satisfaction On the inherent level of local consistency in constraint networks Characterising tractable constraints Experimental evaluation of preprocessing algorithms for constraint satisfaction problems On the minimality and global consistency of row-convex constraint networks Local and global relational consistency A Sufficient Condition for Backtrack-Free Search Synthesizing constraint expressions On the Complexity of Achieving K-Consistency --CTR Yuanlin Zhang , Roland H. C. Yap, Erratum: P. van Beek and R. Dechter's theorem on constraint looseness and local consistency, Journal of the ACM (JACM), v.50 n.3, p.277-279, May Yuanlin Zhang , Roland H. C. Yap, Consistency and set intersection, Eighteenth national conference on Artificial intelligence, p.971-972, July 28-August 01, 2002, Edmonton, Alberta, Canada Amnon Meisels , Andrea Schaerf, Modelling and Solving Employee Timetabling Problems, Annals of Mathematics and Artificial Intelligence, v.39 n.1-2, p.41-59, September Paolo Liberatore, Monotonic reductions, representative equivalence, and compilation of intractable problems, Journal of the ACM (JACM), v.48 n.6, p.1091-1125, November 2001 Moshe Y. Vardi, Constraint satisfaction and database theory: a tutorial, Proceedings of the nineteenth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, p.76-85, May 15-18, 2000, Dallas, Texas, United States Henry Kautz , Bart Selman, The state of SAT, Discrete Applied Mathematics, v.155 n.12, p.1514-1524, June, 2007

relations;constraint networks;constraint-based reasoning;constraint satisfaction problems;local consistency

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Effective erasure codes for reliable computer communication protocols.

Reliable communication protocols require that all the intended recipients of a message receive the message intact. Automatic Repeat reQuest (ARQ) techniques are used in unicast protocols, but they do not scale well to multicast protocols with large groups of receivers, since segment losses tend to become uncorrelated thus greatly reducing the effectiveness of retransmissions. In such cases, Forward Error Correction (FEC) techniques can be used, consisting in the transmission of redundant packets (based on error correcting codes) to allow the receivers to recover from independent packet losses.Despite the widespread use of error correcting codes in many fields of information processing, and a general consensus on the usefulness of FEC techniques within some of the Internet protocols, very few actual implementations exist of the latter. This probably derives from the different types of applications, and from concerns related to the complexity of implementing such codes in software. To fill this gap, in this paper we provide a very basic description of erasure codes, describe an implementation of a simple but very flexible erasure code to be used in network protocols, and discuss its performance and possible applications. Our code is based on Vandermonde matrices computed over GF(pr), can be implemented very efficiently on common microprocessors, and is suited to a number of different applications, which are briefly discussed in the paper. An implementation of the erasure code shown in this paper is available from the author, and is able to encode/decode data at speeds up to several MB/s running on a Pentium 133.

Introduction Computer communications generally require reliable 1 data transfers among the communicating parties. This is usually achieved by implementing reliability at different levels in the protocol This paper appears on ACM Computer Communication Review, Vol.27, n.2, Apr.97, pp.24-36. y The work described in this paper has been supported in part by the Commission of European Communities, Esprit Project LTR 20422 - "Moby Dick, The Mobile Digital Companion (MOBYDICK)", and in part by the Ministero dell'Universit'a e della Ricerca Scientifica e Tecnologica of Italy. 1 Throughout this paper, with reliable we mean that data must be transferred with no errors and no losses. stack, either on a link-by-link basis (e.g. at the link layer), or using end-to-end protocols at the transport layer (such as TCP), or directly in the application. ARQ (Automatic Repeat reQuest) techniques are generally used in unicast protocols: missing packets are retransmitted upon timeouts or explicit requests from the receiver. When the bandwidth-delay product approaches the sender's window, ARQ might result in reduced throughput. Also, in multicast communication protocols ARQ might be highly inefficient because of uncorrelated losses at different (groups of) receivers. In these cases, Forward Error Correction possibly combined with ARQ, become useful: the sender prevents losses by transmitting some amount of redundant informa- tion, which allow the reconstruction of missing data at the receiver without further interactions. Besides reducing the time needed to recover the missing packets, such an approach generally simplifies both the sender and the receiver since it might render a feedback channel unnecessary; also, the technique is attractive for multicast applications since different loss patterns can be recovered from using the same set of transmitted data. FEC techniques are generally based on the use of error detection and correction codes. These codes have been studied for a long time and are widely used in many fields of information process- ing, particularly in telecommunications systems. In the context of computer communications, error detection is generally provided by the lower protocol layers which use checksums (e.g. Cyclic Redundancy Checksums (CRCs)) to discard corrupted packets. Error correcting codes are also used in special cases, e.g. in modems, wireless or otherwise noisy links, in order to make the residual error rate comparable to that of dedicated, wired connections. After such link layer processing, the upper protocol layers have mainly to deal with erasures, i.e. missing packets in a stream. Erasures originate from uncorrectable errors at the link layer (but those are not frequent with properly designed and working hardware), or, more frequently, from congestion in the network which causes otherwise valid packets to be dropped due to lack of buffers. Erasures are easier to deal with than errors since the exact position of missing data is known. Recently, many applications have been developed which use multicast communication. Some of these applications, e.g. audio or videoconferencing tools, tolerate segment losses with a relatively graceful degradation of performance, since data blocks are often independent of each other and have a limited lifetime. Others, such as electronic whiteboards or diffusion of circular information over the network ("electronic newspapers", distribution of software, etc), have instead more strict requirements and require reliable delivery of all data. Thus, they would greatly benefit from an increased reliability in the communication. Despite an increased need, and a general consensus on their usefulness [4, 10, 14, 19] there are very few Internet protocols which use FEC techniques. This is possibly due to the existence of a gap between the telecommunications world, where FEC techniques have been first studied and developed, and the computer communications world. In the former, the interest is focused on error correcting codes, operating on relatively short strings of bits and implemented on dedicated hardware; in the latter, erasure codes are needed, which must be able to operate on packet-sized data objects, and need to be implemented efficiently in software using general-purpose processors. In this paper we try to fill this gap by providing a basic description of the principles of operation of erasure codes, presenting an erasure code which is easy to understand, flexible and efficient to implement even on inexpensive architectures, and discussing various issues related to its performance and possible applications. The paper is structured as follows: Section 2 gives a brief introduction to the principles of operation of erasure codes. Section 3 describes our code and discusses some issues related to its implementation on general purpose processors. Finally, Section 4 briefly shows a number of possible applications in computer communication protocols, both in unicast and multicast protocols. A portable C implementation of the erasure code described in this paper is available from the author [16]. An introduction to erasure codes In this section we give a brief introduction to the principle of operation of erasure codes. For a more in-depth discussion of the problem the interested reader is referred to the copious literature on the subject [1, 11, 15, 20]. In this paper we only deal with the so-called linear block codes as they are simple and appropriate for the applications of our interest. The key idea behind erasure codes is that k blocks of source data are encoded at the sender to produce n blocks of encoded data, in such a way that any subset of k encoded blocks suffices to reconstruct the source data. Such a code is called an (n; code and allows the receiver to recover from up to n \Gamma k losses in a group of n encoded blocks. Figure 1 gives a graphical representation of the encoding and decoding process. Encoder source data Decoder reconstructed data encoded data received data Figure 1: A graphical representation of the encoding/decoding process. Within the telecommunications world, a block is usually made of a small number of bits. In computer communications, the "quantum" of information is generally much larger - one packet of data, often amounting to hundreds or thousands of bits. This changes somewhat the way an erasure code can be implemented. However, in the following discussion we will assume that a block is a single data item which can be operated on with simple arithmetic operations. Large packets can be split into multiple data items, and the encoding/decoding process is applied by taking one data item per packet. An interesting class of erasure codes is that of linear codes, so called because they can be analyzed using the properties of linear algebra. Let be the source data, G an n \Theta k matrix, then an (n; linear code can be represented by for a proper definition of the matrix G. Assuming that k components of y are available at the receiver, source data can be reconstructed by using the k equations corresponding to the known components of y. We call G 0 the k \Theta k matrix representing these equations (Figure 2). This of course is only possible if these equations are linearly independent, and, in the general case, this holds if any k \Theta k matrix extracted from G is invertible. If the encoded blocks include a verbatim copy of the source blocks, the code is called a systematic code. This corresponds to including the identity matrix I k in G. The advantage of a systematic code is that it simplifies the reconstruction of the source data in case one expects very few losses. G n0Decoder Encoder G Figure 2: The encoding/decoding process in matrix form, for a systematic code (the top k rows of G constitute the identity matrix I k ). y 0 and G 0 correspond to the grey areas of the vector and matrix on the right. 2.1 The generator matrix G is called the generator matrix of the code, because any valid y is a linear combination of columns of G. Since G is an n \Theta k matrix with rank k, any subset of k encoded blocks should convey information on all the k source blocks. As a consequence, each column of G can have at most k \Gamma 1 zero elements. In the case of a systematic code G contains the identity matrix I k , which consumes all zero elements. Thus the remaining rows of the matrix must all contain non-zero elements. Strictly speaking, the reconstruction process needs some additional information - namely, the identity of the various blocks - to reconstruct the source data. However, this information is generally derived by other means and thus might not need to be transmitted explicitly. Also, in the case of computer communications, this additional information has a negligible size when compared to the size of a packet. There is however another source of overhead which cannot be neglected, and this is the precision used for computations. If each x i is represented using b bits, representing the y i 's requires more bits if ordinary arithmetic is used. In fact, if each coefficient g ij of G is represented on b 0 bits, the y i 's need b+b bits to be represented without loss of precision. That is a significant overhead, since those excess bits must be transmitted to reconstruct the source data. Rounding or truncating the representation of the y i 's would prevent a correct reconstruction of the source data. 2.2 Avoiding roundings: computations in finite fields Luckily the expansion of data can be overcome by working in a finite field. Roughly speaking, a field is a set in which we can add, subtract, multiply and divide, in much the same way we are used to work on integers (the interested reader is referred to some textbook on algebra [6] or coding theory (e.g. [1, Ch.2 and Ch.4]), where a more formal presentation of finite fields is provided; a relatively simple-to-follow presentation is also given in [2, Chap.2]). A field is closed under addition and multiplication, which means that the result of sums and products of field elements are still field elements. A finite field is characterized by having a finite number of elements. Most of the properties of linear algebra apply to finite fields as well. The main advantage of using a finite field, for our purposes, lies in the closure property which allows us to make exact computations on field elements without requiring more bits to represent the results. In order to work on a finite field, we need to map our data elements into field elements, operate upon them according to the rules of the field, and then apply the inverse mapping to reconstruct the desired results. 2.2.1 Prime fields Finite fields have been shown to exist with is a prime number. Fields with p elements, with p prime, are called prime fields or GF (p), where GF stands for Galois Field. Operating in a prime field is relatively simple, since GF (p) is the set of integers from 0 to under the operations of addition and multiplication modulo p. From the point of view of a software implementation, there are two minor difficulties in using a prime field: first, with the exception of bits to be represented. This causes a slight inefficiency in the encoding of data, and possibly an even larger inefficiency in operating on these numbers since the operand sizes might not match the word size of the processor. The second problem lies in the need of a modulo operation on sums and, especially, multiplications. The modulo is an expensive operation since it requires a division. Both problems, though, can be minimized if 2.2.2 Extension fields Fields with prime and r ? 1, are called extension fields or GF (p r ). The sum and product in extension fields are not done by taking results modulo q. Rather, field elements can be considered as polynomials of degree r \Gamma 1 with coefficients in GF (p). The sum operation is just the sum between coefficients, modulo p; the product is the product between polynomials, computed modulo an irreducible polynomial (i.e. one without divisors in GF (p r of degree r, and with coefficients reduced modulo p. Despite the apparent complexity, operations on extension fields can become extremely simple in the case of 2. In this case, elements of GF(2 r ) require exactly r bits to be represented, a property which simplifies the handling of data. Sum and subtraction become the same operation (a bit-by-bit sum modulo 2), which is simply implemented with an exclusive OR. 2.2.3 Multiplications and divisions An interesting property of prime or extension fields is that there exist at least one special element, usually denoted by ff, whose powers generate all non-zero elements of the field. As an example, a generator for GF (5) is 2, whose powers (starting from 2 0 ) are of ff repeat with a period of length This property has a direct consequence on the implementation of multiplication and division. In fact, we can express any non-zero field element x as can be considered as "logarithm" of x, and multiplication and division can be computed using logarithms, as follows: where jaj b stands for "a modulo b". If the number of field elements not too large, tables can be built off line to provide the "logarithm", the "exponential" and the multiplicative inverse of each non-zero field element. In some cases, it can be convenient to provide a table for multiplications as well. Using the above techniques, operations in extension fields with can be extremely fast and simple to implement. 2.3 Data recovery Recovery of original data is possible by solving the linear system where x is the source data and y 0 is a subset of k components of y available at the receiver. Matrix G 0 is the subset of rows from G corresponding to the components of y 0 . It is useful to solve the problem in two steps: first G 0 is inverted, then This is because the cost of matrix inversion can be amortized over all the elements which are contained in a packet, becoming negligible in many cases. The inversion of G 0 can be done with the usual techniques, by replacing division with multiplication by the inverse field element. The cost of inversion is O(kl 2 ), where l - is the number of data blocks which must be recovered (very small constants are involved in our use of the O() notation). Reconstructing the l missing data blocks has a total cost of O(lk) operations. Provided sufficient resources, it is not impossible to reconstruct the missing data in constant time, although this would be pointless since just receiving the data requires O(k) time. Many implementations of error correcting codes use dedicated hardware (either hardwired, or in the form of a dedicated processor) to perform data reconstruction with the required speed. 3 An erasure code based on Vandermonde matrices A simple yet effective way to build the generator matrix, G, consists in using coefficients of the where the x i 's are elements of GF (p r ). Such matrices are commonly known as Vandermonde matrices, and their determinant is Y If all x i 's are different, the matrix has a non-null determinant and it is invertible. Provided can be constructed, which satisfy the properties required for G. Such matrices can be extended with the identity matrix I k to obtain a suitable generator for a systematic code. Note that there are some special cases of the above code which are of trivial implementation. As an example, an (n; 1) code simply requires the same data to be retransmitted multiple times, hence there is no overhead involved in the encoding. Another simple case is that of a systematic code, where the only redundant block is simply the sum (as defined in GF (p r )) of the k source data blocks, i.e. a simple XOR in case 2. Unfortunately, an (n; 1) code has a low rate and is relatively inefficient compared to codes with higher values of k. Conversely, a code is only useful for small amount of losses. So, in many cases there is a real need for codes with k ? 1 and We have written a portable C implementation of the above code [16] to determine its performance when used within network protocols. Our code supports any r in the range arbitrary packet sizes. The maximum efficiency can be achieved using this allows most operations to be executed using table lookups. The generator matrix has the form indicated above, with x . We can build up to 2 rows in this way, which makes it possible to construct codes up to In our experiments we have used a packet size of 1024 bytes. 3.1 Performance Using a systematic code, the encoder takes groups of k source data blocks to produce redundant blocks. This means that every source data block is used times, and we can expect the encoding time to be a linear function of n \Gamma k. It is probably more practical to measure the time to produce a single data block, which depends on the single parameter k. It is easy to derive that this time is (for sufficiently large packets) linearly dependent on k, hence we can approximate it as encoding c e where the constant c e depends on the speed of the system. The above relation only tells us how fast we can build redundant packets. If we use a systematic code, sending k blocks of source data requires the actual computation of blocks. Thus, the actual encoding speed becomes encoding speed = c e Note that the maximum loss rate that we can sustain is n\Gammak n , which means that, for a given maximum loss rate, the encoding speed also decreases with n. Decoding costs depend on l - min(k; n \Gamma k), the actual number of missing source blocks. Although matrix inversion has a cost O(kl 2 ), this cost is amortized over the size s of a packet; we have found that, for reasonably sized packets (say above 256 bytes), and k up to 32, the cost of matrix inversion becomes negligible compared to the cost of packet reconstruction, which is O(lk). Also for the reconstruction process it is more practical to measure the overall cost per reconstructed block, which is similar to the encoding cost. Then, the decoding speed can be written as decoding speed = c d l with the constant c d slightly smaller than c e because of some additional overheads (including the already mentioned matrix inversion). The accuracy of the above approximations has been tested on our implementation using a packet size of 1024 bytes, and different values of k and l shown in Table 1 (more detailed performance data can be found in [17]). Running times have been determined using a Pentium 133 running FreeBSD, using our code compiled with gcc -O2 and no special optimizations. These experimental results show that the approximation is sufficiently accurate. Also, the values of c e and c d are sufficiently high to allow these codes to be used in a wide range of applications, depending on the actual values of k and l k. The reader will notice that, for a given k, larger values of l (which we have set equal to n \Gamma slightly better performance both in encoding and decoding. On the encoder side this is exclusively due to the effect of caching: since the same source data are used several times to compute multiple redundant blocks, successive computations find the operands already in cache hence running slightly faster. For the decoder, this derives from the amortization of matrix inversion costs over a larger number Encoding Decoding -s MB/s -s MB/s 28 3533 9.06 Table 1: Encoding/decoding times for different values of k and n \Gamma k on a Pentium 133 running FreeBSD of reconstructed blocks 2 . Note that in many cases data exchanged over a network connection are already subject to a small number of copies (e.g. from kernel to user space) and accesses to compute check- sums. Thus, part of the overhead for reconstructing missing data might be amortized by using integrated layer processing techniques [3]. 3.2 Discussion The above results show that a software implementation of erasure codes is computationally expensive, but on today's machines they can be safely afforded with little overhead for low-to- medium speed applications, up to the 100 KB/s range. This covers a wide range of real-time applications including network whiteboards and audio/video conferencing tools, and can even be used to support browsing-type applications. More bandwidth-intensive applications can still make good use of software FEC techniques, with a careful tuning of operating parameters (specifically, our discussion) or provided sufficient processing power is available. The current trend of increasing processing speeds, and the availability of Symmetric MultiProcessor (SMP) desktop computers suggest that, as time goes by, there will likely be plenty of processing power to support these computations (we have measured values for c d and c e in the 30MB/s range on faster machines based on PentiumPRO 200 and UltraSparc processors). Finally, note that in many cases both encoding and decoding can be done offline, so many non-real-time application can use this feature and apply FEC techniques while communicating at much higher speeds than their encoding/decoding ability. 2 and a small overhead existing in our implementation for non reconstructed blocks which are still copied in the reconstruction process Applications Depending on the application, ARQ and FEC can be used separately or together, and in the latter case either on different layers or in a combined fashion. In general, there is a tradeoff between the improved reliability of FEC-based protocols and their higher computational costs, and this tradeoff often dictates the choice. It is beyond the scope of this paper to make an in-depth analysis of the relative advantages of FEC, ARQ or combinations thereof. Such studies are present in some papers in the literature (see, for example, [7, 12, 21]). In this section we limit our interest to computer networks, and present a partial list of applications which could benefit from the use of an encoding technique such as the one described in this paper. The bandwidth, reliability and congestion control requirements of these applications vary widely. Losses in computer networks mainly depend on congestion, and congestion is the network analogue of noise (or interference) in telecommunications systems. Hence, FEC techniques based on a redundant encoding give us similar types of advantages, namely increased resilience to noise and interference. Depending on the amount of redundancy, the residual packet loss rate can be made arbitrarily small, to the point that reliable transfers can be achieved without the need for a feedback channel. Or, one might just be interested in a reduction of the residual loss rate, so that performance is generally improved but feedback from the receiver is still needed. 4.1 Unicast applications In unicast applications, reducing the amount of feedback necessary for reliable delivery is generally useful to overcome the high delays incurred with ARQ techniques in the presence of long delay paths. Also, these techniques can be used in the presence of asymmetrical communication links. Two examples are the following: ffl forward error recovery on long delay paths. TCP communications over long fat pipes suffer badly from random packet losses because of the time needed to get feedback from the receiver. Selective acknowledgements [13] can help improve the situation but only after the transmit window has opened wide enough, which is generally not true during connection startup and/or after an even short sequence of lost packets. To overcome this problem it might be useful to allocate (possibly adaptively, depending on the actual loss rate) a small fraction of the bandwidth to send redundant packets. The sender could compute a small number (1-2) of redundant packets on every group of k packets, and send these packets at the end of the group. In case of a single or double packet loss the receiver could defer the transmission of the dup ack until the expiration of a (possibly fast) timeout 3 . If, by that time, the group is complete and some of the redundant packets are available, then the missing one(s) can be recovered without the need for an explicit retransmission (this this would be equivalent to a fast retransmit). Otherwise, the usual congestion avoidance techniques can be adopted. A variant of RFC1323 timestamps[5] 3 alternatively, the sender could delay retransmissions in the hope that the lost packet can be recovered using the redundant packets. can be used to assign sequence numbers to packets thus allowing the receiver to determine the identity of received packets and perform the reconstruction process (TCP sequence numbers are not adequate for the purpose). ffl power saving in communication with mobile equipment Mobile devices usually adopt wireless communication and have a limited power budget. This results in the need to reduce the number of transmissions. A redundant encoding of data can practically remove the need for acknowledgements while still allowing for reliable communications. As an example, a mobile browser can limit its transmissions to requests only, while incoming responses need not to be explicitly ACKed (such as it is done currently with HTTP over TCP) unless severe losses occur. 4.2 Multicast applications The main field of application of redundant encoding is probably in multicast applications. Here, multiple receivers can experience losses on different packets, and insuring reliability via individual repairs might become extremely expensive. A second advantage derives from the aforementioned reduced need for handling a feedback channel from receivers. Reducing the amount of feedback is an extremely useful feature since it allows protocols to scale well to large numbers of receivers. Applications not depending on a reliable delivery can still benefit from a redundant en- coding, because an improved reliability in the transmission allows for more aggressive coding techniques (e.g. compression) which in turn might result in a more effective usage of the available bandwidth. A list of multicast applications which would benefit from the use of a redundant encoding follows. videoconferencing tools. A redundant encoding with small values of k and can provide an effective protection against losses in videoconferencing applications. By reducing the effective loss rate one can even use a more efficient encoding technique (e.g. fewer "I" frames in MPEG video) which provide a further reduction in the bandwidth. The PET [9] group at Berkeley has done something similar for MPEG video. reliable multicast for groupware. A redundant encoding can be used to greatly reduce the need for retransmissions ("repairs") in applications needing a reliable multicast. One such example is given by the "network whiteboard" type of applications, where reliable transfer is needed for objects such as Postscript files or compound drawings. ffl one-to-many file transfer on LANs. Classrooms using workstations often use this pattern of access to files, either in the booting process (all nodes download the kernel or startup files from a server) or during classes (where students download almost simultaneously the same documents or applications from a centralized server). While these problems can be partly overcome by preloading the software, centralized management is much more convenient and the use of a multicast-FTP type of application can make the system much more scalable. ffl one-to-many file transfer on Wide Area Networks. There are several examples of such an application. Some popular Web servers are likely to have many simultaneous transfers of the same, large, piece of information (e.g. popular software packages). The same applies to, say, a newspaper which is distributed electronically over the network, or video-on-demand type of applications. Unlike local area multicast-FTP, receivers connect to the server at different times, and have different bandwidths and loss rates, and significant congestion control issues exist [8]. By using the encoding presented here, source data can be encoded and transmitted with a very large redundancy (n ?? k). Using such a technique, a receiver basically needs only to collect a sufficient number (k) of packets per block to reconstruct the original file. The RMDP protocol [18] has been designed and implemented by the author using the above technique. Acknowledgements The author wishes to thank Phil Karn for discussions which led to the development of the code described in this paper, and an anonymous referee for comments on an early version of this paper. --R "Theory and Practice of Error Control Codes" "Fast Algorithms for Digital Signal Processing" "Architectural Considerations for a New Generation of Proto- cols" "The Case for packet level FEC" "RFC1323: TCP Extensions for High Performance" "Algebra" "Delay Bounded Type-II Hybrid ARQ for Video Transmission over Wireless Networks" "Receiver-driven Layered Multicast" "Priority Encoding Transmission" "Reliable Broadband Communication Using A Burst Erasure Correcting Code" "Error Control Coding: Fundamentals and Applications" "Automatic-repeat-request error-control schemes" "RFC2018: TCP Selective Acknowledgement Option" "Reliable Multicast: Where to use Forward "Introduction to Error-Correcting Codes" Sources for an erasure code based on Vandermonde matrices. 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FEC;reliable multicast;erasure codes

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Possibilities of using protocol converters for NIR system construction.

Volumes of information available from network information services have been increasing considerably in recent years. Users' satisfaction with an information service depends very much on the quality of the network information retrieval (NIR) system used to retrieve information. The construction of such a system involves two major development areas: user interface design and the implementation of NIR protocols.In this paper we describe and discuss the possibilities of using formal methods of protocol converter design to construct the part of an NIR system client that deals with network communication. If this approach is practicable it can make implementation of NIR protocols more reliable and amenable to automation than traditional designs using general purpose programming languages. This will enable easy implementation of new NIR protocols custom-tailored to specialized NIR services, while the user interface remains the same for all these services.Based on a simple example of implementing the Gopher protocol client we conclude that the known formal methods of protocol converter design are generally not directly applicable for our approach. However, they could be used under certain circumstances when supplemented with other techniques which we propose in the discussion.

Introduction Important terms used in this paper are illustrated in Fig. 1. An information provider makes some information available to a user through a network by means of a network information retrieval (NIR) system, thereby server client user information provider NIR system NIR protocol information service Figure 1: Network information retrieval (NIR) system. providing an information service. An NIR system consists of the provider's part (the server) and the user's part (the client) communicating via an (NIR) protocol. We use the term information retrieval to denote passing information, (a document or other form of informa- tion), from a repository to a user or from a user to a repository in general. We concentrate, however, on passing documents from a repository to a user. Some authors use the term information retrieval to denote a form of getting information from a repository where searching or some more intelligent query processing is involved [9, 15] or distinguish between document retrieval and information retrieval (retrieval of documents and information in other forms) [14]. Usually, a user wants to use a lot of information services available in order to get as much information relevant to his or her needs as possible. At the same time, a user wants to use all information services in a similar way. Good consistency of user interfaces for accessing various information services and for presenting documents of various types will decrease the cognition overhead - the additional mental effort that a user has to make concerning manipulation with a user interface, navigation, and orientation [25]. Consistency can be improved by applying standards for "look and feel" of user interface ele- ments, by using gateways between information services, and, most importantly, by using integrated NIR tools. An integrated NIR tool is a multiprotocol client designed to communicate directly with servers of various information services. This approach has become very popular recently, since it provides a more consistent environment than a set of individual clients and does not have the disadvantages of gateways in the matter of network communication overhead and limited availability. Nevertheless, current integrated NIR tools often suffer from important drawbacks: ffl They support a limited number of NIR protocols and adding a new protocol is not straightforward. ffl Implementation of NIR protocols is not performed systematically, which means that it is not possible to make any formal verification of such a system. ffl Their portability is limited, and porting to another type of a window system is laborious. Constructing such tools involves two major development areas: user interface design and implementation of NIR protocols. In the next two chapters we will briefly describe some of the techniques and formalisms used in these development areas. Then we will mention the role of protocol conversion in network communication and shortly describe three formal methods of protocol converter design. In the rest of the paper, we will demonstrate the application of the protocol conversion techniques to a simple protocol, the Gopher protocol. Despite the acceptance of these techniques, our analysis demonstrates the negative conclusion that these techniques alone are inadequate for a protocol even as simple as the Gopher protocol. We conclude with some suggestions for enhancements. User Interface Design and Layered Models When designing a user interface various models and formal description techniques are used. An important class of models is formed by layered models in which communication between a user and an application is divided into several layers. Messages exchanged at higher layers are conveyed by means of messages exchanged at lower layers. An example of this kind of model is the seven-layer model described by Nielsen in [16] shown in Fig. 2 along with an indication of what kind of information is exchanged at each layer. Various formalisms can be used to describe the behaviour of the system at individual layers. Tradition- ally, most attention has been paid to the syntactic layer. Among the formal description techniques used at this layer the most popular are context-free grammars, finite-state automata, and event-response languages. Taylor [23, 24] proposed a layered model where different layers are conceived as different levels of abstraction in com- Physical communication6417532 Human Goal layer Task layer Lexical layer Physical layer Semantic layer Syntactic layer Alphabetic layer Computer system Virtual communication bered line Delete a num- Delete six lines of the edited text Remove section of my letter Example functionality Tokens Sentences Lexemes Detailed Real-world concepts concepts System Physical information Depressing the "D"-key Exchanged units Figure 2: Seven-layer model for human-computer inter-action munication between the user and the application. Lex- ical, syntactic, and semantic characteristics are spread within all levels. It appears that the layered structure is natural to human-computer interaction and implementations of user interfaces should take this observation into consideration. 3 Network architectures and layered models OSI model The ISO Open System Interconnection (OSI) model [6] divides a network architecture into seven layers (Fig. 3). Each layer provides one or more services to the layer above it. Within a layer, services are implemented by means of protocols. Many current network architectures are based on OSI model layering. Although not all layers are always implemented and functions of multiple layers can be grouped into one layer, dividing communication functionality into layers remains the basic principle. Protocol specification Formal techniques for protocol specification can be classified into three main cate- gories: transition models, programming languages, and combinations of the first two. Transition models are motivated by the observation that protocols consist largely of processing commands from the user (from the higher level), message arrivals (from the lower layer), and internal timeouts. In this paper, we will use finite state machines (graph- ically depicted as state-transition diagrams) which seem Session layer protocols Application layer protocols Presentation Transport layer protocols Presentation layer protocols Network layer protocols Data link protocols Physical layer protocols Presentation Application Transport Network Data link Physical Application Transport Network Data link Physical Figure 3: OSI model. -d0, -ls -tm +ls -d1, -ls -d0, -ls +ls tm, +a0 tm, +a0 -tm Figure 4: The alternating-bit (AB) protocol. to be the most convenient specification technique for our application domain. An example of the state-transition diagram depicting the alternating-bit (AB) protocol [1] is shown in Fig. 4. Data and positive acknowledgment messages exchanged between two entities are stamped with modulo 2 sequence numbers. d0 and d1 denote data messages, a0 and a1 denote acknowledgment messages, a plus sign denotes receiving a message from the other party, a minus sign denotes sending a message to the other party, a double plus sign (++) denotes getting a message from the user (ac- ceptance), a double minus sign (\Gamma\Gamma) denotes putting a message to the user (delivery), ls and tm denote loss of a data message and timeout, respectively. 4 Protocol Converters In network communication we can encounter the following problem. Consider that we have two entities P 0 and a) b) Service Sq Service S Figure 5: a) Communicating entities, b) Using a protocol converter to allow different entities to communicate. communicating by means of a protocol P (thus providing a service S p ) and other two entities Q 0 and Q 1 communicating by means of a protocol Q (thus providing a service S q ), see Fig. 5 a. Then we might want to make thus providing a service S similar to services S p and S q . When protocols P and are compatible enough, this can be achieved by a protocol converter C, which translates messages sent by P 0 into messages of the protocol Q, forwards them to Q 1 , and performs a similar translation in the other direction, see Fig. 5 b. To solve the problem of constructing a protocol converter C, given specifications of P 0 , Q 1 , and S, several more or less formal methods have been developed. Most of them accept specifications of protocol entities in the form of communicating finite-state machines. We give a brief description of the principles used by three important methods. Conversion via projection An image protocol may be derived from a given protocol by partitioning the state set of each communicating entity; states in the same block of the partition are considered to be indistinguishable in the image of that entity. Suppose protocols P and Q can be projected onto the same image protocol, say R. R embodies some of the functionality that is common to both P and Q. The specification of a converter can be derived considering that the projection mapping defines an equivalence between messages of P and Q, just as it does for states. Finding a common image protocol with useful properties requires a heuristic search using intuitive understanding of the protocols. For more details, refer to [13, 5]. The conversion seed approach This approach was first presented in [17]. From the service specification S, a conversion seed X is (heuristically) constructed. The seed X is a finite-state machine whose message set is a subset of the union of the message sets of P 1 and Q A Figure The quotient problem. a partial specification of the converter's behaviour in the form of constraints on the order in which messages may be sent and received by the converter. Then a three-step algorithm [5] is run on P 1 , Q 0 , and X. If a converter C is produced (the algorithm generates a non-empty output), the system comprising P should be analyzed. If this system satisfies the specification S, then C is the desired converter. Otherwise, a different iteration with a different seed could be performed. Unfortunately, if the algorithm fails to produce a converter, it is hard to decide whether the problem was in the seed X used or if there was a hard mismatch between the P and Q protocols. The quotient approach Consider the problem depicted in Fig. 6. Let A be a service specification and let B specify one component of its implementation. The goal is to specify another component C, which interacts with B via its internal interface, so that the behaviour observed at B's external interface implements the service defined by A. This is called the quotient problem. It is clear that Fig. 5 b depicts a form of the quotient problem: correspond to B, S corresponds to A, and the converter to be found is C. An algorithmic solution of the quotient problem for a class of input specifications was presented in [4, 5]. A similar problem has been discussed in [12]. 5 NIR System Design Proposal 5.1 Information Retrieval Cycle A user wants to receive required information easily, quickly, and in satisfactory quality. Although there are many differences in details, the process of obtaining information when using most NIR services can be outlined by the information retrieval cycle shown in Fig. 7. At each step there is an indication whose turn it is, either user's (U) or system's (S). First, the user has an idea about required information, such as "I would like to get some article on X written by Y". Next, the user has to choose an information service and a source of information (a particular site which offers the chosen information service) to exploit. Then, the user has to formulate a query specifying the required information in a form that the chosen informa- Form an idea about required information U Choose a source of information Choose an information service Formulate a query Find required information Present information Is presented information satisfactory ? U U U U yes no Figure 7: Information retrieval cycle. tion service understands and that is sufficient for it to find the information. Now, it is the computer system's task to find the information and present it to the user. There are four possible results: ffl information was found and presented in satisfactory quality ffl information was found and presented, but the user is not satisfied with it ffl no information was found using the given description ffl the given description was not in a form understandable by the given information service When some information was found and presented to the user, but it is not to the user's satisfaction and he or she believes that better information could be obtained, the next iteration in the information retrieval cycle can be exercised. According to the measure of the user's dissatisfaction with the presented information, there are several possible points to return to. In some cases, different presentation of the same information would be satisfac- tory. In other cases, the user has to reformulate the query or choose another source of information or even another information service. Looking at Fig. 7, we can see that the user's role is much easier when a NIR system logically integrates access to various information services. This means that differences between individual information services should be diminished in all steps as much as possible. server client- protocol converter user Figure 8: Client as a protocol converter. menu, forms or event-response library user client server converter Figure 9: Client as a library and a protocol converter. 5.2 Possibilities of Employing Protocol Converters A client part of a NIR system may be seen as communicating via two sets of protocols. It uses one set of protocols to communicate with the user (see Fig. 2) and another set of protocols to communicate with the server (see Fig. 3). This may suggest an idea to us: can the client be constructed as a protocol converter (or a set of protocol converters in the case of a multiprotocol client) using some of the formal methods of protocol converter design? This idea is illustrated in Fig. 8. The first question that arises is: what are the protocols which the converter should transform? In a typical en- vironment, all lower layer protocols up to the transport layer are the same for all NIR services supported by the client. It is the upper layer NIR protocols that differ, usually based on the exchange of textual messages over a transport network connection (e.g., FTP, Gopher proto- col, SMTP, NNTP, HTTP). These protocols seem to be good candidates for the protocol on a server's side of the converter. In the case of a command language user interface, communication with the user can also be regarded as a protocol based on the exchange of textual messages. This could be a protocol on the user's side of the converter. If the user interface uses interaction techniques like menu hierarchy, form filling or currently the most popular direct manipulation, it may be implemented as a library that offers an interface in the form of a protocol similar to that of a command language. Again, a protocol converter could be employed as shown in Fig. 9. While the protocols on the server's side of the converter are given by the information services we want to support, the protocol on the other side is up to us to specify (if it is not directly the user - client protocol as in the configuration shown in Fig. 8). An important decision is the choice of the proper level of communication. Low level communication consisting of requests to display user interface elements and responses about user interac- user user interface . NIR protocol n converter 1 converter n server n server 1 GIR protocol Figure 10: NIR system based on the GIR protocol. tion can ease the development of the user interface but it would make the protocol converter construction more difficult because of a great semantic gap between the two protocols to be converted. We will try to find a level of communication which makes it possible to use one of the formal methods of protocol converter design that we mentioned before. 5.3 General Information Retrieval Pro- tocol Many protocols for client-server communication in current NIR services are similar to some extent. There are common functions that can be identified in most of them such as a request for an information object, sending the requested information object, sending an error message indicating that the requested information object cannot be retrieved, a request to modify an existing or to create a new information object, a request to search the contents of an information object, etc. It seems feasible that a high-level general information retrieval (GIR) protocol providing a high-level (GIR) service can be designed. Such a protocol has to support all major functions of individual NIR services. It would work with a global abstract information space formed by the union of information spaces of individual NIR services. This protocol operating on information objects from the global abstract information space would be converted by a set of protocol converters to particular NIR protocols operating on information objects from information spaces of concrete services. A structure of an NIR system built around the GIR protocol is shown in Fig. 10. Considering the structure of protocols used in current NIR services and respecting that the behaviour of the entity on the left side of the GIR protocol, the user interface, corresponds to the information retrieval cycle depicted in Fig. 7, we can propose a very simple GIR protocol depicted by the state transition diagrams U 0 (the client) and U 1 (the server) shown in Fig. 11. This version is certainly far from being complete and needs to be improved on the basis of later experience, see Chapter 7. The notation used corresponds to that described in Chapter 3. A dashed transition leading to state 1 matches the first step in the information retrieval cycle (Fig. 7). It represents a solely mental process with no interaction over the network and will not be considered in further discussion. A letter u in front of a message name means that it is a GIR protocol message (u stands for universal). Later we will use a letter g for Gopher protocol messages in order to distinguish them. The user chooses both an information service and a source of information in one step. It may be divided into two steps but one step better corresponds to picking up a bookmark or entering a document URL. The choice of information service would select a matched protocol con- verter. Sending all information to the server is delayed in the client until the user enters it completely. This allows backtracking in user input without having to inform the server about it. 5.4 General Window System Interface We may consider another possible usage of protocol converters in an NIR system. There is often a need to port such a system to several platforms - window sys- tems. Some developers try to make their applications more portable by performing the following steps: 1. common functions of all considered window systems.681302 U session session ++ u query ++ u new query - u query - u presenting - u response ++ u idea on required - u inf. service response or error information service ++ u inf. - u error ++ u end session ++ u new or source inf. service Figure 11: General information retrieval protocol. user server server NeWS converter 1 converter 2 NeWS protocol general window system interface application Figure 12: Using protocol converters to construct the general window system interface. 2. Define interface to a general window system that implements functions identified in step 1. 3. Implement the general window system on the top of all considered window systems. 4. Implement the application using the general window system. Some window systems are based on the client-server model (e.g., the X Window System or NeWS). We may try to realize step 3 above with a set of protocol converters converting client-server protocols of considered window systems to the protocol used by the application to communicate with the general window system defined in step 2. This idea is depicted in Fig. 12. Unfortunately, this idea is hardly feasible. The client-server protocols of today's window systems are usually complex and differ significantly from each other (and from a possible protocol of the general window system). Currently known methods of protocol converter design are suitable for protocols that are sufficiently compati- ble. Therefore, the general window system can be more easily implemented as a set of shim libraries built on the top of existing window systems (see Fig. 13). Examples of systems that use this approach are stdwin [20] and SUIT [18]. 5.5 Structure of the Proposed System The possible structure of a multi-service NIR system that uses shim libraries to adapt to various window systems and a set of protocol converters between the GIR protocol and individual NIR protocols is depicted in Fig. 14. The gap between the general window system interface and the GIR protocol is bridged by a module which implements the user interface. It can be designed either as a protocol converter, or in the case of difficulties when applying the formal methods described, as an event-response module written in a general programming language. Integration of information services that are not based on a distributed client-server architecture (with a local lib user Windows "server" general window system interface application lib shim curses lib character terminal shim lib server X lib shim lib sequences programmatic interface Figure 13: Using shim libraries to construct the general window system interface. client and a remote server) with a well-defined NIR proto- col, that have their own remote user interfaces accessible by a remote login, such as library catalogs and databases, can be achieved by using a new front end to the old user interface. This can be a protocol converter that converts the GIR protocol to sequences of characters that would be typed as input by a user when communicating directly with the old user interface, and that performs a similar conversion in the other direction. Quotation marks around the "server" for this type of information service in Fig. 14 indicate that such an information service may or may not be based on the client-server model. Although this front end seems to be subtle and vulnerable to changes of the old user interface, this approach has already been successfully used (but the front end was not implemented as a protocol converter). Examples are SALBIN [8] and IDLE [19]. 5.6 OSI Model Correspondence Fig. 15 illustrates how protocol layering in the proposed system corresponds to the OSI model. Internet information services are based on the TCP/IP protocol fam- user server server system 2 system 3 window system 1 server window window server 1 server 2 lib lib system 2 window window system 3 lib window system 1 lib lib shim shim lib shim module response event- general window system interface GIR protocol front interface old user "server" 3 converter 2 converter 1 Figure 14: A possible structure of a multi-service information retrieval system. Data link protocols Network layer protocols Transport layer protocols Physical layer protocols protocol server Virtual GIR User interface Converters User GIR protocol Application layer protocols Transport Data link Physical Network Transport Network Data link Physical Application Transport interface Figure 15: OSI model correspondence. ily, which provides no explicit support for functions of the session and presentation OSI layers. NIR protocols Gopher protocol, etc.) are, with respect to the OSI model, application layer protocols. Protocol converters in our system use application layer protocols (NIR protocols) on one side and the GIR protocol on the other side. They act as a client side implementation of the application layer and from the client's point of view they create another higher layer based on the GIR protocol. The server side of this higher layer is not actually implemented, it exists as a client side abstraction only. Being application layer implementation, the converters communicate with the transport layer, possibly over a transport interface module which converts an actual transport layer interface to a form suitable for the converter's input and output. On the upper side of the converters, there is the user interface which presents information services to the user. connect query -g response closed close - g closed, close query connected inf. source query error -g ls close, error response Figure Gopher protocol client (left) and server (right). 6 Example As an example of applying the approach described in Chapter 5, we will try to construct a protocol converter implementing the network communication part of the Gopher protocol client. We will try to use all three protocol converter design methods mentioned in Chapter 4 in order to decide which one is the most suitable and whether our approach is feasible at all. The Gopher protocol [2, 3] can be described by two finite-state machines, one for the client side (G 0 ), and the other for the server side (G 1 ), communicating via message passing. Corresponding state-transition diagrams are shown in Fig. 16. The notation used corresponds to that described in Chapter 3. Opening a connection and closing a connection between the client and the server are modeled by the exchange of virtual messages g connect , g connected , close, and g closed . The loss of a message sent by the client is represented by sending a virtual g ls message instead of a "normal" message to the server which then sends back a virtual g tm (timeout) message as a response. The loss of a message sent by the server is represented by sending a virtual g tm message only. If more than one message may be sent in a certain state, one is chosen non-deterministically (of course, with the exception of g response and g error messages, one of them is sent by the Gopher protocol server based on the result of a query processed). Our task is to construct a protocol converter which allows U 0 (the GIR protocol client) and G 1 (the Gopher I U 0 I G 0 2,3 - u query 7,85response or error - u presenting error error response Figure 17: Images of client finite-state machines. I U G I - u error query error 6,8service query Figure 18: Images of server finite-state machines. protocol server) to communicate. 6.1 Conversion via Projection When we try to project the finite-state machines U 0 and onto the common image, some of the best results we can get in terms of achieved similarity and retained functionality are the I U0 and I G0 images shown in Fig. 17. They are close to each other, but they are not exactly the same. To obtain one common image, the functionality would have to be further reduced, which is not acceptable. On the server side, there is a similar problem. We can get images I U1 and I G1 of U 1 and G 1 shown in Fig. 18. These are not exactly the same and, moreover, the common functionality is not sufficient. We would not be able to translate the g connected and g closed messages sent by the Gopher protocol server which the converter needs to receive. The conclusion is that conversion via projection is only suitable for protocols that are close enough, which is not our case. connected response closed Figure 19: Conversion seed S for U 0 - G 1 converter. -g ls response - u - u close connected error closed error response Figure converter produced from the conversion seed S. 6.2 Conversion Seed Approach For the conversion seed approach, G 0 has to be modified so that it contains only transitions that correspond to interaction with the peer entity G 1 (interaction with the user is not included). That is, the transitions from state 4 lead directly to state 7 and the transition from state 8 leads to state 1, which is the starting state. A simple conversion seed S is shown in Fig. 19. It defines constraints on the order in which messages may be received by the converter. Ordering relations between messages being sent and messages being received will be implemented in the converter by the algorithm which constructs the converter as a reduced finite-state machine of communicating with U 0 and a reduced finite-state machine of G 0 when communicating with G 1 [5]. The output of the algorithm for U 1 , G 0 , and S is shown in Fig. 20. In state 8, the converter has to decide whether to send the g response or the g error message to U 0 based on the receiving transition that was used to move from state 5 to state 6. This requires some internal memory and associated decision mechanism in the converter. We can conclude that the conversion seed approach is applicable to our example, but we have to construct a conversion seed heuristically using our knowledge of the converter operation.5 6 G 113+ g connect connected query database query database error response close closed response query connect query - g error close close closed database Figure 21: Gopher protocol server for the quotient approach 6.3 The quotient approach The algorithm based on the solution of the quotient problem described in [4, 5] uses a rendezvous model (as opposed to the message-passing model used in the previous two approaches), in which interaction between two components occurs synchronously via actions. An action can take place when both parties are ready for it. State changes happen simultaneously in both components. Transmission channels between the converter and other communicating entities are modeled explicitly as finite-state machines with internal transitions, which may or may not happen, representing loss of messages. After such a loss, a timeout event occurs at the sender end of the channel. Because of different modeling of message losses in the quotient approach, the state-transition diagram for the Gopher protocol server has to be slightly modified, as shown in Fig. 21. Virtual g ls and g tm messages are removed and new receive transitions are added to cope with duplicate messages sent by the converter. In our case, the converter is collocated with U 0 , meaning there are no losses in U 0 - C communication. We only have to model the C - G 1 channel (shown in Fig. 23), thereby obtaining the configuration shown in Fig. 22. The composition of U 0 , CG 1 chan, and G 1 forms the B part in the quotient problem (Fig. 6). The service specification A is shown in Fig. 24. After applying the algorithm on these inputs we get a converter which has 194 states and 590 transitions, too many to be presented here. Some of the states and their user Figure 22: Quotient approach configuration. close close query response error closed closed error connected response connect Figure ++ u end session database database - u presenting response or error response query database error ++ u new++ u new inf. ++ u end session ++ u end session query service & source source service & ++ u inf. ++ u query Figure 24: Service specification. associated transitions represent alternative sequences in which messages may be sent by the converter. For ex- ample, g close message (request for the Gopher protocol server to close the connection) may be sent by the converter before sending u response message (response for the GIR protocol client) or after it. These alternatives are redundant with respect to the function of the con- verter. Unfortunately, it seems to be difficult to remove them in an automatic manner. A more important problem is that some other states and transitions represent sequences which are not acceptable because the converter has not enough knowledge to form the content of a message to be sent. For exam- ple, g connect message (request for the Gopher protocol server to connect to it) can be sent only when the converter knows where to try to connect, that is after u inf. service and source message has been received, not before it. Transitions leading to such unacceptable sequences have to be eliminated by defining semantic relations between messages and enforcing them in run-time. In our example, two semantic relations are sufficient: service and source connect query The symbol ) means that a message on the right side can be sent or received after a message on the left side has been sent or received, as indicated by the plus and minus signs. Another problem with the quotient seeking algorithm is its computing complexity. Let SX be the set of states of an entity X and let jS X j be the number of states in this set. Then the state set of the quotient can grow exponentially with the upper bound of 2 jS A j\LambdajS B j states. In our configuration, the maximum number of states of B is equal to the product of the numbers of states of U 0 , Thus: In practice, however, the algorithm does not always use exponential time and space. In our example, the maximum number of states of C during computation was 330. We can conclude that the quotient approach, when supplemented with the definition of semantic relations between messages, is applicable to our example. It is more compute-intensive than the conversion seed approach but it is more systematic, and no heuristic constructions are required. In our example, we have shown that even the quotient approach, which is the most systematic of the currently known formal methods of protocol converter design, is not sufficient for our approach on its own. It has to be complemented with the definition of semantic relations between messages. In addition to this, there are several problems with using protocol converters in the proposed way which we have not yet consider: GIR protocol universality, message contents transformation, and covering details of network protocols. GIR protocol universality The GIR protocol suggested in Fig. 11 is too simple. It contains only two methods: set source and get object. A GIR protocol for use in real designs has to incorporate at least the following methods: get object metainformation, modify object, create new object, remove object, and search object. Message content transformation When a converter should send message X in response to receiving message Y, it may have to build the content of message X based on the content of message Y. We call this process the message content transformation. In our example we need to provide for the transformation of the content of the following messages: service and source ) g connect query response ) u response error ) u error There are several ways to formalize message content transformation. We briefly outline four possible approaches translation grammars We can conceive the set of all possible contents of messages on each side of the converter as a language, and individual message contents as sentences of this language. We can then formally describe the languages with two grammars (one for each side of the converter) and the transformation of sentences with two translation grammars (one for each direction). sequential rewriting system We can define a set of rewriting rules that specify how to get the content of an outgoing message beginning with the content of the corresponding incoming message. These rules would be applied sequentially on the message content using string matching in a similar manner as rules for mail header and envelope processing work in the sendmail program [21]. SGML link process The set of all possible contents of messages can be modeled using two SGML-based markup languages, one for each side of the converter. The transformation between them can then be performed as a pair of SGML link processes [10] (one for each direction) or as a pair of SGML tree transformation processes (STTP) defined within DSSSL [7] specifications for both languages. predicate-based rewriting system We can define a set of facts and rules in Prolog or in a similar logical language that specify relations between pieces of information in an incoming message and the corresponding outgoing message. We then begin with a framework of the outgoing message in the form of a term composed of unbound variables. When we apply the defined set of predicates using a rewriting system to this term, its variables become step by step bound to the values from the incoming message content. Further research will be required to find whether and under what conditions the proposed techniques could be used for message content transformation in our approach. Covering details of network protocols Another problem concerns covering details of network communication protocols such as port numbers, parallel connec- tions, and various options and parameters. It seems to be practicable to incorporate them into the protocol on the bottom side of the protocol converter (see Fig. 15) as variables in the content of exchanged messages or even as additional virtual messages recognized by the transport interface used. 8 Conclusion We have shown that formal methods of protocol converter design could under certain circumstances be used to construct the part of an NIR system client that deals with network communication. However, these methods are not sufficient on their own and have to be supplemented with other techniques for our approach to become practicable. In the discussion we have mentioned the most important problems and proposed possible solutions or directions for further research work. If these problems are resolved it will be possible to design many new specialized NIR services with their own custom-tailored protocols since implementation of these protocols can be done easily and reliably. New protocols could also be supported in a different way. Some new NIR services (e.g., HotJava [11]) intend to support new protocols by retrieving the code to implement them by a client from a server. Instead of retrieving the code, only the formal specification could be retrieved and the protocol would be implemented by the converter from the specification. --R "A note on reliable full-duplex transmission over half-duplex lines," The Internet Gopher protocol "Deriving a Protocol Converter: a Top-Down Method," "Formal methods for protocol conversion," "The OSI reference model," Information technology - Text and office systems - Document Style Semantics and Specification Language (DSSSL) "Some network tools for Internet gateway service," "Intelligent information retrieval: An introduction," The SGML Handbook The HotJava Browser: A White Paper "Modeling and optimization of hierarchical synchronous cir- cuits," "Protocol conversion," "Natural Language Processing for Information Retrieval," "A virtual protocol model for computer-human interaction," "A formal protocol conversion method," "Lessons learned from SUIT, the Simple User Interface Toolkit," "IDLE: Unified W3-access to interactive information servers," Designing the User Interface: Strategies for Effective Human-Computer interac- tion "Layered protocols for computer-human dialogue. I: Principles," "Layered protocols for computer-human dialogue. II: Some practical issues," "Hypermedia and Cognition: Designing for Com- prehension," --TR

network information services;network information retrieval;protocol conversion

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Scale-sensitive dimensions, uniform convergence, and learnability.

Learnability in Valiant's PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers. These laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables. Classes of real-valued functions enjoying such a property are also known as uniform Glivenko-Cantelli classes. In this paper, we prove, through a generalization of Sauer's lemma that may be interesting in its own right, a new characterization of uniform Glivenko-Cantelli classes. Our characterization yields Dudley, Gine, and Zinn's previous characterization as a corollary. Furthermore, it is the first based on a Gine, and Zinn's previous characterization as a corollary. Furthermore, it is the first based on a simple combinatorial quantity generalizing the Vapnik-Chervonenkis dimension. We apply this result to obtain the weakest combinatorial condition known to imply PAC learnability in the statistical regression (or agnostic) framework. Furthermore, we find a characterization of learnability in the probabilistic concept model, solving an open problem posed by Kearns and Schapire. These results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner's hypothesis class.

Introduction In typical learning problems, the learner is presented with a finite sample of data generated by an unknown source and has to find, within a given class, the model yielding best predictions on future data generated by the same source. In a realistic scenario, the information provided by the sample is incomplete, and therefore the learner might settle for approximating the actual best model in the class within some given accuracy. If the data source is probabilistic and the hypothesis class consists of functions, a sample size sufficient for a given accuracy has been shown to be dependent on different combinatorial notions of "dimension", each measuring, in a certain sense, the complexity of the learner's hypothesis class. Whenever the learner is allowed a low degree of accuracy, the complexity of the hypothesis class might be measured on a coarse "scale" since, in this case, we do not need the full power of the entire set of models. This position can be related to Rissanen's MDL principle [17], Vapnik's structural minimization method [22], and Guyon et al.'s notion of effective dimension [11]. Intuitively, the "dimension" of a class of functions decreases as the coarseness of the scale at which it is measured increases. Thus, by measuring the complexity at the right "scale" (i.e., proportional to the accuracy) the sample size sufficient for finding the best model within the given accuracy might dramatically shrink. As an example of this philosophy, consider the following scenario. 1 Suppose a meteorologist is requested to compute a daily prediction of the next day's temperature. His forecast is based on a set of presumably relevant data, such as the temperature, barometric pressure, and relative humidity over the past few days. On some special events, such as the day before launching a Space Shuttle, his prediction should have a high degree of accuracy, and therefore he analyzes a larger amount of data to finely tune the parameters of his favorite mathematical meteorological model. On regular days, a smaller precision is tolerated, and thus he can afford to tune the parameters of the model on a coarser scale, saving data and computational resources. In this paper we demonstrate quantitatively how the accuracy parameter plays a crucial role in determining the effective complexity of the learner's hypothesis class. 2 We work within the decision-theoretic extension of the PAC framework, introduced in [12] and also known as agnostic learning. In this model, a finite sample of pairs (x; y) is obtained through independent draws from a fixed distribution P over X \Theta [0; 1]. The goal of the learner is to be able to estimate the conditional expectation of y given x. This quantity is defined by a called the regression function in statistics. The learner is given a class H of candidate regression functions, which may or may not include the true regression function f . This class H is called ffl-learnable if there is a learner with the property that for any distribution P and corresponding regression function f , given a large enough random sample from P , this learner can find an ffl-close approximation 3 to f within the class H, or if f is not in H, an ffl-close approximation to a function in H that best approximates f . (This analysis of learnability is purely information-theoretic, and does not take into account computational complexity.) Throughout the 1 Adapted from [14]. Our philosophy can be compared to the approach studied in [13], where the range of the functions in the hypothesis class is discretized in a number of elements proportional to the accuracy. In this case, one is interested in bounding the complexity of the discretized class through the dimension of the original class. Part of our results builds on this discretization technique. 3 All notions of approximation are with respect to mean square error. paper, we assume that H (and later F) satisfies some mild measurability conditions. A suitable such condition is the "image admissible Suslin" property (see [8, Section 10.3.1, page 101].) The special case where the distribution P is taken over X \Theta f0; 1g was studied in [14] by Kearns and Schapire, who called this setting probabilistic concept learning. If we further demand that the functions in H take only values in f0; 1g, it turns out that this reduces to one of the standard PAC learning frameworks for learning deterministic concepts. In this case it is well known that the learnability of H is completely characterized by the finiteness of a simple combinatorial quantity known as the Vapnik-Chervonenkis (VC) dimension of H [24, 6]. An analogous combinatorial quantity for the probabilistic concept case was introduced by Kearns and Schapire. We call this quantity the P fl -dimension of H, where fl ? 0 is a parameter that measures the "scale" to which the dimension of the class H is measured. They were only able to show that finiteness of this parameter was necessary for probabilistic concept learning, leaving the converse open. We solve this problem showing that this condition is also sufficient for learning in the harder agnostic model. This last result has been recently complemented by Bartlett, Long, and Williamson [4], who have shown that the P fl -dimension characterizes agnostic learnability with respect to the mean absolute error. In [20], Simon has independently proven a partial characterization of (nonagnostic) learnability using a slightly different notion of dimension. As in the pioneering work of Vapnik and Chervonenkis [24], our analysis of learnability begins by establishing appropriate uniform laws of large numbers. In our main theorem, we establish the first combinatorial characterization of those classes of random variables whose means uniformly converge to their expectations for all distributions. Such classes of random variables have been called Glivenko-Cantelli classes in the empirical processes literature [9]. Given the usefulness of related uniform convergence results in combinatorics and randomized algorithms, we feel that this result may have many applications beyond those we give here. In addition, our results rely on a combinatorial result that generalizes Sauer's Lemma [18, 19]. This new lemma considerably extends some previously known results concerning f0; 1; \Lambdag tournament codes [21, 7]. As other related variants of Sauer's Lemma were proven useful in different areas, such as geometry and Banach space theory (see, e.g., [15, 1]), we also have hope to apply this result further. The uniform, distribution-free convergence of empirical means to true expectations for classes of real-valued functions has been studied by Dudley, Gin'e, Pollard, Talagrand, Vapnik, Zinn, and others in the area of empirical processes. These results go under the general name of uniform laws of large numbers. We give a new combinatorial characterization of this phenomenon using methods related to those pioneered by Vapnik and Chervonenkis. Let F be a class of functions from a set X into [0; 1]. (All the results presented in this section can be generalized to classes of functions taking values in any bounded real range.) Let P denote a probability distribution over X such that f is P -measurable for all f 2 F . By P (f) we denote the P-mean of f , i.e., its integral w.r.t. P . By P n (f) we denote the random variable 1 are drawn independently at random according to P . Following Dudley, Gin'e and Zinn [9], we say that F is an ffl-uniform Glivenko-Cantelli class if lim sup Pr sup m-n sup 0: (1) Here Pr denotes the probability with respect to the points x 1 drawn independently at random according to P . 4 The supremum is understood with respect to all distributions P over X (with respect to some suitable oe-algebra of subsets of X ; see [9]). We say that F satisfies a distribution-free uniform strong law of large numbers, or more briefly, that F is a uniform Glivenko-Cantelli class, if F is an ffl-uniform Glivenko-Cantelli class for all We now recall the notion of VC-dimension, which characterizes uniform Glivenko-Cantelli classes of f0; 1g-valued functions. Let F be a class of f0; 1g-valued functions on some domain set, X . We say F C-shatters a set A ' X if, for every E ' A, there exists some f E 2 F satisfying: For every x 2 A n E, 1. Let the V C-dimension of F , denoted V C-dim(F ), be the maximal cardinality of a set A ' X that is V C-shattered by F . (If F V C-shatters sets of unbounded finite sizes, then let V The following was established by Vapnik and Chervonenkis [24] for the "if " part and (in a stronger version) by Assouad and Dudley [2] (see [9, proposition 11, page 504].) Theorem 2.1 Let F be a class of functions from X into f0; 1g. Then F is a uniform Glivenko- Cantelli class if and only if V C-dim(F) is finite. Several generalizations of the V C-dimension to classes of real-valued functions have been previously proposed: Let F be a class of [0; 1]-valued functions on some domain set X . ffl (Pollard [16], see also [12]): We say F P -shatters a set A ' X if there exists a function R such that, for every E ' A, there exists some f E 2 F satisfying: For every Let the P-dimension (denoted by P-dim) be the maximal cardinality of a set A ' X that is -shattered by F . (If F P -shatters sets of unbounded finite sizes, then let -shatters a set A ' X if there exists a constant ff 2 R such that, for every E ' A, there exists some f E 2 F satisfying: For every x 2 A n E, f for every x 2 Let the V -dimension (denoted by V -dim) be the maximal cardinality of a set A ' X that is -shattered by F . (If F V -shatters sets of unbounded finite sizes, then let V It is easily verified (see below) that the finiteness of neither of these combinatorial quantities provides a characterization of uniform Glivenko-Cantelli classes (more precisely, they both provide only a sufficient condition.) Kearns and Schapire [14] introduced the following parametrized variant of the P-dimension. Let F be a class of [0; 1]-valued functions on some domain set X and let fl be a positive real number. We say F -shatters a set A ' X if there exists a function s : A ! [0; 1] such that for every Actually Dudley et al. use outer measure here, to avoid some measurability problems in certain cases. there exists some f E 2 F satisfying: For every x 2 A n E, f E and, for every Let the P fl -dimension of F , denoted P fl -dim(F ), be the maximal cardinality of a set A ' X that is P fl -shattered by F . (If F P fl -shatters sets of unbounded finite sizes, then let P A parametrized version of the V -dimension, which we'll call V fl -dimension, can be defined in the same way we defined the P fl -dimension from the P-dimension. The first lemma below follows directly from the definitions. The second lemma is proven through the pigeonhole principle. Lemma 2.1 For any F and any fl ? 0, P Lemma 2.2 For any class F of [0; 1]-valued functions and for all fl ? 0, The P fl and the V fl dimensions have the advantage of being sensitive to the scale at which differences in function values are considered significant. Our main result of this section is the following new characterization of uniform Glivenko-Cantelli classes, which exploits the scale-sensitive quality of the P fl and the V fl dimensions. Theorem 2.2 Let F be a class of functions from X into [0; 1]. 1. There exist constants a; b ? 0 (independent of F) such that for any fl ? 0 (a) If P fl -dim(F) is finite, then F is an (afl)-uniform Glivenko-Cantelli class. (b) If V fl -dim(F) is finite, then F is a (bfl)-uniform Glivenko-Cantelli class. (c) If P fl -dim(F) is infinite, then F is not a (fl \Gamma -uniform Glivenko-Cantelli class for any (d) If V fl -dim(F) is infinite, then F is not a (2fl \Gamma -uniform Glivenko-Cantelli class for any - ? 0. 2. The following are equivalent: (a) F is a uniform Glivenko-Cantelli class. (b) P fl -dim(F) is finite for all fl ? 0. (c) V fl -dim(F) is finite for all fl ? 0. (In the proof we actually show that a - 24 and b - 48, however these values are likely to be improved through a more careful analysis.) The proof of this theorem is deferred to the next section. Note however that part 1 trivially implies part 2. The following simple example (a special case of [9, Example 4, page 508], adapted to our pur- poses) shows that the finiteness of neither P-dim nor V -dim yields a characterization of Glivenko- Cantelli classes. (Throughout the paper we use ln to denote the natural logarithm and log to denote the logarithm in base 2.) Example 2.1 Let F be the class of all [0; 1]-valued functions f defined on the positive integers and such that f(x) - e \Gammax for all x 2 N and all f 2 F . Observe that, for all . Therefore, F is a uniform Glivenko-Cantelli class by Theorem 2.2. On the other hand, it is not hard to show that the P-dimension and the V -dimension of F are both infinite. Theorem 2.2 provides the first characterization of Glivenko-Cantelli classes in terms of a simple combinatorial quantity generalizing the Vapnik-Chervonenkis dimension to real-valued functions. Our results extend previous work by Dudley, Gin'e, and Zinn, where an equivalent characterization is shown to depend on the asymptotic properties of the metric entropy. Before stating the metric- entropy characterization of Glivenko-Cantelli classes we recall some basic notions from the theory of metric spaces. Let (X; d) be a (pseudo) metric space, let A be a subset of X and ffl ? 0. ffl A set B ' A is an ffl-cover for A if, for every a 2 A, there exists some b 2 B such that ffl. The ffl-covering number of A, N d (ffl; A), is the minimal cardinality of an ffl-cover for A (if there is no such finite cover then it is defined to be 1). ffl A set A ' X is ffl-separated if, for any distinct a; b 2 A, ffl. The ffl-packing number of A, M d (ffl; A), is the maximal size of an ffl-separated subset of A. The following is a simple, well-known fact. Lemma 2.3 For every (pseudo) metric space (X; d), every A ' X, and ffl ? 0 For a sequence of n points x class F of real-valued functions defined on xn (f; g) denote the l 1 distance between f; g 2 F on the points x n , that is l 1 xn 1-i-n As we will often use the l 1 xn distance, let us introduce the notation N (ffl; F ; x n ) and M(ffl; F ; x n ) to stand for, respectively, the ffl-covering and the ffl-packing number of F with respect to l 1 xn . A notion of metric entropy H n , defined by log N (ffl; F ; x n ); has been used by Dudley, Gin'e and Zinn to prove the following. Theorem 2.3 ([9, Theorem 6, page 500]) Let F be a class of functions from X into [0; 1]. Then 1. F is a uniform Glivenko-Cantelli class if and only if lim n!1 H n (ffl; 2. For all ffl ? 0, if lim n!1 H n (ffl; is an (8ffl)-uniform Glivenko-Cantelli class. The results by Dudley et al. also give similar characterizations using l p norms in place of the l 1 norm. Related results were proved earlier by Vapnik and Chervonenkis [24, 25]. In particular, they proved an analogue of Theorem 2.3, where the convergence of means to expectations is characterized for a single distribution P . Their characterization is based on H n (ffl; F) averaged with respect to samples drawn from P . 3 Proof of the main theorem We wish to obtain a characterization of uniform Glivenko-Cantelli classes in terms of their P dimension. By using standard techniques, we just need to bound the fl-packing numbers of sets of real-valued functions by an appropriate function of their P cfl -dimension, for some positive constant c. Our line of attack is to reduce the problem to an analogous problem in the realm of finite- valued functions. Classes of functions into a discrete and finite range can then be analyzed using combinatorial tools. We shall first introduce the discrete counterparts of the definitions above. Our next step will be to show how the real-valued problem can be reduced to a combinatorial problem. The final, and most technical part of our proof, will be the analysis of the combinatorial problem through a new generalization of Sauer's Lemma. Let X be any set and let bg. We consider classes F of functions f from X to B. Two such functions f and g are separated if they are 2-separated in the l 1 metric, i.e., if there exists some x 2 X such that 2. The class F is pairwise separated if f and g are separated for all f 6= g in F . F strongly shatters a set A ' X if A is nonempty and there exists a function s that, for every E ' A, there exists some f E 2 F satisfying: For every x 2 A n E, f E and, for every x 2 E, f E (x) - s(x)+ 1. If s is any function witnessing the shattering of A by F , we shall also say that F strongly shatters A according to s. Let the strong dimension of F , S-dim(F ), be the maximal cardinality of a set A ' X that is strongly shattered by F . (If F strongly shatters sets of unbounded finite size, then let For a function f and a real number ae ? 0, the ae-discretization of f , denoted by f ae , is the function f ae (x) def ae c, i.e. f ae f(x)g. For a class F of nonnegative real-valued functions let Fg. We need the following lemma. Lemma 3.1 For any class F of [0; 1]-valued functions on a set X and for any ae ? 0, 1. for every 2. for every ffl - 2ae and every x Proof. To prove part 1 we show that any set strongly shattered by F ae is also P ae=2 -shattered by F . If A ' X is strongly shattered by F ae , then there exists a function s such that for every E ' A there exists some f (E) 2 F satisfying: for every x 2 A n E, f ae and for every x 2 E, Assume first f ae holds and, by definition of f ae we have f (E) by definition of f ae , we have f (E) (x) - aef ae (x), which implies f (E) (x) - ae \Delta s(x)+ae. Thus A is P ae=2 -shattered by F , as can be seen using the function s defined by s To prove part 2 of the lemma it is enough to observe that, by the definition of F ae , for all f; 2. 2 We now prove our main combinatorial result which gives a new generalization of Sauer's Lemma. Our result extends some previous work concerning f0; 1; \Lambdag tournament codes, proven in a completely different way (see [21, 7]). The lemma concerns the l 1 packing numbers of classes of functions into a finite range. It shows that, if such a class has a finite strong dimension, then its 2-packing number is bounded by a subexponential function of the cardinality of its domain. For simplicity, we arbitrarily fix a sequence x n of n points in X and consider only the restriction of F to this domain, dropping the subscript x n from our notation. Lemma 3.2 If F is a class of functions from a finite domain X of cardinality n to a finite range, Note that for fixed d the bound in Lemma 3.2 is n O(log n) even if b is not a constant but a polynomial in n. Proof of Lemma 3.2. Fix b - 3 (the case b ! 3 is trivial.) Let us say that a class F as above strongly shatters a pair (A; s) (for a nonempty subset A of X and a function s if F strongly shatters A according to s. For all integers h - 2 and n - 1, let t(h; n) denote the maximum number t such that for every set F of h pairwise separated functions f from X to B, F strongly shatters at least t pairs (A; s) where A ' X , A 6= ;, and s : A ! B. If no such F exists, then t(h; n) is infinite. Note that the number of possible pairs (A; s) for which the cardinality of A does not exceed (as for A of size i ? 0 there are strictly less than b i possibilities to choose s.) It follows that, if t(h; n) - y for some h, then M l 1(2; F) ! h for all sets F of functions from X to B and such that S-dim(F) - d. Therefore, to finish the proof, it suffices to show that We claim that t(2; 2. The first part of the claim is readily verified. For the second part, first note that if no set of 2mnb 2 pairwise separated functions from X to B exists, then t(2mnb the claim holds. Assume then that there is a set F of 2mnb 2 pairwise separated functions from X to B. Split it arbitrarily into mnb 2 pairs. For each pair (f; g) find a coordinate x 2 X where 1. By the pigeonhole principle, the same coordinate x is picked for at least mb 2 pairs. Again by the pigeonhole principle, there are at least mb ? 2m of these pairs (f; g) for which the (unordered) set ff(x); g(x)g is the same. This means that there are two sub-classes of F , call them F 1 and F 2 , and there are x 2 X and so that for each for each g 2 F 2 Obviously, the members of F 1 are pairwise separated on X n fxg and the same holds for the members of F 2 . Hence, by the definition of the function t, F 1 strongly shatters at least t(2m; and the same holds for F 2 . Clearly F strongly shatters all pairs strongly shattered by F 1 or F 2 . Moreover, if the same pair (A; s) is strongly shattered both by F 1 and by F 2 , then F also strongly shatters the pair It follows that establishing the claim. Now suppose n ? r - 1. Let repeated application of the above claim, it follows that t(h; n) - 2 r . Since t is clearly monotone in its first argument, and h, this implies t(2(nb 2 . However, since the total number of functions from to B is b n , there are no sets of pairwise separated functions of size larger than this, and hence y in this case. On the other hand, when the result above yields t(2(nb 2 y. Thus in either case y, completing the proof. 2 Before proving Theorem 2.2, we need two more lemmas. The first one is a straightforward adaptation of [22, Section A.6, p. 223]. Lemma 3.3 Let F be a class of functions from X into [0; 1] and let P be a distribution over X. Then, for all ffl ? 0 and all n - 2=ffl 2 , Pr sup \Theta N (ffl=6; F ; x 0 where Pr denotes the probability w.r.t. the sample x drawn independently at random according to P , and E the expectation w.r.t. a second sample x 0 2n also drawn independently at random according to P . Proof. A well-known result (see e.g. [8, Lemma 11.1.5] or [10, Lemma 2.5]) shows that, for all Pr sup sup ffl) where We combine this with a result by Vapnik [22, pp. 225-228] showing that for all ffl ? 0 Pr sup \Theta N (ffl=3; F ; x 0 This concludes the proof. 2 The next result applies Lemma 3.2 to bound the expected covering number of a class F in terms of P fl -dim(F ). Lemma 3.4 Let F be a class of functions from X into [0; 1] and P a distribution over X. Choose where the expectation E is taken w.r.t. a sample x drawn independently at random according to P . Proof. By Lemma 2.3, Lemmas 3.1 and 3.2, and Stirling's approximation, xn xn xn We are now ready to prove our characterization of uniform Glivenko-Cantelli classes. Proof of Theorem 2.2. We begin with part 1.d: If V fl show that F is not a (2fl \Gamma -uniform Glivenko-Cantelli class for any - ? 0. To see this, assume 1. For any sample size n and any d ? n, find in X a set S of d points that are -shattered by F . Then there exists ff ? 0 such that for every E ' S there exists some f E 2 F satisfying: For every x 2 A n E, f E the uniform distribution on S. For any sample x there is a function f 2 F such that f(x i g. Thus, for any large enough we can find some f 2 F such that jP This proves part 1.d. Part 1.c follows from Lemma 2.2. To prove part 1.a we use inequality (2) from Lemma 3.3. Then, to bound the expected covering number we apply Lemma 3.4. This shows that lim sup Pr sup for some a ? 0 whenever P fl -dim(F) is finite. Equation (4) shows that P n (f) ! P (f) in probability for all f 2 F and all distributions P . Furthermore, as Lemma 3.3 and Lemma 3.4 imply that 1, one may apply the Borel-Cantelli lemma and strengthen (4) to almost sure convergence, i.e. lim sup Pr sup m-n sup 0: This completes the proof of part 1.a. The proof of part 1.b follows immediately from Lemma 2.2.The proof of Theorem 2.2, in addition to being simpler than the proof in [9] (see Theorem 2.3 in this paper), also provides new insights into the behaviour of the metric entropy used in that characterization. It shows that there is a large gap in the growth rate of the metric entropy either F is a uniform Glivenko-Cantelli class, and hence, by (3) and by definition of H n , for or F is not a uniform Glivenko-Cantelli class, and hence there exists ffl ? 0 such that P ffl which is easily seen to imply that H n (ffl; n). It is unknown if log 2 n can be replaced by log ff n where 1 - ff ! 2. From the proof of Theorem 2.2 we can obtain bounds on the sample size sufficient to guarantee that, with high probability, in a class of [0; 1]-valued random variables each mean is close to its expectation. Theorem 3.1 Let F be a class of functions from X into [0; 1]. Then for all distributions P over X and all ffl; Pr sup for where d is the P ffl=24 -dimension of F . Theorem 3.1 is proven by applying Lemma 3.3 and Lemma 3.4 along with standard approximations. We omit the proof of this theorem and mention instead that an improved sample size bound has been shown by Bartlett and Long [3, Equation (5), Theorem 9]. In particular, they show that if the P (1=4\Gamma- )ffl -dimension d 0 of F is finite for some - ? 0, then a sample size of order O is sufficient for (5) to hold. 4 Applications to Learning In this section we define the notion of learnability up to accuracy ffl, or ffl-learnability, of statistical regression functions. In this model, originally introduced in [12] and also known as "agnostic learning", the learning task is to approximate the regression function of an unknown distribution. The probabilistic concept learning of Kearns and Schapire [14] and the real-valued function learning with noise investigated by Bartlett, Long, and Williamson [4] are special cases of this framework. We show that a class of functions is ffl-learnable whenever its P affl -dimension is finite for some constant a ? 0. Moreover, combining this result with those of Kearns and Schapire, who show that a similar condition is necessary for the weaker probabilistic concept learning, we can conclude that the finiteness of the P fl -dimension for all fl ? 0 characterizes learnability in the probabilistic concept framework. This solves an open problem from [14]. Let us begin by briefly introducing our learning model. The model examines learning problems involving statistical regression on [0; 1]-valued data. Assume X is an arbitrary set (as above), and be an unknown distribution on Z. Let X and Y be random variables respectively distributed according to the marginal of P on X and Y . The regression function f for distribution P is defined, for all x 2 X , by The general goal of regression is to approximate f in the mean square sense (i.e. in L 2 -norm) when the distribution P is unknown, but we are given z independently generated from the distribution P . In general we cannot hope to approximate the regression function f for an arbitrary distribution . Therefore we choose a hypothesis space H, which is a family of mappings settle for a function in H that is close to the best approximation to f in the hypothesis space H. To this end, for each hypothesis h 2 H, let the function defined by: is the mean square loss of h. The goal of learning in the present context is to find a function b h 2 H such that for some given accuracy ffl ? 0. It is easily verified that if inf h2H P (' h ) is achieved by some h 2 H, then h is the function in H closest to the true regression function f in the L 2 norm. A learning procedure is a mapping A from finite sequences in Z to H. A learning procedure produces a hypothesis b training sample z n . For given accuracy parameter ffl, we say that H is ffl-learnable if there exists a learning procedure A such that lim sup Pr ae oe 0: (7) Here Pr denotes the probability with respect to the random sample z n 2 Z n , each z i drawn independently according to P , and the supremum is over all distributions P defined on a suitable oe-algebra of subsets of Z. Thus H is ffl-learnable if, given a large enough training sample, we can reliably find a hypothesis b h 2 H with mean square error close to that of the best hypothesis in H. Finally, we say H is learnable if and only if it is ffl-learnable for all ffl ? 0. 1g the above definitions of learnability yield the probabilistic concept learning model. In this case, if (7) holds for some ffl ? 0 and some class H, we say that H is ffl-learnable in the p-concept model. We now state and prove the main results of this section. We start by establishing sufficient conditions for ffl-learnability and learnability in terms of the P fl -dimension. Theorem 4.1 There exist constants a; b ? 0 such that for any fl ? 0: 1. If P fl -dim(H) is finite, then H is (afl)-learnable. 2. If V fl -dim(H) is finite, then H is (bfl)-learnable. 3. If P fl -dim(H) is finite for all fl ? 0 or V fl -dim(H) is finite for all fl ? 0, then H is learnable. We then prove the following, which characterizes p-concept learnability. Theorem 4.2 1. If P fl -dim(H) is infinite, then H is not (fl 2 =8 \Gamma -learnable in the p-concept model for any 2. If V fl -dim(H) is infinite, then H is not (fl 2 =2 \Gamma -learnable in the p-concept model for any 3. The following are equivalent: (a) H is learnable in the p-concept model. (b) P fl -dim(H) is finite for all fl ? 0. (c) V fl -dim(H) is finite for all fl ? 0. (d) H is a uniform Glivenko-Cantelli class. Proof of Theorem 4.1. It is clear that part 3 follows from part 1 using Theorem 2.2. Also, by Lemma 2.2, part 1 is equivalent to part 2. Thus, to prove Theorem 4.1 it suffices to establish part 1. We do so via the next two lemmas. Hg. Lemma 4.1 If ' H is an ffl-uniform Glivenko-Cantelli class, then H is (3ffl)-learnable. Proof. The proof uses the method of empirical risk minimization, analyzed by Vapnik [22]. As the empirical loss on the given sample z that is A learning procedure, A ffl , ffl-minimizes the empirical risk if A ffl (z n ) is any b us show that any such procedure is guaranteed to 3ffl-learn H. Fix any n 2 N. If for all h 2 H, then and thus P (' A ffl (zn Hence, since we chose n and ffl arbitrarily, lim sup Pr sup m-n sup implies lim sup Pr ae oe 0:The following lemma shows that bounds on the covering numbers of a family of functions H can be applied to the induced family of loss functions ' H . We formulate the lemma in terms of the square loss but it may be readily generalized to other loss functions. A similar result was independently proven by Bartlett, Long, and Williamson in [4] for the absolute loss L(x; (and with respect to the l 1 metric rather than the l 1 metric used here). Lemma 4.2 For all ffl ? 0, all H, and any z Proof. It suffices to show that, for any f; g 2 H and any 1 - then This follows by noting that, for every s; t; w 2 [0; 1], We end the proof of Theorem 4.1 by proving part 1. By Lemma 4.1, it suffices to show that ' H is (afl)-uniform Glivenko-Cantelli for some a ? 0. To do so we use (2) from Lemma 3.3. Then, to bound the expected covering number, we apply first Lemma 4.2 and then Lemma 3.4. This establishes lim sup Pr sup for some a ? 0 whenever P fl -dim(H) is finite. An application of the Borel-Cantelli lemma to get almost sure convergence yields the proof. 2 We conclude this section by proving our characterization of p-concept learnability. Proof of Theorem 4.2. As ffl-learnability implies ffl-learnability in the p-concept model, we have that part 3 follows from part 1, part 2, and from Theorem 4.1 using Theorem 2.2. The proof of part 2 uses arguments similar to those used to prove part 1.d of Theorem 2.2. Finally note that part 1 follows from part 2 by Lemma 2.2 (we remark that a more restricted version of part 1 was proven in Theorem 11 of [14].) 2 5 Conclusions and open problems In this work we have shown a characterization of uniform Glivenko-Cantelli classes based on a combinatorial notion generalizing the Vapnik-Chervonenkis dimension. This result has been applied to show that the same notion of dimension provides the weakest combinatorial condition known to imply agnostic learnability and, furthermore, characterizes learnability in the model of probabilistic concepts under the square loss. Our analysis demonstrates how the accuracy parameter in learning plays a central role in determining the effective dimension of the learner's hypothesis class. An open problem is what other notions of dimension may characterize uniform Glivenko-Cantelli classes. In fact, for classes of functions with finite range, the same characterization is achieved by each member of a family of several notions of dimension (see [5]). A second open problem is the asymptotic behaviour of the metric entropy: we have already shown that for all ffl ? 0, H n (ffl; is a uniform Glivenko-Cantelli class and We conjecture that for all ffl ? 0, H n (ffl; is a uniform Glivenko-Cantelli class. A positive solution of this conjecture would also affect the sample complexity bound (6) of Bartlett and Long. In fact, suppose that Lemma 3.4 is improved by showing that sup xn M(ffl; F ; x n \Delta cd for some positive constant c and for (note that this implies our conjecture.) Then, combining this with [3, Lemma 10-11], we can easily show a sample complexity bound of O for any 0 ! - ! 1=8 for which is finite. It is not clear how to bring the constant 1=8 down to 1=4 as in (6), which was proven using l 1 packing numbers. Acknowledgments We would like to thank Michael Kearns, Yoav Freund, Ron Aharoni and Ron Holzman for fruitful discussions, and Alon Itai for useful comments concerning the presentation of the results. Thanks also to an anonymous referee for the many valuable comments, suggestions, and references --R Embedding of Minimax nonparametric estimation over classes of sets. More theorems about scale-sensitive dimensions and learning Characterizations of learnability for classes of f0 Learnability and the Vapnik-Chervonenkis dimension A lower bound for f0 A course on empirical processes. Uniform and universal Glivenko-Cantelli classes Some limit theorems for empirical processes. Structural risk minimization for character recognition. Decision theoretic generalizations of the PAC model for neural net and other learning applications. A generalization of Sauer's lemma. Efficient distribution-free learning of probabilistic concepts Some remarks about embedding of Empirical Processes Modeling by shortest data description. On the density of families of sets. A combinatorial problem: Stability and order for models and theories in infinitary languages. Bounds on the number of examples needed for learning functions. Estimation of Dependences Based on Empirical Data. Inductive principles of the search for empirical dependencies. On the uniform convergence of relative frequencies of events to their probabilities. Necessary and sufficient conditions for uniform convergence of means to mathematical expectations. --TR A lower bound for 0,1, * tournament codes Learnability and the Vapnik-Chervonenkis dimension Inductive principles of the search for empirical dependences (methods based on weak convergence of probability measures) Decision theoretic generalizations of the PAC model for neural net and other learning applications Efficient distribution-free learning of probabilistic concepts Characterizations of learnability for classes of {0, MYAMPERSANDhellip;, <italic>n</italic>}-valued functions A generalization of Sauer''s lemma Bounds on the number of examples needed for learning functions More theorems about scale-sensitive dimensions and learning Fat-shattering and the learnability of real-valued functions --CTR Philip M. Long, On the sample complexity of learning functions with bounded variation, Proceedings of the eleventh annual conference on Computational learning theory, p.126-133, July 24-26, 1998, Madison, Wisconsin, United States Martin Anthony , Peter L. Bartlett, Function Learning from Interpolation, Combinatorics, Probability and Computing, v.9 n.3, p.213-225, May 2000 Massimiliano Pontil, A note on different covering numbers in learning theory, Journal of Complexity, v.19 n.5, p.665-671, October John Shawe-Taylor , Robert C. Williamson, A PAC analysis of a Bayesian estimator, Proceedings of the tenth annual conference on Computational learning theory, p.2-9, July 06-09, 1997, Nashville, Tennessee, United States John Shawe-Taylor , Nello Cristianini, Further results on the margin distribution, Proceedings of the twelfth annual conference on Computational learning theory, p.278-285, July 07-09, 1999, Santa Cruz, California, United States Olivier Bousquet , Andr Elisseeff, Stability and generalization, The Journal of Machine Learning Research, 2, p.499-526, 3/1/2002 Shahar Mendelson, On the size of convex hulls of small sets, The Journal of Machine Learning Research, 2, p.1-18, 3/1/2002 Tong Zhang, Covering number bounds of certain regularized linear function classes, The Journal of Machine Learning Research, 2, p.527-550, 3/1/2002 Don Hush , Clint Scovel, Fat-Shattering of Affine Functions, Combinatorics, Probability and Computing, v.13 n.3, p.353-360, May 2004 Don Hush , Clint Scovel, On the VC Dimension of Bounded Margin Classifiers, Machine Learning, v.45 n.1, p.33-44, October 1 2001 Martin Anthony, Generalization Error Bounds for Threshold Decision Lists, The Journal of Machine Learning Research, 5, p.189-217, 12/1/2004 Shahar Mendelson , Petra Philips, On the Importance of Small Coordinate Projections, The Journal of Machine Learning Research, 5, p.219-238, 12/1/2004 Kristin P. Bennett , Nello Cristianini , John Shawe-Taylor , Donghui Wu, Enlarging the Margins in Perceptron Decision Trees, Machine Learning, v.41 n.3, p.295-313, Dec. 2000 Philip M. Long, Efficient algorithms for learning functions with bounded variation, Information and Computation, v.188 n.1, p.99-115, 10 January 2004 John Shawe-Taylor , Peter L. Bartlett , Robert C. Williamson , Martin Anthony, A framework for structural risk minimisation, Proceedings of the ninth annual conference on Computational learning theory, p.68-76, June 28-July 01, 1996, Desenzano del Garda, Italy Barbara Hammer, Generalization Ability of Folding Networks, IEEE Transactions on Knowledge and Data Engineering, v.13 n.2, p.196-206, March 2001 Alberto Bertoni , Carlo Mereghetti , Beatrice Palano, Small size quantum automata recognizing some regular languages, Theoretical Computer Science, v.340 n.2, p.394-407, 27 June 2005 Andrs Antos , Balzs Kgl , Tams Linder , Gbor Lugosi, Data-dependent margin-based generalization bounds for classification, The Journal of Machine Learning Research, 3, p.73-98, 3/1/2003 Yiming Ying , Ding-Xuan Zhou, Learnability of Gaussians with Flexible Variances, The Journal of Machine Learning Research, 8, p.249-276, 5/1/2007 Bernhard Schlkopf , Alexander J. Smola, A short introduction to learning with kernels, Advanced lectures on machine learning, Springer-Verlag New York, Inc., New York, NY, Shahar Mendelson, A few notes on statistical learning theory, Advanced lectures on machine learning, Springer-Verlag New York, Inc., New York, NY, Shahar Mendelson, Learnability in Hilbert spaces with reproducing kernels, Journal of Complexity, v.18 n.1, p.152-170, March 2002 Bin Zou , Luoqing Li, The performance bounds of learning machines based on exponentially strongly mixing sequences, Computers & Mathematics with Applications, v.53 n.7, p.1050-1058, April, 2007

uniform laws of large numbers;vapnik-chervonenkis dimension;PAC learning

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Software Reuse by Specialization of Generic Procedures through Views.

AbstractA generic procedure can be specialized, by compilation through views, to operate directly on concrete data. A view is a computational mapping that describes how a concrete type implements an abstract type. Clusters of related views are needed for specialization of generic procedures that involve several types or several views of a single type. A user interface that reasons about relationships between concrete types and abstract types allows view clusters to be created easily. These techniques allow rapid specialization of generic procedures for applications.

Introduction Reuse of software has the potential to reduce cost, increase the speed of software production, and increase reliability. Facilitating the reuse of software could therefore be of great benefit. G. S. Novak, Jr. is with the Department of Computer Sciences, University of Texas, Austin, An Automatic Programming Server demonstration of this software is available on the World Wide Web via http://www.cs.utexas.edu/users/novak; running the demo requires X windows. Rigid treatment of type matching presents a barrier to reuse. In most languages, the types of arguments of a procedure call must match the types of parameters of the procedure. For this reason, reuse is often found where type compatibility occurs naturally, i.e. where the types are basic or are made compatible by the language (e.g. arrays of numbers). A truly generic procedure should be reusable for any reasonable implementation of its abstract types; a developer of a generic should be able to advertise "my program will work with your data" without knowing what the user's data representation will be. We seek reuse without conformity to rigid standards. We envision two classes of users: developers, who understand the details of abstract types and generic procedures, and users, programmers who reuse the generics but need less expertise and understanding of details. Developers produce libraries of abstract types and generics; by specializing the generics, users obtain program modules for applications. A view provides a mapping between a concrete type and an abstract type 1 in terms of which a generic algorithm is written. Fig. 1 illustrates schematically that a view acts as an interface adapter that makes the concrete type appear as the abstract type 2 . The view provides a clean separation between the semantics of data (as represented by the abstract type) and its implementation, so that the implementation is minimally constrained. Once a view has been made, any generic procedure associated with the abstract type can be automatically specialized for the concrete type, as shown in Fig. 2. In our implementation, the specialized procedure is a Lisp function; if desired, it can be mechanically translated into another language. Tools exist that make it easy to create views; a programmer can obtain a specialized procedure to insert records into an AVL tree (185 lines of C) in a few minutes. Data Procedure Data View Procedure Figure 1: Interfacing with Strong Typing and with Views 1 We consider an abstract type to be a set of basis variables and a set of generic procedures that are written in terms of the basis variables. Goguen [18] and others have used a similar analogy and diagram. Concrete Type View Compiler Generic Procedure Specialized Procedure Figure 2: Specialization of Generic Procedure through View This approach to reuse has several advantages: 1. It provides freedom to select the implementation of data; data need not be designed ab initio to match the algorithms. 2. Several views of a data structure can correspond to different aspects of the data. 3. Several languages are supported. Lisp, C, C++, Java, or Pascal can be generated from a single version of generic algorithms. 4. Tools simplify the specification of views and reduce the learning required to reuse software. 5. Views can be used to automatically: (a) specialize generic procedures from a library, (b) instantiate a program framework from components, (c) translate data from one representation to another, (d) generate methods for object-oriented programming, and (e) interface to programming tools for data display and editing. This paper describes principles of views and specialization of generic algorithms, as well as an implementation of these techniques using the GLISP language and compiler; GLISP is a Lisp-based language with abstract data types. Section 2 describes in conceptual terms how views provide mappings between concrete types and abstract types. Section 3 describes the GLISP compiler and how views are used in specializing generic algorithms. Section 4 discusses clusters of related views that are needed to reuse generic algorithms that involve several types or several views of a type. Section 5 describes the program VIEWAS, which reasons about relations between concrete types and abstract types and makes it easy to create view clusters. Section 6 describes higher-order code and a generic algorithm for finding a convex hull that uses other generics and uses several views of a single data structure. Section 7 describes use of views with object-oriented programming. Section 8 surveys related work, and Section 9 presents conclusions. 2.1 Computation as Isomorphism It is useful to think of all computation as simulation or, more formally, as isomorphism. This idea has a long history; for example, isomorphism is the basis of denotational semantics [20]. Goguen [18] [19] describes views as isomorphisms, with mappings of unary and binary operators. Our views allow broader mappings between concrete and abstract types and include algorithms as well as operators. We use isomorphism to introduce the use of views. Preparata and Yeh [56] give a definition and diagram for isomorphism of semigroups: Given two semigroups G 1 an invertible function is said to be an isomorphism between G 1 and G 2 if, for every a and b in S, '(a a a '(a) \Lambda '(b) '(a) oe \Gamma\Psi \Gamma\Psi Since ' is invertible, there is a computational relationship: a a ffi b is difficult to obtain directly, its value can be computed by encoding a and b using ', performing the computation '(a) \Lambda '(b), and decoding or interpreting the result using the diagram is said to commute [2] if the same result is obtained regardless of which path between two points is followed, as shown in the diagram above. 2.2 Views as Isomorphisms Reuse of generic algorithms through views corresponds to computation as isomorphism. The concrete type corresponds to the left-hand side of an isomorphism diagram; the view maps the concrete type to the abstract type. The generic algorithm corresponds to an operation on the abstract type. By mapping from the concrete type to the abstract type, performing the operation on the abstract type, and mapping back, the result of performing the algorithm on the concrete type is obtained. However, instead of performing the view mapping explicitly Concrete Concrete Type Abstract Generic Algorithm Abstract Type view view oe Compilation, Optimization Concrete Concrete Type Specialized Algorithm Figure 3: Specializing a Generic and materializing the abstract data, the mappings are folded into the generic algorithm to produce a specialized version that operates directly on the concrete data (Fig. 3). As an example, let concrete type pizza contain a value d that represents the diameter of the (circular) pizza. Suppose abstract type circle assumes a radius value r. A view from pizza to circle will specify that r corresponds to d=2. A simple generic procedure that calculates the area of a circle can then be specialized by compilation through the view. Because the view mapping is folded into the generic algorithm and optimized, the specialized algorithm operates directly on the original data and, in this case, does no extra computation (Fig. 4). Code to reference d in data structure pizza is included in the specialized code. area pizza area circle Compilation, Optimization area pizza Figure 4: Example Specialization 2.3 Abstract Data Types and Views An abstract type is an abstraction of a set of concrete types; it assumes an abstract record containing a set of basis variables 3 and has a set of named generic procedures that are written in terms of the basis variables. 4 Any data structure that contains the basis variables, with the same names and types, implements the abstract type. To maximize reuse, constraints on the implementation must be minimized: it should be possible to specialize a generic procedure for any legitimate implementation of its abstract type. diameter radius pizza-as-circle pizza Figure 5: Encapsulation of Concrete Type by View A view encapsulates a concrete type and presents an external interface that consists of the basis variables of the abstract type. Fig. 5 illustrates how view type pizza-as-circle encapsulates pizza and presents an interface consisting of the radius of abstract type circle. The radius is implemented by dividing field diameter of pizza by 2; other fields of pizza are hidden. Code for this example is shown in the following section. In the general case, the interface provides the ability to read and write each basis variable. A read or write may be implemented as access to a variable of the concrete record or by a procedure that emulates the read or write, using the concrete record as storage. 5 A view implements an abstract type if it emulates a record containing the basis variables. Emulation is expressed by two properties: 1. Storage: After a value z is stored into basis variable v, reference to v will yield z. 3 Although it is not required, our abstract types usually specify a concrete record containing the basis variables; this is useful as an example and for testing the generic procedures. 4 This definition of abstract type is different from the algebraic description of abstract types [11] as a collection of abstract sorts, procedure signatures, and axioms. In the algebraic approach, an abstract type is described without regard to its implementation. In our approach, an abstract implementation is assumed because the abstract type has generic procedures that implement operations. 5 Views can be implemented in an object-oriented system by adapter or wrapper objects [16], where a wrapper contains the concrete data, presents the interface of the abstract type, and translates messages between the abstract and concrete types. Our views give the effect of wrappers without creating them. 2. Independence: If a reference to basis variable v yields the value z and a value is then stored into some other basis variable w, a reference to v will still yield z. These properties express the behavior expected of a record: stored values can be retrieved, and storing into one field does not change the values of others. If a view implements an abstract type exactly, as described by the storage and independence properties, then any generic procedure will operate in the same way (produce the same output and have the same side effects) when operating on the concrete data through the view as it does when operating on a record consisting of the basis variables. That is, an isomorphism holds between the abstract type and concrete type, and its diagram commutes. This criterion is satisfied by the following variations of data: 1. Any record structure may be used to contain the variables. 6 2. Names of variables may differ from those of the abstract type: views provide name translation, and the name spaces of the concrete and abstract types are distinct. Some generics use only a subset of basis variables; only those that are used must be defined in a view. An attempt to use an undefined basis variable is detected as an error. A view in effect defines functions to compute basis variables from the concrete variables; if a generic procedure is to "store into" basis variables, these functions must be invertible. Simple functions can be inverted automatically by the compiler. For more complex cases, a procedure can be defined to effect a store into a basis variable. The procedures required for mathematical views may be somewhat complex: in the case of a polar vector (r; '), where the abstract type is a Cartesian vector (x; y), an assignment to basis variable x must update both r and ' so that x will have the new value and y will be unchanged. A program MKV ("make view") [54] allows a user to specify mathematical views graphically by connecting corresponding parts of the concrete type and a diagram associated with the abstract type; MKV uses symbolic algebra to derive view procedures from the correspondences. For wider reuse, the storage and independence properties must be relaxed slightly. Even a simple change of representation, such as division of the diameter value by 2 in the pizza example, changes the point at which numerical overflow occurs; there could also be round-off error. Significant changes of representation should be allowed, such as representing a vector in polar coordinates (r; ') when the basis variables are in Cartesian coordinates (x; y). If a polar vector is viewed as a Cartesian vector using transformations sin('), the mapping is not exact due to round-off error, nor is it one-to-one; however, it is sufficiently accurate for many applications. Ultimately, the user of the system must ensure that the chosen representation is sufficiently accurate. In some cases, a user might want to specify a contents type and let the system define a record using it, e.g. an AVL tree containing strings. This is easily done by substituting the contents type into a prototype record definition with the view mappings predefined. The next section describes how views are implemented and compiled in GLISP. 6 This could include arrays, or sub-records reached by a fixed sequence of pointer traversals. 3 GLISP Language and Compiler GLISP [46, 47, 48, 49] ("Generic Lisp"), a high-level language with abstract data types, is compiled into Lisp; it has a language for describing data in Lisp and in other languages. GLISP is described only briefly here; for more detail, see [49] and [46]. 3.1 Data-independent Code A GLISP type is analogous to a class in object-oriented programming (OOP); it specifies a data structure and a set of methods. For each method, there is a name (selector) and a definition as an expression or function. As in OOP, there is a hierarchy of types; methods can be inherited from ancestor types. The methods of abstract types are generic procedures. In most languages, the syntax of program code depends on the data structures used; this prevents reuse of code for alternative implementations of data. GLISP uses a single Lisp-like syntax. In Lisp, a function call is written inside parentheses: (sqrt x). A similar syntax (feature object) is used in GLISP to access any feature of a data structure [49]: 1. If feature is the name of a field of the type of object, data access is compiled. 2. If feature is a method name (selector) of the type of object, the method call is compiled. 3. If feature is the name of a view of the type of object, the type of object is locally changed to the view type. 4. If feature is a function name, the code is left unchanged. 5. Otherwise, a warning message is issued that feature is undefined. This type-dependent compilation allows variations in data representation: the same code can be used with data that is stored for one type but computed for another type. For example, type circle can assume radius as a basis variable, while a pizza object can store diameter and compute radius. The GLISP compiler performs type inference as it compiles expressions. When the type of an object is known at compile time, reference to a feature can be compiled as in-line code or as a call to a specialized generic. Specialized code depends on the types of arguments to the generic. Compilation by in-line expansion and specialization is recursive at compile time and propagates types during the recursion; this is an important feature. Recursive expansion allows a small amount of source code to expand into large output code; it allows generic procedures to use other generics as subroutines and allows higher-order procedures to be expanded through several levels of abstraction until operations on data are reached. Symbolic optimization folds operations on constants [62], performs partial evaluation [7] [12] and mathematical optimization, removes dead code, and combines operations to improve efficiency. It provides conditional compilation, since a conditional is eliminated when the test can be evaluated at compile time. Optimization often eliminates operations associated with a view, so that use of the view has little or no cost after compilation. 3.2 Views in GLISP A view [46, 49, 50] is expressed as a GLISP type whose record is the concrete type. The abstract type is a superclass of the view type, allowing generics to be inherited 7 . The view type encapsulates the concrete type and defines methods to compute basis variables of the abstract type. As specialized versions of generics are compiled, the compiler caches them in the view type. Examples of abstract type circle, concrete type pizza, and view type pizza-as-circle are shown below; each gives the name of the type followed by its data structure, followed by method (prop), view, and superclass specifications. (circle (list (center vector) (radius real)) (pizza (cons (diameter real) (topping symbol)) views ((circle pizza-as-circle)) ) (pizza-as-circle (p pizza) supers (circle)) pizza-as-circle encapsulates pizza and makes it appear as a circle; its record is named p and has type pizza. It defines basis variable radius as the diameter of p divided by 2 and specifies circle as a superclass; it hides other data and methods of pizza 8 . The following example shows how area defined in circle is compiled through the view; GLISP function t1 is shown followed by compiled code in Lisp. (gldefun t1 (pz:pizza) (area (circle pz))) result type: REAL The code (circle pz) changes the type of pz to the view type pizza-as-circle. The area method is inherited from circle and expanded in-line; basis variable radius is expanded using diameter, which becomes a data access (CAR PZ). If a view defines all basis variables in terms of the concrete type, then any generic procedure of the abstract type can be used through the view. Because compilation by GLISP is recursive, generic procedures can be written using other generics as subroutines, as 7 Only methods are inherited; data structures, and therefore state variables, are not. 8 pizza-as-circle fails to define the basis variable center; this is allowable. An attempt to reference an undefined basis variable is detected as an error. long as the recursion terminates at compile time. 9 A view type may redefine some methods that are generics of the abstract type; this may improve efficiency. For example, a Cartesian vector defines magnitude as a generic, but this value is stored as r in a polar (r; ') vector. When a basis variable is assigned a value, the compiler produces code as follows: 1. If the basis variable corresponds to a field of the concrete type, a store is generated. 2. If the basis variable is defined by an expression that can be inverted algebraically, the compiler does so. For example, assigning a value r to the radius of a pizza-as-circle causes r 2 to be stored into the diameter of record pizza. 3. A procedure can be defined in the view type to accomplish assignment to a basis variable while maintaining the storage and independence properties. MKV [54] produces such procedures automatically. A view can define a procedure to create an instance of the concrete type from a set of basis variables of the abstract type [54]. This is needed for generics that create new data, e.g. when two vectors are added to produce a new vector. Several points about views are worth noting: 1. In general, it is not the case that an object is its view; rather, a view represents some aspect of an object. The object may have other data that are not involved in the view. 2. A view provides name translation. This removes any necessity that concrete data use particular names and eliminates name conflicts. 3. A view can specify representation transformation. 4. There can be several ways of viewing a concrete type as a given abstract type. For example, the same records might be sorted in several ways for different purposes. 4 Clusters of Views Several languages (e.g. Ada, Modula-2, ML, and C++) provide a form of abstract data type that is like a macro: an instance is formed by substituting a concrete type into it, e.g. to make a linked list whose contents is a user type. This technique allows only limited software reuse. We seek to extend the principle that a generic should be reusable for any reasonable implementation of its data to generics that involve several abstract types. Some data structures that might be regarded as a single concept, such as a linked list, involve several types: a linked list has a record type and a pointer type. Many languages finesse the need for two types by providing a pointer type that is derived from the record 9 Recursion beyond a certain depth is trapped and treated as a compilation error. type. In general, however, a pointer can be any data that uniquely denotes a record in a memory: a memory address, a disk address, an array index, an employee number, etc. To maximize generality, the record and pointer must be treated as distinct types. A view maps a single concrete type to a single abstract type. A cluster collects a set of views that are related because they are used in a generic algorithm. For example, a polygon can be represented as a sequence of points; the points could be Cartesian, polar or some type that could be viewed as a point (e.g. a city), and the sequence could be a linked list, array, etc. There should not be a different generic for each combination of types; a single generic should be usable for any combination. A cluster collects the views used by a generic algorithm in a single place, allows inheritance and specialization of generics through the views, and is used in type inference. A cluster has a set of roles, each of which has a name and a corresponding view type; for example, cluster linked-list has roles named record and pointer. A cluster may have super-clusters; each view type that fills a role specifies as a superclass the type that fills the same role in the super-cluster, allowing inheritance of methods from it. The view types also define methods or constants 10 needed by generic procedures; for example, cluster sorted-linked-list requires specification of the field or property of the record on which to sort and whether the sort is ascending or descending. 4.1 Example Cluster: Sorted Linked List This section gives an example record, shows how a cluster is made for it using VIEWAS, and shows how a generic is specialized. We begin by showing the user interaction with VIEWAS to emphasize its ease of use; a later section explains how VIEWAS works. The C structure of example record myrec and its corresponding GLISP type are shown below. 11 struct myrecord - int char *name; int struct myrecord *next; (myrec (crecord myrec (color integer) (name string) (next (- A constant is specified as a method that returns a constant value. 11 The GLISP type could be derived automatically from the C declaration, but this is not implemented. Suppose the user wishes to view myrec as a sorted-linked-list and obtain specialized versions of generics for an application. The user invokes VIEWAS to make the view cluster: (viewas 'sorted-linked-list 'myrec) VIEWAS determines how the concrete type matches the abstract type; it makes some choices automatically and asks the user for other choices: 12 Choice for Specify choice for SORT-VALUE Choices are: (COLOR NAME SIZE) name Specify choice for SORT-DIRECTION Choices are: (ASCENDING DESCENDING) ascending VIEWAS chooses field next as the link of the linked-list record since it is the only possibility; it asks the user for the field on which to sort and the direction of sorting. VIEWAS requires only a few simple choices by the user; the resulting cluster MYREC-AS-SLL and two view types are shown in Fig. 6. Cluster MYREC-AS-SLL has roles named pointer and record that are filled by corresponding view types; MYREC-AS-SLL lists cluster SLL (sorted linked list) as a super-cluster. View type MYREC-AS-SLL-POINTER is a pointer to view type MYREC-AS-SLL-RECORD; it has the corresponding type SLL-POINTER of cluster SLL as a superclass. The generics of SLL are defined as methods of SLL-POINTER. View type MYREC-AS-SLL-RECORD has data named Z16 13 of type MYREC; it lists type SLL-RECORD as a superclass and defines the LINK and After making a view cluster, the user can obtain specialized versions of generics. We do not expect that a user would read the code of generics or the specializations derived from them, but we present a generic and its specialization here to illustrate the process. Fig. 7 shows generic sll-insert; it uses generics rest defined for linked-list (the value of field link) and sort-before. The notation (-. ptr) is short for dereference of pointer ptr. sort-direction is tested in this code; however, since this is constant at compile time, the compiler eliminates the if and keeps only one sort-before test, which is expanded depending on the type of sort-value. converts symbols to upper-case, so upper-case and lower-case representations of the same symbol are equivalent. In general, user input is shown in lower-case, while Lisp output of symbols is upper-case. 13 Names with numeric suffixes are new, unique names generated by the system. The unique name Z16 encapsulates MYREC and prevents name conflicts in the view type because features of MYREC can be accessed only via that name. (GLCLUSTERDEF (ROLES ((POINTER MYREC-AS-SLL-POINTER) (RECORD MYREC-AS-SLL-RECORD)) View type MYREC-AS-SLL-POINTER: GLCLUSTER MYREC-AS-SLL View type MYREC-AS-SLL-RECORD: RESULT lst else new))) Figure 7: Generic: Insert into Sorted Linked List 4.2 Uses of Clusters Clusters serve several goals: 1. Clusters allow independent specification of the several views used in a generic. 2. A generic that performs a given function should be written only once; generics should reuse other generics when possible. Clusters allow generics to be inherited. 3. Clusters are used to derive the correct view types as generics are specialized. 4.2.1 Inheritance through Clusters It is desirable to inherit and reuse generics when possible. In some cases, a cluster can be considered to be a specialization of another cluster, e.g. sorted-linked-list is a specialization of linked-list. Some generics defined for linked-list also work for a sorted-linked-list: the length of a linked-list is the same whether it is sorted or not. Generics should be defined at the highest possible level of abstraction to facilitate reuse. A cluster can specify super-clusters. Fig. 9 shows the inheritance among clusters for the MYREC-AS-SLL example; each cluster can inherit generics from the clusters above. The mechanism of inheritance between clusters is simply inheritance between types. Each type that fills a role in a cluster lists as a superclass the type that fills the corresponding MYREC *sll-insert-1 (lst, new) MYREC *lst, *new; MYREC *ptr, *prev; while ( ptr && strcmp(ptr-?name, new-?name) if (prev != return lst; else return new; Figure 8: Specialized Procedure in C role in the super-cluster. These inheritance paths are specified manually for abstract types; VIEWAS sets up the inheritance paths when it creates view clusters. Inheritance provides defaults for generic procedures and constants. For example, generics of sorted-linked-list use predicate sort-before to compare records; a generic sort-before is defined as ! applied to the sort-value of records. Predicate !, in turn, depends on the type of the sort-value, e.g., string comparison is used for strings. A minimal specification of a sorted-linked-list can use the default sorting predicate, but an arbitrary sort-before predicate can be specified in the record view type if desired. In some cases, inheritance of generics from super-clusters should be preventable. For example, nreverse (destructive reversal of the order of elements in a linked list) is defined for linked-list but should not be available for a sorted-linked-list, since it destroys the sorting order. Prevention of inheritance can be accomplished by defining a method as error, in which case an attempt to use the method is treated as a compilation error. record-with-pointer linked-list sll (sorted-linked-list) Figure 9: Inheritance between Clusters 4.2.2 Type Mappings A cluster specifies a set of related types. A generic procedure is specified in terms of abstract types, but when it is specialized, the corresponding view types must be sub- stituted. For example, at the abstract level, a linked-list-record contains field link whose type is linked-list-pointer, and dereference of a linked-list-pointer yields a linked-list-record. When a generic defined for linked-list is specialized, these types must be replaced by the corresponding view types of the cluster. Care is required in defining generics and view types to ensure that each operation produces a result of the correct type; otherwise, the generic will not compile correctly when specialized. In general, if a generic function f : a 1 , with abstract argument and result types is specialized for concrete types t 1 and t 2 using views v 1 , the specialized function must have signature f s Smith [68] uses the term theory morphism for a similar notion. Dijkstra [15] uses the term coordinate transformation for a similar notion, in which some variables are replaced by others; Gries [23] uses the term coupling invariant for the predicate that describes the relation (between the abstract types or variables and their concrete counterparts) that is to be maintained by functions. Consider the following generic: (gldefun generic-cddr (l:linked-list) (rest (rest l)) ) generic-cddr follows the link of a linked-list record twice: rest is the link value. 14 Now suppose that a concrete record has two pointer fields, so that two distinct linked-list clusters can be made using the two pointer fields. To specialize generic-cddr for both, rest must produce the view type, which defines the correct link, rather than the concrete type. Fig. abstractly illustrates type mappings as used in the generic-cddr example. The figure shows concrete types t i that are viewed as abstract types a i by views v i . Suppose 14 rest (or cdr) and cddr are Lisp functions; we use Lisp names for linked-list generics that are similar. a 1 a 2 a 3 f s Figure 10: Cluster: Views between Type Families is composed with function g : a 2 . The corresponding specialized functions will be f s Because the views are virtual and operations are actually performed on the concrete data, the compiled code will perform g s . However, the result of function f s , as seen by the compiler, must be because function g is defined for abstract type a 2 and is inherited by but is undefined for concrete type t 2 The roles of a cluster are used within generics to specify types that are related to a known view type. Each view type has a pointer to its cluster, and the cluster's roles likewise point to the view types; therefore, it is possible to find the cluster from a view type and then to find the view type corresponding to a given role of that cluster. The GLISP construct (clustertype role code) returns at compile time the type that fills role in the cluster to which the type of code belongs; this construct can be used as a type specifier in a generic, allowing use of any view type of the cluster. For example, a generic's result type can be declared to be the view type that fills a specified role of the cluster to which an argument of the generic belongs. clustertype can also be used to declare local variable types and to create new data of a concrete type through its view. Thus, type signatures and types used within generics can be mapped from the abstract level to the view level so that specialization of generics is performed correctly. 5 View Cluster Construction: VIEWAS A view cluster may be complex, and detailed knowledge of the generic procedures is needed to specify one correctly. We expect that abstract types and view clusters will be defined by experts; however, it should be simple for programmers to reuse the generics. VIEWAS makes it easy to create view clusters without detailed understanding of the abstract types and generics. Its inputs are the name of the view cluster and concrete type(s). VIEWAS determines correspondences between the abstract types of the cluster and the concrete types, asking questions as needed; from these it creates the view cluster and view types. (gldefviewspec '(sorted-linked-list (sorted-sequence) sll t ((record anything)) ((type pointer (- record)) (prop link (partof record pointer) result pointer) (prop copy-contents-names (names-except record (link)) ) (prop sort-value (choose-prop-except record (link))) (prop sort-direction (oneof (ascending descending)) ((pointer pointer) (record record prop (link copy-contents-names sort-value Figure 11: View Specification used by VIEWAS Fig. 11 shows the view specification for a sorted linked list. ((record anything)) is a list of formal parameters that correspond to types given in the call to VIEWAS; argument record can have a type matching anything. Next is a list of names and specifications that are to be matched against the concrete type; following that is a pattern for the output cluster, which is instantiated by substitution of the values determined from the first part. Finally, there is a list of super-clusters of the cluster to be created, (sll). The previous example, (viewas 'sorted-linked-list 'myrec), specifies the name of the target cluster and the concrete type myrec to be matched. VIEWAS first matches the record with the myrec argument; then it processes the matching specifications in order: 1. (type pointer (- record)) The first thing to be determined is the type of the pointer to the record. The pointer defaults to a standard pointer to the record, but a different kind of pointer, such as an array index, will be used if it has been defined. 2. (prop link (partof record pointer) result pointer) The link must be a field of the record, of type pointer. Type filtering restricts the possible matches; if there is only one, it is selected automatically. 3. (prop copy-contents-names (names-except record (link)) ) These are the names of all fields of the record other than the link; the names are used by generics that copy the contents of a record. 4. (prop sort-value (choose-prop-except record (link)) ) The sort-value is compared in sorting; it is chosen from either fields or computed (method) values defined for the record type, excluding the field that is the link. A menu of choices is presented to the user. 5. (prop sort-direction (oneof (ascending descending)) ) This must be ascending or descending; the user is asked to choose. After the items have been matched with the concrete type, the results are substituted into a pattern to form the view type cluster. Fig. 6 above shows cluster myrec-as-sll and view types myrec-as-sll-pointer and myrec-as-sll-record produced by VIEWAS. Properties needed by generics of sorted-linked-list, such as sort-value, are defined in terms of the concrete type. Generics defined for sorted-linked-list explicitly test sort-direction; since this value is constant, only the code for the selected direction is kept. This illustrates that switch values in view types can select optional features of generics. Weide [73] notes that options in reusable procedures are essential to avoid a combinatoric set of versions. For example, the Booch component set [9] contains over 500 components; Batory [5] has identified these as combinations of fewer design decisions. The linked-list library has 28 procedures; one view cluster allows specialization of any of them. VIEWAS requires minimal input; it presents sensible choices, so the user does not need to understand the abstract types in detail. In effect, view specifications use a special-purpose language that guides type matching; this language is not necessarily complete, but is sufficient for a variety of view clusters. Some specifications prevent type errors and often allow a choice to be made automatically, as in the case of the link field. Others, e.g. copy-contents-names, perform bookkeeping to reduce user input. Specifications such as that for sort-value heuristically eliminate some choices; additional restrictions might be helpful, e.g., sort-value could require a type that has an ordering. VIEWAS is not a powerful type matcher, but it tends to be self-documenting, eliminates opportunities for errors, and is effective with minimal input. We assume that the user understands the concrete type and understands the abstract types well enough to make the choices presented. VIEWAS is intended for views of data structures; a companion program MKV [54] uses a graphical interface and algebraic manipulation of equations to make mathematical views. We have also investigated creation of programs from connections of diagrams that represent physical and mathematical models [51]. 6 Higher-order Code 6.1 Compound Structures Abstract types may be used in larger structures. For example, several kinds of queue can be made from a linked-list: front-pointer-queue, with a pointer to the front of a linked list, two-pointer-queue, with pointers to the front and the last record, and end-pointer-queue, with a pointer to the last record in a circularly linked list. A sequence of queue, in turn, can be used for a priority-queue. Generics for compound structures are often small and elegant; for example, insertion in a priority queue is: (gldefun priority-queue-insert (q:priority-queue n:integer new) (insert (index q n) new) ) The code (index q n) indexes the sequence by priority n to yield a queue. insert is interpreted relative to the type of queue. This small function expands into larger code because its operations expand into operations on component structures, which are further expanded. A single definition of a generic covers the combinatoric set of component types. 6.2 Generic Loop Macros A language with abstract types must provide loops over collections of data. Alphard [66] and CLU [38] allow iterators for concrete types; Interlisp [30] provided a flexible looping construct for Lisp lists. SETL [14] provides sets and maps and compiles loops over them, with implementations chosen by the compiler [63]. Generic procedures need loops that are independent of data structure (e.g., array, linked list, or tree); this is done by loop macros. Expansion of generic procedures obeys strict hierarchy and involves independent name spaces. In expanding a loop, however, code specified in the loop statement must be interspersed with the code of the iterator, and the iterator must introduce new variables at the same lexical level; macros are used for these reasons. Names used by the macro are changed when necessary to avoid name conflicts. GLISP provides a generic looping statement of the form: (for item in sequence [when p(item)] verb f(item) ) When this statement is compiled, iterator macros defined for the verb and for the type of sequence are expanded in-line. For example, consider: (gldefun t3 (r:myrec) (for x in (sorted-linked-list r) sum (size x))) This loop iterates through a sequence r using its sorted-linked-list view and inheriting the linked-list iterator; it sums the size of each element x of the linked list. Macros are provided for looping, summation, max, min, averaging, and statistics. Collection macros collect data in a specified form, making it possible to convert from one kind of collection to another (e.g., from an array to a linked list). Data structures may have several levels of structure. For example, a symbol table might be constructed using an array of buckets, where the array is indexed by the first character of a symbol and each array element is a pointer to a bucket, i.e., a sorted linked list of symbols. A double-iterator macro is defined that composes two iterators, allowing a loop over a compound structure: (gldefun t4 (s:symbol-table) (for sym in s sum (name sym))) t4 concatenates the names of all the symbols. The loop expands into nested loops (first through the array of buckets, then through each linked list) and returns a string (since a name is a string and concatenates strings). The compiled code is 23 lines of Lisp. 6.3 Copying and Representation Change The GLISP compiler can recursively expand code through levels of abstraction until operations on data are reached; it interprets code relative to the types to which it is applied. In Lisp, funcall is used to call a function that is determined at runtime. In GLISP, a funcall whose function argument is constant at compile time is treated like other function calls, i.e., it is interpreted relative to its argument types. This makes it possible to write higher-order code that implements compositions of views. The contents of a linked-list record may consist of several items of different types. Generic copy-list makes a new record and copies the contents fields into it, requiring several assignments. This is accomplished by a loop over the copy-contents-names defined in the view type. For each name in copy-contents-names, a funcall of that name on the destination record is assigned the value of a funcall of that name on the source record. (for name in (copy-contents-names (-. l)) do ((funcall name (implementation (-. l))) := (funcall name (implementation (-. m))) Since the list of names is constant, the loop is unrolled. Each funcall then has a constant name that is interpreted as a field or method reference; the result is a sequence of assignments. Since the "function call" on each side of the assignment statement is interpreted relative to the type to which it is applied, this higher-order code can transfer data to a different record type and can change the representation during the transfer, e.g., by converting the radius of a circle in one representation to the area in another representation, or by converting the data to reflect different representations or units of measurement [53]. For example, consider two different types cira and cirb, each of which has a color and lists circle as a superclass: (cira (cons (color symbol) (cons (nxt (- cira)) (radius supers (circle)) (cirb (list (diameter roman) (color (next (- cirb))) prop ((radius (diameter / 2))) supers (circle)) These types have different records, and cira contains an integer radius while cirb contains diameter represented in roman numerals. After viewing each type as a linked-list, it is possible to copy a list from either representation to the other. This illustrates how higher-order code is expanded. First, the loop is unrolled into two assignment statements that transfer color and diameter from source record to destination record; then diameter is inherited from circle for the source record and encoded into Roman numerals for the destination record: (gldefun t5 (u:cira &optional v:cirb) (copy-list-to (linked-list u) (linked-list v))) (t5 '(RED (GREEN (BLUE NIL . 12) . 47) . 9)) 6.4 Several Views Viewing concrete data as a conceptual entity may involve several views; e.g., a polygon can be represented as a sequence of points. Viewing a concrete type as a polygon requires a view of the concrete type t 1 as a sequence of some type t 2 and a view of t 2 as a vector (Fig. 12). View from the element of the concrete sequence to a vector is specified declaratively by giving the name of the view. This view name is used in a funcall inside the polygon procedures; it acts as a type change function that changes the type of the sequence element to its view as a point. This effectively implements composition of views. A single generic can be specialized for a variety of polygon representations. For example, a string of characters can be viewed as a polygon by mapping a character to its position on a keyboard: does the string "car" contain the character "d" on the keyboard? vector iterator iterator Figure 12: Polygon: Sequence of Vector 6.5 Application Languages The Lisp output of our system can be mechanically transformed into other languages, including C, C++, Java, and Pascal. GLISP allows a target language record as a type; accesses to such records are compiled into Lisp code that can be transformed into target language syntax. The code can also run within Lisp to create and access simulated records; this allows the interactive Lisp environment and our programming and data-display tools to be used for rapid prototyping. Conversion of Lisp to the target language is done in stages. Patterns are used to transform idioms into corresponding target idioms and to transform certain Lisp constructs (e.g., returning a value from an if statement) into constructs that will be legal in the target language. These Lisp-to-Lisp transformations are applied repeatedly until no further transformations apply. A second set of patterns transforms the code into nicely-formatted target language syntax. The result may be substantially restructured. A C procedure sll-insert-1 was shown in Fig. 8. This code is readable and has no dependence on Lisp. Versions of generic procedures containing a few hundred lines of code have been created in C, C++, Java, and Pascal. The C version of the convex hull program, described below, runs 20 times faster than the Lisp version. 6.6 A Larger Example: Convex Hull The convex hull of a set of points is the smallest convex polygon that encloses them. Kant studied highly qualified human subjects who wrote algorithms for this task. All subjects took considerable time; some failed or produced an inefficient solution. Although convex hull algorithms are described in textbooks [64] and in the literature, getting an algorithm from such sources is difficult: it is necessary to understand the algorithm, and a published description may omit details of the algorithm or even contain errors [22]. A hand-coded version of a published algorithm requires testing or verification. Fig. 13 illustrates execution of a generic convex hull algorithm. We describe the algorithm Figure 13: Convex Hull of Points and illustrate its use on cities viewed as points. The algorithm uses several views of the same data and reuses other generics; it is similar to the QUICKHULL algorithms [57]. A convex hull is represented as a circularly linked list of vertex points in clockwise order (Fig. 14). An edge is formed by a vertex and its successor. Associated with a vertex is a list of points that may be outside the edge; an edge is split (Fig. 15) by finding the point that is farthest to the left of the edge. If there is such a point, it must be on the convex hull. The edge is split into two edges, from the original vertex to the new point and from the new point to the end vertex, by splicing the new point into the circularly linked list. The subsets of the points that are to the left of each new edge are collected and stored with the corresponding vertex points, and the edges are split again. The algorithm is initialized by making the points with minimum and maximum x values into a two-point polygon with all input points associated with each edge; then each edge is split. Finally, the vertices are collected as a (non-circular) linked list. Fig. 13 shows the successive splittings. The algorithm rapidly eliminates most points from consideration. Fig. 16 shows the type cluster used for convex hull. The line formed by a point and its successor is declared as a virtual line-segment [46]; this is another way of specifying a view. This allows the polygon to be treated simultaneously as a sequence of vertices and as a sequence of edges. Only vertices are represented, but the algorithm deals with edges as well. The internal-record specifies circular-linked-list under property viewspecs; this causes VIEWAS to be called automatically to make the circular-linked-list view \Gamma' A A A A A A A A AU Xy s s s s s s Figure 14: Convex Hull as Circular Linked List of Points s s \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi* d A AU A s s s s s Figure 15: Splitting an Edge so that procedure splice-in and the iterator of that view can be used. Fig. 17 shows generic procedure convex-hull. This procedure initializes the algorithm by finding the two starting points; use of iterators min and max simplifies the code. Next, an initial circularly linked list is made by linking together the starting points, and function split is called for each. Finally, the vertex points are collected as a non-circular list. Fig. shows generic cvh-split, which uses iterator max and signed leftof-distance from a line-segment to a point; this generic is inherited since the line associated with a vertex is a virtual line-segment. leftof-distance expands as in-line code that operates directly on the linked list of vertex records; we can think of a vertex and its successor as a line-segment without materializing one. Operator specifies a push on a list to collect points. cvh-split also uses procedure splice-in of the circular-linked-list view of the points. The split algorithm views the same data in three different ways: as a vertex point, as an edge line-segment, and as a circularly linked list. We believe that use of several views is common and must be supported by reuse technologies. A programmer should not have to understand the algorithm to reuse it. The concrete (gldefclusterc 'convex-hull-cluster '((source-point (convex-hull-source-point vector)) (source-collection (convex-hull-source-collection (listof convex-hull-source-point) prop ((hull convex-hull specialize t)))) (internal-record (convex-hull-internal-record (list (pt convex-hull-source-point) (next (- convex-hull-internal-record)) (points (listof convex-hull-source-point))) prop ((line ((virtual line-segment with msg ((split cvh-split specialize t)) viewspecs Figure data might not be points per se and might have other attributes. To find the convex hull using a traditional algorithm would require making a new data set in the required form, finding the convex hull, and then making a version of the hull in terms of the original data. Specialization of generics is more efficient. For example, consider finding the convex hull of a set of cities, each of which has a latitude and longitude. Fig. 19 shows the city data structure and a hand-written view as a point using a Mercator projection. VIEWAS was used to make a convex hull view of the city-as-point data. Using this view, a specialized version of the convex hull algorithm (229 lines of Lisp) was produced (in about 5 seconds) that operates on the original data. This example illustrates the benefits of our approach to reuse. The generic procedures themselves are relatively small and easy to understand because they reuse other generics. Reuse of the generic procedure for an application has a high payoff: the generated code is much larger and more complex than the few lines that are entered to create the views. 6.7 Testing and Verification Users must have confidence that reused programs will behave as intended. Programmer's Apprentice [61] produced Ada code; the user would read this code and modify it as necessary. We do not believe a programmer should read the output of any reuse system. With our system, in-line code expansion and symbolic optimization can make the output code difficult to read and to relate to the original code sources. Reading someone else's code is difficult, and no less so if the "someone else" is a machine. (gldefun convex-hull (orig-points:(listof vector)) (let (xmin-pt xmax-pt hullp1 hullp2) (if ((length-up-to orig-points then (xmin-pt := (for p in orig-points min (x p))) with with ((next hullp1) := hullp2) ; link circularly ((next hullp2) := hullp1) (split hullp1) (split hullp2) (for p in (circular-linked-list hullp1) collect (pt p)) ) Figure 17: Generic Convex Hull Procedure We believe that a reuse system such as ours will reduce errors. Errors in reusing software components might arise from several sources: 1. The component itself might be in error. 2. The component might be used improperly. 3. The specialization of a component might not be correct. Algorithms that are reused justify careful development and are tested in many applications, so unnoticed errors are unlikely. Humans introduce errors in coding algorithms; Sedgewick notes "Quicksort . is fragile: a simple mistake in the implementation can go unnoticed and can cause it to perform badly for some files." Reuse of carefully developed generics is likely to produce better programs than hand coding. VIEWAS and MKV guide the user by presenting only correct choices. When views are written by hand, type checking usually catches errors. Although GLISP is not strongly typed (because of its Lisp ancestry), there are many error checks that catch nearly all type errors. Our experience with our system has been good, and we have reused generics for new applications; e.g., the generic for distance from a line to a point was reused to test whether a mouse position is close to a line. Ultimately, it must be verified not only that software meets its specification but also that it is what the user really wants. With rapid prototyping based on reuse, developers (gldefun cvh-split (cp:cvhpoint) (let (maxpt pts newcp) (pts := (points cp)) ((points cp) := nil) (if pts is not null then (maxpt := (for p in pts when ((p !? (p1 (line cp))) and (p !? (p2 (line cp)))) (leftof-distance (line cp) p))) (if maxpt and (leftof (line cp) maxpt) then (newcp := (a (typeof cp) with (splice-in (circular-linked-list cp) (circular-linked-list newcp)) (for p in pts do (if (leftof (line cp) p) then ((points cp) +- p) else (if (leftof (line newcp) p) then ((points newcp) +- p))) ) (split cp) Figure can address a program's performance in practice and make modifications easily. Our system allows significant representation changes to be accomplished easily by recompilation. Formal verification might be applied to specialized generics. Gries and Prins [21] suggest a stratified proof of a program obtained by transformation: if a generic is correct and an implementation of its abstract type is correct, the transformed algorithm will be correct. Morgan [42] extends these techniques for proofs of data refinements. Morris [43] provides calculational laws for refinement of programs written in terms of abstract types such as bags and sets. Related methods might be used for proofs of refinements with a system such as ours; a library of proven lemmas about generic components would greatly simplify the task of proving a software system correct. 7 Views and OOP Views can be used to generate methods that allow concrete data to be used with OOP software; this is useful for reuse of OOP software that uses runtime messages to interface to diverse kinds of objects. The GLISP compiler can automatically compile and cache specialized versions of methods based on the definitions given in a type; for example, a method to compute the area of a pizza-as-circle can be generated automatically. (city (list (name symbol) (latitude (units real degrees)) (population integer) (longitude (units real degrees))) views ((point city-as-point)) ) (city-as-point (z17 city) prop ((x ((let (rad:(units real radians)) (rad := (longitude z17)) (y ((signum (latitude z17)) * (log (tan (pi / (abs (latitude z17)) supers (vector)) Figure 19: City and Mercator Projection We have implemented direct-manipulation graphical editors for linked-list and array. A display method and editor for a record can be made interactively using program DISPM, which allows selection of properties to be displayed, display methods to be used, and positions. Given a display method for a record, a generic for displaying and editing structured data containing such records can be used on the concrete data. Figure 20 shows data displayed by the generic linked-list editor. The user can move forward or backward in the displayed list or excise a selected list element; the user can also zoom in on an element to display it in more detail or to edit the contents. This technique allows a single generic editor to handle all kinds of linked-list. The display omits detail such as contents of link fields and shows the data in an easily understood form. Figure 20: Linked List Display 8 Related Work We review closely related work. It is always possible to say "an equivalent program could be written in language x"; however, a system for software reuse must satisfy several criteria simultaneously to be effective [34]. We claim that the system described here satisfies all of these criteria: 1. It has wide applicability: many kinds of software can be expressed as reusable generics. 2. It is easy to use. The amount of user input and the learning required are small. 3. It produces efficient code in several languages. 4. It minimally constrains the representation of data. Generics can be specialized for use with existing data and programs. Brooks [10] contends that there are "no silver bullets" in software development. The system described here is not a silver bullet, but it suggests that significant improvement in software development is possible. 8.1 Software Reuse Krueger [34] is an excellent survey of software reuse, with criteria for practical effectiveness. Biggerstaff and Perlis [8] contains papers on theory and applications of reuse; artificial intelligence approaches are described in [1], [39], and [60]. Mili [41] extensively surveys software reuse, emphasizing technical challenges. 8.2 Software Components The Programmer's Apprentice [61] was based on reuse of clich'es, somewhat analogous to our generics. This project produced some good ideas but had limited success. KBEmacs, a knowledge-based editor integrated with Emacs, helped the user transform clich'e code; unfortunately, KBEmacs was rather slow, and the user had to read and understand the low-level output code. We assume that the user will treat the outputs of our system as "black boxes" and will not need to read or modify the code. Rich [59] describes a plan calculus for representing program and data abstractions; overlays relate program and data plans, analogous to our views. Weide [73] proposed a software components industry based on formally specified and unchangeable components. Because the components would be verified and unchangeable, errors would be prevented; however, the rigidity of the components might make them harder to reuse. Our approach adapts components to fit the application. Zaremski and Wing [77] describe retrieval of reusable ML components based on signature matching of functions and modules; related techniques could be used with our generics. Batory [4] [5] [6] describes a data structure precompiler and construction of software systems from layers of plug-compatible components with standardized interfaces. The use of layers whose interfaces are carefully specified allows the developer to ensure that the layers will interface correctly. We have focused on adapting interfaces so that generics can be reused for independently designed data. 8.3 Languages with Generic Procedures Ada, Modula-2 [28], and C++ [69] allow modules for parameterized abstract types such as STACK[type]. Books of generic procedures [37] [44] contain some of the same procedures that are provided with our system. In Ada and Modula-2, such collections have limited value because such code is easy to write and is only a small part of most applications. The class, template, and virtual function features of C++ allow reuse of generics; however, Stroustrup's examples [69] show that the declarations required are complex and subtle. Our declarations are also complex, but VIEWAS hides this complexity and guides the user in creating correct views. The ideas in VIEWAS might be adapted for other languages. 8.4 Functional and Set Languages ML [74] [55] is like a strongly typed Lisp; it includes polymorphic functions (e.g., functions that operate on lists of an arbitrary type) and functors (functions that map structures of types and functions to structures). ML also includes references (pointers) that allow imperative programming. ML functors can instantiate generic modules such as container types. Our system allows storing into a data structure through a view and composition of views [52]. Miranda [71] is a strongly-typed functional language with higher-order functions. While this allows generics, it is often hard to write functional programs with good performance. provides sets and set operations. [63] describes an attempt to automatically choose data structures in SETL to improve efficiency. Kruchten et al. [35] say "slow is beautiful" to emphasize ease of constructing programs, but inefficient implementations can make even small problems intractable. Transformation Systems Transformation systems repeatedly replace parts of an abstract algorithm specification with code that is closer to an implementation, until executable code is reached. Our views specify transformations from features of abstract types to their implementations. Kant et al. [33] describe Sinapse, which generates programs to simulate spatial differential equations, e.g. for seismic analysis. Sinapse transforms a small specification into a much larger program in Fortran or C; it is written using Mathematica [75] and appears to work well within its domain. Setliff's Elf system [65] automatically generates data structures and algorithms for wire routing on integrated circuits and printed circuit boards. Rules are used to select refinement transformations based on characteristics of the routing task. KIDS [68] transforms problem statements in first-order logic into programs that are highly efficient for certain combinatorial problems. The user must select transformations to be used and must supply a formal domain theory for the application. This system is interesting and powerful, but its user must be mathematically sophisticated. Gries and Prins [21] proposed use of syntactic transformations to specify implementation of abstract algorithms. Volpano [72] and Gries [23] describe systems in which a user specifies transformations for variables, expression patterns, and statement patterns; by performing substitutions on an algorithm, a different version of the algorithm is obtained. This method allows the user to specify detailed transformations for a particular specialization of an algorithm, whereas we rely on type-based transformations and on general optimization patterns. The ability to specify individual transformations in Gries' system gives more flexibility, possibly at the cost of writing more patterns. The Intentional Programming project at Microsoft [67] is based on intentions, which are similar to algorithm fragments expressed as abstract syntax trees. Intentions can be transformed by enzymes at the abstract syntax tree level and can be parsed and unparsed into various surface syntaxes by methods stored with or inherited by the intentions. This work is in progress; its results to date are impressive. Berlin and Weise [7] used partial evaluation to improve efficiency of scientific programs. Given that certain features of a problem are constant, their compiler performs as many constant calculations as possible at compile time, yielding a specialized program that runs faster. Our system includes partial evaluation by in-lining and symbolic optimization. Consel and Danvy [12] survey work on partial evaluation. 8.6 Views Goguen [18] proposes a library interconnection language, LIL. This proposal has much in common with our approach, and Goguen uses the term view similarly; LIL has a stronger focus on mathematical descriptions and axioms. The OBJ language family of Goguen et al. [19] has views that are formal type mappings; operators are mapped by strict isomorphisms. Tracz [70] describes LILEANNA, which implements LIL for construction of Ada packages; views in LILEANNA map types, operations, and exceptions between theories. In our system, views are computational transformations between types; general procedures as well as operators can be reused. Garlan [17] and Kaiser [31] use views to allow tools in a program development environment to access a common set of data. Their MELD system can combine features (collections of classes and methods) to allow "additive" construction of a system from selected features. Meyers [40] discusses problems of consistency among program development tools and surveys approaches including use of files, databases, and views as developed by Garlan. Hailpern and Ossher [25] describe views as subsets of methods of a class, to restrict certain methods to particular clients. Harrison and Ossher [26] argue that OOP is too restrictive for applications that need their own views of objects; they propose subjects that are analogous to class hierarchies. 8.7 Data Translation IDL (Interface Description Language) [36] translates representations, possibly with structure sharing, for exchange of data between parts of a compiler, based on precise data specifications. Herlihy and Liskov [27] describe transmission of data over a network, with representation translation and shared structure; the user writes procedures to encode and decode data for transmission. The Common Object Request Broker Architecture (CORBA) [13] includes an Interface Definition Language and can automatically generate stubs to allow interoperability of objects across distributed systems and across languages and machine architectures. The ARPA Knowledge-Sharing Project [45] addresses the problem of sharing knowledge bases that were developed using different ontologies. Purtilo and Atlee [58] describe a system that translates calling sequences by producing small interface modules that reorder and translate parameters as necessary for the called procedure. Re-representation of data allows reuse of an existing procedure; it requires space and execution time, although [36] found this was a small cost in a compiler. [49] and this paper describe methods for data translation, but these do not handle shared structure. Guttag and Horning [24] describe a formal language for specifying procedure interface signatures and properties. Yellin and Strom [76] describe semi-automatic synthesis of protocol converters to allow interfacing of clients and servers. 8.8 Object-oriented Programming We have described how views can be used to generate methods for OOP. In OOP, messages are interpreted (converted to procedure calls) depending on the type of the receiving object; methods can be inherited from superclasses. The close connection between a class and its requires the user to understand a great deal about a class and its methods. In many OOP systems, a class must include all data of its superclasses, so reuse with OOP restricts implementation of data; names of data and messages must be consistent and must not conflict. Holland [29] uses contracts to specify data and behavioral obligations of composed objects. Contracts are somewhat like our clusters, but require that specializations include certain instance data and implement data in the same way as the generics that are to be specialized. A separate "contract lens" construct is used to disambiguate names where there are conflicts. Our views provide encapsulation that prevents name conflicts; views allow the reuse benefits of OOP with flexibility in implementing data. Some OOP systems are inefficient: since most methods are small, message interpretation overhead can be large, especially in layered systems. C++ [69] has restricted OOP with efficient method dispatching. Opacity of objects prevents optimization across message boundaries unless messages are compiled in-line; C++ allows in-line compilation. Reuse in OOP may require creating a new object to reuse a method of its class; views allow an object to be thought of as another type without having to materialize that type. 9 Conclusions Our approach is based on reuse of programming knowledge: generic procedures, abstract types, and view descriptions. We envision a library of abstract types and generics, developed by experts, that could be adapted quickly for applications. Programmers of ordinary skill should be able to reuse the generics. VIEWAS facilitates making views; easily used interfaces, as opposed to verbose textual specifications with precise syntax, are essential for successful reuse. Systems like VIEWAS might reduce the complexity of the specifications required in other languages. Views also support data translation and runtime message interpretation: a single direct-manipulation editor can handle all implementations of an abstract type. These techniques provide high payoff in generated code relative to the size and complexity of input specifications. They require only modest understanding of the details of library procedures for successful reuse. Our techniques allow restructuring of data to meet new requirements or to improve efficiency. Traditional languages reflect the data implementation in the code [3], making changes costly. Our system derives code from the data definitions; design decisions are stated in a single place and distributed by compilation rather than by hand coding. The ability to produce code in different languages decouples the choice of programming tools from the choice of application language. It allows new tools to extend old systems or to write parts of a system without committing to use of the tool for everything. Just as computation has become a commodity, so that the user no longer cares what kind of CPU chip is inside the box, we may look forward to a time when today's high-level languages become implementation details. Acknowledgements Computer equipment used in this research was furnished by Hewlett Packard and IBM. We thank David Gries, Hamilton Richards, Ben Kuipers, and anonymous reviewers for their suggestions for improving this paper. --R IEEE Trans. Functors: The Categorical Imperative "A 15 Year Perspective on Automatic Programming," "The Design and Implementation of Hierarchical Software Systems with Reusable Components," "Scalable Software Libraries," "Reengineering a Complex Application Using a Scalable Data Structure Compiler," "Compiling Scientific Code Using Partial Eval- uation," Software Reusability (2 vols. Software Components with Ada "No Silver Bullet: Essence and Accidents of Software Engineering," An Introduction to Data Types "Tutorial Notes on Partial Evaluation," "The Common Object Request Broker: Architecture and Specification," The Programming Language A Discipline of Programming Design Patterns: Elements of Reusable Object-Oriented Software "Views for Tools in Integrated Environments," "Reusing and Interconnecting Software Components," "Principles of Parameterized Programming," The Denotational Description of Programming Languages "A New Notion of Encapsulation," "The Transform - a New Language Construct," "Introduction to LCL, a LARCH/C Interface Language," "Extending Objects to Support Multiple Interfaces and Access Control," "Subject-Oriented Programming Critique of Pure Objects)," "A Value Transmission Method for Abstract Data Types," Data Abstraction and Program Development using Modula-2 "Specifying Reusable Components Using Contracts," Xerox Palo Alto Research Center "Synthesizing Programming Environments from Reusable Features," "Understanding and Automating Algorithm Design," "Scientific Programming by Automated Synthesis," "Software Reuse," "Software Prototyping using the SETL Language," "IDL: Sharing Intermediate Representations," The Modula-2 Software Component Library Automating Software Design "Difficulties in Integrating Multiview Development Systems," "Reusing Software: Issues and Research Directions," "Data Refinement by Miracles," "Laws of Data Refinement," The Ada Generic Library "Enabling Technology for Knowledge Sharing," "GLISP: A LISP-Based Programming System With Data Abstraction," "Data Abstraction in GLISP," "Negotiated Interfaces for Software Reuse," "Software Reuse through View Type Clusters," "Generating Programs from Connections of Physical Models," "Composing Reusable Software Components through Views," "Conversion of Units of Measurement," "Creation of Views for Reuse of Software with Different Data Representations," ML for the Working Programmer Introduction to Discrete Structures Computational Geometry "Module Reuse by Interface Adaptation," "A Formal Representation for Plans in the Programmer's Apprentice," Readings in Artificial Intelligence and Software Engineering The Programmer's Apprentice A Mathematical Theory of Global Program Optimization "An Automatic Technique for Selection of Data Representations in SETL Programs," "On the Automatic Selection of Data Structures and Algorithms," "Abstraction and Verification in Alphard: Defining and Specifying Iterators and Generators," "Intentional Programming - Innovation in the Legacy Age," "KIDS: A Semiautomatic Program Development System," "LILEANNA: A parameterized programming language," "An Overview of Miranda," "The Templates Approach to Software Reuse," "Reusable Software Components," Mathematica: a System for Doing Mathematics by Computer "Interfaces, Protocols, and the Semi-Automatic Construction of Software Adaptors," "Signature Matching: A Key to Reuse," --TR --CTR Heinz Pozewaunig , Dominik Rauner-Reithmayer, Support of semantics recovery during code scavenging using repository classification, Proceedings of the 1999 symposium on Software reusability, p.65-72, May 21-23, 1999, Los Angeles, California, United States Hai Zhuge, Component-based workflow systems development, Decision Support Systems, v.35 n.4, p.517-536, July Sanjay Bhansali , Tim J. Hoar, Automated Software Synthesis: An Application in Mechanical CAD, IEEE Transactions on Software Engineering, v.24 n.10, p.848-862, October 1998 Don Batory , Gang Chen , Eric Robertson , Tao Wang, Design Wizards and Visual Programming Environments for GenVoca Generators, IEEE Transactions on Software Engineering, v.26 n.5, p.441-452, May 2000 Fabio Casati , Silvana Castano , Mariagrazia Fugini , Isabelle Mirbel , Barbara Pernici, Using Patterns to Design Rules in Workflows, IEEE Transactions on Software Engineering, v.26 n.8, p.760-785, August 2000 Richard W. Selby, Enabling Reuse-Based Software Development of Large-Scale Systems, IEEE Transactions on Software Engineering, v.31 n.6, p.495-510, June 2005

abstract data type;software reuse;direct-manipulation editor;generic procedure;partial evaluation;generic algorithm;algorithm specialization

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Trading conflict and capacity aliasing in conditional branch predictors.

As modern microprocessors employ deeper pipelines and issue multiple instructions per cycle, they are becoming increasingly dependent on accurate branch prediction. Because hardware resources for branch-predictor tables are invariably limited, it is not possible to hold all relevant branch history for all active branches at the same time, especially for large workloads consisting of multiple processes and operating-system code. The problem that results, commonly referred to as aliasing in the branch-predictor tables, is in many ways similar to the misses that occur in finite-sized hardware caches.In this paper we propose a new classification for branch aliasing based on the three-Cs model for caches, and show that conflict aliasing is a significant source of mispredictions. Unfortunately, the obvious method for removing conflicts --- adding tags and associativity to the predictor tables --- is not a cost-effective solution.To address this problem, we propose the skewed branch predictor, a multi-bank, tag-less branch predictor, designed specifically to reduce the impact of conflict aliasing. Through both analytical and simulation models, we show that the skewed branch predictor removes a substantial portion of conflict aliasing by introducing redundancy to the branch-predictor tables. Although this redundancy increases capacity aliasing compared to a standard one-bank structure of comparable size, our simulations show that the reduction in conflict aliasing overcomes this effect to yield a gain in prediction accuracy. Alternatively, we show that a skewed organization can achieve the same prediction accuracy as a standard one-bank organization but with half the storage requirements.

to the branch-predictor tables. Although this redundancy increases capacity aliasing compared to a standard one-bank structure of comparable size, our simulations show that the reduction in conflict aliasing overcomes this effect to yield a gain in prediction accuracy. Alternatively, we show that a skewed organization can achieve the same prediction accuracy as a standard one-bank organization but with half the storage requirements. Keywords Branch prediction, aliasing, 3 C's classification, skewed branch predictor. Now with Intel Microcomputer Research Lab, Oregon c 1997 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that new copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request Permissions from Publications Dept, ACM Inc., Fax +1 (212) 869-0481, or [email protected]. 1 Introduction and Related Work In processors that speculatively fetch and issue multiple instructions per cycle to deep pipelines, dozens of instructions might be in flight before a branch is resolved. Under these conditions, a mispredicted branch can result in substantial amounts of wasted work and become a bottleneck to exploiting instruction-level parallelism. Accurate branch prediction has come to play an important role in removing this bottleneck. Many dynamic branch prediction schemes have been investigated in the past few years, with each offering certain distinctive features. Most of them, however, share a common characteristic: they rely on a collection of 1- or 2-bit counters held in a predictor table. Each entry in the table records the recent outcomes of a given branch substream [21], and is used to predict the direction of future branches in that substream. A branch substream might be defined by some bits of the branch address, by a bit pattern representing previous branch directions (known as a branch history), by some combination of branch address and branch history, or by bits from target addresses of previous branches [14, 7, 18, 10, 8, 9]. Ideally, we would like to have a predictor table with infinite capacity so that every unique branch substream defined by an (ad- dress, history) pair will have a dedicated predictor. Chen et al. have shown that two-level predictors are close to being optimal, provided unlimited resources for implementing the predictors [3]. Real-world constraints, of course, do not permit this. Chip die-area budgets and access-time constraints limit predictor-table size, and most tables proposed in the literature are further constrained in that they are direct-mapped and without tags. Fixed-sized predictor tables lead to a phenomenon known as aliasing or interference [21, 16], in which multiple (address, his- tory) pairs share the same entry in the predictor table, causing the predictions for two or more branch substreams to intermingle. Aliasing has been classified as either destructive (i.e., a misprediction occurs due to sharing of a predictor-table entry), harmless (i.e., it has no effect on the prediction) or constructive (i.e., aliasing occasionally provides a good prediction, which would have been wrong otherwise) [21]. Young et al. have shown that constructive aliasing is much less likely than destructive aliasing [21]. Recent studies have shown that large or multi-process workloads with a strong OS component exhibit very high degrees of aliasing [11, 5], and require much larger predictor tables than previously thought necessary to achieve a level of accuracy close to an ideal, unaliased predictor table [11]. We therefore expect that new techniques for removing conflict aliasing could provide important gains towards increased branch-prediction accuracy. Branch aliasing in fixed-size, direct-mapped predictor tables is in many ways analogous to instruction-cache or data-cache misses. This suggests an alternative classification for branch aliasing based on the three-Cs model of cache performance first proposed by Hill [6]. As with cache misses, aliasing can be classified as compul- sory, capacity or conflict aliasing. Similarly, as with caches, larger predictor tables reduce capacity aliasing, while associativity in a predictor table could remove conflict aliasing. Unfortunately, a simple-minded adaptation of cache associativity would require the addition of costly tags, substantially increasing the cost of a predictor table. In this paper we examine an alternative approach, called skewed branch prediction, which borrows ideas from skewed-associative caches [12]. A skewed branch predictor is constructed from an odd number (typically 3 or 5) of predictor-table banks, each of which functions like a standard tagless predictor table. When performing a prediction, each bank is accessed in parallel but with a different indexing function, and a majority vote between the resulting lookups is used to predict the direction of the branch. In the next section we explain in greater detail our aliasing clas- sification. In section 3, we quantify aliasing and assess the effect of conflict aliasing on overall branch-prediction accuracy. In section 4, we introduce the skewed branch predictor, a hardware structure designed specifically to reduce conflict aliasing. In section 5, we show how and why the skewed branch predictor removes conflict aliasing effects at the cost of some redundancy. Our analysis includes both simulation and analytical models of performance, and considers a range of possible skewed predictor configurations driven by traces from the instruction-benchmark suite (IBS) [17], which includes complete user and operating-system activity. Section 6 proposes the enhanced skewed branch predictor, a slight modification to the skewed branch predictor, which enables more attractive tradeoffs between capacity and conflict aliasing. Section 7 concludes this study and proposes some future research directions. An Aliasing Classification Throughout this paper, we will focus on global-history prediction schemes for the sake of conciseness. Global-history schemes use both the branch address and a pattern of global history bits, as described in [18, 19, 20, 10, 8]. Previously-proposed global- history predictors are all direct-mapped and tag-less. Given a history length, the distinguishing feature of these predictors is the hashing function that is used to map the set of all (address, history) pairs onto the predictor table. The gshare and gselect schemes [8] have been the most studied global schemes (gselect corresponds to GAs in Yeh and Patt's terminology [18, 19, 20]). In gshare, the low-order address bits and global history bits are XORed together to form an index value 1 , whereas in gselect, low-order address bits and global history bits are concatenated. Aliasing occurs in direct-mapped tag-less predictors when two or more (address, history) pairs map to the same entry. To measure aliasing for a particular global scheme and table, we simulate a structure having the same number of entries and using the same indexing function as the predictor table considered. However, instead of storing 1-bit or 2-bit predictors in the structure, we store the identity of the last (address, history) pair that accessed the en- try. Aliasing occurs when the indexing (address, history) pair is different from the stored pair. The aliasing ratio is the ratio between the number of aliasing occurrences and the number of dynamic conditional branches. When measured in this way, we can see the relationship between branch aliasing and cache misses. Our simulated tagged table is like a cache with a line size of one datum, and an aliasing occurrence corresponds to a cache miss. 1 When the number of history bits is less than the number of index bits, the history bits are XORed with the higher-order end of the section of low-order address bits, as explained in [8] benchmark conditional branch count dynamic static groff 11568181 5634 gs 14288742 10935 mpeg play 8109029 4752 real gcc 13940672 16716 verilog 5692823 3918 Table 1: Conditional branch counts A widely-accepted classification of cache misses is the three-Cs model, first introduced by Hill [6] and later refined by Sugumar and Abraham [15]. The three-Cs model divides cache misses into three groups, depending on their causes. ffl Compulsory misses occur when an address is referenced for the first time. These unavoidable misses are required to fill an empty or "cold" cache. ffl Capacity misses occur when the cache is not large enough to retain all the addresses that will be re-referenced in the future. Capacity misses can be reduced by increasing the total size of the cache. ffl Conflict misses occur when two memory locations contend for the same cache line in a given window of time. Conflict misses can be reduced by increasing the associativity of a cache, or improving the replacement algorithm. Aliasing in branch-predictor tables can be classified in a similar fashion: ffl Compulsory aliasing occurs when a branch substream is encountered for the first time. ffl Capacity aliasing, like capacity cache misses, is due to a pro- gram's working set being too large to fit in a predictor table, and can be reduced by increasing the size of the predictor table ffl Conflict aliasing occurs when two concurrently-active branch substreams map to the same predictor-table entry. Methods for reducing this component of aliasing have not yet, to our knowledge, appeared in the published literature. Quantifying Aliasing 3.1 Experimental Setup We conducted all of our trace-driven simulations using the IBS- Ultrix benchmarks [17]. These benchmarks were traced using a hardware monitor connected to a MIPS-based DECstation running Ultrix 3.1. The resulting traces include activity from all user-level processes as well as the operating-system kernel, and have been determined by other researchers to be a good test of branch-prediction performance [5, 11]. Conditional branch counts 2 derived from these traces are given in Table 1. Although we simulated the sdet and video play benchmarks, they exhibited no special behavior compared with the other bench- marks. We therefore omit sdet and video play results from this paper in the interest of saving space. beq r0,r0 is used as an unconditional relative jump by the MIPS compiler, therefore we did not consider it as conditional. This explains the discrepancy with the branch counts reported in [5, 11] 4-bit history benchmark substream compulsory misprediction ratio aliasing 1-bit 2-bit gs 1.91 0.15 % 7.03 % 5.28 % mpeg play 1.83 0.11 % 9.08 % 7.24 % real gcc 2.36 0.28 % 9.38 % 7.16 % verilog 1.96 0.13 % 6.48 % 4.57 % 12-bit history benchmark substream compulsory misprediction ratio aliasing 1-bit 2-bit groff 7.14 0.35 % 3.63 % 2.56 % gs 7.95 0.61 % 3.71 % 2.77 % mpeg play 6.27 0.37 % 5.85 % 4.52 % real gcc 12.90 1.55 % 4.90 % 3.93 % verilog 9.24 0.64 % 3.74 % 2.66 % Table 2: Unaliased predictor We first simulated an ideal unaliased scheme (i.e., a predictor table of infinite size). The misprediction ratios that we obtained are shown in Table 2 for history lengths of 4 and 12 bits, and for both 1-bit and 2-bit predictors (we include unconditional branches as part of the global-history bits). When an (address, history) pair is encountered for the first time, we do not count it as a misprediction, so compulsory miss contribution to mispredictions is not reported in the last two columns of Table 2. The 2-bit saturating counter gives better prediction accuracy in an unaliased predictor table than the 1-bit predictor. Our intuition is that this difference is due mainly to loop branches. We also measured the substream ratio, which we define as the average number of different history values encountered for a given conditional branch address (see first column of Table 2). The compulsory-aliasing percentage was computed from the number of different (address, history) pairs referenced through-out the trace divided by the total number of dynamic conditional branches. From Table 2, we observe that compulsory aliasing, with a 12-bit history length, generally constitutes less than 1% of the total of all dynamic conditional branches, except in the case of real gcc, which exhibits a compulsory-aliasing rate of 1.55%. 3.2 Quantifying Conflict and Capacity Aliasing To quantify conflict and capacity aliasing, we simulated tagged predictor tables holding (address, history) pairs. Figures 1 and 2 show the miss ratio in direct-mapped (DM) and fully-associative tables using 4 bits and 12 bits of global history, respec- tively. The two direct-mapped tables are indexed with a gshare- and a gselect-like function. The fully-associative table uses a least- recently-used (LRU) replacement policy. The miss ratio for the fully-associative table gives the sum of compulsory and capacity aliasing. The difference between gshare or gselect and the fully-associative table gives the amount of conflict aliasing in the corresponding gshare and gselect predictors. It should be noted that LRU is not an optimal replacement policy [15]. However, because it bases its decisions solely on past information, the LRU policy gives a reasonable base value of the amount of conflict aliasing that can be removed by a hardware-only scheme. It appears that for our benchmarks, gselect has a higher aliasing rate than gshare. This explains why, for a given table size and history length, gshare has a lower misprediction rate than gselect, as claimed in [8]. This difference is very pronounced with 12 bits of global history, because in this case, gselect uses only a very small number of address bits (e.g., only 4 address bits for a 64K-entry table). Figure 1 shows that when the number of entries is larger than or equal to 4K, capacity aliasing nearly vanishes, leaving conflicts as the overwhelming cause of aliasing. The same condition holds in Figure 2 for table sizes greater than about 16K. This leads us to conclude that some amount of associativity in branch prediction tables is needed to limit the impact of aliasing. 3.3 Problems with Associative Predictor Tables Associativity in caches introduces a degree of freedom for avoiding conflicts. In a direct-mapped cache, tag bits are used to determine whether a reference hits or misses. In an associative cache, the tag bits also determine the precise location of the requested data in the cache. Because of its speculative nature, a direct-mapped branch prediction table can be tag-less. To implement associativity, however, we must introduce tags identifying (address, history) pairs. Un- fortunately, the tag width is disproportionately large compared to the width of the individual predictors, which are usually 1 or 2 bits wide. Another method for achieving the benefits of associativity, without having to pay the cost of tags is needed. The skewed branch predictor, described in the next section, is one such method. 4 The Skewed Branch Predictor We have previously noted that the behaviors of gselect and gshare are different even though these two schemes are based on the same (address, history) information. This is illustrated on Figure 3 where we represent a gshare and a gselect table with 16 entries. In this example, there is a conflict both with gshare and gselect, but the (address, history) pairs that conflict are not the same. We can conclude that the precise occurrence of conflicts is strongly related to the mapping function. The skewed branch predictor is based on this observation. The basic principle of the skewed branch predictor is to use several branch-predictor banks (3 banks in the example illustrated in Figure 4), but to index them by different and independent hashing functions computed from the same vector V of information (e.g., branch address and global history). A prediction is read from each of the banks and a majority vote is used to select a final branch direction. The rationale for using different hashing functions for each bank is that two vectors, V and W, that are aliased with each other in one bank are unlikely to be aliased in the other banks. A destructive aliasing of V by W may occur in one bank, but the overall prediction on V is likely to be correct if V does not suffer from destructive aliasing in the other banks. 4.1 Execution Model We consider two policies for updating the predictors across multiple banks: ffl A total update policy: each of the three banks is updated as if it were a sole bank in a traditional prediction scheme. ffl A partial update policy: when a bank gives a bad prediction, it is not updated when the overall prediction is good. This groff gs mpeg play26101418512 1k 2k 4k 8k 16k 32k 64k number of entries DM gselect DM gshare FA LRU51525 number of entries DM gselect DM gshare number of entries DM gselect DM gshare FA LRU nroff real gcc verilog261014 number of entries DM gselect DM gshare FA LRU51525 number of entries DM gselect DM gshare FA LRU51525 number of entries DM gselect DM gshare FA LRU Figure 1: Miss percentages in tables tagged with (address, history) pairs (4-bit history) groff gs mpeg play515254k 8k 16k 32k 64k 128k 256k 512k number of entries DM gselect DM gshare FA LRU515254k 8k 16k 32k 64k 128k 256k 512k number of entries DM gselect DM gshare FA LRU51525 number of entries DM gselect DM gshare FA LRU nroff real gcc verilog51525 number of entries DM gselect DM gshare FA LRU5152535 number of entries DM gselect DM gshare FA LRU515254k 8k 16k 32k 64k 128k 256k 512k number of entries DM gselect DM gshare FA LRU Figure 2: Miss percentages in tables tagged with (address, history) pairs (12-bit history) gselect history Address Figure 3: Conflicts depend on the mapping function history address vote majority Figure 4: A Skewed Branch Predictor wrong predictor is considered to be attached to another (ad- dress, history) pair. When the overall prediction is wrong, all banks are updated as dictated by the outcome of the branch. 4.2 Design Space Chosing the information (branch address, history, etc.) that is used to divide branches into substreams is an open problem. The purpose of this section is not to discuss the relevance of using some combination of information or the other, but to show that most conflict aliasing effects can be removed by using a skewed predictor organization. For the remainder of this paper, the vector of information that will be used for recording branch-prediction information is the concatenation of the branch address and the k bits of global be the set of all V 's. The functions f0 , f1 and f2 used for indexing the three 2 n -entry banks in the experiments are the same as those proposed for the skewed-associative cache in [13]. Consider the decomposition of the binary representation of vector V in bit substrings (V3 ,V2 ,V1 ), such that V1 and V2 are two n-bit strings. Now consider the function H defined as follows: where \Phi is the XOR (exclusive or) operation. We can now define three different mapping functions as follows: Further information about these functions can be found in [13]. The most interesting property of these functions is that if two distinct vectors (V 3; V 2; V 1) and (W3; W2;W 1) map to the same entry in a bank, they will not conflict in the other banks if 1). Any other function family exhibiting the same property might be used. Having defined an implementation of the skewed branch predic- tor, we are now in a position to evaluate it and check its behavior against conventional global-history schemes. For the purposes of comparison, we will use the gshare global scheme for referencing the standard single-bank organization. The skewed branch predictor described earlier will also be referred to as gskewed for the remainder of this paper. 5 Analysis 5.1 Simulation Results The aim of this section is to evaluate the cost-effectiveness of the skewed branch predictor via simulation. In the skewed branch predictor, a prediction associated with a (branch, history) pair is recorded up to three times. It is intuitive that the impact of conflict aliasing is lower in a skewed branch predictor than in a direct-mapped gshare table. However, if the same total number of predictor storage bits is allocated to each scheme, it is not clear that gskewed will yield better results - the redundancy that makes gskewed work also has the effect of increasing the degree of capacity aliasing among a fixed set of predictor entries. Said differently, it may be better to simply build a one-bank predictor table 3 times as large, rather than a 3-bank skewed table. number of entries gshare number of entries gshare gskewed7.47.88.28.699.4 number of entries gshare gskewed nroff real gcc verilog3.844.24.44.62k 4k 8k 16k 32k 64k number of entries gshare number of entries gshare number of entries gshare gskewed Figure 5: Misprediction percentage with 4-bit history groff gs mpeg play3458k 16k 32k 64k 128k 256k number of entries gshare number of entries gshare number of entries gshare gskewed nroff real gcc verilog2.53.54.58k 16k 32k 64k 128k 256k number of entries gshare number of entries gshare number of entries gshare gskewed Figure Misprediction percentage with 12-bit history history length gshare 16k gskewed history length gshare 16k gskewed 3x4k5.56.57.58.5 history length gshare 16k gskewed 3x4k nroff real gcc verilog2.633.43.84.2 history length gshare 16k gskewed history length gshare 16k gskewed history length gshare 16k gskewed 3x4k Figure 7: Misprediction percentage of 3x4k-gskewed vs. 16k-gshare For the direct comparison between gshare and gskewed, we used 2-bit saturating counters and a partial update policy for gskewed. Varying prediction table size The results for a history size of 4 bits and 12 bits are plotted in Figures 5 and 6, respectively, for a large spectrum of table sizes. The interesting region of these graphs is where capacity aliasing for gshare has vanished. In this region, a skewed branch predictor with a partial update policy achieves the same prediction accuracy as a 1-bank pre- dictor, but requires approximately half the storage resources For all benchmarks and for a wide spectrum of predictor sizes, the skewed branch predictor consistently gives better prediction accuracy than the 1-bank predictor. It should be noted that when using the skewed branch predictor and a history length of 4 (12), there is very little benefit in using more than 3x4k (3x16k) entries, while increasing the number of entries to 64k (256k) on gshare still improves the prediction accuracy. Notice that the skewed branch predictor is more able to remove pathological cases. This appears clearly on Figure 6 for nroff. Varying history length For any given prediction table size, some history length is better than others. Figure 7 illustrates the miss rates of a 3x4k-entry gskewed vs. a 16k-entry gshare when varying the history length. The plots show that despite using 25 % less storage resources, gskewed outperforms gshare on all benchmarks except real gcc. Varying number of predictor banks We also considered skewed configurations with five predictor banks. Our simulations results (not reported here) showed that there is very little benefit to increasing the number of banks to five; it appears that a 3-bank skewed branch predictor removes the most significant part of conflict alias- ing, and a more cost-effective use of resources would be to increase the size of the banks rather than to increase their number. Update policy To verify that gskewed is effective in removing conflict aliasing, we compare a 3\LambdaN-entry gskewed branch predictor with a fully-associative N-entry LRU table. Figure 8 illustrates this experiment for a global history length of 4 bits and 2-bit saturating counters. For (address, history) pairs missing in the fully-associative table, a static prediction always taken was assumed. For gskewed, both partial-update and total-update policies are shown. It appears that a 3*N-entry gskewed table with partial update delivers slightly better behavior than the N-entry fully-associative table, but when it uses total-update policy, it exhibits slightly worse behavior. We conclude that a 3xN-entry gskewed predictor with partial update delivers approximately the same performance as an N-entry fully-associative LRU predictor. The reason why partial update is better than total update is in- tuitive. For partial update, when 2 banks give a good prediction and the third bank gives a bad prediction, we do not update the third bank. By not updating the third bank, we enable it to contribute to the correct prediction of a different substream and effectively increase the capacity of the predictor table as a whole. 5.2 Analytical Model Although our simulation results show that a skewed predictor table offers an attractive alternative to the standard one-bank predictor structure, they do not provide much explanation as to why a skewed organization works. In this section, we present an analytical model that helps to better understand why the technique is effective. To make our analytical modeling tractable, we make some simplifying assumptions: we assume 1-bit automatons and the total update policy. We begin by defining the table aliasing probabil- ity. Consider a hashing function F which maps (address, history) FA LRU gskewed TU gskewed PU681012 FA LRU gskewed TU gskewed PU7.58.59.510.5 FA LRU gskewed TU gskewed PU nroff real gcc verilog3.84.24.65512 1k 2k 4k 8k 16k 32k FA LRU gskewed TU gskewed PU8101214 FA LRU gskewed TU gskewed PU56789 FA LRU gskewed TU gskewed PU Figure 8: Misprediction percentage of 3N-entry gskewed vs. N-entry fully-associative LRU pairs onto a N-entry table. The aliasing probability for a dynamic reference (address, history) is defined as follows: Let D be the last-use distance of V, i.e. the number of distinct (address, history) pairs that have been encountered since the last occurrence of V. Assuming F distributes these D vectors equally in the table (i.e., assuming F is a good hashing function), the aliasing probability for dynamic reference V is When N is much greater than 1, we get a good approximation with N (2) The aliasing probability is a function of the ratio between the last-use distance and the number of entries. aliasing probability, and b be the probability that an (address, history) pair is biased taken. With 1- bit predictors, when an entry is aliased, the probability that the prediction given by that entry differs from the unaliased prediction is It should be noted that the aliasing is less likely to be destructive if b is close to 0 or 1 than if b is close to 1=2. Assuming a total update policy, and because we use different hashing functions for indexing the three banks, the events in a bank are not correlated with the events in an other bank. Now consider a particular dynamic reference V. Four cases can occur: 1. With probability not aliased in any of the three banks: the prediction will be the same as the unaliased prediction 2. With probability aliased in one bank, but not in the other two banks: the resulting majority vote will be in the same direction as the unaliased prediction. 3. With probability 3p is aliased in two banks, but not in the remaining one. With probability predictions for both aliased banks are different from the unaliased prediction: the overall prediction is different from the unaliased prediction. 4. With probability p 3 , V is aliased in all three banks. With probability the predictions are different from the unaliased prediction in at least two prediction banks: the skewed prediction is different from the unaliased prediction. In summary, the probability that a prediction in our 3-bank skewed predictor differs from the unaliased prediction is : In contrast, the formula for a direct-mapped 1-bank predictor table is: Pdm and Psk are plotted in Figure 9 for the worst case We have Psk =4 Pdm =2 The main characteristic of the skewed branch predictor is that its mispredict probability is a polynomial function of the aliasing probability. The most relevant region of the curve is where the per- bank aliasing probability, p, is low magnifies the curve for small aliasing probabilities). At comparable storage resources, a 3-bank scheme has a greater per-bank aliasing probability than a 1-bank scheme, because each bank has a smaller number of entries. By taking into account formula (1), we find that for a 3x(N/3)-entry gskewed, Psk is lower than Pdm in a N-entry direct-mapped table when the last-use distance D is less than approximately N, while for D ? N, Psk exceeds Pdm . mispredict overhead per-bank aliasing probability 3 banks Figure 9: destructive aliasing0.5 % mispredict overhead per-bank aliasing probability 3 banks Figure 10: destructive aliasing This highlights the tradeoff that takes place in the skewed branch predictor: a gain on short last-use distance references is traded for a loss on long last-use distance references. Now consider a N-entry fully-associative LRU table. When the last-use distance D is less than N, there is a hit, otherwise there is a miss. Hence, in a predictor table, aliasing for short last-use distance references is conflict aliasing, and aliasing for long last-use distance references is capacity aliasing. In other words, the skewed branch predictor trades conflict aliasing for capacity aliasing. To verify if our mathematical model is meaningful, we extrapolated the misprediction rate for gskewed by measuring D for each dynamic (address, history) pair and applied formulas (1) and (3). When an (address, history) pair was encountered for the first time, we applied formula (3) with 1. The bias probability b was evaluated for the entire trace by measuring the density of static (ad- dress, history) pairs with bias taken, and the value found was then fed back to the simulator when applying formula (3) on the same trace. Finally, we added the unaliased misprediction rate of table 2 (the contribution of compulsory aliasing to mispredictions appears only in the mispredict overhead). The results are shown in Figure 11 for a history length of 4. It should be noted that our model always slightly overestimates the misprediction rate. This can be explained by the constructive aliasing phenomenon that is reported in [21]. As noted above, we made some simplifying assumptions when we devised our analytical model. The difficulty with extending the model to a partial-update policy is that occurrences of aliasing in a bank depend on what happens in the other banks. Modeling the effect of using 2-bit automatons is also difficult because a 2-bit automaton by itself removes some part of aliasing effects on prediction Despite the limitations of the model, it effectively explains why skewed branch prediction works: in a standard one-bank table, the mispredict overhead increases linearly with the aliasing probability, but in an M-bank skewed organization, it increases as an M-th degree polynomial. Because we deal with per-bank aliasing probabili- ties, which range from 0 to 1, a polynomial growth rate is preferable to a linear one. 6 An Enhanced Skewed Branch Predictor Using a short history vector limits the number of (address, his- tory) pairs (see the substream ratio column of Table 2) and therefore the amount of capacity aliasing. On the other hand, using a long history length leads to better intrinsic prediction accuracy on unaliased predictors, but results in a large number of (address, his- tory) pairs. Ideally, given a fixed transistor budget, one would like to benefit from the better intrinsic prediction accuracy associated with a long history, and from the lower aliasing rate associated with a short history. Selecting a good history length is essentially a trade-off between the accuracy of the unaliased predictor and the aliasing probability. While the effect of conflict aliasing on the skewed branch predictor has been shown to be negligible, capacity aliasing remains a major issue. In this section we propose an enhancement to the skewed branch predictor that removes a portion of the capacity- aliasing effects without suffering from increased conflict aliasing. In the enhanced skewed branch predictor, the complete information vector (i.e., branch history and address) is used with the hashing functions f1 and f2 for indexing bank 1 and bank 2, as in the previous gskewed scheme. But for function f0 , which indexes bank 0, we use the usual bit truncation of the branch address (address mod 2 n ). The rationale for this modification is as follows: Consider an enhanced gskewed and gskewed using the same history length L, and (address, history) pair (A; H). has the same last-use distance DL on the three banks of gskewed and on banks 1 and 2 of enhanced gskewed. But for enhanced gskewed, only the address is used for indexing bank 0, so the last-use distance DS of the address A on bank 0 is shorter than DL . Two situations can occur: 1. When DL is small compared with the bank size, the aliasing probability on a bank in either gskewed or enhanced gskewed is small, and both gskewed and enhanced gskewed are likely to deliver the same prediction as the unaliased predictor for history length L, because these predictions will be present in at least two banks. 2. When DL becomes large compared with the bank size, the aliasing probability pL on a any bank of gskewed or banks 1 and 2 of enhanced gskewed becomes close to 1 (formula (2) in the previous section). For both designs, when predictions on banks 1 and 2 differ, the overall prediction is equal to the prediction on bank 0. Now, since DS ! DL , the aliasing probability pS on bank 0 of enhanced gskewed is lower than the aliasing probability pL on bank 0 of gskewed. When DL is too high, the better intrinsic prediction accuracy associated with the long history on bank 0 in gskewed cannot compensate for the increased aliasing probability in bank 0. Our intuition is that when the history length is short, the first situation will dominate and both predictors will deliver equivalent entries/table extrapol. gskewed meas. gskewed8101214 entries/table extrapol. gskewed meas. gskewed9.510.511.512.513.5 entries/table extrapol. gskewed meas. gskewed nroff real gcc verilog5.25.666.46.8512 1k 2k 4k 8k 16k 32k entries/table extrapol. gskewed meas. gskewed1012141618 entries/table extrapol. gskewed meas. gskewed791113 entries/table extrapol. gskewed meas. gskewed Figure Extrapolated vs. measured misprediction percentage prediction, but for a longer history length, the second situation will occur more often and enhanced gskewed will deliver better overall prediction than gskewed. Simulation results: Figure 12 plots the results of simulations that vary the history length for a 3x4K-entry enhanced gskewed, a 3x4K-entry gskewed and a 32K-entry gshare. A partial-update policy was used in these experiments. The curves for gskewed and enhanced gskewed are nearly indistinguishable up to a certain history length. After this point, which is different for each benchmark, the curves begin to diverge, with enhanced gskewed exhibiting lower mispredication rates at longer history lengths. Based on our simulation results, 8 to 10 seems to be a reasonable choice for history length for a 3x4K-entry gskewed table, while for enhanced gskewed, 11 or 12 would be a better choice. Notice that the 3x4K-entry enhanced gskewed performs as well as the 32K-entry gshare on all our benchmarks and for all history lengths, but with less than half of the storage requirements. 7 Conclusions and Future Work Aliasing effects in branch-predictor tables have been recently identified as a significant contributor to branch-misprediction rates. To better understand and minimize this source of prediction error, we have proposed a new branch-aliasing classification, inspired by the three-Cs model of cache performance. Although previous branch-prediction research has shown how to reduce compulsory and capacity aliasing, little has been done to reduce conflict aliasing. To that end, we have proposed skewed branch prediction, a technique that distributes branch predictors across multiple banks using distinct and independent hashing func- tions; multiple predictors are read in parallel and a majority vote is used to arrive at an overall prediction. Our analytical model explains why skewed branch prediction works: in a standard one-bank table, the misprediction overhead increases linearly with the aliasing probability, but in an M-bank skewed organization, it increases as an M-th degree polynomial. Because we deal with per-bank aliasing probabilities, which range from 0 to 1, a polynomial growth rate is preferable to a linear one. The redundancy in a skewed organization increases the amount of capacity aliasing, but our simulation results show that this negative effect is more than compensated for by the reduction in conflict aliasing when using a partial-update policy. For tables of 2-bit predictors and equal lengths of global his- tory, a 3-bank skewed organization consistently outperforms a standard 1-bank organization for all configurations with comparable total storage requirements. We found the update policy to be an important factor, with partial update consistently outperforming total update. Although 5-bank (or greater) configurations are possible, our simulations showed that the improvement over a 3-bank configuration is negligible. We also found skewed branch prediction to be less sensitive to pathological cases (e.g., nroff in Figure 6). To reduce capacity aliasing further, we proposed the enhanced skewed branch predictor, which was shown to consistently reach the performance level of a conventional gshare predictor of more than twice the same size. In addition to these performance advantages, skewed organizations offer a chip designer an additional degree of flexibility when allocating die area. Die-area constraints, for example, may not permit increasing a 1-bank predictor table from 16K to 32K, but a skewed organization offers a middle point: 3 banks of 8K entries apiece for a total of 24K entries. In this paper, we have only addressed aliasing on prediction schemes using a global history vector. The same technique could be applied to remove aliasing in other prediction methods, including per-address history schemes [18, 19, 20], or hybrid schemes [8, 2, 1, 4]. Skewed branch prediction raises some new questions: ffl Update Policies: Are there policies other than partial-update and total-update that offer better performance in skewed or enhanced skewed branch predictors? 4.6 history length enh. gskewed 3x4k gskewed 3x4k gshare 32k4567 history length enh. gskewed 3x4k gskewed 3x4k gshare history length enh. gskewed 3x4k gskewed 3x4k gshare 32k nroff real gcc verilog2.42.83.23.64 history length enh. gskewed 3x4k gskewed 3x4k gshare history length enh. gskewed 3x4k gskewed 3x4k gshare 32k4567 history length enh. gskewed 3x4k gskewed 3x4k gshare 32k Figure 12: Misprediction percentage of enhanced gskewed ffl Distributed Predictor Encodings: In our simulations we adopted the standard 2-bit predictor encodings and simply replicated them across 3 banks. Do there exist alternative "dis- predictor encodings that are more space efficient, and more robust against aliasing? Minimizing Capacity Aliasing: Skewed branch predictors are very effective in reducing conflict-aliasing effects, but they do so at the expense of increased capacity aliasing. Do there exist other techniques, like those used in the enhanced skewed predictor, that could minimize these effects? --R Alternative implementations of hybrid branch predictors. Branch clas- sification: a new mechanism for improving branch predictor performance Analysis of branch prediction via data compression. Using hybrid branch predictors to improve branch prediction accuracy in the presence of context switches. An analysis of dynamic branch prediction schemes on system workloads. Aspects of Cache Memory and Instruction Buffer Performance. Branch prediction strategies and branch target buffer design. Combining branch predictors. Dynamic path-based branch correlation Improving the accuracy of dynamic branch prediction using branch correlation. Correlation and aliasing in dynamic branch predictors. A case for two-way skewed-associative caches Skewed associative caches. A study of branch prediction strategies. Efficient simulation of caches under optimal replacement with applications to miss The influence of branch prediction table interference on branch prediction scheme performance. Coping with code bloat. Alternative implementations of two-level adaptive branch prediction A comparison of dynamic branch predictors that use two levels of branch history. A comparative analysis of schemes for correlated branch prediction. --TR Two-level adaptive training branch prediction Alternative implementations of two-level adaptive branch prediction Improving the accuracy of dynamic branch prediction using branch correlation A case for two-way skewed-associative caches A comparison of dynamic branch predictors that use two levels of branch history Efficient simulation of caches under optimal replacement with applications to miss characterization Branch classification A comparative analysis of schemes for correlated branch prediction Instruction fetching The influence of branch prediction table interference on branch prediction scheme performance Dynamic path-based branch correlation Alternative implementations of hybrid branch predictors Using hybrid branch predictors to improve branch prediction accuracy in the presence of context switches An analysis of dynamic branch prediction schemes on system workloads Correlation and aliasing in dynamic branch predictors Analysis of branch prediction via data compression Skewed-associative Caches A study of branch prediction strategies Aspects of cache memory and instruction buffer performance --CTR Mitchell H. Clifton, Logical conditional instructions, Proceedings of the 37th annual Southeast regional conference (CD-ROM), p.24-es, April 1999 Shlomo Reches , Shlomo Weiss, Implementation and Analysis of Path History in Dynamic Branch Prediction Schemes, IEEE Transactions on Computers, v.47 n.8, p.907-912, August 1998 A. N. Eden , T. Mudge, The YAGS branch prediction scheme, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.69-77, November 1998, Dallas, Texas, United States Marius Evers , Sanjay J. Patel , Robert S. Chappell , Yale N. Patt, An analysis of correlation and predictability: what makes two-level branch predictors work, ACM SIGARCH Computer Architecture News, v.26 n.3, p.52-61, June 1998 Chunrong Lai , Shih-Lien Lu , Yurong Chen , Trista Chen, Improving branch prediction accuracy with parallel conservative correctors, Proceedings of the 2nd conference on Computing frontiers, May 04-06, 2005, Ischia, Italy Alexandre Farcy , Olivier Temam , Roger Espasa , Toni Juan, Dataflow analysis of branch mispredictions and its application to early resolution of branch outcomes, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.59-68, November 1998, Dallas, Texas, United States Toni Juan , Sanji Sanjeevan , Juan J. Navarro, Dynamic history-length fitting: a third level of adaptivity for branch prediction, ACM SIGARCH Computer Architecture News, v.26 n.3, p.155-166, June 1998 Pierre Michaud , Andr Seznec , Stphan Jourdan, An Exploration of Instruction Fetch Requirement in Out-of-Order Superscalar Processors, International Journal of Parallel Programming, v.29 n.1, p.35-58, February 2001 Chih-Chieh Lee , I-Cheng K. Chen , Trevor N. Mudge, The bi-mode branch predictor, Proceedings of the 30th annual ACM/IEEE international symposium on Microarchitecture, p.4-13, December 01-03, 1997, Research Triangle Park, North Carolina, United States Artur Klauser , Srilatha Manne , Dirk Grunwald, Selective Branch Inversion: Confidence Estimation for Branch Predictors, International Journal of Parallel Programming, v.29 n.1, p.81-110, February 2001 E. F. Torres , P. Ibanez , V. Vinals , J. M. Llaberia, Store Buffer Design in First-Level Multibanked Data Caches, ACM SIGARCH Computer Architecture News, v.33 n.2, p.469-480, May 2005 Veerle Desmet , Hans Vandierendonck , Koen De Bosschere, Clustered indexing for branch predictors, Microprocessors & Microsystems, v.31 n.3, p.168-177, May, 2007 Juan C. Moure , Domingo Bentez , Dolores I. Rexachs , Emilio Luque, Wide and efficient trace prediction using the local trace predictor, Proceedings of the 20th annual international conference on Supercomputing, June 28-July 01, 2006, Cairns, Queensland, Australia Renju Thomas , Manoj Franklin , Chris Wilkerson , Jared Stark, Improving branch prediction by dynamic dataflow-based identification of correlated branches from a large global history, ACM SIGARCH Computer Architecture News, v.31 n.2, May Zhijian Lu , John Lach , Mircea R. Stan , Kevin Skadron, Alloyed branch history: combining global and local branch history for robust performance, International Journal of Parallel Programming, v.31 n.2, p.137-177, April Abhas Kumar , Nisheet Jain , Mainak Chaudhuri, Long-latency branches: how much do they matter?, ACM SIGARCH Computer Architecture News, v.34 n.3, p.9-15, June 2006 Adi Yoaz , Mattan Erez , Ronny Ronen , Stephan Jourdan, Speculation techniques for improving load related instruction scheduling, ACM SIGARCH Computer Architecture News, v.27 n.2, p.42-53, May 1999 Jared Stark , Marius Evers , Yale N. Patt, Variable length path branch prediction, ACM SIGPLAN Notices, v.33 n.11, p.170-179, Nov. 1998 Andr Seznec , Stephen Felix , Venkata Krishnan , Yiannakis Sazeides, Design tradeoffs for the Alpha EV8 conditional branch predictor, ACM SIGARCH Computer Architecture News, v.30 n.2, May 2002 Tao Li , Lizy Kurian John , Anand Sivasubramaniam , N. Vijaykrishnan , Juan Rubio, Understanding and improving operating system effects in control flow prediction, ACM SIGPLAN Notices, v.37 n.10, October 2002 Timothy H. Heil , Zak Smith , J. E. Smith, Improving branch predictors by correlating on data values, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.28-37, November 16-18, 1999, Haifa, Israel Alex Ramirez , Josep L. Larriba-Pey , Mateo Valero, Software Trace Cache, IEEE Transactions on Computers, v.54 n.1, p.22-35, January 2005 Andre Seznec, Analysis of the O-GEometric History Length Branch Predictor, ACM SIGARCH Computer Architecture News, v.33 n.2, p.394-405, May 2005 J. Gonzlez , A. Gonzlez, Control-Flow Speculation through Value Prediction, IEEE Transactions on Computers, v.50 n.12, p.1362-1376, December 2001 Gabriel H. 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3 C's classification;aliasing;skewed branch predictor;branch prediction

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Scheduling and data layout policies for a near-line multimedia storage architecture.

Recent advances in computer technologies have made it feasible to provide multimedia services, such as news distribution and entertainment, via high-bandwidth networks. The storage and retrieval of large multimedia objects (e.g., video) becomes a major design issue of the multimedia information system. While most other works on multimedia storage servers assume an on-line disk storage system, we consider a two-tier storage architecture with a robotic tape library as the vast near-line storage and an on-line disk system as the front-line storage. Magnetic tapes are cheaper, more robust, and have a larger capacity; hence, they are more cost effective for large scale storage systems (e.g., videoon-demand (VOD) systems may store tens of thousands of videos). We study in detail the design issues of the tape sub-system and propose some novel tape-scheduling algorithms which give faster response and require less disk buffer space. We also study the disk-striping policy and the data layout on the tape cartridge in order to fully utilize the throughput of the robotic tape system and to minimize the on-line disk storage space.

Introduction In the past few years, we have witnessed tremendous advances in computer technologies, such as storage architectures (e.g. fault tolerant disk arrays and parallel I/O architec- tures), high speed networking systems (e.g., ATM switching technology), compression and coding algorithms. These advances have made it feasible to provide multimedia services, such as multimedia mail, news distribution, advertisement, and entertainment, via high bandwidth networks. Consequently, research in multimedia storage system has received a lot of attention in recent years. Most of the recent research works have emphasized upon the investigation of the design of multimedia storage server systems with magnetic disks as the primary storage. In [2, 7], issues such as real-time playback of multiple audio channels have been studied. In [17], the author presented a technique for storing video and audio streams individually on magnetic disk. The same author proposed in [16] techniques for merging storage patterns of multiple video or audio streams to optimize the disk space utilization and to maximize the number of simultaneous streams. In [9], performance study was carried out on a robotic storage system. In [4, 5], a novel storage structure known as the staggered striping technique was proposed as an efficient way for the delivery of multiple video or audio objects with different bandwidth demands to multiple display stations. In [8], a hierarchical storage server was proposed to support a continuous display of audio and video objects for a personal computer. In [11], the authors proposed a cost model for data placement on storage devices. Finally, a prototype of a continuous media disk storage server was described in [12]. It is a challenging task to implement a cost-effective continuous multimedia storage system that can store many large multimedia objects (e.g., video), and at the same time, can allow the retrieval of these objects at their playback bandwidths. For example, a 100 minutes HDTV video requires at least 2 Mbytes/second display bandwidth and 12 Gbytes of storage[3]. A moderate size video library with 1000 videos would then require TBytes storage. It would not be cost effective to implement and manage such a huge amount of data all on the magnetic disk subsystem. A cost-effective alternative is to store these multimedia objects permanently in a robotic tape library and use a pool of magnetic disks, such as disk arrays [15], for buffering and distribution. In other words, the multimedia objects reside permanently on tapes, and are loaded onto the disks for delivery when requested by the disk server. To reduce the tape access delays, the most actively accessed videos would also be stored in the disks on a long term basis. The disk array functions as a cache for the objects residing in the tape library, as well as a buffer for handling the bandwidth mismatch of the tape drive and the multimedia objects. Given the above architecture, this paper aims at the design of a high performance storage server with the following requirements: ffl Minimal disk buffer space between the robotic tape library and the parallel disk array. Disk space is required for handling the bandwidth mismatch of large multi-media objects, such as video or HDTV, and the tape subsystem. ffl Minimal response time for the request to the multimedia storage system. The response time of a request to a large multimedia object can be greatly reduced by organizing the display unit, the network device, the parallel disk and the robotic tape library as a pipeline such that data flows at the continuous rate of the display bandwidth of the multimedia object along the pipeline. Since multimedia objects reside in the tape subsystem, to minimize the system response time, we have to minimize the tape subsystem response time. Throughout this paper, the tape subsystem response time is defined as the arrival time of the first byte of data of a request to the disk array minus the arrival time of the request to the multimedia storage system. ffl Maximal bandwidth utilization of the tape drives. The current tape library architectures usually have few tape drives. Hence, the bandwidth utilization of tape drives is a major factor of the average response time and throughput of the storage server. A better utilization of the bandwidth of tape drives means a higher throughput of the tape subsystem. The contribution of this paper is twofold. First, we propose a novel scheduling approach for the tape subsystem, and we show that the approach can reduce the system response time, increase the system throughput and lower the disk buffer requirement. Secondly, we study the disk block organization of the disk subsystem and show how it can be incorporated with the tape subsystem to support concurrent upload and playback of large multimedia objects. The organization of the paper is as follows. We describe the architecture of our multimedia storage system and present the tape subsystem scheduling algorithms in Sections 2 and 3 respectively. Then, we discuss the disk buffer requirement for supporting various tape subsystem scheduling algorithms in Section 4. In Section 5, we describe the disk block organization and the data layout on the tape cartridge for supporting concurrent upload and playback of large multimedia objects. In Section 6, we discuss the performance study, and lastly the conclusion is given in Section 7. Our multimedia storage system consists of a robotic tape library and a parallel disk array. The robotic tape library has a robotic arm, multiple tape drives, tape cartridges which multimedia objects reside, and tape cartridge storage cells for placing tape cartridges. Figure 1 illustrates the architectural view of the multimedia storage server. The robotic arm, under computer control, can load and unload tape cartridges. To load a tape cartridge into a tape drive, the system performs the following steps: 1. Wait for a tape drive to become available. 2. If a tape drive is available but occupied by another tape (ex: this is the tape that was uploaded for a previous request), eject the tape in the drive and unload this tape to its storage cell in the library. We call these operations as the drive eject operation and the robot unload operation respectively. 3. Fetch the newly requested tape from its storage cell and load it into the ready tape drive. We call these operations as the robot load operation and the drive load operation respectively. When a multimedia object is requested, the multimedia object is first read from the tape and stored in the disk drives via the memory buffer and the CPU. Then the multimedia object is played back by retrieving the data blocks of the multimedia object from the disk drives, at a continuous rate of the object bandwidth, into the main memory while the storage server sends the data blocks in the main memory to the playback unit via the network interface. Frequently accessed multimedia objects can be cached in the disk drives to reduce tape access and improve system response time as well as throughput. We define the notations for the robotic tape library in Table 1. These notations are useful for the performance study in later sections. display display display or Parallel Disk Array Disk Controller memory buffer Robotic Tape Library Tape Controller cartridge cell robot arm tape cartridge Figure 1: Cost-effective multimedia storage server It is important to point out that the parameter values of a robotic tape library can vary greatly from system to system. For instance, Table 2 shows the typical numbers for two commercial storage libraries. 3 Tape Subsystem and Scheduling Algorithms In this section, we describe several tape drive scheduling algorithms for our multimedia storage system. A typical robotic tape library has one robot arm and a small number of tape drives. A request to the tape library demands reading (uploading) a multimedia object from a tape cartridge. The straight-forward algorithm or the conventional algorithm to schedule a tape drive is to serve request one by one, i.e. the tape drive reads the whole multimedia object of the current request to the disk array before reading the multimedia object of the next request in the queue. Since the number of tape drives is small and the r number of robotic arms. number of tape drives. l tape drive load time. drive eject time. drive rewind time. drive search time. T u robot load or unload time. drive transfer rate. display bandwidth of object O. S(O) size of object O. Table 1: Notations used for the robotic tape library. reading time of a multimedia object is quite long 1 , a new request will often have to wait for an available tape drive. The conventional algorithm performs reasonably well when the tape drive has a bandwidth lower than the display bandwidth of the multimedia objects being requested. However, the conventional algorithm would not result in the good request response time when the tape drive bandwidth is the total display bandwidth of two or more objects. To illustrate this, suppose the tape library is an Ampex DST800 2 with one tape drive. Consider the situation in which two requests of 100 minutes different objects, each with a display bandwidth of 2 Mbytes/second. These two requests arrive at the same time when the tape drive is idle. The video object size is equal to the display duration times the display bandwidth, which is equal to 100 \Theta 60 \Theta 2 Mbytes. With the conventional algorithm, the transfer of the first request starts after a robot load operation and a drive load operation. The response time It takes 1200 seconds to upload a 1 hour HDTV video object by a tape drive with 6 Mbytes/second bandwidth. 2 the parameter values are in Table 2. Parameter Exabyte120 Ampex DST800 average T l 35.4 5 average T e 16.5 seconds 4 seconds seconds 12-13 seconds average T s 45 seconds 15 seconds Number of tapes 116 256 Tape Capacity 5 Gbytes 25 Gbytes Table 2: Typical parameter values of two commercial storage libraries. of the first request is: However, the second request will have to wait for the complete transfer of the first multimedia object request, rewinding that tape (T r ), ejecting that tape from the drive unloading that tape from the tape drive to its cell by the robot (T u ). Then the robot can load the newly requested tape (T u ) and load it into the tape drive (T l ). The response time of the second request is: Hence, the average response time of the two requests is 450 seconds. This scenario is illustrated in Figure 2 TransferR Request 1 Request 2 drive load, drive eject, Figure 2: the conventional tape scheduling algorithm The major problem about the conventional algorithm is that multiple requests can arrive within a short period of time and the average request response time is significantly increased due to the large service time of individual requests. Since the tape drive of the Ampex system is several times the display bandwidth of the multimedia objects, the tape drive can serve the two requests in a time-slice manner such that each request receives about half the bandwidth of the tape drive. Suppose the tape drive serves the two requests in a time-slice manner with a transfer period of 300 seconds as illustrated in Figure 3. The two objects are being uploaded into the disk array at an average rate of 6.5 Mbytes/second 3 . From Figure 3, the response time of the first and second requests are T u seconds and T u +T l +300+T e +T u +T u +T l seconds respectively. Hence, the average response time is (15 seconds or an improvement of 60%. We argue that the time-slice scheduling Request 2 Request 1 drive load, drive eject, Figure 3: the time-slice tape scheduling algorithm algorithm can be implemented with small overheads. In some tape systems, for instance, the D2 tapes used in the Ampex robot system, have the concept of zones[1]. Zones are 3 The overhead of tape switch is approximately10% of the transfer time. Hence the effective bandwidth of the tape drive is 13.05 Mbytes/second or 6.5 Mbytes/second for each object. the places on the tape where the tape can drift to when we stop reading from the tape. The function of the zone is that the tape drive can start reading from the zone rather than rewinding to the beginning of the tape when the tape drive reads the tape again. The time-slice algorithm has the following advantages: ffl The average response time is greatly improved in light load conditions. ffl In the case that the request of a multimedia object can be canceled after uploading some or all parts of the object into the disks (ex: customers may want to cancel the movie due to emergency or the poor entertainment value of the movie), the waste of tape drive bandwidth for uploading unused parts of multimedia objects is reduced. ffl The time-slice algorithm requires less disk buffer space than the conventional algo- rithm. The discussion of disk buffer space requirement is given in Section 4. However, the time-slice algorithm requires more tape switches and therefore has a higher tape switch overheads and a higher chance of robot arm contention. Our goal is to study several versions of the time-slice scheduling algorithm which can minimize the average response time of requests to the multimedia storage system and also, find the point of switch from the time-slice algorithm to the conventional tape scheduling algorithm. In the rest of this section, we will describe each scheduling algorithm in detail. 3.1 Conventional Algorithms The conventional algorithm is any non-preemptive scheduling algorithm, such as the First-Come-First-Served algorithm. As each request arrives, the request joins the request queue. A request in the request queue is said to be ready if the tape cartridge of the request is not being used to serve another request. The simplest scheduling algorithm is the FCFS algorithm. The FCFS algorithm selects the oldest ready request in the queue for reading when a tape drive is available. A disadvantage of the FCFS algorithm is that the response time of a short request can be greatly increased by any preceding long requests [18]. Another possible conventional algorithm is the Shortest-Job-First (SJF) algorithm. The SJF algorithm improves the average response time by serving the ready request with the shortest service time where the service time of a request is the time required to complete the tape switch, the data transfer, and the tape rewind operation of the request. However, a risk of using the SJF algorithm is the possibility of starvation for longer requests as long as there is steady supply of shorter requests. The implementations of the FCFS and SJF algorithms are similar. We have to separate the implementation into two cases, (1) where there is only a single tape drive available for the tape subsystem and, (2) where there are multiple tape drives in the tape subsystem. Single Tape Drive The implementation of the conventional algorithms is straight-forward and is shown as follow: procedure conventional(); begin while true do begin if (there is no ready request) then wait for a ready request; get a ready request from the request queue; serve the request; Multiple Tape Drives The implementation of the conventional algorithms consists of several procedures. The procedure robot is instantiated once and procedure tape is instantiated N t times where each instance of procedure tape corresponds to a physical tape drive and each instance has an unique ID. procedure conventional() begin run robot() as a process; for i := 0 to NUM TAPE -1 do run tape(i) as a process; procedure robot(); begin while true do begin /* accept new request */ if (a request is ready and a tape drive is available) then begin get a request from the request queue; send the request to an idle tape drive; else if (an available drive is occupied) then perform the drive unload operation and the robot unload operation; else wait for a ready request or an occupied available drive; procedure tape(integer id); begin while true do begin wait for a request from robot arm; serve the request; 3.2 Time-slice Algorithms The time-slice algorithms classify requests into two types: (1) non-active requests and (2) active requests. Newly arrived requests are first classified as non-active requests and put into the request queue. A non-active request is said to be ready when the tape cartridge is not being used for serving another request. Active requests are those requests being served by the tape drive in a time-slice manner. Since the time-slice algorithms are viable only if the tape switch overhead is small, we restrict that the tape rewind operation to be performed when a request has been completely served and the tape search operation is performed only at the beginning of the service of a request. This implies that two requests of the same tape cannot be served concurrently. Note that the chance of having two requests of the same tape in the system is very small because (1) the access distribution of objects is highly skewed since video rental statistics suggest some highly skewed access distributions, such as the 80/20 rule, in which 80 percent of accesses go to the most popular 20 percent of the data [6] and, (2) frequently accessed objects are kept in the disk drives. The tape switch time is equal to the total time to complete a tape drive eject opera- tion, a robot unload operation, a robot load operation, a tape drive load operation and a tape search operation. In the remaining of the paper, we let H to be the maximum tape switch time. The time-slice algorithms break a request into many tasks, each with a unique task number. Each task of the same request is served separately in the order of increasing task number. Each request is assigned a time-slice period, s, which is the maximum service time of a task of the request. The service time of a task includes the time required for the tape switch and the data transfer of the task. For the last task of a request, the service time also includes the time required for a tape rewind operation. There are many possible ways to serve several requests in a time-slice manner. We concentrate on two representative time-slice algorithms: the Round-robin (RR) algorithm and the Least Slack (LS) algorithm. 3.2.1 Round-robin Algorithm In this section, we formally describe the Round-robin algorithm. be the active requests and R n+1 ; :::; Rm be the ready non-active requests where m n. Let O i be the video object requested by R i for m. Let be the time-slice periods assigned to R With the Round-robin algorithm, the active requests are served in a round-robin manner. In each round of service, one task of each active request will be served. The active requests are served in the same order in each round of service. In order to satisfy the bandwidth requirement of active request R i , the average transfer bandwidth allocated for R i must be geater than or equal to the bandwidth of R i . Formally speaking, the bandwidth requirement of R i is satisfied if The Round-robin algorithm maintains the following condition: The condition guarantees that the bandwidth requirements of the active requests are satisfied. The efficiency of the algorithm is defined as: (1) When the system is lightly loaded, the tape drive can serve at least one more request in addition to the currently active requests, the average response time is reduced for a smaller time-slice period because a newly arrival is less likely to have to wait for a long period. However, a smaller time-slice period means that a smaller number of active requests can be served simultaneously, thereby increasing the chance that a newly arrived request has to wait for the completion of an active request. Therefore, different time-slice periods or different efficiencies of the time-slice algorithm are required to optimize the average response time at different load conditions. To simplify our discussion, we assume each request has the same time-slice period in the rest of the paper unless we state otherwise. The specification of the Round-robin algorithm is: Simple Round-robin Algorithm. The algorithm assigns each active request a time-slice period of s ? H seconds which has to satisfy the following conditions: Condition 1. The tape drive serves requests R 1 ; :::; R n in a round-robin manner with a time-slice period of s seconds. Condition 2. In each time slice period, the available time for data transfer is the task being served is not the last task of an active request, otherwise, the available time for data transfer is the rewind time of the tape. Condition 3. Request R n+1 which is the oldest ready non-active request becomes active if The straight-forward implementation of the simple Round-robin algorithm is to consider whether more active requests can be served concurrently at the end of a service round, i.e., the algorithm evaluates Condition 3 at the end of each service round. We call this implementation as the RR-1 algorithm. Again, we separate the implementation into two cases, (1) where there is only a single tape drive in the tape subsystem and, (2) where there are multiple tape drives in the tape subsystem. Single Tape Drive procedure RR-1(); begin while true do begin if (there is no active request and ready non-active request) then wait for a ready non-active request; if (the last active request is served and Condition 3 of the Simple Round-robin Algorithm is satisfied) then accept a ready non-active request; get a task from the active task queue; serve the task; With the RR-1 algorithm, a newly arrived request has to wait for one half of the duration of a service round when the tape subsystem can serve at least one more request in addition to the currently active requests. Since the duration of a service round grows linearly with the number of active requests, the average waiting time of a request is high when there are several active requests. To improve the above situation, we can check whether one more request can be served by the tape subsystem after every completion of an active task. We call this improved implementation of the Round-robin algorithm as the RR-2 algorithm. Multiple Tape Drives For the case of multiple tape drives, we have to consider the robot arm contention because the tape drives need to wait for the robot arm to load or unload. In the worst case, each tape switch requires a robot load operation and a robot unload operation. Therefore, the worst case robot waiting time is 2 \Theta T u \Theta (N t \Gamma 1). Hence, Condition 3 can be revised to become: 3.2.2 The Least-slack (LS) Algorithm Let us study another version of time-slice algorithm which can improve on the response time of the multimedia request. In order to maintain the playback continuity of an object, task i of the request of the object must start to transfer data before finishing the playback of the data of the previous task i \Gamma 1. We define the latest start time of transfer (LSTT) of a task of an active request as the latest time that the task has to start to transfer data in order to maintain the playback continuity of the requested object. Formally, the LSTT of task J i is defined as: request arrival time request response time if J i is the first task time of the data of task J The slack time of a task is defined as max(LSTT of task \Gamma current time; 0). Let be the time required to complete the data transfer of J i and the tape rewind operation of J i (if J i is the last task of a request). The deadline of a task J i is defined as: A ready non-active request R can become active when the tasks of R can be served immediately such that each task of an active request can be served at or before its LSTT. LS Algorithm The algorithm serves requests with the following conditions: Condition 1. Each active task can be served in one time-slice period of s seconds. Condition 2. Active tasks are served in ascending order of slack time. Condition 3. In each time slice period, the available time for data transfer is the task being served is not the last task of an active request, otherwise, the available time for data transfer is the rewind time of the tape. Condition 4. The data transfer of each active task can start at or before the LSTT of the active task. Condition 5. A ready non-active request can become active if Condition 4 is not violated after the request has become active. We choose the LS algorithm for tape scheduling because the LS algorithm is optimal for a single tape system [14] 4 in the sense that if scheduling can be achieved by any algorithm, it can be achieved by the optimal algorithm. For the case that the tape subsystem has only one robot arm and one tape drive, Condition 4 of the LS Algorithm can be rewritten as follows. Given a robotic tape library with a single tape drive, let J be the active tasks listed in ascending order of slack time. If no active task is in service, then Condition 4 of the LS algorithm is equivalent to the condition that each active task can be completed at or before its deadline. In other words, Condition 4 of the LS algorithm is equivalent to the following condition: is the tape switch time of J i . 4 The paper discussed scheduling in single and multiple processors. The case of a single tape drive robot library is equivalent to the case of a single processor described in the paper. Proof: Assume there is no active task in service. By Equation (2), an active task can start data transfer at or before its LSTT if and only if it can be completed at or before its deadline. A task J k can be completed at or before its deadline if and only if the time between the current time and the deadline of J k is enough to complete J k and its preceding tasks. Therefore, Condition 4 of the LS algorithm is equivalent to Again, we separate the implementation into two cases, (1) where there is only a single tape drive in the tape subsystem and, (2) where there are multiple tape drives in the tape subsystem. The implementation of the LS algorithm for the single tape case is as follows: Single Tape Drive procedure LS(); begin while true do begin if (there is no request) then wait for a new request; if (there is a ready request is non-empty and acceptnew()) then begin get the oldest ready request; put the tasks of the request into the active task queue; get the active task with the least slack time; serve the task; begin float work; pointer x; if (the active task queue is empty) then work := 0.0; save the active task queue; put the tasks of the oldest ready request into the active task queue; while task queue is not empty do begin x := next active task; work current time) then begin restore the active task queue; return(false) restore the active task queue; Multiple Tape Drives This implementation consists of two procedures: robot and tape drive. Procedure robot performs the following steps repeatedly: accept a ready request if the request can be accepted to become active immediately; if there are active tasks and an idle tape, then send the active task with the least slack time to an idle tape, else wait for an idle tape or an active task. Procedure tape repeatedly waits for an active task and performs the sequence of a drive eject operation, a drive load operation, a data transfer, and a tape rewind operation (for the last task of a request). Procedure robot is instantiated once and procedure tape is instantiated N t times. Each instance of procedure tape has an unique ID. 4 Disk Buffer Space Requirement In this section, we study the disk buffer requirement for the various scheduling algorithms that we have described. First, we show that the conventional algorithm (the FCFS or the SJF algorithm) requires a huge amount of buffer space to achieve the maximum through- put. The following theorem states the buffer space requirement for the conventional algorithm. Theorem 1 If each object of a request is of the same size S and same display bandwidth then the conventional algorithm requires O( B buffer space in order to achieve its maximum throughput where the sustained tape throughput is B t and it is equal to SB t Proof: The tape subsystem achieves its maximum throughput when (1) there is infinite number of ready requests and (2) each request does not have a search time, i.e., the requested object resides at the beginning of the tape cartridge and the tape drive can start to read the object right after the drive load operation has been done. The sustained bandwidth of tape subsystem is: the tape subsystem is idle and starts to serve requests one by one. In time interval (0, S data are consumed at the rate of B d (O) and uploaded at the rate of t . Hence, at time t \GammaB d (O))S buffer space is required to hold the accumulated data. In time interval [ S data are consumed at the rate of 2B d (O) and uploaded at the rate of B t . Therefore, at time t \GammaB d (O))S t \Gamma2B d (O))S buffer space is required to hold the accumulated data. This argument continues until the total object display throughput matches with the tape sustained throughput. To obtain the upper bound buffer requirement, assume we have a tape system whose sustained throughput satisfies the following criteria: then the upper bound buffer requirement is: obtain the lower bound buffer requirement, assume we have a tape system whose sustained throughput B l satisfies the following criteria: then the lower bound buffer requirement is: 1)STherefore, the buffer space requirement is O( B For example, the disk buffer size = 38.22 Gbytes. Corollary 1 If there are N t tape drives in the tape library system. The buffer disk buffer requirement is O( N t B In the following theorem, we state the disk buffer requirement for the Round-robin time-slice algorithm. Theorem 2 If R 1 , ., R n are the active requests that satisfy the following condition: then the Round-robin algorithm achieves the bandwidth requirements of the requested objects, O 1 ; :::; O n iff the disk buffer size is Proof: For R i (1 i n), at least two disk buffers of size (s i \Gamma H)B t required for concurrent uploading and display of object O i . Hence the necessary condition is proved. Suppose for each request O i , there are two disk buffers b i1 and b i2 , each with size of While one buffer is used for uploading the multimedia object from the tape library, the other buffer is used for displaying object O i . At steady state, the maximum period between an available buffer till the time of uploading from tape is b i1 has just been available, the system starts to output data from the other buffer b i2 for display. By the condition of the theorem, b i2 will not be emptied before the tape drive starts to upload data to b i1 . Hence, the bandwidth of O i is satisfied. 5 equivalent to 1.5 hours of display time With the same arguments, we have the following corollary for the disk buffer requirement for the LS algorithm. Corollary 2 If R 1 , ., R n are the active requests that satisfy the following condition: then the LS algorithm achieves the bandwidth requirements of the requested objects, O 1 , iff the disk buffer size is By Theorem 2 and Corollary 2, the LS and Round-robin algorithms require less buffer than the conventional algorithm for the same throughput because the transfer time of each time slice period, s can be chosen to be much smaller than the total upload period of the object, S 5 The Disk Subsystem Since the tape drive bandwidth or the object bandwidth can be higher than the band-width of a single disk drive, we have to use striping techniques to achieve the required bandwidth of the tape drive or the object. In [4], a novel architecture known as the Staggered Striping technique was proposed for high bandwidth objects, such as HTDV video objects. It has been shown that Staggered Striping has a better throughput than the simple striping and virtual data replication techniques for various system loads [4]. In this section, we show how to organize the disk blocks in Staggered Striping together with the robotic tape subsystem so that (1) the bandwidths of the disks and the tape drives are matched, and (2) concurrent upload and display of multimedia objects is supported. 5.1 Staggered Striping We first give a brief review of the staggered striping architecture. With this technique, an object O is divided into subobjects, U i , which are further divided into MO fragments. A fragment is the unit of data transferred to and from a single disk drive. The disk drives are clustered into logical groups. The disk drives in the same logical group are accessed concurrently to retrieve a subobject (U i ) at a rate equivalent to B d (O). The Stride, k, is the distance 6 between the first fragment of U i and the first fragment of U i+1 . The relationships of the above parameters are shown below: e where B disk is the bandwidth of a single disk drive. ffl The size of a subobject = MO \Theta the size of a fragment. ffl A unit of time = the time required for reading a fragment from a single disk drive. Note that a subobject can be loaded from the disk drives into the main memory in one time unit. To reduce the seek and rotational overheads, the fragment size is chosen to be a multiple of the size of a cylinder. A typical 1.2 Gbytes disk drive consists of 1635 cylinders which are of size 756000 bytes each and has a peak transfer rate of 24 Mbit/second, a minimum disk seek time of 4 milliseconds, a maximum disk seek time of 35 milliseconds, and a maximum latency of 16.83 milliseconds. For a fragment size of 2 cylinders, the maximum seek and latency delay times of the first cylinder and the second cylinder are milliseconds and milliseconds respectively. The transfer time of two cylinders is 481 milliseconds. The total service time (including disk seek, latency delay, and disk transfer time) of a fragment is 553.66 milliseconds. Hence, the seek and rotational overheads is about 13% of the disk bandwidth 7 . To simplify 6 which is measured in number of disks 7 A further increase in number of cylinders does not result in much reduction of the overhead. Hence, a fragment of 2 cylinders is reasonable assumption. our discussion, we assume the fragment size is two cylinders and one unit of time is 0.55 seconds. To illustrate the idea of Staggered Striping, we consider the following example: Example 1 Figure 4 shows the retrieval pattern of a 5.0 Mbytes/second object in five 2.5 Mbytes/second disk drives. The stride is 1 and MO is 2. When the object is read for display, subobject U 0 is read from disk drives 0 and 1 and so on. disk time 4 U4.1 U4.0 6 U6.0 U6.1 9 U9.1 U9.0 Figure 4: Retrieval pattern of an object. 5.2 Layout of Storage on the Tape In the following discussion, we assume that (1) staggered striping is used for the storage and retrieval of objects in the disk drives and, (2) the memory buffer between the tape drives and the disk drives is much smaller in size than a fragment. Let the effective bandwidth for the time slice algorithm be B t which is equal to s\GammaH We show that the storage layout of an object on the tape must match the storage layout on the disk drives so as to achieve maximum throughput of the tape drive. When the object is displayed, each fragment requires a bandwidth of B d (O) MO . Therefore, the tape drive produces NO fragments where c in a unit of time. The blocks of NO fragments are stored in a round-robin manner such that the NO fragments are produced as NO continuous streams of data at the same time. Consider the case described in Example 1. Suppose B 3. If the subobjects are stored in the following In the first time unit, U 0:0 , U 0:1 , U 1:0 are read from the tape drive. At the same time, U 0:0 and U 0:1 are stored in disk drive 0 and disk drive 1. Fragment U 1:0 has to be discarded and re-read in the next time unit because disk drive 1 can only store either U 0:1 or U 1:0 . Since the output rate of tape drive must match the input rate of disk drives, the effective bandwidth of the tape drive is 5 Mbytes/second and the tape drive bandwidth cannot be fully utilized. On the other hand, if the storage layout of the object is as follows: fU 0:0 ; U 0:1 ; U 1:1 g, In each time unit, the output fragments from the tape drive can be stored in 3 consecutive disk drives. Hence, the bandwidth of the tape drive is fully utilized. Figure 5 shows the timing diagram for the upload of the object from the tape drive. From time 2, subobject U 0 can be read from disk drives 0 and 1. Hence, the object can be displayed at time 2 while the remaining subobjects are being uploaded into the disk drives from the tape drive. Both the bandwidth of the disk drives and the tape drive are fully utilized. Now we should derive the conditions of matching the way that the fragments are retrieved from the disk and the way that the fragments are uploaded from the tape. Let D and k be the number of disk drives of the disk array and the stride respectively. In the Zg is a representation which shows that the blocks of X, Y, and Z are stored in a round-robin manner. disk time 3 U4.1 U7.1 U4.0 6 U10.1 U11.1 U8.0 9 U14.1 U11.0 U14.0 Figure 5: Upload pattern of an object. rest of the section, we assume that the bandwidth of tape drive is at least (MO+1) \Theta B d (O) MO Given an object O which has been uploaded from a tape drive into the disk array, the retrieval pattern RO of O is an L \Theta D matrix where L is the number of time units required for the retrieval of O from the disk drives and RO (i; j) is equal to "U a:b " if fragment U a:b of O is read at time i from disk drive j. RO (i; contains a blank entry if no fragment is read from disk drive j at time i. Given an object O, the upload pattern PO of O is an L \Theta D matrix where L is the number of time units required for uploading O and PO (i; j) from a tape drive into the disk array is equal to "U a:b " if fragment U a:b is read at time i and stored in disk drive j. RO (i; contains a blank entry if no fragment is stored in disk drive j at time i. Definition 3 The storage pattern LP of a retrieval or upload pattern P is an L \Theta D matrix where L is an integer and LP (i; j) the i-th non-blank entry of column j of P, i.e. LP is obtained by replacing all the blank entries of P by lower non-blanking entries of the same column with the preservation of the row-order of the entries, i.e., 8LP (a; b) and Examples of retrieval and upload patterns are shown in Figures 4 and 5 respectively. The retrieval and upload patterns of Figure 4 and Figure 5 have the same storage pattern which is shown in Figure 6. disk Figure An example of storage pattern. staggered striping, when an object O is uploaded from a tape drive into the disk array, the tape drive bandwidth can be fully utilized if ffl the tape drive reads NO fragments of O into NO different disk drives in each unit of time; and ffl the storage patterns of the retrieval pattern and upload pattern of O are the same, Proof: Assume that the retrieval pattern and the upload pattern of O have the same storage pattern and the tape drive reads NO fragments into NO different disk drives. Since the retrieval pattern and the upload pattern has the same storage pattern, each uploaded fragment (from the tape drive) can be retrieved from its storage disk for display. Since the tape drive reads NO fragments in each unit of time and all uploaded fragments (from the tape drive) can be retrieved from the storage disks, the bandwidth of the tape drive is fully utilized. Definition 4 An object is said to be uniformly distributed over a set of disk drives if each disk drive contains the same number of fragments of the object. Theorem 3 With staggered striping, when an object O is uploaded from a tape drive to the disk array, the tape drive bandwidth can be fully utilized if 1. k and D do not have a common factor greater than 1, i.e. the greatest common divisor (GCD) of k and D is 1, and 2. the data transfer period, s \Gamma H, is a multiple of LCM(D;MO ;N O ) is the least common multiple of integers x, y, z. Proof: Suppose the GCD of k and D is 1 and s \Gamma H is a multiple of LCM(D;MO ;N O ) units. Consider the case that the object starts to be uploaded at time 0. At time i, NO fragments has been stored and uniformly distributed into disk drives (i \Theta (i D. Since the GCD of k and D is 1, mod D is a one-one mapping. If we extend the domain of f to the set of natural numbers N , then This implies that fragments can be uniformly distributed over the disk drives. Hence, at time LCM(D;MO ;N O ) stored and uniformly distributed over the disk drives of the disk buffer. Consider the case that the object is playbacked at time 0. At time i, MO fragments are retrieved from disk drives (i \Theta At time LCM(D;MO ;N O ) retrieved and LCM(D;MO ;N O ) fragments have been retrieved from each disk drive. Let O 0 be the object consisting of the fragments. The following procedure finds the upload pattern PO 0 which has the same storage pattern of the retrieval pattern RO procedure upload(var upattern : upload pattern; rpattern : retrieval pattern); var begin initialize all the entries in count to 0; initialize all the entries in upattern to blank; spattern := storage pattern of rpattern; for for begin c := (i*k+j) mod D; With upload pattern PO 0 , the tape reads NO different fragments into NO different disk drives in each time unit and the storage pattern of the retrieval pattern and the upload pattern of O 0 are the same. By Lemma 2, O 0 can be retrieved with the maximum throughput of the tape drive. Hence, an object of a multiple of the size of O 0 , i.e. can be uploaded with the maximum throughput of the tape drive. Thus, if the data transfer period is a multiple of LCM(D;MO ;N O ) NO , the tape drive bandwidth can be fully utilized. For the case of Example 1, the data transfer period is a multiple of LCM(5;2;3)= 10 time units or 5:5 seconds. For the case that seconds, a reasonable time-slice period is from 200 to 300 seconds 9 . A video on demand system with a capacity of 1000 100-minutes HDTV videos of 2 Mbytes/second bandwidth requires a storage space of 1000 \Theta 12 Mbytes = 12 TBytes. If 10% of the videos reside on disks, 1.2 TBytes disk space is required. The number of 1.2 Gbytes disk drives of the disk array is 1200, and the data transfer period is a multiple of LCM(1200;2;3)= 400 time units = 400 \Theta 0.55 seconds seconds or the time-slice period is 250 seconds. Hence, the disk array of 1200 disk drives can be used as a disk buffer as well as a disk cache. To maximize the tape drive throughput, the maximum output rate of the disk buffer 9 For this range of time-slice period, the tape switch overhead is about 10-15% of the tape drive bandwidth. must be at least the maximum utilized bandwidth of the tape drive. The maximum utilized bandwidth of the tape drive is given by NOB d (O) MO . To have an output rate of at least the maximum utilized bandwidth of the tape drive, the disk buffer must support concurrent retrieval of at least d NO MO e subobjects. For each tape drive, the minimum number of required disk drive for buffering is NO MO e \Theta MO . Video uploading from the tape drive is first stored in the disk array. The playback of the video object can start when the cluster of disk drives for uploading does not overlap with the cluster of disk drives of the first subobject. Hence, the minimum delay, d, of the disk buffer is defined as the smallest integer n such that 80 MO . The stride k should be carefully chosen to minimize the disk buffer delay and improve the overall response time of the storage server. 6 Performance Evaluation We evaluate the the performance of the scheduling algorithms for two values of the tape drive Mbytes/second and 15 Mbytes/second by computer simulation. We assume that (1) each tape contains only one object, and hence the search time of each request is 0 seconds and (2) a request never waits for a tape. Since frequently accessed objects are kept in disk drives, the probability that a request has to wait for a tape which is being used to serve another request is very low 10 . Hence, the second assumption causes negligible errors in the simulation results. We assume that the disk contention between disk reads (generated by playback of objects) and disk writes (generated by upload of objects) is resolved by delaying disk writes [13] as follows. A fragment uploaded from the tape is first stored in the memory buffer and written into its storage disk in an idle period of the disk. This technique smoothes out the bursty data traffic from the disk subsystem the probability is in the order of 0.001 for the parameters of the simulation and hence improves request response time. In practice, the additional memory buffer space required by this technique is small because the aggregate transfer rate of the tape subsystem is much lower than that of the disk subsystem [13]. The storage size of each object is uniformly distributed between (7200; 14400) Mbytes. Table 3 shows the major simulation parameters. The results are presented with 95% confidence intervals where the length of each confidence interval is bounded by 1%. Parameter Case 1 Case 2 seconds 12 seconds Table 3: simulation parameters 6.1 Single Tape Drive We first study the performance of the algorithms in a system with one robot arm and one tape drive. Here, the request arrival process is Poisson. Case 1. Tape drive The maximum throughput of the tape subsystem is 1.95 requests/hour. Table 4 presents the average response time of the FCFS, SJF, RR, and LS algorithms. Blank entries in the table show that the tape subsystem has reached the maximum utilization and the system cannot sustain the input requests. The efficiency of RR and LS algorithms is defined as the percentage of time spent in data transfer. An efficiency of 90% means that 10% of time is spent in tape switches. We define the relative response time to be the ratio of the scheduling algorithm response time divided by the FCFS algorithm response time. The relative response times of the SJF, RR, and LS algorithms are shown in Figure 7. Req. Arr. FCFS SJF RR-1 RR-1 RR-1 LS LS LS Rate (req./hr.) (sec) (sec) (sec) (sec) (sec) (sec) (sec) (sec) 1.00 1022.84 936.15 867.36 1323.80 2426.81 652.23 1059.95 2069.68 Table 4: Response time vs request arrival rate. Case 2. Tape drive bandwidth = 15 Mbytes/second. The maximum throughput of the tape subsystem is 4.72 requests/hour. Here, we consider a tape subsystem with a higher performance tape drive. The average response time of the FCFS, SJF, RR, and LS algorithms are tabulated in Table 5. Again, those Request Arrival Rate (req/hour) Relative Average Response Time Figure 7: The relative response time of SJF, RR, and LS scheduling algorithms. blank entries in the table represent a case whereby the tape subsystem has reached the maximum utilization and the system cannot sustain the input requests. The relative response time of RR and LS algorithms is shown in Figure 8. Req. Arr. FCFS SJF RR-1 RR-1 RR-1 RR-2 RR-2 RR-2 LS LS LS Rate (req./hr.) (sec) (sec) (sec) (sec) (sec) (sec) (sec) (sec) (sec) (sec) (sec) 2.0 298.70 280.84 145.82 110.13 163.25 104.66 75.62 132.80 106.30 73.92 102.24 2.5 447.42 410.77 250.22 226.21 500.59 162.93 171.68 466.91 169.01 170.92 380.94 Table 5: Response time vs request arrival rate In both cases, the LS algorithm has the best performance in a wide range of request arrival rates. The simulation result shows that the time-slice algorithm (especially the LS algorithm) performs better than the FCFS algorithm and the SJF algorithm under a RR-2 RR-2 RR-2 Request Arrival Rate (req/hour) Relative Average Response Time Figure 8: Relative response time of SJF, RR-1, RR-2, and LS algorithms. wide range of request arrival rates. The SJF algorithm performs better than the FCFS algorithm for all request arrival rates. 6.2 Multiple Tape Drives Previous experiments have shown that LS and RR algorithms outperform the FCFS and SJF algorithms in a wide range of load conditions. We study the effect of robot arm contention of the LS algorithm in this experiment. The system contains 4 tape drives which have a bandwidth of 15.0 Mbytes/second. The maximum throughput of the tape subsystem is 18.90 requests/hour. The results are shown in Table 6. A plot of the relative response time vs arrival rate is shown in Figure 9. In this simulation experiment, we found out that for the large range of request arrival rates, the utilization of robot arm is very small. For example, the robot arm utilization is only 0:215 when the request arrival rate is 14:0 requests/hour. Hence, the effect of Request Arrival Rate FCFS SJF LS (E=0.9) 2.0 19.19 19.18 15.13 4.0 25.11 25.02 16.22 6.0 36.05 35.60 20.60 Table Multiple tape drives case: response time vs request arrival rate. robot arm contention is not a major factor in determining the average response time. 6.3 Throughput Under Finite Disk Buffer In this section, we study the maximum throughput of the FCFS, SJF, and LS algorithms with finite disk buffer space. The maximum throughput of the scheduling algorithm is found by a close queueing network in which there are 200 clients and each client initiates a new request immediately after its previous request has been served. Hence, there are always 200 requests in the system. The maximum throughput of the LS, FCFS, and SJF algorithms are evaluated for Cases 1 and 2. In each case, the size of each disk buffer is chosen to be large enough to store the data uploaded from a tape drive in one time-slice period. The efficiency of the LS algorithm is chosen to be 0.9, and therefore, the time-slice period is 300 seconds. The disk buffer sizes of Case 1 and 2 are 1.582 Gbytes and 3.955 Gbytes respectively. The results for Case 1 and Case 2 are shown in Figure 10 and Request Arrival Rate (req/hour) Relative Average Response Time Figure 9: Relative response time of the SJF and LS algorithms Figure respectively. From the figures, we observe that the LS algorithm has much higher throughput (in some cases, we have 50 % improvement) than the FCFS and SJF algorithms in a wide range of number of disk buffers. The throughput of each algorithm grows with the number of disk buffers but the LS algorithm reaches its maximumpossible throughput with about half of the buffer requirement that the FCFS algorithm needs to achieve its maximum possible throughput. The SJF algorithm performs slightly better than the FCFS algorithm. The FCFS (or SJF) algorithm performs better than the LS algorithm for about 10% when the disk buffer space is large enough. 6.4 Discussion of Results The results show that the LS and Round-robin algorithms outperform the conventional algorithms (FCFS and SJF) in a wide range of request arrival rates. In all the cases, the LS algorithm with 90% efficiency outperforms the FCFS algorithm and the SJF algorithm when the request arrival rate is below 60% of the maximum throughput of the tape subsystem. The conventional algorithms have a better response time when the request FCFS Number of Disk Buffers Maximum Throughput (req./hr.) Figure 10: Maximum throughput of the FCFS, SJF, and LS algorithms arrival rate is quite high (above 70% of the maximum throughput of the conventional algorithms). For the LS or Round-robin algorithm, the algorithm performs better with a lower efficiency factor at low request arrival rate and better with a higher efficiency factor at high request arrival rate. The results also show that the relative response time of the LS and Round-robin algorithms reach a minimum at certain request arrival rate. This is because the response time is the sum of the waiting time W and the tape switch time H. At low request arrival rate, H is the major component of the response time. As the request arrival rate increases from zero, the waiting time of the conventional algorithms grows faster than that of the LS and Round-robin algorithms because the LS and Round-robin algorithms can serve several requests at the same time and hence reduce the chance of waiting for available tape drive. Therefore, the relative response time of the LS and Round-robin algorithms decreases with the increase of request arrival rate when the request arrival is low. When the request arrival rate is high enough, the waiting time of LS and Round-robin algorithms becomes higher than that of the conventional algorithms because the conventional algorithms have a better utilization of the tape drive bandwidth which covers the high load conditions. FCFS Number of Disk Buffers Maximum Throughput (req./hr.) Figure Maximum throughput of the FCFS, SJF, and LS algorithms 7 Concluding Remarks In this paper, we have proposed a cost-effective near-line storage system for a large scale multimedia storage server using a robotic tape library. We have studied a class of novel time-slice scheduling algorithms for the tape subsystem and have shown that under light to moderate workload, this class of tape scheduling algoritms has better response time and requires less disk buffer space than the conventional algorithm. Also, we have complemented our work to the proposed Staggered Striping architecture [4] and showed that using our proposed scheduling algorithms, how we can organize the data layout on disks and tape cartridges for concurrent upload and display of large multimedia objects. From the performance results, the selection of time-slice value is often more important than the choice of the time-slice algorithm used. If the request arrival process is known in advance (i.e. the average request arrival rate and the inter-arrival time distribution are known), the time-slice value can be adjusted by using pre-computed results (obtained by either analytical methods or simulations). In practical environments, the request arrival process is usually not known in advance. One simple method that can be used is to adjust the time-slice value according to the length of the queue of waiting requests, i.e., a larger time-slice value is required if the length of the queue is longer. The function from the queue length to the time-slice value can be pre-determined by empirical studies. In general, the optimal time-slice value depends on the request arrival process, the number of requests waiting for service, and the states of the currently active requests. Further work is required to find the best way to determine the optimal time-slice value. --R The Ampex DST800 Robotic Tape Library Technical Marketing Document. "A File System for Continuous Media," "Channel Coding for Digital HDTV Terrestrial Broadcasting," "Staggered Striping in Multi-media Information Systems," "A Fault Tolerant Design of a Multimedia Server," An Evaluation of New Applications "Principles of Delay-Sensitive Multimedia Data Storage and Retrieval," "On Multimedia Repositories, Personal Com- puters, and Hierarchical Storage Systems," "Analysis of Striping Techniques in Robotic Storage Libraries," "Video On Demand: Architecture, Systems, and Applications," "Using "The Design of a Storage Server for Continuous Me- dia," "Scheduling and Replacement Policies for a Hierarchical Multimedia Storage Server," "Multiprocessor Scheduling in a Hard Real-Time Environment," "A case for Redundant Arrays of Inexpensive Disks (RAID)," "Efficient Storage Techniques for Digital Continuous Multimedia," "Designing an On-Demand Multimedia Service," Operating Systems "Designing a Multi-User HDTV Storage Server," --TR A case for redundant arrays of inexpensive disks (RAID) Principles of delay-sensitive multimedia data storage retrieval A file system for continuous media Staggered striping in multimedia information systems On multimedia repositories, personal computers, and hierarchical storage systems Tertiary storage Fault tolerant design of multimedia servers Efficient Storage Techniques for Digital Continuous Multimedia Using tertiary storage in video-on-demand servers --CTR Kien A. Hua , Ying Cai , Simon Sheu, Patching M. Y. Y. Leung , J. C. S. Lui , L. Golubchik, Use of Analytical Performance Models for System Sizing and Resource Allocation in Interactive Video-on-Demand Systems Employing Data Sharing Techniques, IEEE Transactions on Knowledge and Data Engineering, v.14 n.3, p.615-637, May 2002 S.-H. Gary Chan , Fouad A. Tobagi, Modeling and Dimensioning Hierarchical Storage Systems for Low-Delay Video Services, IEEE Transactions on Computers, v.52 n.7, p.907-919, July

multimedia storage;scheduling;data layout

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Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer.

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

Introduction One of the first results in the mathematics of computation, which underlies the subsequent development of much of theoretical computer science, was the distinction between computable and non-computable functions shown in papers of Church [1936], Turing [1936], and Post [1936]. Central to this result is Church's thesis, which says that all computing devices can be simulated by a Turing machine. This thesis greatly simplifies the study of computation, since it reduces the potential field of study from any of an infinite number of potential computing devices to Turing machines. Church's thesis is not a mathematical theorem; to make it one would require a precise mathematical description of a computing device. Such a description, however, would leave open the possibility of some practical computing device which did not satisfy this precise mathematical description, and thus would make the resulting mathematical theorem weaker than Church's original thesis. With the development of practical computers, it has become apparent that the distinction between computable and non-computable functions is much too coarse; computer scientists are now interested in the exact efficiency with which specific functions can be computed. This exact efficiency, on the other hand, is too precise a quantity to work with easily. The generally accepted compromise between coarseness and precision distinguishes efficiently and inefficiently computable functions by whether the length of the computation scales polynomially or superpolynomially with the input size. The class of problems which can be solved by algorithms having a number of steps polynomial in the input size is known as P. For this classification to make sense, we need it to be machine-independent. That is, we need to know that whether a function is computable in polynomial time is independent of the kind of computing device used. This corresponds to the following quantitative version of Church's thesis, which Vergis et al. [1986] have called the ``Strong Church's Thesis" and which makes up half of the "Invariance Thesis" of van Emde Boas [1990]. Thesis (Quantitative Church's thesis). Any physical computing device can be simulated by a Turing machine in a number of steps polynomial in the resources used by the computing device. In statements of this thesis, the Turing machine is sometimes augmented with a random number generator, as it has not yet been determined whether there are pseudorandom number generators which can efficiently simulate truly random number generators for all purposes. Readers who are not comfortable with Turing machines may think instead of digital computers having an amount of memory that grows linearly with the length of the computation, as these two classes of computing machines can efficiently simulate each other. There are two escape clauses in the above thesis. One of these is the word "physical." Researchers have produced machine models that violate the above quantitative Church's thesis, but most of these have been ruled out by some reason for why they are not "phys- ical," that is, why they could not be built and made to work. The other escape clause in the above thesis is the word "resources," the meaning of which is not completely specified above. There are generally two resources which limit the ability of digital computers to solve large problems: time (computation steps) and space (memory). There are more resources pertinent to analog computation; some proposed analog machines that seem able to solve NP-complete problems in polynomial time have required the machining of FACTORING WITH A QUANTUM COMPUTER 3 exponentially precise parts, or an exponential amount of energy. (See Vergis et al. [1986] and Steiglitz [1988]; this issue is also implicit in the papers of Canny and Reif [1987] and Choi et al. [1995] on three-dimensional shortest paths.) For quantum computation, in addition to space and time, there is also a third potentially important resource, precision. For a quantum computer to work, at least in any currently envisioned implementation, it must be able to make changes in the quantum states of objects (e.g., atoms, photons, or nuclear spins). These changes can clearly not be perfectly accurate, but must contain some small amount of inherent impreci- sion. If this imprecision is constant (i.e., it does not depend on the size of the input), then it is not known how to compute any functions in polynomial time on a quantum computer that cannot also be computed in polynomial time on a classical computer with a random number generator. However, if we let the precision grow polynomially in the input size (that is, we let the number of bits of precision grow logarithmically in the input size), we appear to obtain a more powerful type of computer. Allowing the same polynomial growth in precision does not appear to confer extra computing power to classical mechanics, although allowing exponential growth in precision does [Hartmanis and Simon 1974, Vergis et al. 1986]. As far as we know, what precision is possible in quantum state manipulation is dictated not by fundamental physical laws but by the properties of the materials and the architecture with which a quantum computer is built. It is currently not clear which architectures, if any, will give high precision, and what this precision will be. If the precision of a quantum computer is large enough to make it more powerful than a classical computer, then in order to understand its potential it is important to think of precision as a resource that can vary. Treating the precision as a large constant (even though it is almost certain to be constant for any given machine) would be comparable to treating a classical digital computer as a finite automaton - since any given computer has a fixed amount of memory, this view is technically correct; however, it is not particularly useful. Because of the remarkable effectiveness of our mathematical models of computation, computer scientists have tended to forget that computation is dependent on the laws of physics. This can be seen in the statement of the quantitative Church's thesis in van Emde Boas [1990], where the word "physical" in the above phrasing is replaced with the word "reasonable." It is difficult to imagine any definition of "reasonable" in this context which does not mean "physically realizable," i.e., that this computing machine could actually be built and would work. Computer scientists have become convinced of the truth of the quantitative Church's thesis through the failure of all proposed counter-examples. Most of these proposed counter-examples have been based on the laws of classical mechanics; however, the universe is in reality quantum mechanical. Quantum mechanical objects often behave quite differently from how our intuition, based on classical mechanics, tells us they should. It thus seems plausible that the natural computing power of classical mechanics corresponds to Turing machines, 1 while the natural computing power of quantum mechanics might be greater. I believe that this question has not yet been settled and is worthy of further investigation. See Vergis et al. [1986], Steiglitz [1988], and Rubel [1989]. In particular, turbulence seems a good candidate for a counterexample to the quantitative Church's thesis because the non-trivial dynamics on many length scales may make it difficult to simulate on a classical computer. 4 P. W. SHOR The first person to look at the interaction between computation and quantum mechanics appears to have been Benioff [1980, 1982a, 1982b]. Although he did not ask whether quantum mechanics conferred extra power to computation, he showed that reversible unitary evolution was sufficient to realize the computational power of a Turing machine, thus showing that quantum mechanics is at least as powerful computationally as a classical computer. This work was fundamental in making later investigation of quantum computers possible. Feynman [1982,1986] seems to have been the first to suggest that quantum mechanics might be more powerful computationally than a Turing machine. He gave arguments as to why quantum mechanics might be intrinsically expensive computationally to simulate on a classical computer. He also raised the possibility of using a computer based on quantum mechanical principles to avoid this problem, thus implicitly asking the converse question: by using quantum mechanics in a computer can you compute more efficiently than on a classical computer? Deutsch [1985, 1989] was the first to ask this question explicitly. In order to study this question, he defined both quantum Turing machines and quantum circuits and investigated some of their properties. The question of whether using quantum mechanics in a computer allows one to obtain more computational power was more recently addressed by Deutsch and Jozsa [1992] and Berthiaume and Brassard [1992a, 1992b]. These papers showed that there are problems which quantum computers can quickly solve exactly, but that classical computers can only solve quickly with high probability and the aid of a random number generator. However, these papers did not show how to solve any problem in quantum polynomial time that was not already known to be solvable in polynomial time with the aid of a random number generator, allowing a small probability of error; this is the characterization of the complexity class BPP, which is widely viewed as the class of efficiently solvable problems. Further work on this problem was stimulated by Bernstein and Vazirani [1993]. One of the results contained in their paper was an oracle problem (that is, a problem involving a "black box" subroutine that the computer is allowed to perform, but for which no code is accessible) which can be done in polynomial time on a quantum Turing machine but which requires super-polynomial time on a classical computer. This result was improved by Simon [1994], who gave a much simpler construction of an oracle problem which takes polynomial time on a quantum computer but requires exponential time on a classical computer. Indeed, while Bernstein and Vaziarni's problem appears contrived, Simon's problem looks quite natural. Simon's algorithm inspired the work presented in this paper. Two number theory problems which have been studied extensively but for which no polynomial-time algorithms have yet been discovered are finding discrete logarithms and factoring integers [Pomerance 1987, Gordon 1993, Lenstra and Lenstra 1993, Adleman and McCurley 1995]. These problems are so widely believed to be hard that several cryptosystems based on their difficulty have been proposed, including the widely used RSA public key cryptosystem developed by Rivest, Shamir, and Adleman [1978]. We show that these problems can be solved in polynomial time on a quantum computer with a small probability of error. Currently, nobody knows how to build a quantum computer, although it seems as though it might be possible within the laws of quantum mechanics. Some suggestions have been made as to possible designs for such computers [Teich et al. 1988, Lloyd 1993, FACTORING WITH A QUANTUM COMPUTER 5 1994, Cirac and Zoller 1995, DiVincenzo 1995, Sleator and Weinfurter 1995, Barenco et al. 1995b, Chuang and Yamomoto 1995], but there will be substantial difficulty in building any of these [Landauer 1995a, Landauer 1995b, Unruh 1995, Chuang et al. 1995, Palma et al. 1995]. The most difficult obstacles appear to involve the decoherence of quantum superpositions through the interaction of the computer with the environment, and the implementation of quantum state transformations with enough precision to give accurate results after many computation steps. Both of these obstacles become more difficult as the size of the computer grows, so it may turn out to be possible to build small quantum computers, while scaling up to machines large enough to do interesting computations may present fundamental difficulties. Even if no useful quantum computer is ever built, this research does illuminate the problem of simulating quantum mechanics on a classical computer. Any method of doing this for an arbitrary Hamiltonian would necessarily be able to simulate a quantum computer. Thus, any general method for simulating quantum mechanics with at most a polynomial slowdown would lead to a polynomial-time algorithm for factoring. The rest of this paper is organized as follows. In x2, we introduce the model of quantum computation, the quantum gate array, that we use in the rest of the paper. In xx3 and 4, we explain two subroutines that are used in our algorithms: reversible modular exponentiation in x3 and quantum Fourier transforms in x4. In x5, we give our algorithm for prime factorization, and in x6, we give our algorithm for extracting discrete logarithms. In x7, we give a brief discussion of the practicality of quantum computation and suggest possible directions for further work. Quantum computation In this section we give a brief introduction to quantum computation, emphasizing the properties that we will use. We will describe only quantum gate arrays, or quantum acyclic circuits, which are analogous to acyclic circuits in classical computer science. For other models of quantum computers, see references on quantum Turing machines [Deutsch 1989, Bernstein and Vazirani 1993, Yao 1993] and quantum cellular automata [Feynman 1986, Margolus 1986, 1990, Lloyd 1993, Biafore 1994]. If they are allowed a small probability of error, quantum Turing machines and quantum gate arrays can compute the same functions in polynomial time [Yao 1993]. This may also be true for the various models of quantum cellular automata, but it has not yet been proved. This gives evidence that the class of functions computable in quantum polynomial time with a small probability of error is robust, in that it does not depend on the exact architecture of a quantum computer. By analogy with the classical class BPP, this class is called BQP. Consider a system with n components, each of which can have two states. Whereas in classical physics, a complete description of the state of this system requires only n bits, in quantum physics, a complete description of the state of this system requires To be more precise, the state of the quantum system is a point in a 2 n -dimensional vector space. For each of the 2 n possible classical positions of the components, there is a basis state of this vector space which we represent, for example, by j011 meaning that the first bit is 0, the second bit is 1, and so on. Here, the ket notation jxi means that x is a (pure) quantum state. (Mixed states will 6 P. W. SHOR not be discussed in this paper, and thus we do not define them; see a quantum theory book such as Peres [1993] for this definition.) The Hilbert space associated with this quantum system is the complex vector space with these 2 n states as basis vectors, and the state of the system at any time is represented by a unit-length vector in this Hilbert space. As multiplying this state vector by a unit-length complex phase does not change any behavior of the state, we need only numbers to completely describe the state. We represent this superposition of states as a where the amplitudes a i are complex numbers such that each jS i i is a basis vector of the Hilbert space. If the machine is measured (with respect to this basis) at any particular step, the probability of seeing basis state jS i i is ja however, measuring the state of the machine projects this state to the observed basis vector jS i i. Thus, looking at the machine during the computation will invalidate the rest of the computation. In this paper, we only consider measurements with respect to the canonical basis. This does not greatly restrict our model of computation, since measurements in other reasonable bases could be simulated by first using quantum computation to perform a change of basis and then performing a measurement in the canonical basis. In order to use a physical system for computation, we must be able to change the state of the system. The laws of quantum mechanics permit only unitary transformations of state vectors. A unitary matrix is one whose conjugate transpose is equal to its inverse, and requiring state transformations to be represented by unitary matrices ensures that summing the probabilities of obtaining every possible outcome will result in 1. The definition of quantum circuits (and quantum Turing machines) only allows local unitary transformations; that is, unitary transformations on a fixed number of bits. This is physically justified because, given a general unitary transformation on n bits, it is not at all clear how one would efficiently implement it physically, whereas two-bit transformations can at least in theory be implemented by relatively simple physical systems [Cirac and Zoller 1995, DiVincenzo 1995, Sleator and Weinfurter 1995, Chuang and Yamomoto 1995]. While general n-bit transformations can always be built out of two-bit transformations [DiVincenzo 1995, Sleator and Weinfurter 1995, Lloyd 1995, Deutsch et al. 1995], the number required will often be exponential in n [Barenco et al. 1995a]. Thus, the set of two-bit transformations form a set of building blocks for quantum circuits in a manner analogous to the way a universal set of classical gates (such as the AND, OR and NOT gates) form a set of building blocks for classical circuits. In fact, for a universal set of quantum gates, it is sufficient to take all one-bit gates and a single type of two-bit gate, the controlled NOT, which negates the second bit if and only if the first bit is 1. Perhaps an example will be informative at this point. A quantum gate can be expressed as a truth table: for each input basis vector we need to give the output of the gate. One such gate is: FACTORING WITH A QUANTUM COMPUTER 7 Not all truth tables correspond to physically feasible quantum gates, as many truth tables will not give rise to unitary transformations. The same gate can also be represented as a matrix. The rows correspond to input basis vectors. The columns correspond to output basis vectors. The (i; when the ith basis vector is input to the gate, the coefficient of the jth basis vector in the corresponding output of the gate. The truth table above would then correspond to the following matrix: p: A quantum gate is feasible if and only if the corresponding matrix is unitary, i.e., its inverse is its conjugate transpose. Suppose our machine is in the superposition of statespj10i and we apply the unitary transformation represented by (2.2) and (2.3) to this state. The resulting output will be the result of multiplying the vector (2.4) by the matrix (2.3). The machine will thus go to the superposition of states2 This example shows the potential effects of interference on quantum computation. Had we started with either the state j10i or the state j11i, there would have been a chance of observing the state j10i after the application of the gate (2.3). However, when we start with a superposition of these two states, the probability amplitudes for the state j10i cancel, and we have no possibility of observing j10i after the application of the gate. Notice that the output of the gate would have been j10i instead of j11i had we started with the superposition of statespj10i which has the same probabilities of being in any particular configuration if it is observed as does the superposition (2.4). If we apply a gate to only two bits of a longer basis vector (now our circuit must have more than two wires), we multiply the gate matrix by the two bits to which the gate is 8 P. W. SHOR applied, and leave the other bits alone. This corresponds to multiplying the whole state by the tensor product of the gate matrix on those two bits with the identity matrix on the remaining bits. A quantum gate array is a set of quantum gates with logical "wires" connecting their inputs and outputs. The input to the gate array, possibly along with extra work bits that are initially set to 0, is fed through a sequence of quantum gates. The values of the bits are observed after the last quantum gate, and these values are the output. To compare gate arrays with quantum Turing machines, we need to add conditions that make gate arrays a uniform complexity class. In other words, because there is a different gate array for each size of input, we need to keep the designer of the gate arrays from hiding non-computable (or hard to compute) information in the arrangement of the gates. To make quantum gate arrays uniform, we must add two things to the definition of gate arrays. The first is the standard requirement that the design of the gate array be produced by a polynomial-time (classical) computation. The second requirement should be a standard part of the definition of analog complexity classes, although since analog complexity classes have not been widely studied, this requirement is much less widely known. This requirement is that the entries in the unitary matrices describing the gates must be computable numbers. Specifically, the first log n bits of each entry should be classically computable in time polynomial in n [Solovay 1995]. This keeps non-computable (or hard to compute) information from being hidden in the bits of the amplitudes of the quantum gates. 3 Reversible logic and modular exponentiation The definition of quantum gate arrays gives rise to completely reversible computation. That is, knowing the quantum state on the wires leading out of a gate tells uniquely what the quantum state must have been on the wires leading into that gate. This is a reflection of the fact that, despite the macroscopic arrow of time, the laws of physics appear to be completely reversible. This would seem to imply that anything built with the laws of physics must be completely reversible; however, classical computers get around this fact by dissipating energy and thus making their computations thermodynamically irreversible. This appears impossible to do for quantum computers because superpositions of quantum states need to be maintained throughout the computation. Thus, quantum computers necessarily have to use reversible computation. This imposes extra costs when doing classical computations on a quantum computer, as is sometimes necessary in subroutines of quantum computations. Because of the reversibility of quantum computation, a deterministic computation is performable on a quantum computer only if it is reversible. Luckily, it has already been shown that any deterministic computation can be made reversible [Lecerf 1963, Bennett 1973]. In fact, reversible classical gate arrays have been studied. Much like the result that any classical computation can be done using NAND gates, there are also universal gates for reversible computation. Two of these are Toffoli gates [Toffoli 1980] and Fredkin gates [Fredkin and Toffoli 1982]; these are illustrated in Table 3.1. The Toffoli gate is just a controlled controlled NOT, i.e., the last bit is negated if and only if the first two bits are 1. In a Toffoli gate, if the third input bit is set to 1, then the third output bit is the NAND of the first two input bits. Since NAND is a FACTORING WITH A QUANTUM COMPUTER 9 Table 3.1: Truth tables for Toffoli and Fredkin gates. Toffoli Gate INPUT OUTPUT Fredkin Gate INPUT OUTPUT universal gate for classical gate arrays, this shows that the Toffoli gate is universal. In a Fredkin gate, the last two bits are swapped if the first bit is 0, and left untouched if the first bit is 1. For a Fredkin gate, if the third input bit is set to 0, the second output bit is the AND of the first two input bits; and if the last two input bits are set to 0 and 1 respectively, the second output bit is the NOT of the first input bit. Thus, both AND and NOT gates are realizable using Fredkin gates, showing that the Fredkin gate is universal. From results on reversible computation [Lecerf 1963, Bennett 1973], we can compute any polynomialtime function F (x) as long as we keep the input x in the computer. We do this by adapting the method for computing the function F non-reversibly. These results can easily be extended to work for gate arrays [Toffoli 1980, Fredkin and Toffoli 1982]. When AND, OR or NOT gates are changed to Fredkin or Toffoli gates, one obtains both additional input bits, which must be preset to specified values, and additional output bits, which contain the information needed to reverse the computation. While the additional input bits do not present difficulties in designing quantum computers, the additional output bits do, because unless they are all reset to 0, they will affect the interference patterns in quantum computation. Bennett's method for resetting these bits to 0 is shown in the top half of Table 3.2. A non-reversible gate array may thus be turned into a reversible gate array as follows. First, duplicate the input bits as many times as necessary (since each input bit could be used more than once by the gate array). Next, keeping one copy of the input around, use Toffoli and Fredkin gates to simulate non-reversible gates, putting the extra output bits into the RECORD register. These extra output bits preserve enough of a record of the operations to enable the computation of the gate array to be reversed. Once the output F (x) has been computed, copy it into a register that has been preset to zero, and then undo the computation to erase both the first OUTPUT register and the RECORD register. To erase x and replace it with F (x), in addition to a polynomial-time algorithm for F , we also need a polynomial-time algorithm for computing x from F (x); i.e., we need that F is one-to-one and that both F and F \Gamma1 are polynomial-time computable. The method for this computation is given in the whole of Table 3.2. There are two stages to this computation. The first is the same as before, taking x to (x; F (x)). For the second stage, shown in the bottom half of Table 3.2, note that if we have a method to compute non-reversibly in polynomial time, we can use the same technique to reversibly map F (x) to However, since this is a reversible computation, P. W. SHOR Table 3.2: Bennett's method for making a computation reversible. we can reverse it to go from (x; F (x)) to F (x). Put together, these two pieces take x to F (x). The above discussion shows that computations can be made reversible for only a constant factor cost in time, but the above method uses as much space as it does time. If the classical computation requires much less space than time, then making it reversible in this manner will result in a large increase in the space required. There are methods that do not use as much space, but use more time, to make computations reversible [Bennett 1989, Levine and Sherman 1990]. While there is no general method that does not cause an increase in either space or time, specific algorithms can sometimes be made reversible without paying a large penalty in either space or time; at the end of this section we will show how to do this for modular exponentiation, which is a subroutine necessary for quantum factoring. The bottleneck in the quantum factoring algorithm; i.e., the piece of the factoring algorithm that consumes the most time and space, is modular exponentia- tion. The modular exponentiation problem is, given n, x, and r, find x r (mod n). The best classical method for doing this is to repeatedly square of x (mod n) to (mod n) for i - log 2 r, and then multiply a subset of these powers (mod n) to get x r (mod n). If we are working with l-bit numbers, this requires O(l) squar- ings and multiplications of l-bit numbers (mod n). Asymptotically, the best classical result for gate arrays for multiplication is the Sch-onhage-Strassen algorithm [Sch-onhage and Strassen 1971, Knuth 1981, Sch-onhage 1982]. This gives a gate array for integer multiplication that uses O(l log l log log l) gates to multiply two l-bit numbers. Thus, asymptotically, modular exponentiation requires O(l 2 log l log log l) time. Making this reversible would na-ively cost the same amount in space; however, one can reuse the space used in the repeated squaring part of the algorithm, and thus reduce the amount of space needed to essentially that required for multiplying two l-bit numbers; one simple method for reducing this space (although not the most versatile one) will be given later in this section. Thus, modular exponentiation can be done in O(l 2 log l log log l) time and O(l log l log log l) space. While the Sch-onhage-Strassen algorithm is the best multiplication algorithm discovered to date for large l, it does not scale well for small l. For small numbers, the best gate arrays for multiplication essentially use elementary-school longhand multiplication in binary. This method requires O(l 2 ) time to multiply two l-bit numbers, and thus modular exponentiation requires O(l 3 time with this method. These gate arrays can be made reversible, however, using only O(l) space. We will now give the method for constructing a reversible gate array that takes only FACTORING WITH A QUANTUM COMPUTER 11 O(l) space and O(l 3 ) time to compute (a; x a (mod n)) from a, where a, x, and n are l-bit numbers. The basic building block used is a gate array that takes b as input and outputs n). Note that here b is the gate array's input but c and n are built into the structure of the gate array. Since addition (mod n) is computable in O(log n) time classically, this reversible gate array can be made with only O(logn) gates and O(logn) work bits using the techniques explained earlier in this section. The technique we use for computing x a (mod n) is essentially the same as the classical method. First, by repeated squaring we compute x 2 i (mod n) for all i ! l. Then, to obtain x a (mod n) we multiply the powers x 2 i (mod n) where 2 i appears in the binary expansion of a. In our algorithm for factoring n, we only need to compute x a (mod n) where a is in a superposition of states, but x is some fixed integer. This makes things much easier, because we can use a reversible gate array where a is treated as input, but where x and n are built into the structure of the gate array. Thus, we can use the algorithm described by the following pseudocode; here, a i represents the ith bit of a in binary, where the bits are indexed from right to left and the rightmost bit of a is a 0 . power power := power x 2 i (mod n) endif endfor The variable a is left unchanged by the code and x a (mod n) is output as the variable power . Thus, this code takes the pair of values (a; 1) to (a; x a (mod n)). This pseudocode can easily be turned into a gate array; the only hard part of this is the fourth line, where we multiply the variable power by x 2 i (mod n); to do this we need to use a fairly complicated gate array as a subroutine. Recall that x 2 i (mod n) can be computed classically and then built into the structure of the gate array. Thus, to implement this line, we need a reversible gate array that takes b as input and gives bc (mod n) as output, where the structure of the gate array can depend on c and n. Of course, this step can only be reversible if gcd(c; n) = 1, i.e., if c and n have no common factors, as otherwise two distinct values of b will be mapped to the same value of bc (mod n); this case is fortunately all we need for the factoring algorithm. We will show how to build this gate array in two stages. The first stage is directly analogous to exponentiation by repeated multiplication; we obtain multiplication from repeated addition (mod n). Pseudocode for this stage is as follows. result := 0 result endif endfor n) can be precomputed and built into the structure of the gate array. P. W. SHOR The above pseudocode takes b as input, and gives (b; bc (mod n)) as output. To get the desired result, we now need to erase b. Recall that gcd(c; n) = 1, so there is n). Multiplication by this c \Gamma1 could be used to reversibly take bc (mod n) to (bc (mod n); bcc \Gamma1 (mod b). This is just the reverse of the operation we want, and since we are working with reversible computing, we can turn this operation around to erase b. The pseudocode for this follows. endif endfor As before, result i is the ith bit of result. Note that at this stage of the computation, b should be 0. However, we did not set b directly to zero, as this would not have been a reversible operation and thus impossible on a quantum computer, but instead we did a relatively complicated sequence of operations which ended with which in fact depended on multiplication being a group (mod n). At this point, then, we could do something somewhat sneaky: we could measure b to see if it actually is 0. If it is not, we know that there has been an error somewhere in the quantum computation, i.e., that the results are worthless and we should stop the computer and start over again. However, if we do find that b is 0, then we know (because we just observed it) that it is now exactly 0. This measurement thus may bring the quantum computation back on track in that any amplitude that b had for being non-zero has been eliminated. Further, because the probability that we observe a state is proportional to the square of the amplitude of that state, depending on the error model, doing the modular exponentiation and measuring b every time that we know that it should be 0 may have a higher probability of overall success than the same computation done without the repeated measurements of b; this is the quantum watchdog (or quantum Zeno) effect [Peres 1993]. The argument above does not actually show that repeated measurement of b is indeed beneficial, because there is a cost (in time, if nothing else) of measuring b. Before this is implemented, then, it should be checked with analysis or experiment that the benefit of such measurements exceeds their cost. However, I believe that partial measurements such as this one are a promising way of trying to stabilize quantum computations. Currently, Sch-onhage-Strassen is the algorithm of choice for multiplying very large numbers, and longhand multiplication is the algorithm of choice for small numbers. There are also multiplication algorithms which have efficiencies between these two al- gorithms, and which are the best algorithms to use for intermediate length numbers [Karatsuba and Ofman 1962, Knuth 1981, Sch-onhage et al. 1994]. It is not clear which algorithms are best for which size numbers. While this may be known to some extent for classical computation [Sch-onhage et al. 1994], using data on which algorithms work better on classical computers could be misleading for two reasons: First, classical computers need not be reversible, and the cost of making an algorithm reversible depends on the algorithm. Second, existing computers generally have multiplication for 32- or 64-bit numbers built into their hardware, and this will increase the optimal changeover FACTORING WITH A QUANTUM COMPUTER 13 points to asymptotically faster algorithms; further, some multiplication algorithms can take better advantage of this hardwired multiplication than others. Thus, in order to program quantum computers most efficiently, work needs to be done on the best way of implementing elementary arithmetic operations on quantum computers. One tantalizing fact is that the Sch-onhage-Strassen fast multiplication algorithm uses the fast Fourier transform, which is also the basis for all the fast algorithms on quantum computers discovered to date; it is tempting to speculate that integer multiplication itself might be speeded up by a quantum algorithm; if possible, this would result in a somewhat faster asymptotic bound for factoring on a quantum computer, and indeed could even make breaking RSA on a quantum computer asymptotically faster than encrypting with RSA on a classical computer. 4 Quantum Fourier transforms Since quantum computation deals with unitary transformations, it is helpful to be able to build certain useful unitary transformations. In this section we give a technique for constructing in polynomial time on quantum computers one particular unitary transfor- mation, which is essentially a discrete Fourier transform. This transformation will be given as a matrix, with both rows and columns indexed by states. These states correspond to binary representations of integers on the computer; in particular, the rows and columns will be indexed beginning with 0 unless otherwise specified. This transformations is as follows. Consider a number a with 0 - a ! q for some q where the number of bits of q is polynomial. We will perform the transformation that takes the state jai to the stateq 1=2 That is, we apply the unitary matrix whose (a; c) entry is 1 exp(2-iac=q). This Fourier transform is at the heart of our algorithms, and we call this matrix A q . Since we will use A q for q of exponential size, we must show how this transformation can be done in polynomial time. In this paper, we will give a simple construction for A q when q is a power of 2 that was discovered independently by Coppersmith [1994] and Deutsch [see Ekert and Jozsa 1995]. This construction is essentially the standard fast Fourier transform (FFT) algorithm [Knuth 1981] adapted for a quantum computer; the following description of it follows that of Ekert and Jozsa [1995]. In the earlier version of this paper [Shor 1994], we gave a construction for A q when q was in the special class of smooth numbers with small prime power factors. In fact, Cleve [1994] has shown how to construct A q for all smooth numbers q whose prime factors are at most O(logn). us represent an integer a in binary as ja l\Gamma1 a For the quantum Fourier transform A q , we only need to use two types of quantum gates. These gates are R j , which operates on the jth bit of the quantum computer: 14 P. W. SHOR and S j;k , which operates on the bits in positions j and k with To perform a quantum Fourier transform, we apply the matrices in the order (from left to right) R that is, we apply the gates R j in reverse order from R l\Gamma1 to R 0 , and between R j+1 and R j we apply all the gates S j;k where k ? j. For example, on 3 bits, the matrices would be applied in the order R 2 S 1;2 R 1 S 0;2 S 0;1 R 0 . To take the Fourier transform A q when thus need to use l(l \Gamma 1)=2 quantum gates. Applying this sequence of transformations will result in a quantum stateq 1=2 b exp(2-iac=q) jbi, where b is the bit-reversal of c, i.e., the binary number obtained by reading the bits of c from right to left. Thus, to obtain the actual quantum Fourier transform, we need either to do further computation to reverse the bits of jbi to obtain jci, or to leave these bits in place and read them in reverse order; either alternative is easy to implement. To show that this operation actually performs a quantum Fourier transform, consider the amplitude of going from First, the factors of 1= 2 in the R matrices multiply to produce a factor of 1=q 1=2 overall; thus we need only worry about the exp(2-iac=q) phase factor in the expression (4.1). The matrices S j;k do not change the values of any bits, but merely change their phases. There is thus only one way to switch the jth bit from a j to b j , and that is to use the appropriate entry in the matrix R j . This entry adds - to the phase if the bits a j and b j are both 1, and leaves it unchanged otherwise. Further, the matrix S j;k adds -=2 k\Gammaj to the phase if a j and b k are both 1 and leaves it unchanged otherwise. Thus, the phase on the path from jai to jbi is X 0-j!l 0-j!k!l This expression can be rewritten as 0-j-k!l Since c is the bit-reversal of b, this expression can be further rewritten as 0-j-k!l Making the substitution l in this sum, we get 0-j+k!l l a j c k (4.8) FACTORING WITH A QUANTUM COMPUTER 15 Now, since adding multiples of 2- do not affect the phase, we obtain the same phase if we sum over all j and k less than l, obtaining where the last equality follows from the distributive law of multiplication. Now, so the above expression is equal to 2-ac=q, which is the phase for the amplitude of jai ! jci in the transformation (4.1). large in the gate S j;k in (4.3), we are multiplying by a very small phase factor. This would be very difficult to do accurately physically, and thus it would be somewhat disturbing if this were necessary for quantum computation. Luckily, Coppersmith [1994] has shown that one can define an approximate Fourier transform that ignores these tiny phase factors, but which approximates the Fourier transform closely enough that it can also be used for factoring. In fact, this technique reduces the number of quantum gates needed for the (approximate) Fourier transform considerably, as it leaves out most of the gates S j;k . 5 Prime factorization It has been known since before Euclid that every integer n is uniquely decomposable into a product of primes. Mathematicians have been interested in the question of how to factor a number into this product of primes for nearly as long. It was only in the 1970's, however, that researchers applied the paradigms of theoretical computer science to number theory, and looked at the asymptotic running times of factoring algorithms [Adleman 1994]. This has resulted in a great improvement in the efficiency of factoring algorithms. The best factoring algorithm asymptotically is currently the number field sieve [Lenstra et al. 1990, Lenstra and Lenstra 1993], which in order to factor an integer takes asymptotic running time exp(c(log n) 1=3 (log log n) 2=3 ) for some constant c. Since the input, n, is only log n bits in length, this algorithm is an exponential-time algorithm. Our quantum factoring algorithm takes asymptotically O((log n) 2 (log log n) (log log log n)) steps on a quantum computer, along with a polynomial (in log n) amount of post-processing time on a classical computer that is used to convert the output of the quantum computer to factors of n. While this post-processing could in principle be done on a quantum computer, there is no reason not to use a classical computer if they are more efficient in practice. Instead of giving a quantum computer algorithm for factoring n directly, we give a quantum computer algorithm for finding the order of an element x in the multiplicative group (mod n); that is, the least integer r such that x r j 1 (mod n). It is known that using randomization, factorization can be reduced to finding the order of an element [Miller 1976]; we now briefly give this reduction. To find a factor of an odd number n, given a method for computing the order r of x, choose a random x (mod n), find its order r, and compute gcd(x gcd(a; b) is the greatest common divisor of a and b, i.e., the largest integer that divides both a and b. The Euclidean algorithm [Knuth 1981] can be used to compute gcd(a; b) in polynomial time. Since n), the gcd(x P. W. SHOR fails to be a non-trivial divisor of n only if r is odd or if x n). Using this criterion, it can be shown that this procedure, when applied to a random x (mod n), yields a factor of n with probability at least 1 \Gamma 1=2 is the number of distinct odd prime factors of n. A brief sketch of the proof of this result follows. Suppose that be the order of x (mod p a i r is the least common multiple of all the r i . Consider the largest power of 2 dividing each r i . The algorithm only fails if all of these powers of 2 agree: if they are all 1, then r is odd and r=2 does not exist; if they are all equal and larger than 1, then x for every i. By the Chinese remainder theorem [Knuth 1981, Hardy and Wright 1979, Theorem 121], choosing an x (mod n) at random is the same as choosing for each i a number x i (mod p a i i ) at random, where p a i i is the ith prime power factor of n. The multiplicative group (mod p ff ) for any odd prime power p ff is cyclic [Knuth 1981], so for any odd prime power p a i i , the probability is at most 1=2 of choosing an x i having any particular power of two as the largest divisor of its order r i . Thus each of these powers of 2 has at most a 50% probability of agreeing with the previous ones, so all k of them agree with probability at most 1=2 k\Gamma1 , and there is at least a chance that the x we choose is good. This scheme will thus work as long as n is odd and not a prime finding factors of prime powers can be done efficiently with classical methods. We now describe the algorithm for finding the order of x (mod n) on a quantum computer. This algorithm will use two quantum registers which hold integers represented in binary. There will also be some amount of workspace. This workspace gets reset to after each subroutine of our algorithm, so we will not include it when we write down the state of our machine. Given x and n, to find the order of x, i.e., the least r such that x r j 1 (mod n), we do the following. First, we find q, the power of 2 with We will not include when we write down the state of our machine, because we never change these values. In a quantum gate array we need not even keep these values in memory, as they can be built into the structure of the gate array. Next, we put the first register in the uniform superposition of states representing numbers a (mod q). This leaves our machine in stateq 1=2 This step is relatively easy, since all it entails is putting each bit in the first register into the superposition 1 Next, we compute x a (mod n) in the second register as described in x3. Since we keep a in the first register this can be done reversibly. This leaves our machine in the stateq 1=2 jai jx a (mod n)i : (5.2) We then perform our Fourier transform A q on the first register, as described in x4, mapping jai toq 1=2 FACTORING WITH A QUANTUM COMPUTER 17 That is, we apply the unitary matrix with the (a; c) entry equal to 1 exp(2-iac=q). This leaves our machine in stateq Finally, we observe the machine. It would be sufficient to observe solely the value of jci in the first register, but for clarity we will assume that we observe both jci and jx a (mod n)i. We now compute the probability that our machine ends in a particular state ff , where we may assume Summing over all possible ways to reach the state ff , we find that this probability is a: x a jx k where the sum is over all a, 0 - a ! q, such that x a j x k (mod n). Because the order of x is r, this sum is over all a satisfying a j k (mod r). Writing a that the above probability is We can ignore the term of exp(2-ikc=q), as it can be factored out of the sum and has magnitude 1. We can also replace rc with frcg q , where frcg q is the residue which is congruent to rc (mod q) and is in the range \Gammaq=2 ! frcg q - q=2. This leaves us with the expression fi fi fi fi fi fiq exp(2-ibfrcg q =q) We will now show that if frcg q is small enough, all the amplitudes in this sum will be in nearly the same direction (i.e., have close to the same phase), and thus make the sum large. Turning the sum into an integral, we obtainq r cexp(2-ibfrcg q =q)db If jfrcg q j - r=2, the error term in the above expression is easily seen to be bounded by O(1=q). We now show that if jfrcg q j - r=2, the above integral is large, so the probability of obtaining a state ff is large. Note that this condition depends only on c and is independent of k. Substituting in the above integral, we getr Z r r cexp r u du: approximating the upper limit of integration by 1 results in only a O(1=q) error in the above expression. If we do this, we obtain the integralr Z 1exp r u du: (5.10) P. W. SHOR0.020.060.100 c Figure 5.1: The probability P of observing values of c between 0 and 255, given and Letting frcg q =r vary between \Gamma 1and 1, the absolute magnitude of the integral (5.10) is easily seen to be minimized when frcg q which case the absolute value of expression (5.10) is 2=(-r). The square of this quantity is a lower bound on the probability that we see any particular state ff with frcg q - r=2; this probability is thus asymptotically bounded below by 4=(- 2 r 2 ), and so is at least 1=3r 2 for sufficiently large n. The probability of seeing a given state ff will thus be at least 1=3r 2 if \Gammar i.e., if there is a d such that \Gammar Dividing by rq and rearranging the terms gives r We know c and q. Because q ? n 2 , there is at most one fraction d=r with r ! n that satisfies the above inequality. Thus, we can obtain the fraction d=r in lowest terms by rounding c=q to the nearest fraction having a denominator smaller than n. This fraction can be found in polynomial time by using a continued fraction expansion of c=q, which FACTORING WITH A QUANTUM COMPUTER 19 finds all the best approximations of c=q by fractions [Hardy and Wright 1979, Chapter X, Knuth 1981]. The exact probabilities as given by equation (5.7) for an example case with and are plotted in Figure 5.1. The value could occur when factoring 33 if x were chosen to be 5, for example. Here q is taken smaller than 33 2 so as to make the values of c in the plot distinguishable; this does not change the functional structure of P(c). Note that with high probability the observed value of c is near an integral multiple of If we have the fraction d=r in lowest terms, and if d happens to be relatively prime to r, this will give us r. We will now count the number of states ff which enable us to compute r in this way. There are OE(r) possible values of d relatively prime to r, where OE is Euler's totient function [Knuth 1981, Hardy and Wright 1979, x5.5]. Each of these fractions d=r is close to one fraction c=q with There are also r possible values for x k , since r is the order of x. Thus, there are rOE(r) states ff which would enable us to obtain r. Since each of these states occurs with probability at least 1=3r 2 , we obtain r with probability at least OE(r)=3r. Using the theorem that OE(r)=r ? log r for some constant ffi [Hardy and Wright 1979, Theorem 328], this shows that we find r at least a fraction of the time, so by repeating this experiment only O(log log r) times, we are assured of a high probability of success. In practice, assuming that quantum computation is more expensive than classical computation, it would be worthwhile to alter the above algorithm so as to perform less quantum computation and more postprocessing. First, if the observed state is jci, it would be wise to also try numbers close to c such as c \Sigma 1, c \Sigma since these also have a reasonable chance of being close to a fraction qd=r. Second, if c=q - d=r, and d and r have a common factor, it is likely to be small. Thus, if the observed value of c=q is rounded off to d 0 =r 0 in lowest terms, for a candidate r one should consider not only r 0 but also its small multiples 2r 0 , 3r 0 , . , to see if these are the actual order of x. Although the first technique will only reduce the expected number of trials required to find r by a constant factor, the second technique will reduce the expected number of trials for the hardest n from O(log log n) to O(1) if the first (log n) 1+ffl multiples of r 0 are considered [Odylzko 1995]. A third technique is, if two candidate r's have been found, say r 1 and r 2 , to test the least common multiple of r 1 and r 2 as a candidate r. This third technique is also able to reduce the expected number of trials to a constant [Knill 1995], and will also work in some cases where the first two techniques fail. Note that in this algorithm for determining the order of an element, we did not use many of the properties of multiplication (mod n). In fact, if we have a permutation f mapping the set f0; itself such that its kth iterate, f (k) (a), is computable in time polynomial in log n and log k, the same algorithm will be able to find the order of an element a under f , i.e., the minimum r such that f (r) (a) = a. 6 Discrete logarithms For every prime p, the multiplicative group (mod p) is cyclic, that is, there are generators g such that 1, g, g 2 , . , g p\Gamma2 comprise all the non-zero residues (mod p) [Hardy and Wright 1979, Theorem 111, Knuth 1981]. Suppose we are given a prime p and such P. W. SHOR a generator g. The discrete logarithm of a number x with respect to p and g is the integer r with p). The fastest algorithm known for finding discrete logarithms modulo arbitrary primes p is Gordon's [1993] adaptation of the number field sieve, which runs in time exp(O(log p) 1=3 (log log p) 2=3 )). We show how to find discrete logarithms on a quantum computer with two modular exponentiations and two quantum Fourier transforms. This algorithm will use three quantum registers. We first find q a power of 2 such that q is close to p, i.e., with Next, we put the first two registers in our quantum computer in the uniform superposition of all jai and jbi (mod compute g a x \Gammab (mod p) in the third register. This leaves our machine in the statep \Gamma 1 a x \Gammab (mod p) As before, we use the Fourier transform A q to send jai ! jci and jbi ! jdi with probability amplitude 1 q exp(2-i(ac+bd)=q). This is, we take the state ja; bi to the stateq exp This leaves our quantum computer in the state(p \Gamma 1)q c;d=0 exp a x \Gammab (mod p) Finally, we observe the state of the quantum computer. The probability of observing a state jc; d; yi with y j g k (mod p) is a;b a\Gammarbjk exp where the sum is over all (a; b) such that a \Gamma rb 1). Note that we now have two moduli to deal with, q. While this makes keeping track of things more confusing, it does not pose serious problems. We now use the relation and substitute (6.5) in the expression (6.4) to obtain the amplitude on ff , which exp The absolute value of the square of this amplitude is the probability of observing the state ff . We will now analyze the expression (6.6). First, a factor of FACTORING WITH A QUANTUM COMPUTER 21 exp(2-ikc=q) can be taken out of all the terms and ignored, because it does not change the probability. Next, we split the exponent into two parts and factor out b to obtain(p \Gamma 1)q exp exp where and Here by fzg q we mean the residue of z (mod q) with \Gammaq=2 ! fzg q - q=2, as in equation (5.7). We next classify possible outputs (observed states) of the quantum computer into "good" and "bad." We will show that if we get enough "good" outputs, then we will likely be able to deduce r, and that furthermore, the chance of getting a "good" output is constant. The idea is that if where j is the closest integer to T=q, then as b varies between 0 and 2, the phase of the first exponential term in equation (6.7) only varies over at most half of the unit circle. Further, if then jV j is always at most q=12, so the phase of the second exponential term in equation (6.7) never is farther than exp(-i=6) from 1. If conditions (6.10) and (6.11) both hold, we will say that an output is "good." We will show that if both conditions hold, then the contribution to the probability from the corresponding term is significant. Furthermore, both conditions will hold with constant probability, and a reasonable sample of c's for which condition (6.10) holds will allow us to deduce r. We now give a lower bound on the probability of each good output, i.e., an output that satisfies conditions (6.10) and (6.11). We know that as b ranges from 0 to the phase of exp(2-ibT=q) ranges from 0 to 2-iW where and j is as in equation (6.10). Thus, the component of the amplitude of the first exponential in the summand of (6.7) in the direction is at least cos(2- jW=2 \Gamma W b=(p \Gamma 2)j). By condition (6.11), the phase can vary by at most -i=6 due to the second exponential exp(2-iV=q). Applying this variation in the manner that minimizes the component in the direction (6.13), we get that the component in this direction is at least 22 P. W. SHOR Thus we get that the absolute value of the amplitude (6.7) is at least(p \Gamma 1)q cos Replacing this sum with an integral, we get that the absolute value of this amplitude is at leastq From condition (6.10), jW j - 1, so the error term is O( 1 pq ). As W varies between \Gamma 1and 1, the integral (6.16) is minimized when jW 1. Thus, the probability of arriving at a state jc; d; yi that satisfies both conditions (6.10) and (6.11) is at least Z 2-=3 cos u du or at least :054=q 2 ? 1=(20q 2 ). We will now count the number of pairs (c; d) satisfying conditions (6.10) and (6.11). The number of pairs (c; d) such that (6.10) holds is exactly the number of possible c's, since for every c there is exactly one d such that (6.10) holds. Unless gcd(p \Gamma 1; q) is large, the number of c's for which (6.11) holds is approximately q=6, and even if it is large, this number is at least q=12. Thus, there are at least q=12 pairs (c; d) satisfying both conditions. Multiplying by which is the number of possible y's, gives approximately pq=12 good states jc; d; yi. Combining this calculation with the lower bound 1=(20q 2 ) on the probability of observing each good state gives us that the probability of observing some good state is at least p=(240q), or at least 1=480 (since q ! 2p). Note that each good c has a probability of at least (p of being observed, since there of y and one value of d with which c can make a good state jc; d; yi. We now want to recover r from a pair c; d such that (mod 1); (6.18) where this equation was obtained from condition (6.10) by dividing by q. The first thing to notice is that the multiplier on r is a fraction with denominator evenly divides . Thus, we need only round d=q off to the nearest multiple of by the integer to find a candidate r. To show that the quantum calculation need only be repeated a polynomial number of times to find the correct r requires only a few more details. The problem is that we cannot divide by a number c 0 which is not relatively prime to p \Gamma 1. For the discrete log algorithm, we do not know that all possible values of c 0 are generated with reasonable likelihood; we only know this about one-twelfth of them. This additional difficulty makes the next step harder than the corresponding step in the FACTORING WITH A QUANTUM COMPUTER 23 algorithm for factoring. If we knew the remainder of r modulo all prime powers dividing could use the Chinese remainder theorem to recover r in polynomial time. We will only be able to prove that we can find this remainder for primes larger than 18, but with a little extra work we will still be able to recover r. Recall that each good (c; d) pair is generated with probability at least 1=(20q 2 ), and that at least a twelfth of the possible c's are in a good (c; d) pair. From equation (6.19), it follows that these c's are mapped from c=q to c 0 by rounding to the nearest integral multiple of 1=(p \Gamma 1). Further, the good c's are exactly those in which c=q is close to c 0 =(p \Gamma 1). Thus, each good c corresponds with exactly one c 0 . We would like to show that for any prime power p ff i dividing is unlikely to contain p i . If we are willing to accept a large constant for our algorithm, we can just ignore the prime powers under 18; if we know r modulo all prime powers over 18, we can try all possible residues for primes under with only a (large) constant factor increase in running time. Because at least one twelfth of the c's were in a good (c; d) pair, at least one twelfth of the c 0 's are good. Thus, for a prime power p ff i i , a random good c 0 is divisible by p ff i with probability at most 12=p ff i . If we have t good c 0 's, the probability of having a prime power over that divides all of them is therefore at most where ajb means that a evenly divides b, so the sum is over all prime powers greater than goes down by at least a factor of 2=3 for each further increase of t by 1; thus for some constant t it is less than 1=2. Recall that each good c 0 is obtained with probability at least 1=(40q) from any experiment. Since there are q=12 good c 0 's, after 480t experiments, we are likely to obtain a sample of t good c 0 's chosen equally likely from all good c 0 's. Thus, we will be able to find a set of c 0 's such that all prime powers p ff i are relatively prime to at least one of these c 0 's. To obtain a polynomial time algorithm, all one need do is try all possible sets of c 0 's of size t; in practice, one would use an algorithm to find sets of c 0 's with large common factors. This set gives the residue of r for all primes larger than 18. For each prime p i less than 18, we have at most possibilities for the residue modulo p ff i is the exponent on prime p i in the prime factorization of 1. We can thus try all possibilities for residues modulo powers of primes less than 18: for each possibility we can calculate the corresponding r using the Chinese remainder theorem and then check to see whether it is the desired discrete logarithm. If one were to actually program this algorithm there are many ways in which the efficiency could be increased over the efficiency shown in this paper. For example, the estimate for the number of good c 0 's is likely too low, especially since weaker conditions than (6.10) and (6.11) should suffice. This means that the number of times the experiment need be run could be reduced. It also seems improbable that the distribution of bad values of c 0 would have any relationship to primes under 18; if this is true, we need not treat small prime powers separately. This algorithm does not use very many properties of Z p , so we can use the same algorithm to find discrete logarithms over other fields such as Z p ff , as long as the field P. W. SHOR has a cyclic multiplicative group. All we need is that we know the order of the generator, and that we can multiply and take inverses of elements in polynomial time. The order of the generator could in fact be computed using the quantum order-finding algorithm given in x5 of this paper. Boneh and Lipton [1995] have generalized the algorithm so as to be able to find discrete logarithms when the group is abelian but not cyclic. 7 Comments and open problems It is currently believed that the most difficult aspect of building an actual quantum computer will be dealing with the problems of imprecision and decoherence. It was shown by Bennett et al. [1994] that the quantum gates need only have precision O(1=t) in order to have a reasonable probability of completing t steps of quantum computation; that is, there is a c such that if the amplitudes in the unitary matrices representing the quantum gates are all perturbed by at most c=t, the quantum computer will still have a reasonable chance of producing the desired output. Similarly, the decoherence needs to be only polynomially small in t in order to have a reasonable probability of completing t steps of computation successfully. This holds not only for the simple model of decoherence where each bit has a fixed probability of decohering at each time step, but also for more complicated models of decoherence which are derived from fundamental quantum mechanical considerations [Unruh 1995, Palma et al. 1995, Chuang et al. 1995]. How- ever, building quantum computers with high enough precision and low enough decoherence to accurately perform long computations may present formidable difficulties to experimental physicists. In classical computers, error probabilities can be reduced not only though hardware but also through software, by the use of redundancy and error-correcting codes. The most obvious method of using redundancy in quantum computers is ruled out by the theorem that quantum bits cannot be cloned [Peres 1993, x9-4], but this argument does not rule out more complicated ways of reducing inaccuracy or decoherence using software. In fact, some progress in the direction of reducing inaccuracy [Berthiaume et al. 1994] and decoherence [Shor 1995] has already been made. The result of Bennett et al. [1995] that quantum bits can be faithfully transmitted over a noisy quantum channel gives further hope that quantum computations can similarly be faithfully carried out using noisy quantum bits and noisy quantum gates. Discrete logarithms and factoring are not in themselves widely useful problems. They have only become useful because they have been found to be crucial for public-key cryp- tography, and this application is in turn possible only because they have been presumed to be difficult. This is also true of the generalizations of Boneh and Lipton [1995] of these algorithms. If the only uses of quantum computation remain discrete logarithms and factoring, it will likely become a special-purpose technique whose only raison d'-etre is to thwart public key cryptosystems. However, there may be other hard problems which could be solved asymptotically faster with quantum computers. In particular, of interesting problems not known to be NP-complete, the problem of finding a short vector in a lattice [Adleman 1994, Adleman and McCurley 1995] seems as if it might potentially be amenable to solution by a quantum computer. In the history of computer science, however, most important problems have turned out to be either polynomial-time or NP-complete. Thus quantum computers will likely not become widely useful unless they can solve NP-complete problems. Solving NP- FACTORING WITH A QUANTUM COMPUTER 25 complete problems efficiently is a Holy Grail of theoretical computer science which very few people expect to be possible on a classical computer. Finding polynomial-time algorithms for solving these problems on a quantum computer would be a momentous discovery. There are some weak indications that quantum computers are not powerful enough to solve NP-complete problems [Bennett et al. 1994], but I do not believe that this potentiality should be ruled out as yet. Acknowledgements I would like to thank Jeff Lagarias for finding and fixing a critical error in the first version of the discrete log algorithm. 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Papageorgiou , H. Woniakowski, Classical and Quantum Complexity of the Sturm--Liouville Eigenvalue Problem, Quantum Information Processing, v.4 n.2, p.87-127, June 2005 Harry Buhrman , Richard Cleve , Avi Wigderson, Quantum vs. classical communication and computation, Proceedings of the thirtieth annual ACM symposium on Theory of computing, p.63-68, May 24-26, 1998, Dallas, Texas, United States Andris Ambainis, A new protocol and lower bounds for quantum coin flipping, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.134-142, July 2001, Hersonissos, Greece Cristopher Moore , Daniel Rockmore , Alexander Russell, Generic quantum Fourier transforms, ACM Transactions on Algorithms (TALG), v.2 n.4, p.707-723, October 2006 Scott Aaronson, Quantum lower bound for the collision problem, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada Andrew M. Childs , Richard Cleve , Enrico Deotto , Edward Farhi , Sam Gutmann , Daniel A. Spielman, Exponential algorithmic speedup by a quantum walk, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA Cristopher Moore , Daniel Rockmore , Alexander Russell, Generic quantum Fourier transforms, Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, January 11-14, 2004, New Orleans, Louisiana Amr Sabry, Modeling quantum computing in Haskell, Proceedings of the ACM SIGPLAN workshop on Haskell, p.39-49, August 28-28, 2003, Uppsala, Sweden A. 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Chong , Isaac L. Chuang , John Kubiatowicz, Building quantum wires: the long and the short of it, ACM SIGARCH Computer Architecture News, v.31 n.2, May Holger Spakowski , Mayur Thakur , Rahul Tripathi, Quantum and classical complexity classes: separations, collapses, and closure properties, Information and Computation, v.200 n.1, p.1-34, 1 July 2005 An introduction to quantum computing for non-physicists, ACM Computing Surveys (CSUR), v.32 n.3, p.300-335, Sept. 2000 Robert Beals , Harry Buhrman , Richard Cleve , Michele Mosca , Ronald de Wolf, Quantum lower bounds by polynomials, Journal of the ACM (JACM), v.48 n.4, p.778-797, July 2001 R. Srikanth, A Computational Model for Quantum Measurement, Quantum Information Processing, v.2 n.3, p.153-199, June Dagmar Bruss , Gbor Erdlyi , Tim Meyer , Tobias Riege , Jrg Rothe, Quantum cryptography: A survey, ACM Computing Surveys (CSUR), v.39 n.2, p.6-es, 2007 Scott Aaronson, Guest Column: NP-complete problems and physical reality, ACM SIGACT News, v.36 n.1, March 2005 Andrew Odlyzko, Discrete Logarithms: The Past and the Future, Designs, Codes and Cryptography, v.19 n.2-3, p.129-145, March 2000 David S. Johnson, The NP-completeness column, ACM Transactions on Algorithms (TALG), v.1 n.1, p.160-176, July 2005 Jrg Rothe, Some facets of complexity theory and cryptography: A five-lecture tutorial, ACM Computing Surveys (CSUR), v.34 n.4, p.504-549, December 2002 Rodney Van Meter , Mark Oskin, Architectural implications of quantum computing technologies, ACM Journal on Emerging Technologies in Computing Systems (JETC), v.2 n.1, p.31-63, January 2006

spin systems;quantum computers;algorithmic number theory;church's thesis;fourier transforms;foundations of quantum mechanics;prime factorization;discrete logarithms

264407

Strengths and Weaknesses of Quantum Computing.

Recently a great deal of attention has been focused on quantum computation following a sequence of results [Bernstein and Vazirani, in Proc. 25th Annual ACM Symposium Theory Comput., 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339], [Simon, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 116--123, SIAM J. Comput., 26 (1997), pp. 1340--1349], [Shor, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 124--134] suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of $\NP$ can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$. We also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class $\NP \cap \coNP$ cannot be solved on a QTM in time $o(2^{n/3})$. The former bound is tight since recent work of Grover [in {\it Proc.\ $28$th Annual ACM Symposium Theory Comput.}, 1996] shows how to accept the class $\NP$ relative to any oracle on a quantum computer in time $O(2^{n/2})$.

Introduction Quantum computational complexity is an exciting new area that touches upon the foundations of both theoretical computer science and quantum physics. In the early eighties, Feynman [12] pointed out that straightforward simulations of quantum mechanics on a classical computer appear to require a simulation overhead that is exponential in the size of the system and the simulated time; he asked whether this is inherent, and whether it is possible to design a universal quantum computer. Deutsch [9] defined a general model of quantum computation - the quantum Turing machine. Bernstein and Vazirani [4] proved that there is an efficient universal quantum Turing machine. Yao [17] extended this by proving that quantum circuits (introduced by Deutsch [10]) are polynomially equivalent to quantum Turing machines. The computational power of quantum Turing machines (QTMs) has been explored by several researchers. Early work by Deutsch and Jozsa [11] showed how to exploit some inherently quantum mechanical features of QTMs. Their results, in conjunction with subsequent results by Berthiaume and Brassard [5, 6], established the existence of oracles under which there are computational problems that QTMs can solve in polynomial time with cer- tainty, whereas if we require a classical probabilistic Turing machine to produce the correct answer with certainty, then it must take exponential time on some inputs. On the other hand, these computational problems are in BPP 1 relative to the same oracle, and therefore efficiently solvable in the classical sense. The quantum analogue of the class BPP is 1 BPP is the class of decision problems (languages) that can be solved in polynomial time by probabilistic Turing machines with error probability bounded by 1/3 (for all inputs). Using standard boosting techniques, the error probability can then be made exponentially small in k by iterating the algorithm k times and returning the majority answer. the class BQP 2 [5]. Bernstein and Vazirani [4] proved that BPP ' BQP ' PSPACE, thus establishing that it will not be possible to conclusively prove that BQP 6= BPP without resolving the major open problem P ? PSPACE. They also gave the first evidence that BQP 6= BPP (polynomial-time quantum Turing machines are more powerful than polynomial-time probabilistic Turing machines), by proving the existence of an oracle relative to which there are problems in BQP that cannot be solved with small error probability by probabilistic machines restricted to running in n o(log n) steps. Since BPP is regarded as the class of all "efficiently computable" languages (computational problems), this provided evidence that quantum computers are inherently more powerful than classical computers in a model-independent way. Simon [16] strengthened this evidence by proving the existence of an oracle relative to which BQP cannot even be simulated by probabilistic machines allowed to run for 2 n=2 steps. In addition, Simon's paper also introduced an important new technique which was one of the ingredients in a remarkable result proved subsequently by Shor [15]. Shor gave polynomial-time quantum algorithms for the factoring and discrete logarithm problems. These two problems have been well-studied, and their presumed intractability forms the basis of much of modern cryptography. In view of these results, it is natural to ask whether NP ' BQP; i.e. can quantum computers solve NP-complete problems in polynomial time? 3 In this paper, we address this question by proving that relative to an oracle chosen uniformly at random [3], with probability 1, the class NP cannot be solved on a quantum 2 BQP is the class of decision problems (languages) that can be solved in polynomial time by quantum Turing machines with error probability bounded by 1/3 (for all inputs)-see [4] for a formal definition. We prove in Section 4 of this paper that, as is the case with BPP, the error probability of BQP machines can be made exponentially small. 3 Actually it is not even clear whether BQP ' BPP NP ; i.e. it is unclear whether nondeterminism together with randomness is sufficient to simulate quantum Turing machines. In fact, Bernstein and Vazi- rani's [4] result is stronger than stated above. They actually proved that relative to an oracle, the recursive Fourier sampling problem can be solved in BQP, but cannot even be solved by Arthur-Merlin games [1] with a time bound of n o(log n) (thus giving evidence that nondeterminism on top of probabilism does not help). They conjecture that the recursive Fourier sampling cannot even be solved in the unrelativized polynomial-time hierarchy. Turing machine in time o(2 n=2 ). We also show that relative to a permutation oracle chosen uniformly at random, with probability 1, the class NP " co-NP cannot be solved on a quantum Turing machine in time o(2 n=3 ). The former bound is tight since recent work of Grover [13] shows how to accept the class NP relative to any oracle on a quantum computer in time O(2 n=2 ). See [7] for a detailed analysis of Grover's algorithm. What is the relevance of these oracle results? We should emphasize that they do not rule out the possibility that NP ' BQP. What these results do establish is that there is no black-box approach to solving NP-complete problems by using some uniquely quantum-mechanical features of QTMs. That this was a real possibility is clear from Grover's [13] result, which gives a black-box approach to solving NP-complete problems in square-root as much time as is required classically. One way to think of an oracle is as a special subroutine call whose invocation only costs unit time. In the context of QTMs, subroutine calls pose a special problem that has no classical counterpart. The problem is that the subroutine must not leave around any bits beyond its computed answer, because otherwise computational paths with different residual information do not interfere. This is easily achieved for deterministic subroutines since any classical deterministic computation can be carried out reversibly so that only the input and the answer remain. However, this leaves open the more general question of whether a BQP machine can be used as a subroutine. Our final result in this paper is to show how any BQP machine can be modified into a tidy BQP machine whose final superposition consists almost entirely of a tape configuration containing just the input and the single bit answer. Since these tidy BQP machines can be safely used as subroutines, this allows us to show that BQP BQP. The result also justifies the definition of oracle quantum machines that we now give. Oracle Quantum Turing Machines In this section and the next, we shall assume without loss of generality that the Turing machine alphabet (for each track or tape) is f0; 1; #g, where "#" denotes the blank symbol. Initially all tapes are blank except that the input tape contains the actual input surrounded by blanks. We shall use \Sigma to denote f0; 1g. In the classical setting, an oracle may be described informally as a device for evaluating some Boolean function A : \Sigma ! \Sigma, on arbitrary arguments, at unit cost per evaluation. This allows to formulate questions such as "if A were efficiently computable by a Turing machine, which other functions (or languages) could be efficiently computed by Turing machines?". In the quantum setting, an equivalent question can be asked, provided we define oracle quantum Turing machines appropriately-which we do in this section-and Turing machines can be composed-which we show in Section 4 of this paper. An oracle QTM has a special query tape (or track), all of whose cells are blank except for a single block of non-blank cells. In a well-formed oracle QTM, the Turing machine rules may allow this region to grow and shrink, but prevent it from fragmenting into non-contiguous blocks. 4 Oracle QTMs have two distinguished internal states: a pre-query state q q and a post-query state q a . A query is executed whenever the machine enters the pre-query state. If the query string is empty, a no-op occurs, and the machine passes directly to the post- query state with no change. If the query string is nonempty, it can be written in the form denotes concatenation. In that case, the result of a call on oracle A is that internal control passes to the post-query state while the contents of 4 This restriction can be made without loss of generality and it can be verified syntactically by allowing only machines that make sure they do not break the rule before writing on the query tape. the query tape changes from jx ffi bi to jx ffi (b \Phi A(x))i, where "\Phi" denotes the exclusive-or (addition modulo 2). Except for the query tape and internal control, other parts of the oracle QTM do not change during the query. If the target bit jbi is supplied in initial state j0i, then its final state will be jA(x)i, just as in a classical oracle machine. Conversely, if the target bit is already in state jA(x)i, calling the oracle will reset it to j0i. This ability to "uncompute" will often prove essential to allow proper interference among computation paths to take place. Using this fact, it is also easy to see that the above definition of oracle Turing machines yields unitary evolutions if we restrict ourselves to machines that are well-formed in other respects, in particular evolving unitarily as they enter the pre-query state and leave the post-query state. The power of quantum computers comes from their ability to follow a coherent superposition of computation paths. Similarly oracle quantum machines derive great power from the ability to perform superpositions of queries. For example, oracle A might be called when the query tape is in state j/ ffi x ff x jx ffi 0i, where ff x are complex coef- ficients, corresponding to an arbitrary superposition of queries with a constant j0i in the target bit. In this case, after the query, the query string will be left in the entangled state x ff x jx ffi A(x)i. It is also useful to be able to put the target bit b into a superposition. For example, the conditional phase inversion used in Grover's algorithm can be achieved by performing queries with the target bit b in the nonclassical superposition 2. It can readily be verified that an oracle call with the query tape in state x ffi fi leaves the entire machine state, including the query tape, unchanged if leaves the entire state unchanged while introducing a phase factor \Gamma1 if It is often convenient to think of a Boolean oracle as defining a length-preserving function on \Sigma . This is easily accomplished by interpreting the oracle answer on the pair (x; i) as the i th bit of the function value. The pair (x; i) is encoded as a binary string using any standard pairing function. A permutation oracle is an oracle which, when interpreted as a length-preserving function, acts for each n - 0 as a permutation on \Sigma n . Henceforth, when no confusion may arise, we shall use A(x) to denote the length-preserving function associated with oracle A rather than the Boolean function that gives rise to it. Let us define BQTime(T (n)) A as the sets of languages accepted with probability at least 2=3 by some oracle QTM M A whose running time is bounded by T (n). This bound on the running time applies to each individual input, not just on the average. Notice that whether or not M A is a BQP-machine might depend upon the oracle A-thus M A might be a BQP-machine while M B might not be one. Note: The above definition of a quantum oracle for an arbitrary Boolean function will suffice for the purposes of the present paper, but the ability of quantum computers to perform general unitary transformations suggests a broader definition, which may be useful in other contexts. For example, oracles that perform more general, non-Boolean unitary operations have been considered in computational learning theory [8] and for hiding information against classical queries [14]. Most broadly, a quantum oracle may be defined as a device that, when called, applies a fixed unitary transformation U to the current contents jzi of the query tape, replacing it by U jzi. Such an oracle U must be defined on a countably infinite-dimensional Hilbert space, such as that spanned by the binary basis vectors jffli; j0i; j1i; j00i; j01i; j10i; where ffl denotes the empty string. Clearly, the use of such general unitary oracles still yields unitary evolution for well-formed oracle Turing machines. Naturally, these oracles can map inputs onto superpositions of outputs, and vice versa, and they need not even be length-preserving. However, in order to obey the dictum that a single machine cycle ought not to make infinite changes in the tape, one might require that U jzi have amplitude zero on all but finitely many basis vectors. (One could even insist on a uniform and effective version of the above restriction.) Another natural restriction one may wish to impose upon U is that it be an involution, U so that the effect of an oracle call can be undone by a further call on the same oracle. Again this may be crucial to allow proper interference to take place. Note that the special case of unitary transformation considered in this paper, which corresponds to evaluating a classical Boolean function, is an involution. 3 On the Difficulty of Simulating Nondeterminism on QTMs The computational power of QTMs lies in their ability to maintain and compute with exponentially large superpositions. It is tempting to try to use this "exponential parallelism" to simulate non-determinism. However, there are inherent constraints on the scope of this parallelism, which are imposed by the formalism of quantum mechanics. 5 In this section, we explore some of these constraints. To see why quantum interference can speed up NP problems quadratically but not exponentially, consider the problem of distinguishing the empty oracle an oracle containing a single random unknown string y of known length n (i.e. A(y)=1, but 8 x6=y A(x)=0). We require that the computer never answer yes on an empty oracle, and seek to maximize its "success probability" of answering yes on a nonempty oracle. A classical computer can do no better than to query distinct n-bit strings at random, giving a success probability 1=2 n after one query and k=2 n after k queries. How can a quantum computer do 5 There is a superficial similarity between this exponential parallelism in quantum computation and the fact that probabilistic computations yield probability distributions over exponentially large domains. The difference is that in the probabilistic case, the computational path is chosen by making a sequence of random choices-one for each step. In the quantum-mechanical case, it is possible for several computational paths to interfere destructively, and therefore it is necessary to keep track of the entire superposition at each step to accurately simulate the system. better, while respecting the rule that its overall evolution be unitary, and, in a computation with a nonempty oracle, all computation paths querying empty locations evolve exactly as they would for an empty oracle? A direct quantum analog of the classical algorithm would start in an equally-weighted superposition of 2 n computation paths, query a different string on each path, and finally collapse the superposition by asking whether the query had found the nonempty location. This yields a success probability 1=2 n , the same as the classical computer. However, this is not the best way to exploit quantum parallelism. Our goal should be to maximize the separation between the state vector j/ k i after k interactions with an empty oracle, and the state vector j/ k (y)i after k interactions with an oracle nonempty at an unknown location y. Starting with a uniform superposition x it is easily seen that the separation after one query is maximized by a unitary evolution to x This is a phase inversion of the term corresponding to the nonempty location. By testing whether the post-query state agrees with j/ 0 i we obtain a success probability approximately four times better than the classical value. Thus, if we are allowed only one query, quantum parallelism gives a modest improvement, but is still overwhelmingly likely to fail because the state vector after interaction with a nonempty oracle is almost the same as after interaction with an empty oracle. The only way of producing a large difference after one query would be to concentrate much of the initial superposition in the y term before the query, which cannot be done because that location is unknown. Having achieved the maximum separation after one query, how best can that separation be increased by subsequent queries? Various strategies can be imagined, but a good one (called "inversion about the average" by Grover [13]) is to perform an oracle-independent unitary transformation so as to change the phase difference into an amplitude difference, leaving the y term with the same sign as all the other terms but a magnitude approximately threefold larger. Subsequent phase-inverting interactions with the oracle, alternating with oracle-independent phase-to-amplitude conversions, cause the distance between j/ 0 i and j/ k (y)i to grow linearly with k, approximately as 2k= N=2. This results in a quadratic growth of the success probability, approximately as 4k 2 =2 n for small k. The proof of Theorem 3.5 shows that this approach is essentially optimal: no quantum algorithm can gain more than this quadratic factor in success probability compared to classical algorithms, when attempting to answer NP-type questions formulated relative to a random oracle. 3.1 Lower Bounds on Quantum Search We will sometimes find it convenient to measure the accuracy of a simulation by calculating the Euclidean distance 6 between the target and simulation superpositions. The following theorem from [4] shows that the simulation accuracy is at most 4 times worse than this Euclidean distance. Theorem 3.1 If two unit-length superpositions are within Euclidean distance " then observing the two superpositions gives samples from distributions which are within total variation distance 7 at most 4". 6 The Euclidean distance between x ff x jxi and x fijxi is defined as ( x 7 The total variation distance between two distributions D and D 0 is x Definition 3.2 Let jOE i i be the superposition of M A on input x at time i. We denote by the sum of squared magnitudes in jOE i i of configurations of M which are querying the oracle on string y. We refer to q y (jOE i i) as the query magnitude of y in jOE i i. Theorem 3.3 Let jOE i i be the superposition of M A on input x at time i. be a set of time-strings pairs such that T . Now suppose the answer to each query (i; y) 2 F is modified to some arbitrary fixed a i;y (these answers need not be consistent with an oracle). Let jOE 0 i be the time i superposition of M on input x with oracle A modified as stated above. Then jjOE ". Proof. Let U be the unitary time evolution operator of M A . Let A i denote an oracle such that if (i; y) 2 F then A i be the unitary time evolution operator of M A i . Let jOE i i be the superposition of M A on input x at time i. We define jE i i to be the error in the i th step caused by replacing the oracle A with A i . Then So we have Since all of the U i are unitary, jU T The sum of squared magnitudes of all of the E i is equal to (i;y)2F q y (jOE i i) and therefore at most " 2 In the worst case, the U T could interfere constructively; however, the squared magnitude of their sum is at most T times the sum of their squared magnitudes, Corollary 3.4 Let A be an oracle over alphabet \Sigma. For y 2 \Sigma , let A y be any oracle such that 8x 6= y A y be the time i superposition of M A on input x and be the time i superposition of M Ay on input x. Then for every " ? 0, there is a set S of cardinality at most 2T 2 Proof. Since each jOE t i has unit length, y q y (jOE i i) - T . Let S be the set of strings y such that . . Therefore by Theorem 3.3 ".Theorem 3.5 For any T (n) which is o(2 n=2 ), relative to a random oracle, with probability does not contain NP. Proof. Recall from Section 2 that an oracle can be thought of as a length-preserving function: this is what we mean below by A(x). Let yg. Clearly, this language is contained in NP A . Let T We show that for any bounded-error oracle QTM M A running in time at most T (n), with probability 1, M A does not accept the language LA . The probability is taken over the choice of a random length-preserving oracle A. Then, since there are a countable number of QTMs and the intersection of a countable number of probability 1 events still has probability 1, we conclude that with probability 1, no bounded error oracle QTM accepts LA in time bounded by T (n). pick n large enough so that T (n) - 2 n=2. We will show that the probability that M gives the wrong answer on input 1 n is at least 1=8 for every way of fixing the oracle answers on inputs of length not equal to n. The probability is taken over the random choices of the oracle for inputs of length n. Let us fix an arbitrary length-preserving function from strings of lengths other than n over alphabet \Sigma. Let C denote the set of oracles consistent with this arbitrary function. Let A be the set of oracles in C such that 1 n has no inverse (does not belong to LA ). If the oracle answers to length n strings are chosen uniformly at random, then the probability that the oracle is in A is at least 1=4. This is because the probability that 1 n has no inverse is which is at least 1=4 (for n sufficiently large). Let B be the set of oracles in C such that 1 n has a unique inverse. As above, the probability that a randomly chosen oracle is in B is ( which is at least 1=e. Given an oracle A in A, we can modify its answer on any single input, say y, to 1 n and therefore get an oracle A y in B. We will show that for most choices of y, the acceptance probability of M A on input 1 n is almost equal to the acceptance probability of M Ay on input 1 n . On the other hand, M A must reject 1 n and M Ay must accept 1 n . Therefore M cannot accept both LA and LAy . By working through the details more carefully, it is easy to show that M fails on input 1 n with probability at least 1=8 when the oracle is a uniformly random function on strings of length n, and is an arbitrary function on all other strings. Let A y be the oracle such that A y 3.4 there is a set S of at most 338T 2 (n) strings such that the difference between the th superposition of M Ay on input 1 n and M A on input 1 n has norm at most 1=13. Using Theorem 3.1 we can conclude that the difference between the acceptance probabilities of M Ay on input 1 n and M A on input 1 n is at most 1=13 \Theta 4 ! 1=3. Since M Ay should accept with probability at least 2=3 and M A should reject 1 n with probability at least 2=3, we can conclude that M fails to accept either LA or LAy . So, each oracle A 2 A for which M correctly decides whether 1 n 2 LA can, by changing a single answer of A to 1 n , be mapped to at least (2 different oracles which M fails to correctly decide whether 1 n 2 LA f . Moreover, any particular is the image under this mapping of at most 2 since where it now answers 1 n , it must have given one of the possible answers. Therefore, the number of oracles in B for which M fails must be at least 1=2 the number of oracles in A for which M succeeds. So, calling a the number of oracles in A for which M fails, M must fail for at least a Therefore M fails to correctly decide whether with probability at least (1=2)P [A] - 1=8. It is easy to conclude that M decides membership in LA with probability 0 for a uniformly chosen oracle A. 2 Note: Theorem 3.3 and its Corollary 3.4 isolate the constraints on "quantum parallelism" imposed by unitary evolution. The rest of the proof of the above theorem is similar in spirit to standard techniques used to separate BPP from NP relative to a random oracle [3]. For example, these techniques can be used to show that, relative to a random oracle A, no classical probabilistic machine can recognize LA in time o(2 n ). However, quantum machines can recognize this language quadratically faster, in time O( using Grover's algorithm [13]. This explains why a substantial modification of the standard technique was required to prove the above theorem. The next result about NP " co-NP relative to a random permutation oracle requires a more subtle argument; ideally we would like to apply Theorem 3.3 after asserting that the total query magnitude with which A \Gamma1 (1 n ) is probed is small. However, this is precisely what we are trying to prove in the first place. Theorem 3.6 For any T (n) which is o(2 n=3 ), relative to a random permutation oracle, with probability 1, BQTime(T (n)) does not contain NP " co-NP. Proof. For any permutation oracle A, let first bit of A \Gamma1 (y) is 1g. Clearly, this language is contained in (NP " co-NP) A . Let T We show that for any bounded-error oracle QTM M A running in time at most T (n), with probability 1, M A does not accept the language LA . The probability is taken over the choice of a random permutation oracle A. Then, since there are a countable number of QTMs and the intersection of a countable number of probability 1 events still has probability 1, we conclude that with probability 1, no bounded error oracle QTM accepts LA in time bounded by T (n). pick n large enough so that T (n) - 2 n=3. We will show that the probability that M gives the wrong answer on input 1 n is at least 1=8 for every way of fixing the oracle answers on inputs of length not equal to n. The probability is taken over the random choices of the permutation oracle for inputs of length n. Consider the following method of defining random permutations on f0; 1g be a sequence of strings chosen uniformly at random in f0; 1g n . Pick - 0 uniformly at random among permutations such that -(x 0 - is the transposition each - i is a random permutation on f0; 1g n . Consider a sequence of permutation oracles A i , such that A i and A i . Denote by jOE i i the time i superposition of M A T (n) on input 1 n , and by jOE 0 i the time i superposition of M A T (n)\Gamma1 on input 1 n . By construction, with probability exactly 1=2, the string 1 n is a member of exactly one of the two languages and LA T . We will show that E[ Here the expectation is taken over the random choice of the oracles. By Markov's bound, P [ 3=4. Applying Theorem 3.1 we conclude that if then the acceptance probability of M A T (n) and M A T (n)\Gamma1 differ by at most 8=25 ! 1=3, and hence either both machines accept input 1 n or both reject that input. Therefore M A T (n) and give the same answers on input 1 n with probability at least 3=4. By construction, the probability that the string 1 n belongs to exactly one of the two languages LA T (n) and is equal to P [first bit of x T first bit of x T (n) Therefore, we can conclude that with probability at least 1=4, either M A T (n) or M A T (n)\Gamma1 gives the wrong answer on input 1 n . Since each of A T (n) and A T (n)\Gamma1 are chosen from the same distribution, we can conclude that M A T (n) gives the wrong answer on input 1 n with probability at least 1=8. To bound E[ we show that jOE T (n) i and jOE 0 are each close to a certain superposition j/ T (n) i. To define this superposition, run M on input 1 n with a different oracle on each step: on step i, use A i to answer the oracle queries. Denote by j/ i i, the time i superposition that results. Consider the set of time-string pairs Tg. It is easily checked that the oracle queries in the computation described above and those of M A T (n) and M A T (n)+1 differ only on the set S. We claim that the expected query magnitude of any pair in the set is at most 1=2 n , since for j - i, we may think of x j as having been randomly chosen during step j, after the superposition of oracle queries to be performed has already been written on the oracle tape. Let ff be the sum of the query magnitudes for time-string pairs in S. Then for " be a random variable such that (n). Then by Theorem 3.3, showed above that But E["= s Therefore E[ that E[ Finally, it is easy to conclude that M decides membership in LA with probability 0 for a uniformly random permutation oracle A. 2 Note: In view of Grover's algorithm [13], we know that the constant ``1=2'' in the statement of Theorem 3.5 cannot be improved. On the other hand, there is no evidence that the constant "1=3" in the statement of Theorem 3.6 is fundamental. It may well be that Theorem 3.6 would still hold (albeit not its current proof) with 1=2 substituted for 1=3. Corollary 3.7 Relative to a random permutation oracle, with probability 1, there exists a quantum one-way permutation. Given the oracle, this permutation can be computed efficiently even with a classical deterministic machine, yet it requires exponential time to invert even on a quantum machine. Proof. Given an arbitrary permutation oracle A for which A \Gamma1 can be computed in time n=3 ) on a quantum Turing machine, it is just as easy to decide LA as defined in the proof of Theorem 3.6. It follows from that proof that this happens with probability 0 when A is a uniformly random permutation oracle. 2 4 Using a Bounded-Error QTM as a Subroutine The notion of a subroutine call or an oracle invocation provides a simple and useful abstraction in the context of classical computation. Before making this abstraction in the context of quantum computation, there are some subtle considerations that must be thought through. For example, if the subroutine computes the function f , we would like to think of an invocation of the subroutine on the string x as magically writing f(x) in some designated spot (actually xoring it to ensure unitarity). In the context of quantum algorithms, this abstraction is only valid if the subroutine cleans up all traces of its intermediate calculations, and leaves just the final answer on the tape. This is because if the subroutine is invoked on a superposition of x's, then different values of x would result in different scratch-work on the tape, and would prevent these different computational paths from interfering. Since erasing is not a unitary operation, the scratch-work cannot, in general, be erased post-facto. In the special case where f can be efficiently computed deterministically, it is easy to design the subroutine so that it reversibly erases the scratch-work-simply compute f(x), copy f(x) into safe storage, and then uncompute f(x) to get rid of the scratch work [2]. However, in the case that f is computed by a BQP machine, the situation is more complicated. This is because only some of the computational paths of the machine lead to the correct answer f(x), and therefore if we copy f(x) into safe storage and then uncompute f(x), computational paths with different values of f(x) will no longer interfere with each other, and we will not reverse the first phase of the computation. We show, nonetheless, that if we boost the success probability of the BQP machine before copying f(x) into safe storage and uncomputing f(x), then most of the weight of the final superposition has a clean tape with only the input x and the answer f(x). Since such tidy BQP machines can be safely used as subroutines, this allows us to show that BQP BQP. The result also justifies our definition of oracle quantum machines. The correctness of the boosting procedure is proved in Theorems 4.13 and 4.14. The proof follows the same outline as in the classical case, except that we have to be much more careful in simple programming constructs such as looping, etc. We therefore borrow the machinery developed in [4] for this purpose, and present the statements of the relevant lemmas and theorems in the first part of this section. The main new contribution in this section is in the proofs of Theorems 4.13 and 4.14. The reader may therefore wish to skip directly ahead to these proofs. 4.1 Some Programming Primitives for QTMs In this subsection, we present several definitions, lemmas and theorems from [4]. Recall that a QTM M is defined by a triplet (\Sigma; Q; ffi) where: \Sigma is a finite alphabet with an identified blank symbol #, Q is a finite set of states with an identified initial state q 0 and final state q f 6= q 0 , and ffi , the quantum transition function, is a function where ~ C is the set of complex numbers whose real and imaginary parts can be approximated to within 2 \Gamman in time polynomial in n. Definition 4.1 A final configuration of a QTM is any configuration in state q f . If when QTM M is run with input x, at time T the superposition contains only final configurations and at any time less than T the superposition contains no final configuration, then M halts with running time T on input x. The superposition of M at time T is called the final superposition of M run on input x. A polynomial-time QTM is a well-formed QTM which on every input x halts in time polynomial in the length of x. Definition 4.2 A QTM M is called well-behaved if it halts on all input strings in a final superposition where each configuration has the tape head in the same cell. If this cell is always the start cell, we call the QTM stationary. We will say that a QTM M is in normal form if all transitions from the distinguished state q f lead to the distinguished state q 0 , the symbol in the scanned cell is left unchanged, and the head moves right, say. Formally: Definition 4.3 A QTM is in normal form if Theorem 4.4 If f is a function mapping strings to strings which can be computed in deterministic polynomial time and such that the length of f(x) depends only on the length of x, then there is a polynomial-time, stationary, normal form QTM which given input x, produces output x; f(x), and whose running time depends only on the length of x. If f is a one-to-one function from strings to strings that such that both f and f \Gamma1 can be computed in deterministic polynomial time, and such that the length of f(x) depends only on the length of x, then there is a polynomial-time, stationary, normal form QTM which given input x, produces output f(x), and whose running time depends only on the length of x. Definition 4.5 A multi-track Turing machine with k tracks is a Turing machine whose alphabet \Sigma is of the form \Sigma 1 \Theta \Sigma 2 \Theta \Delta \Delta \Delta \Theta \Sigma k with a special blank symbol # in each \Sigma i so that the blank in \Sigma is (#). We specify the input by specifying the string on each "track" (separated by ';'), and optionally by specifying the alignment of the contents of the tracks. Lemma 4.6 Given any QTM and any set \Sigma 0 , there is a QTM behaves exactly as M while leaving its second track unchanged Lemma 4.7 Given any QTM there is a QTM M such that the M 0 behaves exactly as M except that its tracks are permuted according to -. Lemma 4.8 If M 1 and M 2 are well-behaved, normal form QTMs with the same alphabet, then there is a normal form QTM M which carries out the computation of M 1 followed by the computation of M 2 . Lemma 4.9 Suppose that M is a well-behaved, normal form QTM. Then there is a normal such that on input x; k with k ? 0, the machine M 0 runs M for k iterations on its first track. Definition 4.10 If QTMs M 1 and M 2 have the same alphabet, then we say that the computation of M 1 if the following holds: for any input x on which M 1 halts, let c x and OE x be the initial configuration and final superposition of M 1 on input x. Then M 2 on input the superposition OE x , halts with final superposition consisting entirely of configuration c x . Note that for M 2 to reverse M 1 , the final state of M 2 must be equal to the initial state of 1 and vice versa. Lemma 4.11 If M is a normal form QTM which halts on all inputs, then there is a normal that reverses the computation of M with slowdown by a factor of 5. Finally, recall the definition of the class BQP. Definition 4.12 Let M be a stationary, normal form, multi-track QTM M whose last track has alphabet f#; 0; 1g. We say that M accepts x if it halts with a 1 in the last track of the start cell. Otherwise we say that M rejects x. A QTM accepts the language L ' (\Sigma \Gamma #) with probability accepts with probability at least p every string x 2 L and rejects with probability at least p every string We define the class BQP (bounded-error quantum polynomial time) as the set of languages which are accepted with probability 2=3 by some polynomial-time QTM. More generally, we define the class BQTime(T (n)) as the set of languages which are accepted with probability 2=3 by some QTM whose running time on any input of length n is bounded by T (n). 4.2 Boosting and Subroutine Calls Theorem 4.13 If QTM M accepts language L with probability 2=3 in time T (n) ? n, with T (n) time-constructible, then for any " ? 0, there is a QTM M 0 which accepts L with is polynomial in log 1=" but independent of n. Proof. Let M be a stationary QTM which accepts the language L in time T (n). We will build a machine that runs k independent copies of M and then takes the majority vote of the k answers. On any input x, M will have some final superposition of strings If we call A the set of i for which x i has the correct answer M(x) then running M on separate copies of its input k times will produce i. Then the probability of seeing jx i such that the majority have the correct answer M(x) is the sum of jff i 1 2 such that the majority of lie in A. But this is just like taking the majority of k independent coin flips each with probability at least 2=3 of heads. Therefore there is some constant b such that log 1=", the probability of seeing the correct answer will be at least 1 \Gamma ". So, we will build a machine to carry out the following steps. 1. Compute 2. Write out k copies of the input x spaced out with 2n blank cells in between, and write down k and n on other tracks. 3. Loop k times on a machine that runs M and then steps n times to the right. 4. Calculate the majority of the k answers and write it back in the start cell. We construct the desired QTM by building a QTM for each of these four steps and then dovetailing them together. Since Steps 1, 2, and 4 require easily computable functions whose output length depend only on k and the length of x, we can carry them out using well-behaved, normal form QTMs, constructed using Theorem 4.4, whose running times also depend only on k and the length of x. So, we complete the proof by constructing a QTM to run the given machine k times. First, using Theorem 4.4 we can construct a stationary, normal form QTM which drags the integers k and n one square to the right on its work track. If we add a single step right to the end of this QTM and apply Lemma 4.9, we can build a well-behaved, normal form QTM moves which n squares to the right, dragging k and n along with it. Dovetailing this machine after M , and then applying Lemma 4.9 gives a normal form QTM that runs M on each of the k copies of the input. Finally, we can dovetail with a machine to return with k and n to the start cell by using Lemma 4.9 two more times around a QTM which carries k and n one step to the left. 2 The extra information on the output tape of a QTM can be erased by copying the desired output to another track, and then running the reverse of the QTM. If the output is the same in every configuration in the final superposition, then this reversal will exactly recover the input. Unfortunately, if the output differs in different configurations, then saving the output will prevent these configurations from interfering when the machine is reversed, and the input will not be recovered. We show is the same in most of the final superposition, then the reversal must lead us close to the input. Theorem 4.14 If the language L is contained in the class BQTime(T (n)), with T (n) ? n and T (n) time-constructible, then for any " ? 0, there is a QTM M 0 which accepts L with " and has the following property. When run on input x of length n, runs for time bounded by cT (n), where c is a polynomial in log 1=", and produces a final superposition in which jxijL(x)i, with otherwise, has squared magnitude at least 1 \Gamma ". Proof. Let be a stationary, normal form QTM which accepts language L in time bounded by T (n). According to Theorem 4.13, at the expense of a slowdown by factor which is polynomial in log 1=" but independent of n, we can assume that M accepts L with probability on every input. Then we can construct the desired M 0 by running M , copying the answer to another track, and then running the reverse of M . The copy is easily accomplished with a simple two-step machine that steps left and back right while writing the answer on a clean track. Using Lemma 4.11, we can construct a normal form QTM M R which reverses M . Finally, with appropriate use of Lemmas 4.6 and 4.7, we can construct the desired stationary QTM by dovetailing machines M and M R around the copying machine. To see that this M 0 has the desired properties, consider running M 0 on input x of length n. M 0 will first run M on x producing some final superposition of configurations y ff y jyi of M on input x. Then it will write a 0 or 1 in the extra track of the start cell of each configuration, and run M R on this superposition y ff y jyijb y i. If we were to instead run M R on the superposition jOE y ff y jyijM(x)i we would after T (n) steps have the superposition consisting entirely of the final configuration with output x; M(x). Clearly, hOEjOE 0 i is real, and since M has success probability at least Therefore, since the time evolution of M R is unitary and hence preserves the inner product, the final superposition of M 0 must have an inner product with jxijM(x)i which is real and at least 1 \Gamma "=2. Therefore, the squared magnitude in the final superposition of M 0 of the final configuration with output x; M(x) must be at least (1 \Gamma Corollary 4.15 BQP Acknowledgement We wish to thank Bob Solovay for several useful discussions. --R "Arthur - Merlin games: A randomized proof system, and a hierarchy of complexity classes" "Logical reversibility of computation" "Relative to a random oracle A, P A 6= NP A 6= co-NP A with probability 1" "Quantum complexity theory" "The quantum challenge to structural complexity theory" "Oracle quantum computing" "Tight bounds on quantum searching" "Learning DNF over uniform distribution using a quantum example oracle" "Quantum theory, the Church-Turing principle and the universal quantum computer" "Quantum computational networks" "Rapid solution of problems by quantum computation" "Simulating physics with computers" "A fast quantum mechanical algorithm for database search" "Phase information in quantum oracle computing" "Algorithms for quantum computation: Discrete logarithms and factoring" "On the power of quantum computation" "Quantum circuit complexity" --TR --CTR Feng Lu , Dan C. 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quantum Turing machines;quantum polynomial time;oracle quantum Turing machines

264525

A Prioritized Multiprocessor Spin Lock.

AbstractIn this paper, we present the PR lock, a prioritized spin lock mutual exclusion algorithm. The PR lock is a contention-free spin lock, in which blocked processes spin on locally stored or cached variables. In contrast to previous work on prioritized spin locks, our algorithm maintains a pointer to the lock holder. As a result, our spin lock can support operations on the lock holder (e.g., for abort ceiling protocols). Unlike previous algorithms, all work to maintain a priority queue is done while a process acquires a lock when it is blocked anyway. Releasing a lock is a constant time operation. We present simulation results that demonstrate the prioritized acquisition of locks, and compare the performance of the PR lock against that of the best alternative prioritized spin lock.

Introduction Mutual exclusion is a fundamental synchronization primitive for exclusive access to critical sections or shared resources on multiprocessors [17]. The spin-lock is one of the mechanisms that can be used to provide mutual exclusion on shared memory multiprocessors [2]. A spin-lock usually is implemented using atomic read- modify-write instructions such as Test&Set or Compare&Swap, which are available on most shared-memory multiprocessors [16]. Busy waiting is effective when the critical section is small and the processor resources are not needed by other processes in the interim. However, a spin-lock is usually not fair, and a naive implementation can severely limit performance due to network and memory contention [1, 11]. A careful design can avoid contention by requiring processes to spin on locally stored or cached variables [19]. In real time systems, each process has timing constraints and is associated with a priority indicating the urgency of that process [26]. This priority is used by the operating system to order the rendering of services among competing processes. Normally, the higher the priority of a process, the faster it's request for services gets honored. When the synchronization primitives disregard the priorities, lower priority processes may block the execution of a process with a higher priority and a stricter timing constraint [24, 23]. This priority may cause the higher priority process to miss its deadline, leading to a failure of the real time system. Most of the work done in synchronization is not based on priorities, and thus is not suitable for real time systems. Furthermore, general purpose parallel processing systems often have processes that are "more important" than others (kernel processes, processes that hold many locks, etc. The performance of such systems will benefit from prioritized access to critical sections. In this paper, we present a prioritized spin-lock algorithm, the PR-lock. The PR-lock algorithm is suitable for use in systems which either use static-priority schedulers, or use dynamic-priority schedulers in which the relative priorities of existing tasks do not change while blocked (such as Earliest Deadline First [26] or Minimum Laxity [15]). The PR-lock is a contention-free lock [19], so its use will not create excessive network or memory contention. The PR-lock maintains a queue of records, with one record for each process that has requested but not yet released the lock. The queue is maintained in sorted order (except for the head record) by the acquire lock operations, and the release lock operation is performed in constant time. As a result, the queue order is maintained by processes that are blocked anyway, and a high priority task does not perform work for a low priority task when it releases the lock. The lock keeps a pointer to the record of the lock holder, which aids in the implementation of priority inheritance protocols [24, 23]. A task's lock request and release are performed at well-defined points in time, which makes the lock predictable. We present a correctness proof, and simulation results which demonstrate the prioritized lock access, the locality of the references, and the improvement over a previously proposed prioritized spin lock. We organize this paper as follows. In Section 1.1 we describe previous work in this area and in Section 2, we present our algorithm. In Section 3 we argue the correctness of our algorithm. In Section 4 we discuss an extension to the algorithm presented in Section 2. In Section 5 we show the simulation results which compare the performance of the PR-lock against that of other similar algorithms. In Section 6 we conclude this paper by suggesting some applications and future extensions to the PR-lock algorithm. 1.1 Previous Work Our PR-lock algorithm is based on the MCS-lock algorithm, which is a spin-lock mutual exclusion algorithm for shared-memory multiprocessors [19]. The MCS-lock grants lock requests in FIFO order, and blocked processes spin on locally accessible flag variables only, avoiding the contention usually associated with busy-waiting in multiprocessors [1, 11]. Each process has a record that represents its place in the lock queue. The MCS-lock algorithm maintains a pointer to the tail of the lock queue. A process adds itself to the queue by swapping the current contents of the tail pointer for the address of its record. If the previous tail was nil, the process acquired the lock. Otherwise, the process inserts a pointer to its record in the record of the previous tail, and spins on a flag in its record. The head of the queue is the record of the lock holder. The lock holder releases the lock by reseting the flag of its successor record. If no successor exists, the lock holder sets the tail pointer to nil using a Compare&Swap instruction. Molesky, Shen, and Zlokapa [20] describe a prioritized spin lock that uses the test-and-set instruction. Their algorithm is based on Burn's fair test-and-set mutual exclusion algorithm [5]. However, this lock is not contention-free. Markatos and LeBlanc [18] presents a prioritized spin-lock algorithm based on the MCS-lock algorithm. Their acquire lock algorithm is almost the same as the MCS acquire lock algorithm, with the exception that Markatos' algorithm maintains a doubly linked list. When the lock holder releases the lock, it searches for the highest priority process in the queue. This process' record is moved to the head of the queue, and its flag is reset. However, the point at which a task requests or releases a lock is not well defined, and the lock holder might release the lock to a low priority task even though a higher priority task has entered the queue. In addition, the work of maintaining the priority queue is performed when a lock is released. This choice makes the time to release a lock unpredictable, and significantly increases the time to acquire or release a lock (as is shown in section 5). Craig [10] proposes a modification to the MCS lock and to Markatos' lock that substitutes an atomic Swap for the Compare&Swap instruction, and permits nested locks using only one lock record per process. Goscinski [12] develops two algorithms for mutual exclusion for real time distributed systems. The algorithms are based on token passing. A process requests the critical section by broadcasting its intention to all other processes in the system. One algorithm grants the token based on the priorities of the processes, whereas the other algorithm grants the token to processes based on the remaining time to run the processes. The holder of the token enters the critical section. The utility of prioritized locks is demonstrated by rate monotonic scheduling theory [9, 24]. Suppose there are N periodic processes on a uniprocessor. Let E i and C i represent the execution time and the cycle time (periodicity) of the process T i . We assume that C 1 - C 2 - CN . Under the assumption that there is no blocking, [9] show that if for each j Then all processes can meet their deadlines. Suppose that B j is the worst case blocking time that process T j will incur. Then [24] show that all tasks can meet their deadlines if Thus, the blocking of a high priority process by a lower priority process has a significant impact on the ability of tasks to meet their deadlines. Much work has been done to bound the blocking due to lower priority processes. For example, the Priority Ceiling protocol [24] guarantees that a high priority process is blocked by a lower priority process for the duration of at most one critical section. The Priority Ceiling protocol has been extended to handle dynamic-priority schedulers [7] and multiprocessors [23, 8]. Our contribution over previous work in developing prioritized contention-free spin locks ([18] and [10]) is to more directly implement the desired priority queue. Our algorithm maintains a pointer to the head of the lock queue, which is the record of the lock holder. As a result, the PR-lock can be used to implement priority inheritance [24, 23]. The work of maintaining priority ordering is performed in the acquire lock operation, when a task is blocked anyway. The time required to release a lock is small and predictable, which reduces the length and the variance of the time spent in the critical section. The PR-lock has well-defined points in time in which a task joins the lock queue and releases its lock. As a result, we can guarantee that the highest priority waiting task always receives the lock. Finally, we provide a proof of correctness. Our PR-lock algorithm is similar to the MCS-lock algorithm in that both maintain queues of blocked processes using the Compare&Swap instruction. However, while the MCS-lock and Markatos' lock maintain a global pointer to the tail of the queue, the PR-lock algorithm maintains a global pointer to the head of the queue. In both the MCS-lock and the Markatos' lock, the processes are queued in FIFO order, whereas in the PR-lock, the queue is maintained in priority order of the processes. 2.1 Assumptions We make the following assumptions about the computing environment: 1. The underlying multiprocessor architecture supports an atomic Compare&Swap instruction. We note that many parallel architectures support this instruction, or a related instruction [13, 21, 3, 28]. 2. The multiprocessor has shared memory with coherent caches, or has locally-stored but globally- accessible shared memory. 3. Each processor has a record to place in the queue for each lock. In a NUMA architecture, this record is allocated in the local, but globally accessible, memory. This record is not used for any other purpose for the lifetime of the queue. In Section 4, we allow the record to be used among many lock queues. 4. The higher the actual number assigned for priority, the higher the priority of a process (we can also assume the opposite). 5. The relative priorities of blocked processes do not change. Acceptable priority assignment algorithms include Earliest Deadline First and Minimum Laxity. It should be noted that each process p i participating in the synchronization can be associated with an unique processor P i . We expect that the queued processes will not be preempted, though this is not a requirement for correctness. 2.2 Implementation The PR-lock algorithm consists of two operations. The acquire lock operation acquires a designated lock and the release lock operation releases the lock. Each process uses the acquire lock and release lock operations to synchronize access to a resource: acquire lock(L, r) critical section release lock(L) The following sub-sections present the required version of Compare&Swap, the needed data structures, and the acquire lock and release lock procedures. 2.2.1 The Compare&Swap The PR-lock algorithms make use of the Compare&Swap instruction, the code for which is shown in Figure 1. Compare&Swap is often used on pointers to object records, where a record refers to the physical memory space and an object refers to the data within a record. Current is a pointer to a record, Old is a previously sampled value of Current, and New is a pointer to a record that we would like to substitute for *Old (the record pointed to by Old). We compute the record *New based on the object in *Old (or decide to perform the swap based on the object in *Old), so we want to set Current equal to New only if Current still points to the record *Old. However, even if Current points to *Old, it might point to a different object than the one originally read. This will occur if *Old is removed from the data structure, then re-inserted as Current with a new object. This sequence of events cannot be detected by the Compare&Swap and is known as the A-B-A problem. Following the work of Prakash et al. [22] and Turek et al. [27], we make use of a double-word Com- pare&Swap instruction [21] to avoid this problem. A counter is appended to Current which is treated as a part of Current. Thus Current consists of two parts: the value part of Current and the counter part of Current. This counter is incremented every time a modification is made to *Current. Now all the variables Procedure CAS(structure pointer *Current, *Old, *New) /* Assume CAS operates on double words */ atomicf *Current == *Old else f Figure 1: CAS used in the PR-lock Algorithm Current, Old , and New are twice their original size. This approach reduces the probability of occurrence of the A-B-A problem to acceptable levels for practical applications. If a double-word Compare&Swap is not available, the address and counter can be packed into 32 bits by restricting the possible address range of the lock records. We use a version of the Compare&Swap operation in which the current value of the target location is returned in old, if the Compare&Swap fails. The semantics of the Compare&Swap used is given in Figure 1. A version of the Compare&Swap instruction that returns only TRUE or FALSE can be used by performing an additional read. 2.2.2 Data Structures The basic data structure used in the PR-lock algorithm is a priority queue. The lock L contains a pointer to the first record of the queue. The first record of the queue belongs to the process currently using the lock. If there is no such process, then L contains nil. Each process has a locally-stored but globally-accessible record to insert into the lock queue. If process inserts record q into the queue, we say that q is p's record and p is q's process. The record contains the process priority, the next-record pointer, a boolean flag Locked on which the process owning the element busy-waits if the lock is not free, and an additional field Data that can be used to store application-dependent information about the lock holder. The next-record pointer is a double sized variable: one half is the actual pointer and the other half is a counter to avoid the A-B-A problem. The counter portion of the pointer itself has into two parts: one bit of the counter called the Dq bit is used to indicate whether the queuing element is in the queue. The rest of the bits are used as the actual counter. This technique is similar to the one used by Prakash et al. [22] and Turek et al. [27]. Their counter refers to the record referenced by the pointer. In our algorithm, the counter refers to the record that contains the pointer, not the record that is pointed to. If the Dq bit of a record q is FALSE, then the record is in the queue for a lock L. If the Dq bit is TRUE, then the record is probably not in the queue (for a short period of time, the record might be in the queue with its Dq bit set TRUE). The Dq bit lets the PR-lock avoid garbage accesses. Each process keeps the address of its record in a local variable (Self). In addition, each process requires two local pointer variables to hold the previous and the next queue element for navigating the queue during the enqueue operation (Prev Node and Next Node). The data structures used are shown in Figure 2. The Dq bit of the Pointer field is initialized to TRUE, and the Ctr field is initialized to 0 before the record is first used. A typical queue formed by the PR-lock algorithm is shown in Figure 3 below. Here L points to the record q 0 of the current process holding the lock. The record q 0 has a pointer to the record q 1 of the next process having the highest priority among the processes waiting to acquire the lock L. Record q 1 points to record q 2 of the next higher priority waiting process and so on. The record q n belongs to the process with the least priority among waiting processes. 2.2.3 Acquire Lock Operation The acquire lock operation is called by a process - before using the critical section or resource guarded by lock L. The parameters of the acquire lock operation are the lock pointer L and the record - q of the process (passed to local variable Self). An acquire lock operation searches for the correct position to insert - q into the queue using Prev Node and Next Node to keep track of the current position. In Figure 4, Prev Node and Next Node are abbreviated to P and N. The records pointed by P and N are q i and q i+1 , belonging to processes p i and p i+1 . Process positions itself so that P r(p i r is a function which maps a process to its priority. Once such a position is found, - q is prepared for insertion by making - q point to q i+1 . Then, the insertion is committed by making q i to point to - q by using the Compare&Swap instruction. The various stages and final result are shown in Figure 4. The acquire lock algorithm is given in Figure 5. Before the acquire lock procedure is called, the Data and the Priority fields of the process' record are initialized appropriately. In addition, the Dq bit of the Next pointer is implicitly TRUE. The acquire lock operation begins by assuming that the lock is currently free (the lock pointer L is structure Pointer f structure Object *Ptr; boolean Dq; structure Record f structure structure of data Data; boolean Locked; integer Priority; structure Pointer Next; Shared Variable structure Pointer L; Private Variables structure Pointer Self, Prev Node, Next node; boolean Success, Failure; constant TRUE, FALSE, NULL, MAX PRIORITY; Data Priority Next.Ctr Next.Ptr Locked Next.Dq Record Structure Figure 2: Data Structures used in the PR-lock Algorithm Figure 3: Queue data structure used in PR-lock algorithm Start Position Prepare Commit Figure 4: Stages in the acquire lock operation null). It attempts to change L to point to its own record with the Compare&Swap instruction. If the Compare&Swap is successful, the lock is indeed free, so the process acquires the lock without busy-waiting. In the context of the composite pointer structures that the algorithm uses, a NULL pointer is all zeros. If the swap is unsuccessful, then the acquiring process traverses the queue to position itself between a higher or equal priority process record and a lower priority process record. Once such a junction is found, will point to the record of the higher priority process and Next Node will point to the record of the lower priority process. The process first sets its link to Next Node. Then, it attempts to change the previous record's link to its own record by the atomic Compare&Swap. If successful, the process sets the Dq flag in its record to FALSE indicating its presence in the queue. The process then busy-waits until its Locked bit is set to FALSE, indicating that it has been admitted to the critical section. There are three cases for an unsuccessful attempt at entering the queue. Problems are detected by examining the returned value of the failed Compare&Swap marked as F in the algorithm. Note that the returned value is in the Next Node. In addition, a process might detect that it has misnavigated while searching the queue. When we read Next Node, the contents of the record pointed to by Prev Node are fixed because the record's counter is read into Next Node. 1. A concurrent acquire lock operation may overtake the acquire lock operation and insert its own Procedure acquire lock(L, Self) f do f else f /* Lock in Use */ do f Next Node=Prev Node.Ptr-?Next; if((Next Node.Dq==TRUE) /* Deque, Try Again */ ii or (Prev Node.Ptr-?Priority!Self.Ptr-?Priority)) f iii else f if(Next Node.Ptr==NULL or (Next Node.Ptr!=NULL and Next Node.Ptr-?Priority!Self.Ptr-?Priority))f use lock */ else f if((Next Node.Dq==TRUE) /* Deque, Try Again */ ii or Prev Node.Ptr-?Priority ! else Next Node=Prev Node; i gwhile(!Success and !Failure); Figure 5: The acquire lock operation procedure record immediately after Prev Node, as shown in Figure 10. In this case the Compare&Swap will fail at the position marked F in Figure 5. The correctness of this operation's position is not affected, so the operation continues from its current position (line marked by i in Figure 5). 2. A concurrent release lock operation may overtake the acquire lock operation and removes the record pointed to by Prev Node, as shown in Figure 11. In this case, the Dq bit in the link pointer of this record will be TRUE. The algorithm checks for this condition when it scans through the queue and when it tries to commit its modifications. The algorithm detects the situation in the two places marked by ii in the Figure 5. Every time a new record is accessed (by Prev Node), its link pointer is read into Next Node and the Dq bit is checked. In addition, if the Compare&Swap fails, the link pointer is saved in Next Node and the Dq bit is tested. If the Dq bit is TRUE, the algorithm starts from the beginning. 3. A concurrent release lock operation may overtake the acquire lock operation and remove the record pointed to by Prev Node, and then the record is put back into the queue, as shown in Figure 12. If the record returns with a priority higher than or equal to Self's priority, then the position is still correct and the operation can continue. Otherwise, the operation cannot find the correct insertion point, so it has to start from the beginning. This condition is tested at the lines marked iii in Figure 5. The spin-lock busy waiting of a process is broken by the eventual release of the lock by the process which is immediately ahead of the waiting process. 2.2.4 Release Lock Operation The release lock operation is straight forward and the algorithm is given in Figure 6. The process p releasing the lock sets the Dq bit in its record's Link pointer to TRUE, indicating that the record is no longer in the queue. Setting the Dq bit prevents any acquire lock operation from modifying the link. The releasing process copies the address of the successor record, if any, to L. The process then releases the lock by setting the Locked boolean variable in the record of the next process waiting to be FALSE. To avoid testing special cases in the acquire lock operation, the priority of the head record is set to the highest possible priority. 3 Correctness of PR-lock Algorithm In this section, we present an informal argument for the correctness properties of our PR-lock algorithm. We prove that the PR-lock algorithm is correct by showing that it maintains a priority queue, and the head Procedure release lock(L, Self)f L=Self.Ptr-?Next; /* Release Lock */ if(Self.Ptr-?Next!=NULL)f Figure The release lock operation procedure of the priority queue is the process that holds the lock. The PR-lock is decisive-instruction serializable [25]. Both operations of the PR-lock algorithm have a single decisive instruction. The decisive instruction for the acquire lock operation is the successful Compare&Swap and the decisive instruction for the release lock operation is setting the Dq bit. Corresponding to a concurrent execution C of the queue operations, there is an equivalent (with respect to return values and final states) serial execution S d such that if operation O 1 executes its decisive instruction before operation O 2 does in C, then O 1 ! O 2 in S d . Thus, the equivalent priority queue of a PR-lock is in a single state at any instant, simplifying the correctness proof (a concurrent data structure that is linearizable but not decisive-instruction serializable might be in several states simultaneously [14]). We use the following notation in our discussion. PR-lock L has lock pointer L, which points to the first record in the lock queue (and the record of the process that holds the lock). Let there be N processes p 1 , that participate in the lock synchronization for a priority lock L, using the PR-lock algorithm. As mentioned earlier, each process p i allocates a record q i to enqueue and dequeue. Thus, each process p i participating in the lock access is associated with a queue record q i . Let P r(p i ) be a function which maps a process to its priority, a number between 1 and N. We also define another function P r(q i ) which maps a record belonging to a process p i to its priority. A priority queue is an abstract data type that consists of: ffl A finite set Q of elements . For simplicity, we assume that every n i is unique. This assumption is not required for correctness, and in fact processes of the same priority will obtain the lock in FCFS order. ffl Two operations enqueue and dequeue At any instant, the state of the queue can be defined as We call q 0 the head record of priority queue Q. The head record's process is the current lock holder. Note that the non-head records are totally ordered. The enqueue operation is defined as enqueue where The dequeue operation on a non-empty queue is defined as where the return value is q 0 . A dequeue operation on an empty queue is undefined. For every PR-lock L, there is an abstract priority queue Q. Initially, both L and Q are empty. When a process - p with a record - q performs the decisive instruction for the acquire lock operation, Q changes state to enqueue (Q; - q). Similarly, when a process executes the decisive instruction for a release lock operation, Q changes state to dequeue(Q). We show that when we observe L, we find a structure that is equivalent to Q. To observe L, we take a consistent snapshot [6] of the current state of the system memory. Next, we start at the lock pointer L and observe the records following the linked list. If the head record has its Dq bit set and its process has exited the acquire lock operation, then we discard it from our observation. If we observe the same records in the same sequence in both L and Q, then we say that L and Q are equivalent, and we write L , Q. Theorem 1 The representative priority queue Q is equivalent to the observed queue of the PR-lock L. Proof: We prove the theorem by induction on the decisive instructions, using the following two lemmas. before a release lock decisive instruction, then Q , L after the release lock decisive instruction. Proof: Let release lock decisive instruction. A release lock operation is equivalent to a dequeue operation on the abstract queue. By definition, q1 qn Before After Figure 7: Observed queue L before and after a release lock Before After Figure 8: Observed queue L before and after an acquire lock The before and after states of L are shown in Figure 7. If L points to the record q 0 before the release lock decisive instruction, the release lock decisive instruction sets the Dq bit in q 0 to TRUE, removing q 0 from the observable queue. Thus, Q , L after the release lock operation. Note that L will point to q 1 before the next release lock decisive instruction. 2 before an acquire lock decisive instruction, then Q , L after the acquire lock decisive instruction. Proof: There are two different cases to consider: Case 1: before the acquire lock decisive instruction. The equivalent operation on the abstract queue Q is the enqueue operation. Thus, If the lock L is empty, - q's process executes a successful decisive Compare&Swap instruction to make L to point to - q and acquires the lock (Figure 8). Clearly, Q , after the acquire lock decisive instruction. Case 2: before the acquire lock decisive instruction. The state of the queue Q after the acquire lock is given by The corresponding L before and after the acquire lock is shown in Figure 9. The pointers P and N are the Prev Node and Next Node pointers by which - q's acquire lock operation positions its record such that the process observes P r(q the Next pointer in - q is is set to the address of q i+1 . The Before After Figure 9: Observed queue L before and after an acquire lock Compare&Swap instruction, marked F in Figure 5, attempts to make the Next pointer in q i point to - q. If the Compare&Swap instruction succeeds, then it is the decisive instruction of - q's process and the resulting queue L is illustrated in the Figure 9. This is equivalent to Q after the enqueue operation. If the Compare&Swap succeeds only when q i is in the queue, q i+1 is the successor record, and P r(q If there are no concurrent operations on the queue, we can observe that the P and N are positioned correctly and the Compare&Swap succeeds. If there are other concurrent operations, they can interfere with the execution of an acquire lock operation, A. There are three possibilities: Case a: Another acquire lock A' enqueued its record q 0 between q i and q i+1 , but q i has not yet been dequeued. If P r(q q's process will attempt to insert - q between q i and q i+1 . Process A 0 has modified q i 's next pointer, so that - q's Compare&Swap will fail. Since q i has not been dequeued, process should continue its search from q i , which is what happens. If q's process can skip over q 0 and continue searching from q i+1 , which is what happens. This scenario is illustrated in Figure 10. Case b: A release lock operation R overtakes A and removes q i from the queue (i.e., R has set q i 's Dq bit), and q i has not yet been returned to the queue (its Dq bit is still false). Since q i is not in the lock queue, A is lost and must start searching again. Based on its observations of q i and q i+1 , A may have decided to continue searching the queue or to commit its operation. In either case A sees the Dq bit set and fails, so A starts again from the beginning of the queue. This scenario is illustrated in Figure 11 Case c: A release lock operation R overtakes A and removes q i from the queue, and then q i is put back in the queue by another acquire lock A'. If A tries to commit its operation, then the pointer in q i is changed, so the Compare&Swap fails. Note that even if q i is pointing to q i+1 , the version numbers prevent the decisive instruction from succeeding. If A continues searching, then there are two possibilities based on the new value of P r(q lost and cannot find the correct place to insert - q. This condition is detected when the priority of q i is examined (the lines marked iii in Figure 5), and operation A restarts from the head of the queue. If P can still find a correct place to insert - past q' q' Before A' After A' Continue A Figure 10: A concurrent acquire lock A' succeeds before A F Before R After R Restart A Figure 11: A concurrent release lock R succeeds before A continues searching. This scenario is illustrated in Figure 12. what interference occurs, A always takes the right action. Therefore, Q , L after the acquire lock decisive instruction. 2 To prove the theorem we use induction. Initially, points to nil. So, Q , L is trivially true. Suppose that the theorem is true before the i th decisive instruction. If the i th decisive instruction is for an acquire lock operation, Lemma after the i th decisive instruction. If the i th decisive instruction is for a release lock operation, Lemma 1 after the i th decisive instruction. Therefore, the inductive step holds, and hence, Q , L. 2 Extensions In this section we discuss a couple of simple extensions that increase the utility of the PR-lock algorithm. 4.1 Multiple Locks As described, a record for a PR-lock can be used for one lock queue only (otherwise, a process might obtain a lock other than the one it desired). If the real-time system has several critical sections, each with their own locks (which is likely), each process must have a lock record for each lock queue, which wastes space. Fortunately, a simple extension of the PR-lock algorithm allows a lock record to be used in many different lock queues. We replace the Dq bit by a Dq string of l bits. If the Dq string evaluates to i ? 0 when interpreted ri+1 ri+1 rm rm Restart A if Pr(q^) > Pr(qi) Continue A if Pr(q^) <= Pr(qi) After R and A' Before R Figure 12: Release lock R and acquire lock A' succeed before A as a binary number, then the record in in the queue for lock i. If the Dq string evaluates to 0, then the record is (probably) not in any queue. The acquire lock and release lock algorithms carry through by modifying the test for being or not being in queue i appropriately. We note that if a process sets nested locks, a new lock record must be used for each level of nesting. Craig [10] presents a method for reusing the same record for nested locks. 4.2 Backing Out If a process does not obain the lock after a certain deadline, it might wish to stop waiting and continue processing. The process must first remove its record from the lock queue. To do so, the process follows these steps: 1. Find the preceding record in the lock queue, using the method from the algorithm for the acquire lock operation. If the process determines that its record is at the head of the lock queue, return with a "lock obtained" value. 2. Set the Dq bit (Dq string) of the process' record to ``Dequeued''. 3. Perform a compare and swap of the predecessor record's next pointer with the process' next pointer. If the Compare&swap fails, go to 1. If the Compare&swap succeeds, return with a "lock released" value. the value of the process's successor. If the process removes itself from the queue without obtaining the lock, the Compare&swap is the decisive instruction. If the Compare&swap fails, the predecessor might have released the lock, or third process has enqueued itself as the predecessor. The process can't distinguish between these possibilities, so it must re-search the lock queue. 5 Simulation Results We simulated the execution of the PR-lock algorithm in PROTEUS, which is a configurable multiprocessor simulator [4]. We also implemented the MCS-lock and Markatos' lock to demonstrate the difference in the acquisition and release time characteristics. In the simulation, we use a multiprocessor model with eight processors and a global shared memory. Each processor has a local cache memory of 2048 bytes size. In PROTEUS, the units of execution time are cycles. Each process executes for a uniformly randomly distributed time, in the range 1 to 35 cycles, before it issues an acquire-lock request. After acquiring the lock, the process stays in the critical section for a fixed number of cycles (150) plus another uniformly randomly distributed number (1 to 400) of cycles before releasing the lock. This procedure is repeated fifty times. The average number of cycles taken to acquire a lock by a process is then computed. PROTEUS simulates parallelism by repeatedly executing a processor's program for a time quanta, Q. In our simulations, 10. The priority of a process is set equal to the process/processor number and the lower the number, the higher the priority of a process. Figures 13 and 14 show the average time taken for a process to acquire a lock using the MCS-lock algorithm and the PR-lock algorithm, respectively. A process using MCS-lock algorithm has to wait in the FIFO queue for all other processes in every round. However, a process using the PR-lock algorithm will wait for a time that is proportional to the number of higher priority processes. As an example, the highest and second highest priority process on the average waits for about one critical section period. We note that the two highest priority processes have about the same acquire lock execution time because they alternate in acquiring the lock. Only after both of these processes have completed their execution can the third and fourth highest priority processes obtain the lock. Figure 14 clearly demonstrates that the average acquisition time for a lock using PR-lock is proportional to the process priorities, whereas the average acquisition time is proportional to the number of processes in case of the MCS-lock algorithm. This feature makes the PR-lock algorithm attractive for use in real time systems. In Figure 15, we show the average time taken for a process to acquire the lock using Markatos' algorithm. The same prioritized lock-acquisition behavior is shown, but the average time to acquire a lock is 50% greater than when the PR-lock is used. At first this result is puzzling, because Markatos' lock performs the majority of its work when the lock is released and the PR-lock performs its work when the lock is acquired. However, the time to release a lock is part of the time spent in the critical section, and the time to acquire a lock depends primarily on time spent in the critical section by the preceding lock holders. Thus, the PR-lock allows much faster access to the critical section. As we will see, the PR-lock also allows more predictable access to the critical section. Figure shows the cache hit ratio at each instance of time on all the processors. Most of the time the cache-hit ratio is 95% or higher on each of the processors, and we found an average cache hit rage of 99.72% to 99.87%. Thus, the PR-lock generates very little network or memory contention in spite of the processes using busy-waiting. Finally, we compared the time required to release a lock using both the PR-lock and Markatos' lock. These results are shown in Figure 17 and for Markatos' lock in Figure 18. The time to release a lock using PR-lock is small, and is consistent for all of the processes. Releasing a lock using Markatos' lock requires significantly more time. Furthermore, in our experiments a high priority process is required to spend significantly more Average Time(Cycles) x 100 Processor(Priority) Figure 13: Lock acquisition time for the MCS-lock Algorithm time releasing a lock than is required for a low priority process. This behavior is a result of the way that the simulation was run. When high priority processes are executing, all low priority processes are blocked in the queue. As a result, many records must be searched when a high priority process releases a lock. Thus, a high priority process does work on behalf of low priority processes. The time required for a high priority process to release its lock depends on the number of blocked processes in the queue. The result is a long and unpredictable amount of time required to release a lock. Since the lock must be released before the next process can acquire the lock, the time required to acquire a lock is also made long and unpredictable. 6 Conclusion In this paper, we present a priority spin-lock synchronization algorithm, the PR-lock, which is suitable for real-time shared-memory multiprocessors. The PR-lock algorithm is characterized by a prioritized lock ac- quisition, a low release overhead, very little bus-contention, and well-defined semantics. Simulation results show that the PR-lock algorithm performs well in practice. This priority lock algorithm can be used as presented for mutually exclusive access to a critical section or can be used to provide higher level synchronization constructs such as prioritized semaphores and monitors. The PR-lock maintains a pointer to the record of the lock holder, so the PR-lock can be used to implement priority inheritance protocols. Finally, the PR-lock algorithm can be adapted for use as a single-dequeuer, multiple-enqueuer parallel priority queue. Average Time(Cycles) x 100 Processor(Priority) 28 3213579 Figure 14: Lock acquisition time for the PR-lock Algorithm Average Time(Cycles) x 100 Processor(Priority) Figure 15: Lock acquisition time for the Markatos' Algorithm Time x 10000 Processor Figure Cache hit ratio for the PR-lock Algorithm Average Time(Cycles) Processor(Priority) Figure 17: Lock release time for the PR-Lock Algorithm Average Processor(Priority) Figure 18: Lock release time for Markatos' Algorithm While several prioritized spin locks have been proposed, the PR-lock has the following advantages: ffl The algorithm is contention free. ffl A higher priority process does not have to work for a lower priority process while releasing a lock. As a result, the time required to acquire and release a lock is fast and predictable. ffl The PR-lock has a well-defined acquire-lock point. ffl The PR-lock maintains a pointer to the process using the lock that facilitates implementing priority inheritance protocols. For future work, we are interested in prioritizing access to other operating system structures to make them more appropriate for use in a real-time parallel operating system. --R The performance of spin lock alternatives for shared memory multiprocessors. Concurrent Programming Principles and Practice. Mutual exclusion with linear waiting using binary shared variables. Distributed snapshots: Determining global states of distributed systems. Dynamic priority ceiling: A concurrency control protocol for real-time systems A priority ceiling protocol for multiple-instance resources Scheduling algorithms for multiprogramming in a hard real-time environ- ment Queuing spin lock alternatives to support timing predictability. Characterizing memory hotspots in a shared memory mimd machine. Two algorithms for mutual exclusion in real-time distributed computer systems A methodology for implementing highly concurrent data objects. A correctness condition for concurrent objects. A performance analysis of minimum laxity and earliest deadline in a real-time system Efficient synchronization on multiprocessors with shared memory. Multiprocessor synchronization primitives with priorities. Algorithms for scalable synchronization on shared-memory multiprocessors Predictable synchronization mechanisms for real-time systems Priority inheritance protocols: An approach to real-time synchronization Concurrent search structure algorithms. Tutorial Hard Real-Time Systems Locking without blocking: Making lock based concurrent data structure algorithms nonblocking. --TR --CTR Prasad Jayanti, f-arrays: implementation and applications, Proceedings of the twenty-first annual symposium on Principles of distributed computing, July 21-24, 2002, Monterey, California James H. Anderson , Yong-Jik Kim , Ted Herman, Shared-memory mutual exclusion: major research trends since 1986, Distributed Computing, v.16 n.2-3, p.75-110, September

spin lock;priority queue;mutual exclusion;parallel processing;real-time system

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An Optimal Algorithm for the Angle-Restricted All Nearest Neighbor Problem on the Reconfigurable Mesh, with Applications.

AbstractGiven a set S of n points in the plane and two directions $r_1$ and $r_2,$ the Angle-Restricted All Nearest Neighbor problem (ARANN, for short) asks to compute, for every point p in S, the nearest point in S lying in the planar region bounded by two rays in the directions $r_1$ and $r_2$ emanating from p. The ARANN problem generalizes the well-known ANN problem and finds applications to pattern recognition, image processing, and computational morphology. Our main contribution is to present an algorithm that solves an instance of size n of the ARANN problem in O(1) time on a reconfigurable mesh of size nn. Our algorithm is optimal in the sense that $\Omega\;(n^2)$ processors are necessary to solve the ARANN problem in O(1) time. By using our ARANN algorithm, we can provide O(1) time solutions to the tasks of constructing the Geographic Neighborhood Graph and the Relative Neighborhood Graph of n points in the plane on a reconfigurable mesh of size nn. We also show that, on a somewhat stronger reconfigurable mesh of size $n\times n^2,$ the Euclidean Minimum Spanning Tree of n points can be computed in O(1) time.

Introduction Recently, in an effort to enhance both its power and flexibility, the mesh-connected architecture has been endowed with various reconfigurable features. Examples include the bus automaton [21, 22], the reconfigurable mesh [15], the mesh with bypass capability [8], the content addressable array processor [29], the reconfigurable network [2], the polymorphic processor array [13, 14], the reconfigurable bus with shift switching [11], the gated-connection network [23, 24], and the polymorphic torus [9, 10]. Among these, the reconfigurable mesh has emerged as a very attractive and versatile architecture. In essence, a reconfigurable mesh (RM) consists of a mesh augmented by the addition of a dynamic bus system whose configuration changes in response to computational and communication needs. More precisely, a RM of size n \Theta m consists of nm identical SIMD processors positioned on a rectangular array with n rows and m columns. As usual, it is assumed that every processor knows its own coordinates within the mesh: we let P (i; denote the processor placed in row i and column j, with P (1; 1) in the north-west corner of the mesh. Each processor P (i; j) is connected to its four neighbors P (i \Gamma and P exist, and has 4 ports denoted by N, S, E, and W in Figure 1. Local connections between these ports can be established, under program control, creating a powerful bus system that changes dynamically to accommodate various computational needs. We assume that the setting of local connection is destructive in the sense that setting a new pattern of connections destroys the previous one. Most of the results in this paper assume a model that allows at most two connections to be set in each processor at any one time. Furthermore, these two connections must involve disjoint pairs of ports as illustrated in Figure 2. Some other models proposed in the literature allow more than two connections to be set in every processor [9, 10]. One of our results uses such a model. In accord with other workers [9, 10, 13-16, 21] we assume that communications along buses take O(1) time. Although inexact, recent experiments with the YUPPIE and the GCN reconfigurable multiprocessor system [16, 23, 24] seem to indicate that this is a reasonable working hypothesis. It is worth mentioning that at least Figure 1: A reconfigurable mesh of size 4 \Theta 5 Figure 2: Examples of allowed connections and corresponding buses two VLSI implementations have been performed to demonstrate the feasibility and benefits of the two-dimensional reconfigurable mesh: one is the YUPPIE (Yorktown Ultra-Parallel Polymorphic Image Engine) chip [9, 10, 16] and the other is the GCN (Gated-Connection Network) chip [23, 24]. These two implementations suggested that the broadcast delay, although not constant, is very small. For example, only 16 machine cycles are required to broadcast on a 10 6 -processor YUPPIE. The GCN has further shortened the delay by adopting pre-charged circuits. Newer developments seem to suggest the feasibility of implementations involving the emerging optical technology. One of the fundamental features that contributes to a perceptionally relevant description useful in shape analysis is the distance properties among points in a planar set. In this context, nearest- and furthest-neighbor computations are central to pattern recognition classification techniques, image processing, computer graphics, and computational morphology [4, 20, 26, 27]. In image processing, for example, proximity is a simple and important metric for potential similarities of objects in the image space. In pattern recognition, the same concept appears in clustering, and computing similarities between sets [4]. In mor- phology, closeness is often a valuable tool in devising efficient algorithms for a number of seemingly unrelated problems [26]. A classic problem in this domain involves computing for every point in a given set S, a point that is closest to it: this problem is known as the All-Nearest Neighbor problem (ANN, for short) and has been well studied in both sequential and parallel [1, 4, 20, 26]. Recently, Jang and Prasanna [5] provided an O(1) time algorithm for solving the ANN problem for n points in the plane on a RM of size n \Theta n. In this paper we address a generalization of the ANN problem, namely the the Angle- Restricted All Nearest Neighbor problem (ARANN, for short). Just as the ANN problem, the ARANN problem has wide-ranging applications in pattern recognition, image processing, and morphology. For points p and q in the plane, we let d(p; q) stand for the Euclidean distance between p and q. Further, we say that q is (r )-dominated by p if q lies inside of the closed planar region determined by two rays in directions r 1 and r 2 emanating from p. In this terminology, a point q in S is said to be the (r )-nearest neighbor of p if q is (r )-dominated by p and )-dominated by pg. The ARANN problem involves determining the (r )-nearest neighbor of every point in S. Refer to Figure 3 for an illustration. Here, p 2 is (0; -=3)-dominated by p 1 , but is not (0; -=3)- dominated by p 4 . The (0; -=3)-nearest neighbor of p 1 is p 2 . A class of related problems in pattern recognition and morphology involves associating a certain graph with the set S. This graph is, of course, application-specific. For example, in pattern recognition one is interested in the Euclidean Minimum Spanning Tree of S, the Relative Neighborhood Graph of S, the Geographic Neighborhood Graph, the Symmetric Furthest Neighbor Graph, the Gabriel Graph of S, and the Delaunay Graph of S, to name a few [20, 25-27]. The Euclidean Minimum Spanning Tree of S, denoted by EMST(S), is the minimum Figure 3: Illustrating (0; -)-domination and the corresponding ARANN graph Figure 4: Lune of p and q spanning tree of the weighted graph with vertices S and with the edges weighted by the corresponding Euclidean distance. In other words, the edge-set is Sg where (p; q; d(p; q)) is the edge connecting points p and q having weight d(p; q). The Relative Neighborhood Graph, RNG(S) of a set S of points in the plane has been introduced by Toussaint [26] in an effort to capture many perceptually relevant features of the set S. Specifically, given a set S of points in the plane, RNG(S) has for vertices the points of S together with an edge between p and q whenever d(p; q) - max s2S fd(p; s),d(q; s)g. An equivalent definition states that two vertices p, q are joined by an edge in RNG(S) if no other points of S lie inside LUNE(p; q), the lune of p, q defined as the set of points in the GNG Figure 5: Illustrating EMST, RNG, GNG, and GNG 1 plane enclosed in the region determined by two disks with radius d(p; q) centered at p and q, respectively. Refer to Figure 4 for an illustration. be the undirected graph with vertex-set S and with edge-set f(p; q)jq is the i-=3)-nearest neighbor of pg. The GNG of S, denoted GNG(S), is the graph with vertex-set S and whose edges are [ 1-i-6 E i . Refer to Figure 5 for an example of the concepts defined above. Given set S of n points in the plane and two directions r 1 and r 2 , the Angle-Restricted All Nearest Neighbor graph of S, denoted ARANN(S) is the directed graph whose vertices are the points in S; the points p and q are linked by a directed edge from p to q whenever q is the (r )-nearest neighbor of p. The reader will not fail to note that the problem of computing the (r )-closest neighbor of each point in S and the problem of computing the graph ARANN(S) of S are intimately related, in the sense that the solution to either of them immediately yields a solution to the other. For this reason in the remaining part of this work we shall focus on the problem of computing the graph ARANN(S) and we shall refer to this task, informally, as solving the ARANN problem. Referring again to Figure 3, the corresponding ARANN graph contains the directed edges is the (0; -)-closest neighbor of p 1 and p 3 is the (0; -)-closest neighbor of p 2 . The points p 3 and p 4 have no (0; -)-closest neighbor and so, in the ARANN graph they show up as isolated vertices. Several sequential algorithms for the computing the EMST, the GNG, the ARANN, and the RNS graphs of a set of points have been proposed in the literature [3, 7, 25, 26]. In particular, [3] has shown that the ARANN graph of a set of n points in the plane can be computed sequentially in O(n log n) time. The main contribution of this work is to present an algorithm to compute the ARANN graph of n points in O(1) time on a RM of size n \Theta n. We also show that our algorithm is optimal in the sense that n 2 processors are necessary to compute the ARANN of n points in O(1) time. It is not hard to see that the ANN problem is easier than the ARANN, because the ANN corresponds to the computation of ARANN for the particular directions 0 and 2-. Our second main contribution is to extend our ARANN algorithm to solve in O(1) time the problems of computing the Geographic Neighborhood Graph, the Relative Neighborhood Graph, and the Euclidean Minimum Spanning Tree of a set S of n points in the plane. As we already mentioned, Jang and Prasanna [5] have shown that the All Nearest Neighbor problem, of a set S of n points in the plane, can be solved in O(1) time on a RM of size n \Theta n. The key idea of the algorithm in [5] is as follows: In the first stage, the points are partitioned into n 1=4 horizontal groups each of n 3=4 points by n 1=4 \Gamma 1 horizontal lines and into n 1=4 vertical groups of n 3=4 points by n vertical lines. For each point, the nearest neighbor over the all points that are in the same horizontal or vertical group is retained as a candidate for the nearest neighbor over the whole set of points. Having computed the set of candidates, the second stage of the algorithm in [5] uses the fact that the candidates of at most 8 n points are not the correct nearest neighbors over all the points. So, by computing the nearest neighbor of these exceptional 8 n points, the ANN problem can be solved. If the angle is restricted, then this algorithm does not work, because it is possible that none of the candidates retained in stage 1 is the actual angle restricted nearest neighbor. This situation is depicted in Figure 6. We will develop new tools for dealing with the ARANN problem. These tools are interesting in their own right and may be of import in the resolution of other related problems. At the same time, it is clear that by using the ARANN algorithm, the ANN problem and the task of computing the GNG can be solved in O(1) time on an n \Theta n RM. Furthermore, the RNG can be computed in O(1) time on an n \Theta n RM and the EMST can be computed in O(1) time on an n \Theta n 2 RM. These algorithms are based on the fact that the RNG is a subgraph of GNG and EMST is a subgraph of RNG [25,26]. In Section 2 we demonstrate a lower bound on the size of a reconfigurable mesh necessary to compute the ARANN, GNG, RNG, and Figure None of the candidates of stage 1 are the true (0; -)-nearest neighbors EMST in O(1) time. Section 3 presents basic algorithms used by our ARANN algorithm. Section 4 presents our optimal ARANN algorithm and Section 5 presents algorithms for the GNG, RNG, and EMST. Finally, Section 6 offers concluding remarks and poses open problems. Let us consider the ARANN problem for the directions \Gamma-=2 and -=2. Consider a set of points on the x-axis such that point (a is assigned to the i-th column of an RM of size m \Theta n. After the computation of the ARANN, the processors of the i-th column know the (\Gamma-=2; -=2)-nearest neighbor of (a that a 1 ! a n=2 ! a for each point -=2)-nearest neighbor is the point (a i+n=2 ; 0). Therefore, information about n=2 points (a n=2 ; 0); (a n=2+1 must be transferred through the m links that connect the (n=2 \Gamma 1)-th column and the n=2-th column of the RM. Hence, time is required to solve the ARANN problem. Therefore, we have the following result. Theorem 2.1 processors are necessary to solve an instance of size n of the ARANN problem on the RM in O(1) time. Since the proof above can be applied to the GNG, RNG, and EMST, we have Corollary 2.2 processors are necessary to compute the GNG, RNG, and the EMST of n points in O(1) time on a RM. 3 Basic Algorithms This section reviews basic computational results on reconfigurable meshes that will be used in our subsequent algorithms. Recently, Lin et al. [12], Ben-Asher et al. [2], Jang and Prasanna [6], and Nigam and Sahni [17] have proved variants of the following result. Lemma 3.1 A set of n items stored one per processor in one row or one column of a reconfigurable mesh of size n \Theta n can be sorted in O(1) time. For a sequence of n numbers, its prefix-maxima is a sequence gg. The prefix-minima can be defined in the same way. For the prefix-maxima and prefix-minima, we have Lemma 3.2 Given a sequence of n numbers stored one per processor in one row of a reconfigurable mesh of size n \Theta n ffl , its prefix-maxima (resp. prefix-minima) can be computed in time for every fixed ffl ? 0. Proof. Olariu et al. [18] have shown to compute the maximum of n items in O(1) time on reconfigurable mesh of size n \Theta n. Essentially, if we assign n processors to each number, we can determine if there is a number larger than it. If such a number is not found, it is the maximum. Since an n \Theta n RM can find the maximum of n elements in O(1) time, an easy extension shows that an n \Theta n 2 RM can compute the prefix-maxima of n numbers in O(1) time. Based on this idea, we can devise an O(1) time algorithm for an n \Theta n ffl RM. Assume that each number is assigned to a row of the platform. Partition the sequence of n numbers into n ffl=2 sequences A 1 ; A ffl=2 each of which contains n 1\Gammaffl=2 numbers. Next, compute the (local) prefix-maxima of each A i (1 - i - n ffl=2 ) on an n 1\Gammaffl=2 \Theta n ffl submesh recursively. Let be the maximum within A i . Further, compute the (global) prefix- maxima of a sequence )g. This can be done in O(1) time by the O(1) time algorithm discussed above. Finally, for each number a j (2 A i ), compute the maximum of its local prefix-maximum and the global prefix-maximum maxfA that corresponds to the prefix-maximum of a j . Since the depth of the recursion is O(1=ffl), the maximum can be computed in O(1) time. 2 The prefix-sums of n binary values can be computed in a similar fashion. The reader is referred to [19] for the details. Lemma 3.3 For every fixed ffl ? 0, the prefix-sums of a binary sequence can be computed in O(1) time on a reconfigurable mesh of size n \Theta n ffl . Our next result assumes a reconfigurable mesh wherein a processor can connect, or fuse, an arbitrary number of ports [28]. On this platform, we will show basic graph algorithms that are essentially the same as those previously presented [28]. However, the number of processors is reduced by careful implementation of the algorithms to the RM. A graph E) consists of a set V of n vertices and an edge-set e edges where u E) is said to be numbered if the vertex set is ng. A graph E) is weighted if the edge-set is positive number. The reachability problem [28] for a vertex u of G involves determining all the vertices of G that can be reached by a path from u. Here, u is referred to as the source. Lemma 3.4 Given a numbered graph E) and a node u 2 V , the single source reachability problem can be solved in O(1) time on an n \Theta e RM, if each edge is assigned to a column of the RM. Proof. Let n), be the edge assigned to the i-th column of a RM of size n \Theta e. For each i, P connect their four ports into one. All the other Figure 7: Bus configuration for the reachability problem processors connect their E and W ports as well as their N and S ports in pairs. We note that the bus configuration thus obtained corresponds to the graph in the sense that the horizontal buses in row u i and row v i row are connected through the vertical bus in column i, as illustrated in Figure 7. Next, processor P (u; 1) sends a signal from its E port and every processor P (v; 1) reads its E port. It is not hard to see that vertex v is reachable from u if and only if processor (v; 1) has received the signal sent by P (u; 1). Therefore, the reachability problem can be solved in O(1) time on an n \Theta e reconfigurable mesh that allows all ports to be fused together.By using the single source reachability algorithm, we have the following lemma. Lemma 3.5 Given a numbered weighted graph G with n vertices and e edges, its Minimum Spanning Tree can be computed in O(1) time on a reconfigurable mesh of size e \Theta ne provided that every processor can fuse all its ports together. Proof. For each i, (1 - i - e), let G be the graph such that lexicographically larger than (w for each i, it is determined whether u i and v i are reachable in the graph G i . If so, then not an edge of MST, otherwise it is an MST edge. By Lemma 3.4 the reachability can be determined in O(1) time on an e \Theta n RM, so the MST edges can be determined in O(1) time on an e \Theta ne RM, as claimed. 2 4 An Optimal Algorithm for the ARANN Problem Consider a collection S of n points in the plane and directions r 1 and r 2 with 2-. The main goal of this section is to present an optimal O(1) time algorithm for computing the corresponding graph ARANN(S). We begin our discussion by pointing out a trivial suboptimal solution to the problem at hand. Lemma 4.1 For every fixed ffl ? 0, the task of solving an arbitrary instance of size n of the ARANN problem can be performed in O(1) time on a RM of size n \Theta n 1+ffl . Proof. Partition the RM into n submeshes of size n \Theta n ffl and assign each submesh to a point. Each point p find, in its own submesh, the nearest neighbor q over all the points )-dominated by p, and report the edge (p; q) as an edge of ARANN(S). We note that the task of finding the nearest neighbor of every point in S can be seen as an instance of the (prefix) minimum problem and can be solved in O(1) time by the algorithm of Lemma 3.2. 2 In the remainder of this section we will show how to improve this naive algorithm to run in O(1)-time on a RM of size n \Theta n. First, assume that the given directions are 0 and r, with that is, the angle of the closed region is acute. Consider a set of points in the plane stored one per processor in the first row of a RM of size n \Theta n such that for all i, (1 - i - n), processor P (1; i) stores point p i . The details of our algorithm are spelled out as follows. Step 1. Sort the points in S by y-coordinate and partition them into n 1=3 subsets Y of n 2=3 points each, such that the y-coordinate of all points in Y i is smaller than the y-coordinate of all points in Y Step 2. For each point p in S compute x 0 and sort the points by are x- and y-coordinate of p. Next, partition the points into Figure 8: Partitioning into X's and Y 's subsets points each, such that for every choice of points p in Step 3. For each point p in X i , r)-nearest neighbor X(p) over all points in X Step 4. For each point p in Y i , r)-nearest neighbor Y (p) over all points in Y Step 5. For each i and j, (1 - For each point its (0; r)-nearest neighbor Z(p) over all points in Z i;j ; Step 6. For each point p, find the closest of the three points X(p), Y (p), and Z(p), and return it as the (0; r)-nearest neighbor of p. We refer the reader to Figure 8 for an example, where (Actually, 4, but this inconsistency is nonessential.) Our next goal is to show that the algorithm we just presented can be implemented to run in O(1) time on a RM of size n \Theta n. By virtue of Lemma 3.1, Steps 1 and 2 can be completed in O(1) time on a RM of size n \Theta n. In Step 3, a submesh of size n \Theta n 2=3 can Minima(P )-=3 Figure 9: Illustrating Maxima(S) and Minima(S) for directions 0 and -=3 be assigned to each X i . Consequently, Step 3 can be completed in O(1) time by the naive algorithm of Lemma 4.1. In the same way, Step 4 can be implemented to run in O(1) time. Step 6 involves only local computation and can be performed, in the obvious way, in O(1) time. The remainder of this section is devoted to showing that with a careful implementation Step 5 will run in O(1) time. We shall begin by presenting a few technical results that are key in understanding why our implementation works. Consider, as before, a set of points. A point p i in S is a (0; r)-maximal (resp. minimal) point of S if p i is not (0; r)-dominated by (resp. does not (0; r)-dominate) any other point in S. We shall use Maxima(S) (resp. Minima(S)) to denote the chain of all maximal points in S specified in counter-clockwise order (resp. all minimal points in clockwise order). These concepts are illustrated in Figure 9 for the directions 0 and -=3. Next, we propose to show that only points in Maxima(X may have their (0; r)- nearest neighbor in Z i;j . Moreover, if the (0; r)-nearest neighbor of a point in Maxima(X i "Y j ) lies in Z i;j , then it can only be in Minima(Z i;j ). Lemma 4.2 For three points direction r (0 Proof. Consider the triangle let u be the point where the edge 3 of this triangle cuts one of the rays in the directions 0 or r emanating from p 2 . We shall assume, without loss of generality, that the point u lies on the ray in the direction 0. must be larger than -=2. This implies that that the angle (which is larger that the angle larger than -=2. Hence, Note that Lemma 4.2 does not hold for a direction r larger than -=2. This is the reason we restricted the angle r to be less than -=2. Consider two sets of points P and Q such that all points in P (0; r)-dominate all the points in Q. Let ARANN r (P; Q) be a set of edges (p; q) such that p belongs to P , q belongs to Q, and q is the (0; r)-nearest neighbor of p. We have the following lemma. 4.3 No two edges in ARANN r (P; Q) intersect. Proof. Suppose not: some two edges (p; q) and (p intersect at a point u. ?From the triangle inequality applied to the triangles puq 0 and p 0 uq we have But now, (1) implies that either must hold. However, this contradicts the assumption that both (p; q) and (p are edges in ARANN r Hence, the graph ARANN r (P; Q) is planar. Lemma 4.4 If (p; Proof. If p does not belong to Minima(P ), then there exists a point p 0 in Minima(P ) that dominates q. But now, Lemma 4.2 guarantees that a contradiction. In case q does not belong to Maxima(Q) we can show a contradiction in an essentially similar fashion. 2 Lemma 4.4 has the following important consequence. Corollary 4.5 ARANN r Corollary 4.5 guarantees that in order to compute ARANN r (P; Q) examining all the pairs of points p in P and q in Q is not necessary: all that is needed is to compute ARANN r (Minima(P ); Maxima(Q)). To pursue this idea further, write Minima(P and assume that for some fixed ffl, (0 motivates the following approach to compute ARANN r (Minima(P ); Maxima(Q)). Let sample(Minima(P )) be a subset of n ffl=2 points fp n ffl=2 of Minima(P ) and partition Minima(P ) into n ffl=2 chains in such a way that and ffl for every k, g. Refer to Figure 10 for an illustration. For each k (1 (2 Q) be the (0; r)-nearest neighbor of p kn ffl=2 over all Q. Observe that the points q j k thus defined induce a partition of the set Q into n ffl=2 chains ffl=2 such that and ffl for every k, g. We note that q j k 4.3 guarantees that in order to compute the (0; r)- nearest neighbor of a point p in P k with respect to Maxima(Q), we can restrict ourselves to computing the (0; r)-nearest neighbor of p over Q k . In other words, ARANN r ARANN r (P k ; Maxima(Q)). Therefore, the task of computing ARANN r (Minima(P ); Maxima(Q)) reduces to that of computing ARANN r The sampling strategy outlined above leads to the following algorithm for computing ARANN r (Minima(P ); Maxima(Q)) in O(1) time on a RM of size n \Theta n ffl , for some fixed We assume that each point in Minima(P ) has been assigned to one column and each point in Maxima(Q) has been assigned to one row of the RM. Partition columnwise the original RM of size n \Theta n ffl into n ffl=2 submeshes of size n \Theta n ffl=2 each. In the k-th such submesh, 1 - k - n ffl=2 , compute d(p kn ffl=2 ; q) for all points q in Maxima(Q); Step 2 For every k, (1 - k - n ffl=2 ), use the k-th submesh to compute Figure 10: Partitioning Minima(P ) and Maxima(Q) ffl the (0; r)-nearest neighbor q j k of p kn ffl=2 by finding the smallest of the distances computed above; ffl using the point q determine Q k . If then the (0; r)-nearest neighbor of every point p in P k is precisely q j k , and thus We shall, therefore, assume that jQ k j - 2 for all k; Step 3 Partition the given RM of size n\Thetan ffl rowwise into n ffl=2 submeshes as follows. For each the k-th submesh has size (jQ k through j k of the original mesh. In the k-th submesh compute ARANN r follows: Step 3.1 Partition the k-th submesh of size (jQ k submeshes each of size (jQ k Assign each point p in P k to each such submesh, and compute d(p; q) for each point q in Q k . Step 3.2 By using the algorithm of Lemma 3.2, compute the minimum of all d(p; q) over all q in Q k in each submesh assigned to p and return the (0; r)-nearest neighbor of p. The reader should have no difficulty to confirm that Steps 1 and 2 can be performed in constant time. By using the prefix-sums algorithm of Lemma 3.3, the partitioning in Step 3 can be performed in constant time. Steps 3.1 and 3.2 can also be implemented to run in constant time. To summarize, we have proved the following result. Lemma 4.6 If Minima(P ) and Maxima(Q) have been assigned to the columns and the rows, respectively, of a RM of size n \Theta n ffl , then ARANN r (Minima(P ); Maxima(Q)) can be computed in O(1) time for every fixed We are now in a position to discuss an O(1) time implementation of Step 5 that computes Z(p) for all p. For simplicity, assume that the RM has n \Theta 2n processors. Step 5.1 Partition the n \Theta 2n RM columnwise into n 2=3 submeshes. For be of size n \Theta (jX Step 5.2 In each submesh R(i; Step 5.3 In each submesh R(i; Step 5.4 In each submesh R(i; i;j )). If Step 5.1 is complicated, because the number of columns of each submesh is different. The partitioning specified in Step 5.1 can be completed in O(1) time by using the sorting algorithm of Lemma 3.1: sort the n points in lexicographical order of (x 0 (p); y(p)). Clearly, for each i and j, all points in X are consecutive in the sorted points. If the smallest point in and the largest one are s-th and t-th in the sorted order, then the submesh assigned columns s Step 5.1 can be completed in O(1) time. Steps 5.2 can be completed as follows: Let k. Compute the postfix-maxima of g, by the algorithm of Lemma 3.2. Then, and only if maxfy(p k+1 ); y(p k+2 ); y(p k+3 can be computed in O(1) time. Step 5.3 can be completed in the same way. To apply the algorithm of Lemma 4.6 to Step 5.4, a serial number must be assigned to the points in Minima(X and those in Minima(Z 0 i;j ). These numbering can be obtained in the obvious way by using the prefix-sum algorithm of Lemma 3.3. Then, by executing the algorithm of Lemma 4.6, Step 5.4, can be completed in O(1) time. Therefore, ARANN r (S) can be computed in O(1) time on an n \Theta 2n RM. Since the algorithm above that uses n \Theta 2n processors, can be implemented on an n \Theta n RM by a simple scheduling technique, ARANN r (S) can also be computed in O(1) time on an n \Theta n RM. Furthermore, the ARANN r (S) for r -=2 can be computed by partition the angle into several acute angles. For example, the ARANN 2-=3 (S) can be computed as follows: 1. compute ARANN -=2 (S), 2. rotate all the points in S by an angle of -=2 clockwise about the origin, 3. compute ARANN -=6 (S), and 4. for each point, determined the nearest of the two points computed in 1 and 2 that corresponds to the nearest point for ARANN 2-=3 (S). Therefore, we have proved the following result. Theorem 4.7 Given an arbitrary set of n points in the plane and a direction r, (0 ! r - 2-), the corresponding instance of the ARANN problem can be solved in O(1) time on a reconfigurable mesh of size n \Theta n. 5 Application to Proximity Problems The goal of this section is to show that the result of Theorem 4.7 leads to O(1) time algorithms for the GNG and RNG and the EMST. To begin, from Theorem 4.7, can be computed in O(1) time on an n \Theta n RM. Therefore, we have Corollary 5.1 Given n points in the plane, its GNG can be computed in O(1) time on a reconfigurable mesh of size n \Theta n. Since each GNG i is planar, GNG i has at most 3n \Gamma 6 edges and thus GNG has at most edges. Furthermore, RNG is a subgraph of GNG [7]. Therefore, we have Theorem 5.2 Given n points in the plane, the corresponding RNG can be computed in O(1) time on an n \Theta n RM. Proof. For each edge in GNG, check whether there is a point in its lune. If such a point does not exist, this edge is an RNG edge, and vice versa. This checking can be done in O(1) time by n processors for each edge. Since the GNG of n points has at most 18n \Gamma 36 edges, the RNG can be computed in O(1) time on an n \Theta n RM. 2 Furthermore, by using the MST algorithm for Lemma 3.5, we have the following theorem. Theorem 5.3 Given n points in the plane, its EMST can be computed in O(1) time on a reconfigurable mesh of size n \Theta n 2 . Proof. By applying the MST algorithm for a graph to the RNG, the EMST of n points can be computed, because EMST is a subgraph of RNG. Since the RNG has at most 3n \Gamma 6 edges, an n \Theta n 2 RM is sufficient to compute EMST in O(1) time. 2 6 Concluding Remarks We have shown an optimal algorithm on a reconfigurable mesh for computing the Angle Restricted All Nearest Neighbor problem. By using this algorithm, we have also shown optimal algorithms on a reconfigurable mesh for computing the Geographical Neighborhood Graph and the Relative Neighborhood Graph. These algorithms are optimal in the sense that there is no O(1)-time algorithm that solves instances of size n of these problems on an reconfigurable mesh. Furthermore, we have also shown that the Euclidean Minimum Spanning Tree of a set of n points in the plane can be computed in O(1) time on an n \Theta n 2 reconfigurable mesh, provided that every processor can fuse its ports. It remains open to find an O(1)-time EMST algorithm on an n \Theta n reconfigurable mesh that matches the lower bound. --R Parallel Computational Geometry The power of reconfiguration Voronoi diagrams based on convex functions Pattern Classification and Scene Analysis Parallel geometric problems on the reconfigurable mesh An optimal sorting algorithm on reconfigurable meshes Computing relative neighborhood graphs in the plane IEEE Transactions on Computers Reconfigurable buses with shift switching - concepts and appli- cations Sorting in O(1) time on a reconfigurable mesh of size N IEEE Transactions on Parallel and Distributed Systems Hardware support for fast reconfigurability in processor arrays Parallel Computations on Reconfigurable Meshes Connection autonomy in SIMD computers: a VLSI implemen- tation Sorting n numbers on n Fundamental Data Movement International Journal of High Speed Computing Fundamental Algorithms on Reconfigurable Meshes Computational Geometry - An Introduction On the ultimate limitations of parallel processing Bus automata bit serial associate processor The gated interconnection network for dynamic programming the relative neighborhood graph with an application to minimum spanning trees The relative neighborhood graph of a finite planar set The symmetric all-furthest neighbor problem Constant time algorithms for the transitive closure problem and its applications IEEE Transactions on Parallel and Distributed Systems The image understanding architecture --TR --CTR Hongga Li , Hua Lu , Bo Huang , Zhiyong Huang, Two ellipse-based pruning methods for group nearest neighbor queries, Proceedings of the 13th annual ACM international workshop on Geographic information systems, November 04-05, 2005, Bremen, Germany Ramachandran Vaidyanathan , Jerry L. Trahan , Chun-ming Lu, Degree of scalability: scalable reconfigurable mesh algorithms for multiple addition and matrix-vector multiplication, Parallel Computing, v.29 n.1, p.95-109, January Dimitris Papadias , Yufei Tao , Kyriakos Mouratidis , Chun Kit Hui, Aggregate nearest neighbor queries in spatial databases, ACM Transactions on Database Systems (TODS), v.30 n.2, p.529-576, June 2005

reconfigurable mesh;mobile computing;ARANN;proximity problems;lower bounds;ANN

264559

Optimal Registration of Object Views Using Range Data.

AbstractThis paper deals with robust registration of object views in the presence of uncertainties and noise in depth data. Errors in registration of multiple views of a 3D object severely affect view integration during automatic construction of object models. We derive a minimum variance estimator (MVE) for computing the view transformation parameters accurately from range data of two views of a 3D object. The results of our experiments show that view transformation estimates obtained using MVE are significantly more accurate than those computed with an unweighted error criterion for registration.

Introduction An important issue in the design of 3D object recognition systems is building models of physical objects. Object models are extensively used for synthesizing and predicting object appearances from desired viewpoints and also for recognizing them in many applications such as robot navigation and industrial inspection. It becomes necessary on many occasions to construct models from multiple measurements of 3D objects, especially when a precise geometric model such as a CAD description is not available and cannot be easily obtained. This need is felt particularly with 3D free-form objects, such as sculptures and human faces that may not possess simple analytical shapes for representation. With growing interest in creating virtual museums and virtual reality functions such as walk throughs, creating computer images corresponding to arbitrary views of 3D scenes and objects remains a challenge. Automatic construction of 3D object models typically involves three steps: (i) data acquisition from multiple viewpoints, (ii) registration of views, and (iii) integration. Data acquisition involves obtaining either intensity or depth data of multiple views of an object. Integration of multiple views is dependent on the representation chosen for the model and requires knowledge of the transformation relating the data obtained from different viewpoints. The intermediate step, registration, is also known as the correspondence problem [1] and its goal is to find the transformations that relate the views. Inaccurate registration leads to greater difficulty in seamlessly integrating the data. It ultimately affects surface classification since surface patches from different views may be erroneously merged, resulting in holes and discontinuities in the merged surface. For smooth merging of data, accurate estimates of transformations are vital. In this paper we focus on the issue of pairwise registration of noisy range images of an object obtained from multiple viewpoints using a laser range scanner. We derive a minimum variance estimator to compute the transformation parameters accurately from range data. We investigate the effect of surface measurement noise on the registration of a pair of views and propose a new method that improves upon the approach of Chen and Medioni [1]. We have not seen any work that reports to date, establishing the dependencies between the orientation of a surface, noise in the sensed surface data, and the accuracy of surface normal estimation and how these dependencies can affect the estimation of 3D transformation parameters that relate a pair of object views. We present a detailed analysis of this "orientation effect" with geometrical arguments and experimental results. Previous Work There have been several research efforts directed at solving the registration problem. While the first category of approaches relies on precisely calibrated data acquisition device to determine the transformations that relate the views, the second kind involves techniques to estimate the transformations from the data directly. The calibration-based techniques are inadequate for constructing a complete description of complex shaped objects as views are restricted to rotations or to some known viewpoints only and therefore, the object surface geometry cannot be exploited in the selection of vantage views to obtain measurements. With the second kind, inter-image correspondence has been established by matching the data or the surface features derived from the data [2]. The accuracy of the feature detection method employed determines the accuracy of feature correspondences. Potmesil [3] matched multiple range views using a heuristic search in the view transformation space. Though quite general, this technique involves searching a huge parameter space, and even with good heuristics, it may be computationally very expensive. Chen and Medioni avoid the search by assuming an initial approximate transformation for the registration, which is improved with an iterative algorithm [1] that minimizes the distance from points in a view to tangential planes at corresponding points in other views. Besl and McKay [4], Turk and Levoy [5], Zhang [6] employ variants of the iterated closest-point algo- rithm. Blais and Levine [7] propose a reverse calibration of the range-finder to determine the point correspondences between the views directly and use stochastic search to estimate the transforma- tion. These approaches, however, do not take into account the presence of noise or inaccuracies in the data and its effect on the estimated view-transformation. Our registration technique also uses a distance minimization algorithm to register a pair of views, but we do not impose the requirement that one surface has to be strictly a subset of the other. While our approach studies in detail the effect of noise on the objective function [1] that is being minimized and proposes an improved function to register a pair of views, Bergevin et al. [8, 9] propose to register all views simultaneously to avoid error accumulation due to sequential registration. Haralick et al. [10] have also showed that a weighted least-squares technique is robust under noisy conditions under various scenarios such as 2D-2D, 3D-3D image registration. 3 A Non-Optimal Algorithm for Registration Two views, P and Q of a surface are said to be registered when any pair of points, p and q from the two views representing the same object surface point can be related to each other by a single rigid 3D spatial transformation T , so that 8p 2 obtained by applying the transformation T to p, and T is expressed in homogeneous coordinates as a function of three rotation angles, ff, fi and fl about the x, y and z axes respectively, and three translation parameters, t x , t y and t z . The terms "view" and "image" are used interchangeably in this paper. The approach of [1] is based on the assumption that an approximate transformation between two views is already known and the goal is to refine the initial estimate to obtain more accurate global registration. The following objective function was used to minimize the distances from surface points in one view to another iteratively: where T k is the 3D transformation applied to a control point at the kth iteration, 0g is the line normal to P at is the intersection point of surface Q with the transformed line T k l i , n k is the normal to Q at q k is the tangent plane to Q at q k i and d s is the signed distance from a point to a plane as given in Eq. (2). Note that '\Delta' stands for the scalar product and `\Theta' for the vector product. Figure 1 illustrates the distance measure d s between surfaces P and Q. This registration algorithm thus finds a T that minimizes e k , using a least squares method iteratively. The tangent plane S k serves as a local linear approximation to the surface Q at a point. The intersection point q k i is an approximation to the actual corresponding point q i that is unknown at each iteration k. An initial T 0 that approximately registers the two views is used to start the iterative process. The signed distance d s , from a transformed point T to a tangential (b) (a) Figure 1: Point-to-plane distance: (a) Surfaces P and Q before the transformation T k at iteration k is applied; (b) distance from the point p i to the tangent plane S k i of Q. define the transformed point and the tangential plane respectively. Note that (x; is the transpose of the vector (x; z). By minimizing the distance from a point to a plane, only the direction in which the distance can be reduced is constrained. The convergence of the process can be tested by verifying that the difference between the errors e k at any two consecutive iterations is less than a pre-specified threshold. The line-surface intersection given by the intersection of the normal line l i and Q is found using an iterative search near the neighborhood of prospective intersection points. 4 Registration and Surface Error Modeling Range data are often corrupted by measurement errors and sometimes lack of data. The errors in surface measurements of an object include scanner errors, camera distortion, and spatial quanti- zation, and the missing data can be due to self-occlusion or sensor shadows. Due to noise, it is generally impossible to obtain a solution for a rigid transformation that fits two sets of noisy three-dimensional points exactly. The least-squares solution in [1] is non-optimal as it does not handle the errors in z measurements and it treats all surface measurements with different reliabilities equally. Our objective is to derive a transformation that globally registers the noisy data in some optimal sense. With range sensors that provide measurements in the form of a graph surface z = f(x; y), it is assumed that the error is present along the z axis only, as the x and y measurements are usually laid out in a grid. There are different uncertainties along different surface orientations and they need to be handled appropriately during view registration. Furthermore, the measurement error is not uniformly distributed over the entire image. The error may depend on the position of a point, relative to the object surface. A measurement error model dealing with the sensor's viewpoint has been previously proposed [11] for surface reconstruction where the emphasis was to recover straight line segments from noisy single scan 3D surface profiles. In this paper, we show that the noise in z values affects the estimation of the tangential plane parameters differently depending on how the surface is oriented. Since the estimated tangential plane parameters play a crucial role in determining the distance d s which is being minimized to estimate T , we study the effect of noise on the estimation of the parameters of the plane fitted and on the minimization of d s . The error in the iterative estimation of T is a combined result of errors in each control point (x; z) T from view 1 and errors in fitting tangential planes at the corresponding control points in view 2. 4.1 Fitting Planes to Surface Data with Noise Surface normal to the plane uncertainty region in Z Horizontal plane Surface normal to the plane uncertainty region in Z Effective uncertainty region affecting the normal Inclined plane (a) (b) Figure 2: Effect of noise in z measurements on the fitted normal: (a) when the plane is horizontal; (b) when it is inclined. The double-headed arrows indicate the uncertainty in depth measurements. Figure 2 illustrates the effect of noise in the values of z on the estimated plane parameters. For the horizontal plane shown in Figure 2(a), an error in z (the uncertainty region around z) directly affects the estimated surface normal. In the case of an inclined plane, the effect of errors in z on the surface normal to the plane is much less pronounced as shown in Figure 2(b). Here, even if the error in z is large, only its projected error along the normal to the plane affects the normal estimation. This projected error becomes smaller than the actual error in z as the normal becomes more and more inclined with respect to the vertical axis. Therefore, our hypothesis is that as the angle between the vertical (Z) axis and the normal to the plane increases, the difference between the fitted plane parameters and the actual plane parameters should decrease. We carried out simulations to study the actual effect of the noise in the z measurements on estimating the plane parameters and to verify our hypothesis. The conventional method for fitting planes to a set of 3D points uses a linear least squares algorithm. This linear regression method implicitly assumes that two of the three coordinates are measured without errors. However, it is possible that in general, surface points can have errors in all three coordinates, and surfaces can be in any orientation. Hence, we used a classical eigenvector method (principal components analysis) [12] that allows us to extract all linear dependencies. Let the plane equation be Ax +By set of surface measurements used in fitting a plane at a point on a surface. Let and be the vector containing the plane parameters. We solve for the vector h such that kAhk is minimized. The solution of h is a unit eigenvector of A T A associated with the smallest eigenvalue. We renormalize h such that (A; B; C) T is the unit normal to the fitted plane and D is the distance of the plane from the origin of the coordinate system. This planar fit minimizes the sum of the squared perpendicular distances between the data points and the fitted plane and is independent of the choice of the coordinate frame. In our computer simulations, we used synthetic planar patches as test surfaces. The simulation data consisted of surface measurements from planar surfaces at various orientations with respect to the vertical axis. Independent and identically distributed (i.i.d.) Gaussian and uniform noise with zero mean and different variances were added to the z values of the synthetic planar data. The standard deviation of the noise used was in the range 0.001-0.005 in. as this realistically models the error in z introduced by a Technical Arts 100X range scanner [13] that was employed to obtain the range data for our experiments. The planar parameters were estimated using the eigenvector method at different surface points with a neighborhood of size 5 \Theta 5. The error E fit in fitting the plane was defined as the norm of the difference between the actual normal to the plane and the normal of the fitted plane estimated with the eigenvector method. Figure 3(a) shows the plot of versus the orientation (with respect to the vertical axis) of the normal to the simulated plane at different noise variances. The plot shows E fit averaged over 1,000 trials at each orientation. It can be seen from Figure 3(a) that in accordance with our hypothesis, the error in fitting a plane decreases with an increase in the angle between the vertical axis and the normal to the plane. When the plane is nearly horizontal (i.e., the angle is small), the error in z entirely contributes to fit as indicated by Figure 2(a). The error plots for varying amounts of variance were observed Squared difference between the actual and estimated normals Angle between the normal to the plane and vertical axis "std_dev_of_noise=0.0" "std_dev_of_noise=0.001" "std_dev_of_noise=0.002" "std_dev_of_noise=0.003" "std_dev_of_noise=0.004" Squared Difference betweeb the estimated and actual normals Angle between the normal to the plane and vertical axis "std_dev_of_noise=0.0" "std_dev_of_noise=0.001" "std_dev_of_noise=0.002" "std_dev_of_noise=0.003" "std_dev_of_noise=0.004" "std_dev_of_noise=0.005" (a) (b) Figure 3: Effect of i.i.d. Gaussian noise in z measurements on the plane: (a) estimated using eigenvector approach; (b) estimated using linear regression. to have the same behavior with orientation as shown in Figure 3(a). Similar curves were obtained with a uniform noise model also [14]. These simulations confirm our hypothesis about the effect of noise in z on the fitted plane parameters as the surface orientation changes. We repeated the simulations using the linear regression method to fit planes to surface data. We refer the reader to [14] for details. Figure 3(b) shows the error E fit between the fitted and actual normals to the plane at various surface orientations when i.i.d. Gaussian noise was added to the z values. Our hypothesis is well supported by this error plot also. 4.2 Proposed Optimal Registration Algorithm Since the estimated tangential plane parameters are affected by the noise in z measurements, any inaccuracies in the estimates in turn, influence the accuracy of the estimates of d s , thus affecting the error function being minimized during the registration. Further, errors in z themselves affect d s estimates (see Eq. 2). Therefore, we characterize the error in the estimates of d s by modeling the uncertainties associated with them using weights. Our approach is inspired by the Gauss-Markov theorem [15] which states that an unbiased linear minimum variance estimator of a parameter vector m when is the one that minimizes (y y is a random noise vector with zero mean and covariance matrix \Gamma y . Based on this theorem, we formulate an optimal error function for registration of two object views as ds ds is the estimated variance of the distance d s . When the reliability of a z value is low, the variance of the distance oe 2 ds is large and the contribution of d s to the error function is small, and when the reliability of the z measurement is high, oe 2 ds is small, and the contribution of d s is large; d s with a minimum variance affects the error function more. One of the advantages of this minimum variance criterion is that we do not need the exact noise distribution. What we require only is that the noise distribution be well-behaved and have short tails. In our simulations, we employ both Gaussian and uniform noise distributions to illustrate the effectiveness of our method. We need to know only the second-order statistics of the noise distribution, which in practice can often be estimated. 4.3 Estimation of the Variance oeds We need to estimate oe 2 ds to model the reliability of the computed d s at each control point, which can then be used in our optimal error function in Eq. (4). Let the set of all the surface points be denoted by P and the errors in the measurements of these points be denoted by a random vector ffl. The error e ds in the distance computed is due to the error in the estimated plane parameters and the error in the z measurement, and therefore is a function of P and ffl. Since we do not know ffl, if we can estimate the standard deviation of e ds (with ffl as a random vector) from the noise-corrupted surface measurements P , we can use it in Eq. (4). 4.3.1 Estimation of oe 2 ds Based on Perturbation Analysis Perturbation analysis is a general method for analyzing the effect of noise in data on the eigenvectors obtained from the data. It is general in the sense that errors in x, y and z can all be handled. This analysis is also related to the general eigenvector method that we studied for plane estimation. The analysis for estimating oe 2 ds is simpler if we use linear regression method to do plane fitting [14]. Since we fit a plane with the eigenvector method that uses the symmetric matrix computed from the (x; measurements in the neighborhood of a surface point, we need to analyze how a small perturbation in the matrix C caused by the noise in the measurements can affect the eigenvectors. Recall that these eigenvectors determine the plane parameters (A; B; C; D) T which in turn determine the distance d s . We assume that the noise in the measurements has zero mean and some variance and that the latter can be estimated empirically. The correlation in noise at different points is assumed to be negligible. Estimation of correlation in noise is very difficult but even if we estimate it, its impact may turn out to be insignificant. We estimate the standard deviation of errors in the plane parameters and in d s on the basis of the first-order perturbations, i.e., we estimate the "linear terms" of the errors. Before we proceed, we discuss some of the notational conventions that are used: I m is an m \Theta m identity matrix; diag(a; b) is a 2 \Theta 2 diagonal matrix with a and b as its diagonal elements. Given a noise-free matrix A, its noise-matrix is denoted by \Delta A and the noise-corrupted version of A is denoted by . The vector ffi is used to indicate the noise vector, We use \Gamma with a corresponding subscript to specify the covariance matrix of the noise vector/matrix. For a given matrix vector A can be associated with it as A thus consists of the column vectors of A that are lined up together. As proved in [16], if C is a symmetrical matrix formed from the measurements and h is the parameter vector (A; B; C; D) T given by the eigenvector of C associated with the smallest eigenvalue, say - then the first-order perturbation in the parameter vector h is given by where and H is an orthonormal matrix such that A is a 4 \Theta 4 noise or perturbation matrix associated with A T A. If \Delta A T A can be estimated, then the perturbation ffi h in h can be estimated by a first-order approximation as in Eq. (5). We estimate \Delta A T A from the perturbation in the surface measurements. We assume for the sake of simplicity of analysis that only the z component of a surface measurement errors, with this general model. This analysis is easily and directly extended to include errors in x and y if their noise variances are known. Let z i have additive errors ffi z i We then get If the errors in z at different points on the surface have the same variance oe 2 , we get the covariance matrix where Now, consider the error in h. As stated before, we have In the above equation, we have rewritten the matrix \Delta A T A as a vector ffi A T A and moved the perturbation to the extreme right of the expression. Then the perturbation of the eigenvector is the linear transformation (by matrix G h ) of the perturbation vector ffi A T A . Since we have \Gamma A T A ), we need to relate ffi A T A to ffi A T . Using a first-order approximation [16], we get A A: (12) Letting A where G A T A is easily determined from the equation G A T are matrices with 4 \Theta n submatrices F ij and G ij a ji I 4 , and G ij is a 4 \Theta 4 matrix with the ith column being the column vector A j and all other columns being zero. Thus, we get Then the covariance matrix of h is given by The distance d s is affected by the errors in the estimation of the plane parameters, and the z measurement in Therefore, the error variance in d s is ds @ds @A @ds @ds @ds @D @ds @z \Theta [\Gamma hz @ds @A @ds @ds @ds @D @ds The covariance matrix \Gamma hz is given by Once the variance of d s , oe 2 ds is estimated, we employ it in our optimal error function: ds 4.3.2 Simulation Results Figure 4(a) shows the plot of the actual standard deviation of the distance d s versus the orientation of the plane with respect to the vertical axis. Note that the mean of d s is zero when the Actual standard deviation of distance d_s Angle between the normal to the plane and vertical axis "std_dev_of_noise=0.0" "std_dev_of_noise=0.001" "std_dev_of_noise=0.002" "std_dev_of_noise=0.003" "std_dev_of_noise=0.004" Actual standard deviation of distance d_s Angle between the normal to the plane and vertical axis "std_dev_of_noise=0.0" "std_dev_of_noise=0.001" "std_dev_of_noise=0.002" "std_dev_of_noise=0.003" "std_dev_of_noise=0.004" Estimated standard deviation of distance d_s Angle between the normal to the plane and vertical axis "std_dev_of_noise=0.0" "std_dev_of_noise=0.001" "std_dev_of_noise=0.002" "std_dev_of_noise=0.003" "std_dev_of_noise=0.004" "std_dev_of_noise=0.005" (a) (b) Figure 4: Standard deviation of d s versus the planar orientation with a Gaussian noise model: (a) actual oe ds ; (b) estimated oe ds using the perturbation analysis. points are in complete registration and when there is no noise. We generated two views of synthetic planar surfaces with the view transformation between them being an identity transformation. We experimented with the planar patches at various orientations. We added uncorrelated Gaussian noise independently to the two views. Then we estimated the distance d s at different control points using Eq. (2) and computed its standard deviation. The plot shows the values averaged over 1,000 trials. As indicated by our hypothesis, the actual standard deviation of d s decreases as the planar orientation goes from being horizontal to vertical. As the variance of the added Gaussian noise to the z measurements increases, oe ds also increases. Similar results were obtained when we added uniform noise to the data [14]. We compared the actual variance with the estimated variance of the distance (Eq. (16)) in order to verify whether our modeling of errors in z values at various surface orientations is correct. We computed the estimated variance of the distance d s using our error model using Eq. (16) with the same experimental setup as described above. Figure 4(b) illustrates the behavior of the estimated standard deviation of d s as the inclination of the plane (the surface orientation) changes. A comparison of Figures 4(a) and 4(b) shows that both the actual and the estimated standard deviation plots have similar behavior with varying planar orientation and their values are proportional to the amount of noise added. This proves the correctness of our error model of z and its effect on the distance d s . Simulation results, when we repeated the experiments to compute both the actual and the estimated oe ds using the planar parameters estimated with the linear regression method, were similar to those shown in Figure 4. This also demonstrates the important fact that the method used for planar fitting does not bias our results. 5 View Registration Experiments In this section we demonstrate the improvements in the estimation of view transformation parameters on real range images using our MVE. We will henceforth refer to Chen and Medioni's technique [1] as C-M method. We obtained range images of complex objects using a Technical Arts laser range scanner. We performed uniform subsampling of the depth data to locate the control points in view 1 that were to be used in the registration. From these subsampled points we chose a fixed number of points that were present on smooth surface patches. The local smoothness of the surface was verified using the value of residual standard deviation resulting from the least-squares fitting of a plane in the neighborhood of a point. A good initial guess for the view transformation was determined automatically when the range images contained the entire object surface and the rotations of the object in the views were primarily in the plane. Our method is based on estimating an approximate rotation and translation by aligning the major (principal) axes of the object views [14]. Figures 5(a) and 5(c) depict the two major axes of the objects. We used this estimated transformation as an initial guess for the iterative procedure in our experiments, so that no prior knowledge of the sensor placement was needed. Experimental results show the effectiveness of our method in refining such rough estimates. The same initial guess was used with the C-M method and the proposed MVE. We employed Newton's method for minimizing the error function iteratively. In order to measure the error in the estimated rotation parameters, we utilize an error measure that does not depend on the actual rotation parameters. The relative error of rotation matrix R, ER is defined [16] to be R is an estimate of R. Since R, the geometric sense of ER is the square root of the mean squared distance between the three unit vectors of the rotated orthonormal frames. Since the frames are orthonormal, 3. The error in translation, E t is defined as the square root of the sum of the squared differences between the estimated and actual t x , t y and t z values. 5.1 Results Figure 5 shows the range data of a cobra head and a Big-Y pipe. The figure renders depth as pseudo intensity and points almost vertically oriented are shown darker. View 2 of the cobra head was obtained by rotating the surface by 5 ffi about the X axis and 10 ffi about the Z axis. Table 1 shows (a) (b) (c) (d) Figure 5: Range images and their principal axes: (a) View 1 of a Cobra head; (b) View 2 of the cobra head; (c) Big-Y pipe data generated from its CAD model; (d) View 2 of Big-Y pipe. the values of ER and E t for the cobra views estimated using only as few as 25 control points. It can be seen that the transformation parameters obtained with the MVE are closer to the ground truth than those estimated using the unweighted objective function of C-M method. Even when more control points (about 156) were used, the estimates using our method were closer to the ground truth than those obtained with the C-M method [14]. We also show the performance of our method when the two viewpoints are substantially different and the depth values are very noisy. Figure 5 shows two views of the Big-Y pipe generated from the CAD model. The second view was generated by rotating the object about the Z axis by 45 ffi . We also added Gaussian noise with mean zero and standard deviation of 0:5 mm to the z values of the surfaces in view 2. Table 2 shows ER and E t computed with 154 control points. It can be seen from these tables that the transformation matrix, especially the rotation matrix obtained with the MVE is closer to the ground truth than that obtained using C-M method. The errors in translation components of the final transformation estimates are mainly due to the approximate initial guess. Parameters Actual C-M MVE value method Parameters Actual C-M MVE value method Table 1: Estimated transformation for Table 2: Estimated transformation for the cobra views. the Big-Y views. Our method refined these initial values to provide a final solution very close to the actual values. Our method also handled large transformations between views robustly. With experiments on range images of facial masks, we found that even when the depth data were quite noisy owing to the roughness of the surface texture of the objects and also due to self-occlusion, more accurate estimates of the transformation were obtained with the MVE. When the overlapping object surface between the views is quite small, the number of control points available for registration tends to be small and in such situations also the MVE has been found to have substantial improvement in the accuracy of the transformation estimate. Note that measurement errors are random and we minimize the expected error in the estimated solution. However, our method does not guarantee that every component in the solution will have a smaller error in every single case. We also used the MVE for refining the pose estimated using cosmos-based recognition system for free-form objects [14]. The rotational component of the transformation of a test view of Vase2 (View 1 shown in Figure 6(a)) relative to its best-matched model view (View 2 shown in Figure was estimated using surface normals of corresponding surface patch-groups determined by the recognition system. A total of 10 pairs of corresponding surface patch-groups was used to estimate the average rotation axis and the angle of rotation. These rotation parameters were used to compute the 3 \Theta 3 rotation matrix which was then used as an initial guess to register the model view (View 2) with the test view (View 1) of Vase2 using the MVE. We note here that the computational procedure for MVE was augmented using a verification mechanism for checking the validity of the control points during its implementation [17]. We derived the results presented in this section using this augmented procedure. Figures 6(c)-(g) show the iterative registration of the model view with the scene view. It can be seen that the views are in complete registration with one another at the end of seven iterations. Figure 7 shows the registration of a model view with a scene view of Phone through several iterations of the algorithm. The registration scheme converged with the lowest error value at the (a) (b) (c) (d) (e) (f) (g) Figure Pose estimation: (a) View1 of a vase; (b) view2; (c)-(f) model view registered with the test view of Vase2 at the end of the first, third, fourth and fifth iterations; (g) registered views at the convergence of the algorithm. (a) (b) (c) (d) (e) (f) (g) Figure 7: Registration of views of Phone: (a) View 1; (b) view 2; (c)-(f) model view registered with the test view of Phone at the end of first, second, third and fourth iterations; (g) registered views at the convergence of the algorithm. sixth iteration. It can be seen that even with a coarse initial estimate of the rotation, the registration technique can align the two views successfully within a few iterations. Given a coarse correct initial guess, registration, on the average, takes about seconds to register two range images whose sizes are 640 \Theta 480 on a SPARCstation 10 with 32MB RAM. 5.2 Discussion In general, all the orientation parameters of an object will be improved by the proposed MVE method if the object surface covers a wide variety of orientations which is true with many natural objects. This is because each locally flat surface patch constrains the global orientation estimate of the object via its surface normal direction. For example, if the object is a flat surface, then only the global orientation component that corresponds to the surface normal can be improved, but not the other two components that are orthogonal to it. For the same reason, the surface normal of a cylindrical surface (without end surfaces) covers only a great circle of the Gaussian sphere, and thus, only two components of its global orientation can be improved. The more surface orientations that an object covers, the more complete the improvement in its global orientation can be, by the proposed MVE method. An analysis of the performance of the MVE and unweighted registration algorithms with surfaces of various geometries can be found in [14]. When more than two views have to be registered, our algorithm for registering a pair of object views can be used either sequentially (with the risk of error accumulation) or in parallel, e.g., with the star-network scheme [9]. Note however, that we have not extended our weighted approach to the problem of computing the transformation between n views simultaneously. When there is a significant change in the object depth the errors in z at different points on the surface may no longer have the same variance; the variance typically increases with greater depth. In such situations our perturbation analysis still holds, except for the covariance matrix \Gamma A T in Eq. 9. The diagonal elements of this matrix will no longer be identical as we assumed. Each element, which is a summary of the noise variance at the corresponding point in the image, must reflect the combined effect of variation due to depth, measurement unreliability due to surface inclination, etc., and therefore a suitable noise model must be assumed or experimentally created. 6 Summary Noise in surface data is a serious problem in registering object views. The transformation that relates two views should be estimated robustly in the presence of errors in surface measurements for seamless view integration. We established the dependency between the surface orientation and the accuracy of surface normal estimation in the presence of error in range data, and its effect on the estimation of transformation parameters with geometrical analysis and experimental results. We proposed a new error model to handle uncertainties in z measurements at different orientations of the surface being registered. We presented a first-order perturbation analysis of the estimation of planar parameters from surface data. We derived the variance of the point-to-plane distance to be minimized to update the view transformation during registration. We employed this variance as a measure of the uncertainty in the distance resulting from noise in the z value and proposed a minimum variance estimator to estimate transformation parameters reliably. The results of our experiments on real range images have shown that the estimates obtained using our MVE generally are significantly more reliable than those computed with an unweighted distance criterion. Acknowledgments This work was supported by a grant from Northrop Corporation. We thank the reviewers for their helpful suggestions for improvement. --R "Object modelling by registration of multiple range images," "Integrating information from multiple views," "Generating models for solid objects by matching 3D surface segments," "A method for registration of 3-D shapes," "Zippered polygon meshes from range images," "Iterative point matching for registration of free-form curves and surfaces," "Registering multiview range data to create 3D computer graphics," "Registering range views of multipart objects," "Towards a general multi-view registration technique," "Pose estimation from corresponding point data," "Scene reconstruction and description: Geometric primitive extraction from multiple view scattered data," "Surface classification: Hypothesis testing and parameter estima- tion," Experiments in 3D CAD-based Inpection using Range Images cosmos: A Framework for Representation and Recognition of 3D Free-Form Objects "Motion and structure estimation from stereo image sequences," "Motion and structure from two perspective views: Algorithms, error analysis, and error estimation," "From images to models: Automatic 3D object model construction from multiple views," --TR --CTR Alberto Borghese , Giancarlo Ferrigno , Guido Baroni , Antonio Pedotti , Stefano Ferrari , Riccardo Savar, Autoscan: A Flexible and Portable 3D Scanner, IEEE Computer Graphics and Applications, v.18 n.3, p.38-41, May 1998 Byung-Uk Lee , Chul-Min Kim , Rae-Hong Park, An Orientation Reliability Matrix for the Iterative Closest Point Algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.10, p.1205-1208, October 2000 Gregory C. Sharp , Sang W. Lee , David K. Wehe, ICP Registration Using Invariant Features, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.1, p.90-102, January 2002 Xianfeng Wu , Dehua Li, Range image registration by neural network, Machine Graphics & Vision International Journal, v.12 n.2, p.257-266, February Zonghua Zhang , Xiang Peng , David Zhang, Transformation image into graphics, Integrated image and graphics technologies, Kluwer Academic Publishers, Norwell, MA, 2004 Ross T. Whitaker , Jens Gregor, A Maximum-Likelihood Surface Estimator for Dense Range Data, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.10, p.1372-1387, October 2002 Luca Lucchese , Gianfranco Doretto , Guido Maria Cortelazzo, A Frequency Domain Technique for Range Data Registration, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.11, p.1468-1484, November 2002

3D free-form objects;automatic object modeling;view transformation estimation;image registration;range data;view integration

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Datapath scheduling with multiple supply voltages and level converters.

We present an algorithm called MOVER (Multiple Operating Voltage Energy Reduction) to minimize datapath energy dissipation through use of multiple supply voltages. In a single voltage design, the critical path length, clock period, and number of control steps limit minimization of voltage and power. Multiple supply voltages permit localized voltage reductions to take up remaining schedule slack. MOVER initially finds one minimum voltage for an entire datapath. It then determines a second voltage for operations where there is still schedule slack. New voltages con be introduced and minimized until no schedule slack remains. MOVER was exercised for a variety of DSP datapath examples. Energy savings ranged from 0% to 50% when comparing dual to single voltage results. The benefit of going from two to three voltages never exceeded 15%. Power supply costs are not reflected in these savings, but a simple analysis shows that energy savings can be achieved even with relatively inefficient DC-DC converters. Datapath resource requirements were found to vary greatly with respect to number of supplies. Area penalties ranged from 0% to 170%. Implications of multiple voltage design for IC layout and power supply requirements are discussed.

INTRODUCTION A great deal of current research is motivated by the need for decreased power dissipation while satisfying requirements for increased computing capacity. In portable An earlier abbreviated version of this work was reported in the Proceedings of the 1997 IEEE International Symposium on Circuits and Systems, Hong Kong. This research was supported in part by ARPA (F33615-95-C-1625), NSF CAREER award (9501869-MIP), ASSERT program (DAAH04-96-1-0222), IBM, AT&T/Lucent, and Rockwell. Authors' address: School of Electrical and Computer Engineering Purdue University, West Lafayette, Indiana, 47907-1285, USA Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 869-0481, or [email protected]. c 1997 by the Association for Computing Machinery, Inc. systems, battery life is a primary constraint on power. However, even in non-portable systems such as scientific workstations, power is still a serious constraint due to limits on heat dissipation. One design technique that promises substantial power reduction is voltage scaling. The term "voltage scaling" refers to the trade-off of supply voltage against circuit area and other CMOS device parameters to achieve reduced power dissipation while maintaining circuit performance. The dominant source of power dissipation in a conventional CMOS circuit is due to the charging and and discharging of circuit capacitances during switching. For static CMOS, the switching power is proportional to dd [Rabaey 1996]. This relationship provides a strong incentive to lower supply voltage, especially since changes to any other design parameter can only achieve linear savings with respect to the parameter change. The penalty of voltage reduction is a loss of circuit performance. The propagation delay of CMOS is approximately proportional to Vdd [Rabaey 1996], where VT is the transistor threshold voltage. A variety of techniques are applied to compensate for the loss of performance with respect to V dd including reduction of threshold voltages, increasing transistor widths, optimizing the device technology for a lower supply voltage, and shortening critical paths in the data path by means of parallel architectures and pipelining. Data path designs can benefit from voltage scaling even without changes in device technologies. Algorithm transformations and scheduling techniques can be used to increase the latency available for some or all data path operations. The increased latency allows an operation to execute at a lower supply voltage without violating schedule constraints. "Architecture-Driven Voltage Scaling" is a name applied to this approach. A number of researchers have developed systems or proposed methods that incorporate architecture driven voltage scaling [Chandrakasan et al. 1995; Raghunathan and Jha 1994; Raghunathan and Jha 1995; Goodby et al. 1994; Kumar et al. 1995; SanMartin and Knight 1995; Raje and Sarrafzadeh 1995; Gebotys 1995]. HYPER- LP [Chandrakasan et al. 1995] is a system that applies transformations to the data flow graph of an algorithm to optimize it for low power. Other systems accept the algorithm as given and apply a variety of techniques during scheduling, module selection, resource binding, etc. to minimize power dissipation. All of the systems mentioned above try to exploit parallelism in the algorithm to shorten critical paths so that reduced supply voltages can be used. Most systems [Chandrakasan et al. 1995; Raghunathan and Jha 1994; Raghunathan and Jha 1995; Goodby et al. 1994; Kumar et al. 1995; Gebotys 1995] also minimize switched capacitance in the data path. Most voltage scaling approaches require that the IC operate at a single supply voltage. Although substantial energy savings can be realized with a single minimum supply voltage, one cannot always take full advantage of available schedule slack to reduce the voltage. Non-uniform path lengths, a fixed clock period, and a fixed number of control steps can all result in schedule slack that is not fully exploited. Figure provides examples of each type of bottleneck. When there are non-uniform path lengths, the critical (longest) path determines the minimum supply voltage even though the shorter path could execute at a still lower voltage and meet timing constraints. When the clock period is a bottleneck, some operations only use part of a clock period. The slack within these clock periods goes to waste. Additional voltages would permit such operations to use the entire clock period. Finally, a fixed number of control steps (resulting from a fixed clock period and latency constraint) may lead to unused clock cycles if the sequence of operations does not match the number of available clock cycles. A3 Unused Slack Unused Slack A3 Unused Slack Non-Uniform Path Length Period Number of Control Steps A3 A4Fig. 1. Examples of scheduling bottlenecks Literature on multiple voltage synthesis is limited, but this is changing. Publications that address the topic include [Raje and Sarrafzadeh 1995], [Gebotys 1995], and [Johnson and Roy 1996]. Raje and Sarrafzadeh [Raje and Sarrafzadeh 1995] schedule the data path and assign voltages to data path operators so as to minimize power given a predetermined set of supply voltages. Logic level conversions are not explicitly modeled in their formulation. Gebotys [Gebotys 1995] used an integer programming approach to scheduling and partitioning a VLSI system across multiple chips operating at different supply voltages. Johnson [Johnson and Roy 1996] used an integer program to choose voltages from a list of candidates, schedule datapath operations, model logic level conversions, and assign voltages to each operation. Chang and Pedram [Chang and Pedram 1996] address nearly the same problem, applying a dynamic programming approach to optimize non-pipelined datapaths and a modified list scheduler to handle functionally pipelined datapaths. 2. DATAPATH SPECIFICATIONS A datapath is specified in the form of a data flow graph (DFG) where each vertex represents an operation and each arc represents a data flow or latency constraint. This DFG representation is similar to the "sequencing graph" representation described by DeMicheli [DeMicheli 1994] except that hierarchical and conditional graph entities are not supported. The DFG is a directed acyclic graph, G(V; E), with vertex set V and edge set Each vertex corresponds one-to-one with an operator in the data path. Each edge corresponds one-to-one with a dependency between two operators: a data flow, a latency constraint, or both. Associated with each vertex is an attribute that specifies the operator type such as adder, multiplier, or null operation (NO- OP). Associated with each edge is an attribute that indicates a latency constraint between the start times of the source and destination operations. A positive value indicates a minimum delay between operation start times. The magnitude of a negative value specifies a maximum allowable delay from the destination to the source. Figure 2 provides a simple example of a datapath specification and defines elements of the DFG notation. Maximum Latency of 1 Sample Period Data Flow [Min. Latency Clock Cycles] Adder 2's Complement Multiplier source sink Fig. 2. Sample datapath specification and key to notation Two types of NO-OP's are used which we will refer to as ``transitive'' and "non- NO-OP's. The term ``transitive'' is used to indicate that a NO-OP propagates signals without any delay or cost. Neither type of NO-OP introduces delay or power dissipation. Both serve as vertices in the DFG to which latency constraints can be attached. The transitive NO-OP is treated as if signals and their logic levels are propagated through the NO-OP. 3. MOVER SCHEDULING ALGORITHM MOVER will generate a schedule, select a user specified number of supply voltage levels, and assign voltages to each operation. MOVER uses an ILP method to evaluate the feasibility of candidate supply voltage selections, to partition operations among different power supplies, and to produce a minimum area schedule under latency constraints once voltages have been selected. The algorithm proceeds in several phases. First, MOVER determines maximum and minimum bounds on the time window in which each operation must execute. It then searches for a minimum single supply voltage. Next, MOVER partitions datapath operations into two groups: those which will be assigned to a higher supply voltage and those which will be assigned to a lower supply voltage. The high voltage group is initially fixed to a voltage somewhat above the minimum single voltage. MOVER then searches for a minimum voltage for the lower group. The voltage of the lower group is fixed. A new minimum voltage for the upper group is sought. To find a three supply schedule, partition the lower voltage group and search for new minimum voltages for bottom, middle, and upper groups. 3.1 ILP Formulation At the core of MOVER is an integer linear program (ILP) that is used repeatedly to evaluate possible supply voltages, partition operations between different power supplies, and produce a schedule that minimizes resource usage. In each case, MOVER analyzes the DFG and generates a collection of linear inequalities that represent precedence constraints, timing constraints, and resource constraints for the datapath to be scheduled. A weighted sum of the energy dissipation for each operation is used as the optimization objective when partitioning operations or evaluating the feasibility of a supply voltage. A weighted sum of resource usage serves as the optimization objective when minimizing resources. The inequalities and the objective function are packed into a matrix of coefficients that are fed into an ILP program solver (CPLEX). MOVER interprets the results from CPLEX and annotates the DFG to indicate schedule times and voltage assignments. The architectural model assumed by MOVER is depicted in Figure 3. All operator outputs have registers. Each operator output feeds only one register. That register operates at the same voltage as the operator supplying its input. All level conversions, when needed, are performed at operator inputs. operator operator operator register level converter Fig. 3. MOVER architectural model MOVER's ILP formulation works on a DFG where voltage assignments for some operations may already be fixed. For operations not already fixed to a voltage, the formulation chooses between two closely spaced voltages so as to minimize energy. The voltages are chosen to be close enough together that level conversions from one to the other can be ignored. Consequently, level conversions only need to be accounted between operations fixed to different voltages and on interfaces between fixed and unfixed operations. 3.2 ILP Decision Variables Three categories of decision variables are used in the MOVER ILP formulation. One set of variables of the form x i;l;s indicates the start time and supply voltage assignment for each operator that has not already been fixed to a particular supply voltage. x begins execution on clock cycle l using supply voltage s. Under any other condition, x i;l;s will equal zero. The supply voltage selection is limited to two values where selects the lower and selects the higher candidate voltage. Another set of variables, x i;l , indicates the start time of operations for which the supply voltage has been fixed. x indicates that operation i starts at clock cycle l. Under any other condition, x i;l will equal zero. The last group of variables, a m;s , indicates the allocation of operator resources to each possible supply voltage. a m;s will be greater than or equal to the number of resources of type m that are allocated to supply voltage s. In this case, s can be an integer in the range (1; # fixed supplies corresponds to the new candidate supply voltages. s ? 2 corresponds to supply voltages that have already been fixed. 3.3 Objective Functions The objective function (equation 1) estimates the energy required for one execution of the data path as a function of the voltage assigned to each operation. Consider the energy expression split into two parts. The first nested summation counts the total energy contribution associated with operations not already fixed to a supply voltage. The second nested summation counts the total energy contribution of operations that are already fixed to a particular supply voltage. For each operation j that has not been fixed to a supply voltage (e.g., the first nested summation accumulates the energy of operation j (onrg(j; s the register at the output of operation j (rnrg(s j ; fanout j )), and any level conversions required at the input to is the index of the supply voltage assigned to operation j. fanout j is the fanout capacitive load on operation j. c reg is the input capacitance of a register to which the operation output is connected. The decision variables x j;l;s are used to select which lookup table values for operator, register, and level conversion energy are added into the total energy. We must sum over both candidate supply voltages s j and all clock cycles l in the possible execution time window R j of operation j. E conv is the set of DFG arcs that may require a level conversion, depending on voltage assignments. V oper is the set of DFG vertices that are not NO-OPs. V fix is the set of DFG vertices (operations) that have been fixed to a particular voltage. V free is the set of vertices that have not previously been fixed to a voltage. For each operation j that has been fixed to a supply voltage, we again accumulate the energy of each operation, register, and level conversion. The only difference from the expression for free operations is that now all voltages in the expression are constants determined prior to solving the ILP formulation. Consequently, the index s j can be removed from the summation and the decision variable x. j2Vfree "Voper x j;l;s \Theta j2Vfix "Voper x j;l \Theta conversion energy at the input of free and fixed operations respectively. c in i is the input capacitance of operation i. ij(i;j)2Econv and i2Vfix ij(i;j)2Econv and i2Vfree cnrg fix ij(i;j)2Econv and i2Vfix ij(i;j)2Econv and i2Vfree Equation 4 is the objective function used when minimizing resource usage. Here, a m;s indicates the minimum number of operators of type m with supply voltage s needed to implement a datapath. Each operation of type m is considered to have an area of aream . M oper represents the set of all operation types excluding NO-OPs. The summation accumulates an estimate of the total circuit resources required to implement a datapath. m2Moper aream \Theta a m;s (4) 3.4 ILP Constraint Inequalities Equation 5 guarantees that only one start time l is assigned to each operation i for which the supply voltage is already fixed. Equation 6 guarantees that only one start time l and supply voltage s can be assigned to each operation i that does not have a supply voltage assignment. Equation 7 guarantees that the voltage of a transitive NO-OP j matches the voltage of all operations supplying an input to the transitive NO-OP. V trnoop is the set of vertices in the DFG corresponding to transient NO-OP's. E is the set of all arcs in the DFG. l Equation 8 enforces precedence constraints specified in the DFG. Simplified versions of the constraint can be used if the source or destination operations are fixed to a voltage. This constraint is an adaptation of the structured precedence constraint shown by Gebotys [Gebotys 1992] to produce facets of the scheduling polytope. Each arc (i; j) with a latency lat i;j - 0 specifies a minimum latency from the start of operation i to the start of operation j. Equation 8 defines the set of precedence constraint inequalities corresponding to DFG arcs where the source and destination operations are both free (not fixed to a voltage). Simplified versions of this constraint are used when source or destination operations are fixed to a voltage.X del i;s i +l l 1 =0 l 2 =l Equation 9 enforces maximum latency constraints specified in the DFG. Each arc (i; j) with a latency lat i;j ! 0 specifies a maximum delay from operation j to operation i. Equation 9 defines the set of maximum latency constraint inequalities corresponding to arcs where the source and destination operations are both fixed to a voltage). Simplified versions of this constraint are used when source or destination operations are fixed to a voltage. The remaining equations are simplifications of equation 9.X l 2 =l\Gammalat i;j +1 Equations 10 and 11 ensure that resource usage during each time step does not exceed the resource allocation given by a m;s . The expressions on the left computes the number operations of type m with supply voltage s that are executing concurrently during clock cycle l. a m;s indicates the number of type m resources that have been allocated to supply voltage s. Equation 10 enforces the resource constraint enforces the constraint for fixed operations. Free operations are allowed to take on one of two candidate voltages. These resource constraints can be easily modified to support functional pipelining with a sample period of l samp by combining the left hand sides for l, l l l 1 =l\Gammadel i;s i +1 - a m;s i l l 1 =l\Gammadel i;s i +1 - a m;s i (11) Table I. Voltage search algorithm 1. Choose starting voltages V2 and 2. Create matrix of ILP constraint inequalities. 3. Obtain minimum energy solution to inequalities. The solution will provide a schedule, a mapping of V1 or V2 to each operator, an energy estimate, and an area estimate for the datapath. 4a. If a solution was found, then If most operations were assigned to V1 , then Choose new candidate voltages midway between V1 and V lo . Go to step 2. else There must be little or no benefit to assigning operations to V1 Fix all operations to V2 4b. else (if the problem was infeasible) Choose new candidate voltages midway between V2 and Vhi . Go to step 2. Equation 12 enforces the user specified resource constraints. maxres(m) represents the total number of resources of type m (regardless of voltage) that can be permitted. The left side expression accumulates the number of resources of type m that have been allocated to all supply voltages. The total is not allowed to exceed the user specified number of resources. a m;s - maxres(m) 8m 2 M oper (12) 3.5 Voltage search MOVER searches a continuous range of voltages when seeking a minimum voltage one, two, or three power supply design. The user must specify a convergence threshold V conv that is used to determine when a voltage selection is acceptably close to minimum. Let V hi and V lo represent the current upper and lower bound on the supply voltage. When searching for a minimum single supply voltage, all operations are initially considered to be free (not fixed to a voltage). When searching for a minimum set of two or three supply voltages, MOVER considers one power supply at a time. The voltage will be fixed for any operations not allocated to the supply voltage under consideration. Table I outlines the voltage search algorithm. 3.6 Partitioning Partitioning is the process by which MOVER takes all free operations in the DFG and allocates each to one of two possible power supplies. Partitioning is not performed until a single minimum supply voltage is known for the group of operations. supply voltage for the free operations. Choose two candidate supply voltages (V a and V b ) one slightly above V 1 and the other slightly below. Set up the ILP constraint inequalities. Obtain a minimum energy schedule. Operations will only be assigned to V a if there is schedule slack available. There may be several ways that the operations can be partitioned. In such a case, the optimal ILP solution will maximize the energy dissipation of the lower voltage group (i.e., put the most energy hungry operations in the lower voltage group). This will tend to maximize the benefit from reducing the voltage of the lower group. Given a successful partition, operations assigned to V a will be put into the lower supply voltage group and operations assigned to V b will be put into the higher supply voltage group. The partition will fail if all operations are allocated to the lower supply voltage, all operations are allocated to the higher supply voltage, or the ILP solver exceeds some resource limit. The first situation indicates that the minimum single voltage could be a bit lower. In this event, MOVER lowers the values of V a and V b by Vconvand tries the partition again. Lowering V a and V b too far leads to a completely infeasible ILP problem. The second situation indicates that there is not enough schedule slack available for any operations to bear a further reduction in voltage. In this case, MOVER terminates. The only remedies for the third situation are to either increase resource and time limits on the ILP solver or make the problem smaller. 4. CHARACTERIZATION OF DATAPATH RESOURCES The results presented in this paper make use of four types of circuit resources: an adder, multiplier, register, and level converter. MOVER requires models of the energy and delay of each type of resource as a function of supply voltage, load capacitance, and average switching activity. Each type of resource was simulated in HSPICE using 0.8 micron MOSIS library models with the level 3 MOS model. Energy dissipation, worst case delay, and input capacitances were measured from the simulation. All resources were 16 bits wide. Load capacitance on each output was 0.1pF. Input vectors were generated to provide 50% switching activities. 4.1 Datapath operators and registers During optimization, operation energies and delays are scaled as a function of the voltage assignment being evaluated. Energy dissipation (E) for each operator and register scales with respect to supply voltage as Table II. Nominal energy and delay values used by MOVER Resource Energy d dC Energy Delay d Delay Cin Type [pJ] [pJ/pF] [ns] [ns/pF] [pF] ADDER 84 200 12.0 3.5 0.021 MULTIPLIER 2966 200 18.5 3.33 0.095 REGISTER 312 200 0.48 2.25 0.045 is the energy dissipation of the operator or register measured at the nominal supply voltage V 0 . Delay each operator and register scale with respect to supply voltage as \Theta where t p0 is the propagation delay measured at the nominal supply voltage V 0 . The energy and delay scaling factors were derived directly from the CMOS energy and delay equations described by Rabaey [Rabaey 1996]. Energy and delay are also scaled linearly with respect to the estimated load capacitance on output signals. Table II gives the model parameters used by MOVER for each type of resource. Note that the register delay given here is just the propagation time relative to a clock edge. Register setup time is treated as part of the datapath operator delays. 4.2 Level conversion Whenever one resource has to drive an input of another resource operating at a higher voltage, a level conversion is needed at the interface. Four alternatives were considered to accomplish this: omit the level converter, use a chain of inverters at successively higher voltages, use an active or passive pullup, or use a differential cascode voltage switch (DCVS) circuit as a level converter [Chandrakasan et al. 1994; Usami and Horowitz 1995]. We omit the level converter for step-down conversions and use the DCVS circuit for step-up conversions. Given appropriate transistor sizes, this circuit exhibits no static current paths and it can operate over a full 1.5V to 5.0V range of input and output supply voltages. A model was needed that could accurately indicate the power dissipation and propagation delay of the DCVS level converter as a function of the input logic supply voltage V 1 , output logic supply voltage V 2 , and load capacitance. The circuit was studied both analytically and from HSPICE simulation results to determine a suitable form for the model equations. Coefficients of the equations were then calibrated so that the model equations would produce families of curves closely matching simulation results for V 1 ranging from 1:5V to 5V and These are the ranges of supply voltages for which a level converter is needed. Typical energy dissipation of the level converter was found to be on the order of 5 to 15pJ per switching event per bit, given a 0.1pF load. Typical propagation delays range were approximately 1ns for level conversions such as 3.3V to 5V or 2.4V to 3.3V. Propagation delays become large as the input voltage of the level converter falls towards 2V T . A 2.5V to 5V conversion had a delay of about 2.5ns. A 2V to 5V conversion had a delay of nearly 5ns. All transistors 0.8u length and 4.0u width except where noted M2N M3N M1N IN Fig. 4. DCVS Level Converter 5. RESULTS 5.1 Datapath examples ILP schedule optimization results are presented for six example data paths: a four point FFT (FFT4), the 5th order elliptic wave filter benchmark (ELLIP) [Rao 1992], a 6th order Auto-Regressive Lattice filter (LATTICE), a frequency sampled filter (FSAMP) with three 2nd order stages and one 1st order stage, a direct form 9 tap linear phase FIR filter (LFIR9), and a 5th order state-space realization of an IIR filter (SSIIR). In the FFT data path, complex signal paths are split into real and imaginary data flows. For all other data paths, the signals are modeled as non-complex integer values. All data flows were taken to be 16 bits wide. Switching activities at all nodes were assumed to be 50%, i.e., the probability of a transition on any selected 1 bit signal is 50% in any one sample interval. Each example was modeled for one sample period with data flow and latency constraints specified for any feedback signals. Any loops that start and finish within the same sample period were completely unrolled. Any loops spanning multiple sample periods were broken. A data flow passing from one sample period to the next was represented by input and output nodes in the DFG connected by a backward arc to specify a maximum latency constraint from the input to the output. A 20ns clock was specified for all examples. Latency constraints were specified so that the data introduction interval equals the maximum delay from the input to the output of the data path. 5.2 MOVER Results Figure presents energy reduction results. The left-most column identifies the particular datapath topology and indicates the number of operations (additions, Name Lat/Clks +/x 1 2 3 [min] Datapath Max Max Voltages Exec(host) Min Lat.,Unlim. Resources Energy ratio vs. 1 supply, Energy adds NR NR - 0.27(1) 2.3 3.6 - 0.23(1) 1144 2.3 3.6 0.40(1) 1148 2.3 3.6 - 0.48(1) 1233 1.9 2.4 3.6 0.72(1) 1235 26 adds 2.3 3.6 - 1.53(2) 2206 1.9 2.4 3.6 2.65(2) 2181 2.3 3.6 - 3.43(2) 1631 1.9 2.4 3.6 5.30(2) 1600 2.3 3.6 - 3.77(2) 1963 NR NR NR 6.02(2) 2.3 3.6 - 50.6(2) 1237 NR NR NR 101.(2) adds 3.5 4.9 - 0.72(2) 13904 mults 3.0 3.5 4.9 1.40(2) 12538 NR NR NR 4.10(2) 3.0 3.5 4.9 6.88(2) 14342 2.3 3.0 3.6 59.9(2) 8480 14 adds 4.2 4.8 - 2.28(1) 15882 9 mults NR NR NR 4.57(1) 38 del 11/11 - 3.6 - 2.10(1) 8828 2.4 3.0 3.6 8.85(1) 5401 2.3 3.0 3.6 9.82(1) 6263 2.4 3.0 - 6.00(1) 5768 1.6 3.0 3.0 9.60(1) 6191 8 adds 3.5 4.9 - 0.72(1) 6344 5 mults 3.0 3.5 4.9 1.38(1) 5683 8 del 8/8 - 3.6 - 0.43(1) 4923 2.3 3.6 - 0.98(1) 2415 2.3 3.0 3.6 1.48(1) 3434 2.4 3.0 - 1.47(1) 3213 1.6 3.0 3.1 2.37(1) 3717 adds 3.0 4.9 - 0.52(1) 15770 mults NR NR NR 0.83(1) 2.4 3.0 3.6 4.67(1) 6250 NR NR NR 1.25(1) Fig. 5. Multi-voltage Energy Savings multiplications, and sample period delays) performed in one iteration of the data- path. "Max Lat/Clks" specifies the maximum latency (equal to the data sample rate) and the maximum number of control steps (Clks), both given in terms of the number of clock cycles. "Max +/-" specifies the maximum numbers of adder and multiplier circuits permitted in the design. Values of "-" indicate that unlimited resources were permitted. The columns headed by "Voltages 1 2 3" indicate the supply voltages selected by MOVER. A "-" is used to fill voltage columns "2" or "3" in those cases where a one or two supply voltage result is presented. The string "NR" in voltage columns "1" and "2" indicates that a solution with two supply voltages could not be obtained. "NR" in all three columns indicates that a solution with three supply voltages could not be obtained. The "Exec" column reports the minutes of execution time (Real, not CPU) required to obtain the result. The number in parenthesis identifies the type of machine used to obtain the result. "(1)" indicates a SPARCserver 1000 with 4 processors and 320MB of RAM. "(2)" indicates a Sparc 5 with 64MB of RAM. The bar graph down the center represents the normalized energy consumption of each test case. Each energy result is divided by the single supply voltage, unlimited resource, minimum latency result to obtain a normalized value. Single supply voltage results are shown with black bars. All other results are shown in gray. This style of presentation is intended to visually emphasize the effect of different latency, resource, and supply voltage constraints on the energy estimate. The right-most column presents the absolute energy estimate in units of Figure 6 presents area penalty results. All but two columns have the same meaning as the corresponding columns in figure 5. The only exceptions are the bar graph and the "area" column on the right. The "area" value is a weighted sum of the minimum circuit resources required to implement the datapath schedule. The resources (all bits wide) were weighted as follows: adder=1, multiplier=16, register=0.75, and level converter=0.15. These weights are proportional to the transistor count of each resource. Each area value was divided by the area estimate for the corresponding single voltage result. Each single voltage result is shown as a black bar. Two and three voltage results are shown in gray. 5.3 Observations The preceding results permit several observations to be made regarding the effect of latency, circuit resource, and supply voltage constraints on energy savings, area costs, and execution time. Because our primary objective has been to minimize energy dissipation through use of multiple voltages, we are especially interested in the comparison of multiple supply voltage results to minimum single supply voltage results. Energy savings ranging from 0% to 50% were observed when comparing multiple to single voltage results. Estimated area penalties ranged from a slight improvement to a 170% increase in area. Actual area penalties could be higher, since our estimate only considers the number of circuit resources used. There is not a clear correlation between energy savings and area penalty when looking at the complete set of results. Sometimes a substantial energy savings was achieved with minimal increased circuit resources, other times even a small energy savings incurred a large area cost. Name Lat/Clks +/x 1 2 3 Area ratio vs. 1 supply [adder=1] Area adds NR NR - 2.3 3.6 - 30.4 2.3 3.6 30.4 2.3 3.6 - 22.8 1.9 2.4 3.6 22.8 4/4 12/- 2.3 - 14 26 adds 2.3 3.6 - 12.65 1.9 2.4 3.6 12.8 2.3 3.6 - 13.3 1.9 2.4 3.6 16.9 2.3 3.6 - 12.85 1.9 2.4 3.6 2.3 3.6 - 11.8 1.9 2.4 3.6 adds 3.5 4.9 - 88.8 mults 3.0 3.5 4.9 89.85 NR NR NR 3.0 3.5 4.9 39.1 2.3 3.0 3.6 41.85 14 adds 4.2 4.8 - 136.2 9 mults NR NR NR 169.4 38 del 11/11 - 3.6 - 83.75 2.4 3.0 3.6 120.15 2.3 3.0 3.6 89.95 2.4 3.0 - 88.2 1.6 3.0 3.0 90.1 8 adds 3.5 4.9 - 94.8 5 mults 3.0 3.5 4.9 98.1 8 del 8/8 - 3.6 - 42.75 2.3 3.6 - 87.3 2.3 3.0 3.6 47 2.4 3.0 - 45.55 1.6 3.0 3.1 47.3 adds 3.0 4.9 - 174 mults NR NR NR 2.4 3.0 3.6 181.75 NR NR NR Fig. 6. Multi-voltage Area Penalties If we consider the impact of latency constraints alone, effects on area and energy are easier to observe. In most cases, multiple voltage area penalties were greatest for the minimum latency unlimited resource test cases. We can also observe that increasing latency constraints always led to the same or lower energy for a given number of supply voltages. However, the effect of latency constraints on the single vs. multiple voltage trade-off varied greatly from one example to another. Results for multiple voltages are most favorable in situations where the single supply voltage solution did not benefit from increased latency, perhaps due to a control step bottleneck such as illustrated earlier in figure 1. The effect of resource constraints on energy savings are also relatively easy to observe. Not surprisingly, resource constraints tended to produce the lowest area penalties. The only reason for any area penalty at all in the resource constrained case is that sometimes the minimum single supply solution does not require all of the resources that were permitted. Energy estimates based on resource constrained schedules were consistently the same or higher than estimates based on unlimited resource schedules. The results presented previously do not include energy or area costs associated with multiplexers that would be required to support sharing of functional units and registers. However, an analysis of multiplexer requirements for most of these schedules indicated that multiplexers would not have changed the relative trade-off between number of voltages, energy dissipation, and circuit area. In a few cases the energy and area costs were increased substantially (up to 50% for energy and 108% for area), but the comparison between one, two, and three voltages was always either similar to the earlier results or shifted somewhat in favor of multiple voltages. The maximum energy savings was 54%, and the average was 32% when comparing two supply voltages to one. The maximum area penalty was 132% and the average was 42%. Results for three supply voltages, at best, were only slightly better than the two supply results. Multiplexer costs were estimated in the following manner. A simple greedy algorithm was used to assign a functional unit to each operation and a register to each data value. Given this resource binding, we determined the fan-in to each functional unit and register. Assuming a pass-gate multiplexer implementation, we estimated worst case capacitance on signal paths, total gate capacitance switched by control lines, and relative circuit area as a function of the fan-in and data bus width. A single pass gate, turned on, was estimated to add a 5fF load to each data input bit and 5fF to the control inputs of a multiplexer. The circuit area for a pass gate was taken to be 0:07\Theta (the area of one bit slice of a full adder). Multiplexer capacitances and area were added to the costs already used by MOVER. MOVER was then used to generate a new datapath schedule that accounts for these costs. In some cases, supply voltages had to be elevated slightly relative to previous results in order to compensate for increased propagation delays. 6. DESIGN ISSUES There are several design issues that a designer will need to take into consideration when a multiple voltage design is targeted for fabrication. In particular, the effects of multiple voltage operation on IC layout and power supply requirements should be considered. In this section, we will discuss the issues and identify improvements that would allow MOVER to more completely take them into account. 6.1 Layout Following are some ways that multiple voltage design may affect IC layout. (1) If the multiple supplies are generated off-chip, additional power and ground pins will be required. (2) It may be necessary to partition the chip into separate regions, where all operations in a region operate at the same supply voltage. (3) Some kind of isolation will be needed between regions operated at different voltages. There may be some limit on the voltage difference that can be tolerated between regions. Protection against latch-up may be needed at the logic interfaces between regions of different voltage. design rules for routing may be needed to deal with signals at one voltage passing through a region at another voltage. Isolation requirements between different voltage regions can probably be adequately addressed by increased use of substrate contacts, separate routing of power and ground, increased minimum spacing between routes (for example, between one signal having a 2V swing and another with a 5V swing), and slightly increased spacing between wells. While these practices will increase circuit area somewhat, the effect should be small in comparison to increased circuitry (adders, multipliers, registers, etc.) needed to support parallel operations at reduced supply voltages. Area for isolation will be further mitigated by grouping together resources at a particular voltage into a common region. Isolation is then only needed at the periphery of the region. Some of these layout issues can be incorporated into multiple voltage scheduling. Perhaps the greatest impact will be related to grouping operations of a particular supply voltage into a common region. Closely intermingled operations at different voltages could lead to complex routing between regions, increased need for level conversions, and increased risk of latch-up. Assigning highly connected operations to the same voltage could not only improve routing, but should also lead to fewer voltage regions on the chip, less space lost to isolation between voltage regions, and fewer signals passing between regions operating at different voltages. 6.2 Circuit Design There are some circuit design issues that still need to be addressed by MOVER including alternative level converter designs and control logic design. Alternative level converter designs such as the combined register and level converter should be considered. The DCVS converter design considered in this paper does not exhibit static power consumption, but short circuit energy is a problem. Delays and energy also increase greatly as the input voltage to the level converter becomes small. MOVER makes assumptions about datapath control and clocking that are convenient for scheduling and energy estimation, but will require support from the control logic. It is assumed that the entire control of the datapath is accomplished through selective clocking of registers and switching of multiplexers. This will require specially gated clocks for each register. 6.3 Power Supplies Before implementing a multiple voltage datapath, some decisions must be made regarding the voltages that can be selected and the type of power supply to be used. Regarding voltage selection, we must decide how many supplies to use and determine whether or not non-standard voltages are acceptable. Regarding the type of power supply, we will only consider the choice between generating the voltage on-chip or off-chip. All of these choices will depend largely on the application. If on chip heat dissipation is a primary constraint, voltages would be generated off chip and DC-DC conversion efficiency would be a low priority. If battery life is the bottleneck, DC-DC conversion efficiency will determine whether or not multiple voltages will reap an energy savings. A simple analysis provides some insight into the conditions under which a new supply voltage could be justified. In a battery powered system, we would need a DC to DC converter to obtain the new voltage. Let - represent the efficiency of the DC to DC converter. The efficiency can be most easily described as the power output to the datapath divided by the power input to the DC-DC converter. This model does not explicitly represent the effect of the amount of loading or choice of voltages on converter efficiency. For now, we are only trying to determine the degree of converter efficiency needed in order to make a new supply voltage viable. Conversely, given a DC-DC converter of known efficiency, we want to know how much voltage reduction is needed to justify use of the converter. Let ff represent the fraction of switched capacitance in the datapath that will be allocated to the new supply voltage. V 1 represents the primary supply voltage. represents the new reduced supply voltage under consideration. E 1 represents the energy dissipation of the datapath operating with the single supply voltage V 1 . The energy E 1 can be split into a portion, ff representing the circuitry that will run at voltage V 2 , and a remaining portion will continue to run at When the new supply voltage V 2 is introduced, the first term in equation be scaled by the factor V 2V 2. The new datapath energy dissipation (ignoring DC-DC becomes: However, the energy lost in the DC-DC converter equals the energy of the circuitry operating at V 2 divided by the efficiency of the converter. A bit of algebraic manipulation will reveal the system energy savings (including converter losses) as a function of ff, -, V 1 , and V 2 . lost Consider a simple example. Let Suppose 60% of the circuit can operate at voltage V 2 . Given an ideal DC-DC converter, the energy savings would be 36%. However, when the converter efficiency is considered, the savings drops more than a half to 17%. The break-even point occurs when 2. For the last example, the converter efficiency has to be at least 41% to avoid losing energy. In practice, the break-even point will be somewhat higher due to logic level conversions that will be required within the datapath. The preceding analysis suggests that a DC to DC converter does not have to be exceedingly efficient in order to achieve energy savings. Had the voltage reduction been merely from 3.3V to 3.0V, DC-DC converter efficiency would have to be at least 83%. Converter designs are available that easily exceed this efficiency requirement. Stratakos et al. [Stratakos et al. 1994] designed a DC-DC converter that achieves better than 90% efficiency for a 6V to 1.5V voltage reduction. 7. CONCLUSIONS In this paper we have presented MOVER, a tool which reduces the energy dissipation of a datapath design through use of multiple supply voltages. An area estimate is produced based on the minimum number of circuit resources required to implement the design. One, two, and three supply voltage designs are generated for consideration by the circuit designer. The user has control over latency constraints, resource constraints, total number of control steps, clock period, voltage range, and number of power supplies. MOVER can be used to examine and trade-off the effects of each constraint on the energy and area estimates. MOVER iteratively searches the voltage range for minimum voltages that will be feasible in a one, two, and three supply solution. An exact ILP formulation is used to evaluate schedule feasibility for each voltage selection. The same ILP formulation is used to determine which operations are assigned to each power supply. MOVER was exercised for six different datapath specifications, each subjected to a variety of latency, resource, and power supply constraints for a total of 70 test cases. The test cases were modest in size, ranging from 13 to 26 datapath operations and 2 to 24 control steps. The results indicate that some but not all datapath specifications can benefit significantly from use of multiple voltages. In many cases, energy was reduced substantially going from one to two supply voltages. Improvements as much as 50% were observed, but 20-30% savings were more typical. Adding a third supply produced relatively little improvement over two supplies, 15% improvement at most. Results from MOVER are comparable and in many cases better than results obtained using the MESVS (Minimum Energy Scheduling with Voltage Selection) ILP formulation presented in [Johnson and Roy 1996]. Behavior with respect to latency, resource, and supply voltage constraints is similar between MOVER and MESVS. The improvement relative to a pure ILP formulation is due to the fact that ILP formulation could only select from a discrete set of voltages, whereas MOVER can select from a continuous range of voltages. ACKNOWLEDGMENTS We would like to thank James Cutler for his programming work, the low power research group at Purdue, and the anonymous reviewers for their critiques. --R Optimizing power using transformations. Design of portable systems. Energy minimization using multiple supply voltages. Synthesis and Optimization of Digital Circuits. Optimal VLSI Architectural Synthesis: Area An ILP model for simultaneous scheduling and partitioning for low power system mapping. Microarchitectural synthesis of performance-constrained Optimal selection of supply voltages and level conversions during data path scheduling under resource constraints. Digital integrated circuits Behavioral synthesis for low power. An iterative improvementalgorithm for low power data path synthesis. Variable voltage scheduling. The fifth order elliptic wave filter benchmark. Clustered voltage scaling technique for low-power design --TR Optimal VLSI architectural synthesis Power-profiler Clustered voltage scaling technique for low-power design Variable voltage scheduling An iterative improvement algorithm for low power data path synthesis Digital integrated circuits Energy minimization using multiple supply voltages Synthesis and Optimization of Digital Circuits Profile-Driven Behavioral Synthesis for Low-Power VLSI Systems Optimal Selection of Supply Voltages and Level Conversions During Data Path Scheduling Under Resource Constraints Behavioral Synthesis for low Power Microarchitectural Synthesis of Performance-Constrained, Low-Power VLSI Designs --CTR Saraju P. Mohanty , N. Ranganathan , Sunil K. Chappidi, Simultaneous peak and average power minimization during datapath scheduling for DSP processors, Proceedings of the 13th ACM Great Lakes symposium on VLSI, April 28-29, 2003, Washington, D. C., USA Tohru Ishihara , Hiroto Yasuura, Voltage scheduling problem for dynamically variable voltage processors, Proceedings of the 1998 international symposium on Low power electronics and design, p.197-202, August 10-12, 1998, Monterey, California, United States Ling Wang , Yingtao Jiang , Henry Selvaraj, Scheduling and optimal voltage selection with multiple supply voltages under resource constraints, Integration, the VLSI Journal, v.40 n.2, p.174-182, February, 2007 Ali Manzak , Chaitali Chakrabarti, A low power scheduling scheme with resources operating at multiple voltages, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.10 n.1, p.6-14, 2/1/2002 Ling Wang , Yingtao Jiang , Henry Selvaraj, Scheduling and Partitioning Schemes for Low Power Designs Using Multiple Supply Voltages, The Journal of Supercomputing, v.35 n.1, p.93-113, January 2006 Dongxin Wen , Ling Wang , Yingtao Jiang , Henry Selvaraj, Power optimization for simultaneous scheduling and partitioning with multiple voltages, Proceedings of the 7th WSEAS International Conference on Mathematical Methods and Computational Techniques In Electrical Engineering, p.156-161, October 27-29, 2005, Sofia, Bulgaria Ashok Kumar , Magdy Bayoumi , Mohamed Elgamel, A methodology for low power scheduling with resources operating at multiple voltages, Integration, the VLSI Journal, v.37 n.1, p.29-62, February 2004 Amitabh Menon , S. K. Nandy , Mahesh Mehendale, Multivoltage scheduling with voltage-partitioned variable storage, Proceedings of the international symposium on Low power electronics and design, August 25-27, 2003, Seoul, Korea Woo-Cheol Kwon , Taewhan Kim, Optimal voltage allocation techniques for dynamically variable voltage processors, Proceedings of the 40th conference on Design automation, June 02-06, 2003, Anaheim, CA, USA Saraju P. Mohanty , N. Ranganathan , Sunil K. Chappidi, ILP models for simultaneous energy and transient power minimization during behavioral synthesis, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.11 n.1, p.186-212, January 2006 Inki Hong , Miodrag Potkonjak , Mani B. Srivastava, On-line scheduling of hard real-time tasks on variable voltage processor, Proceedings of the 1998 IEEE/ACM international conference on Computer-aided design, p.653-656, November 08-12, 1998, San Jose, California, United States Inki Hong , Darko Kirovski , Gang Qu , Miodrag Potkonjak , Mani B. Srivastava, Power optimization of variable voltage core-based systems, Proceedings of the 35th annual conference on Design automation, p.176-181, June 15-19, 1998, San Francisco, California, United States Ling Wang , Yingtao Jiang , Henry Selvaraj, Multiple voltage synthesis scheme for low power design under timing and resource constraints, Integrated Computer-Aided Engineering, v.12 n.4, p.369-378, October 2005 Shaoxiong Hua , Gang Qu, Approaching the Maximum Energy Saving on Embedded Systems with Multiple Voltages, Proceedings of the IEEE/ACM international conference on Computer-aided design, p.26, November 09-13, Woo-Cheol Kwon , Taewhan Kim, Optimal voltage allocation techniques for dynamically variable voltage processors, ACM Transactions on Embedded Computing Systems (TECS), v.4 n.1, p.211-230, February 2005 Gang Qu, What is the limit of energy saving by dynamic voltage scaling?, Proceedings of the 2001 IEEE/ACM international conference on Computer-aided design, November 04-08, 2001, San Jose, California Hsueh-Chih Yang , Lan-Rong Dung, On multiple-voltage high-level synthesis using algorithmic transformations, Proceedings of the 2005 conference on Asia South Pacific design automation, January 18-21, 2005, Shanghai, China Deming Chen , Jason Cong , Yiping Fan , Junjuan Xu, Optimality study of resource binding with multi-Vdds, Proceedings of the 43rd annual conference on Design automation, July 24-28, 2006, San Francisco, CA, USA Saraju P. Mohanty , N. Ranganathan, Energy-efficient datapath scheduling using multiple voltages and dynamic clocking, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.10 n.2, p.330-353, April 2005 Deming Chen , Jason Cong , Junjuan Xu, Optimal module and voltage assignment for low-power, Proceedings of the 2005 conference on Asia South Pacific design automation, January 18-21, 2005, Shanghai, China Liqiong Wei , Zhanping Chen , Mark Johnson , Kaushik Roy , Vivek De, Design and optimization of low voltage high performance dual threshold CMOS circuits, Proceedings of the 35th annual conference on Design automation, p.489-494, June 15-19, 1998, San Francisco, California, United States Deming Chen , Jason Cong , Junjuan Xu, Optimal simultaneous module and multivoltage assignment for low power, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.11 n.2, p.362-386, April 2006 Krishnan Srinivasan , Karam S. 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DSP;low power design;multiple voltage;datapath scheduling;power optimization;scheduling;high-level synthesis;level conversion

265177

On Parallelization of Static Scheduling Algorithms.

AbstractMost static algorithms that schedule parallel programs represented by macro dataflow graphs are sequential. This paper discusses the essential issues pertaining to parallelization of static scheduling and presents two efficient parallel scheduling algorithms. The proposed algorithms have been implemented on an Intel Paragon machine and their performances have been evaluated. These algorithms produce high-quality scheduling and are much faster than existing sequential and parallel algorithms.

Introduction Static scheduling utilizes the knowledge of problem characteristics to reach a global optimal, or near optimal, solution. Although many people have conducted their research in various manners, they all share a similar underlying idea: take a directed acyclic graph representing the parallel program as input and schedule it onto processors of a target machine to minimize the completion time. This is an NP-complete problem in its general form [7]. Therefore, many hueristic algorithms that produce satisfactory performance have been proposed [11, 13, 5, 14, 12, 4, 9]. Although these scheduling algorithms apply to parallel programs, the algorithms themselves are sequential, and are executed on a single processor system. A sequential algorithm is slow. Scalability of static scheduling is restricted since a large memory space is required to store the task graph. A natural solution to this problem is using multiprocessors to schedule tasks to multiprocessors. In fact, without parallelizing the scheduling algorithm and running it on a parallel computer, a scalable scheduler is not feasible. A parallel scheduling algorithm should have the following features: ffl High quality - it is able to minimize the completion time of a parallel program. ffl Low complexity - it is able to minimize the time for scheduling a parallel program. These two requirements contradict each other in general. Usually, a high-quality scheduling algorithm is of a high complexity. The Modified Critical-Path (MCP) algorithm was introduced [13], which offered a good quality with relatively low complexity. In this paper, we propose two parallelized versions of MCP. We will describe the MCP algorithm in the next section. Then, we will discuss different approaches for parallel scheduling, as well as existing parallel algorithms in section 3. In sections 4 and 5, we will present the VPMCP and HPMCP algorithms, respectively. A comparison of the two algorithms will be presented in section 6. 2 The MCP Algorithm A macro dataflow graph is a directed acyclic graph with a starting point and an end point [13]. A macro dataflow graph consists of a set of nodes fn 1 connected by a set of edges, each of which is denoted by e(n Each node represents a task, and the weight of a node is the execution time of the task. Each edge represents a message transferred from one node to another node, and the weight of the edge is equal to the transmission time of the message. When two nodes are scheduled to the same processing element (PE), the weight of the edge connecting them becomes zero. To define this scheduling algorithm succinctly, we will first define the as-late-as-possible time of a node. The ALAP time is defined as T critical is the length of the critical path, and level(n i ) is the length of the longest path from node n i to the end point, including node n i [6]. In fact, high-quality scheduling algorithms more or less rely on the ALAP time or level. The Modified Critical-Path (MCP) algorithm was designed to schedule a macro dataflow graph on a bounded number of PEs. The MCP Algorithm 1. Calculate the ALAP time of each node. 2. Sort the node list in an increasing ALAP order. Ties are broken by using the smallest ALAP time of the successor nodes, the successors of the successor nodes, and so on. 3. Schedule the first node in the list to the PE that allows the earliest start time, considering idle time slots. Delete the node from the list and repeat Step 3 until the list is empty. 2 In step 3, when determining the start time, idle time slots created by communication delays are also considered. A node can be inserted to the first feasible idle time slot. This method is called an insertion algorithm. The MCP algorithm has been compared to four other well-known scheduling algorithms under the same assumption, that are, ISH [10], ETF [8], DLS [12], and LAST [3]. It has been shown that MCP performed the best [2]. The complexity of the MCP algorithm is O(n 2 logn), where n is the number of nodes in a graph. In the second step, the ties can be broken randomly to have a simplified version of MCP. The scheduling quality only varies a little, but the complexity is reduced to O(n 2 ). In the following, we will use this simplified version of MCP. 3 Approaches for Parallelization of Scheduling Algorithms The basic idea behind parallel scheduling algorithms is that instead of identifying one node to be scheduled each time, we identify a set of nodes that can be scheduled in parallel. In the following, the PEs that execute a parallel scheduling algorithm are called the physical PEs (PPEs) in order to distinguish them from the target PEs (TPEs) to which the macro dataflow graph is to be scheduled. The quality and speed of a parallel scheduler depend on data partitioning. There are two major data domains in a scheduling algorithm, the source domain and the target domain. The source domain is the macro-dataflow graph and the target domain is the schedule for target processors. We consider two approaches for parallel scheduling. The first one is called the vertical scheme. Each PPE is assigned a set of graph nodes using space domain partitioning. Also, each maintains schedules for one or more TPEs. The second one is called the horizontal scheme. Each PPE is assigned a set of graph nodes using time domain partitioning. The resultant schedule is also partitioned so that each PPE maintains a portion of the schedule of every TPE. Each PPE schedules its own portion of the graph before all PPEs exchange information with each other to determine the final schedule. The vertical and horizontal schemes are illustrated in Figure 1. The task graph is mapped to the time-space domain. Here, we assume that three PPEs schedule the graph to six TPEs. Thus, in the vertical scheme, each PPE holds schedules of two TPEs. In the horizontal scheme, each PPE holds a portion of schedules of six TPEs. The vertical scheme and the horizontal scheme are outlined in Figure 2 and 3, respectively. In the vertical scheme, PPEs exchange information and schedule graph nodes to TPEs. Frequent information exchange results in large communication overhead. With horizontal partitioning, each PPE can schedule its graph partition without exchanging information with another PPE. In the last step, PPEs exchange information of their sub-schedules and concatenate them to obtain the final schedule. The problem with this method is that the start times of all partitions other than the first one are unknown. The time needs to be estimated, and scheduling quality depends on the estimation. There is almost no work in designing a parallel algorithm for scheduling. In fact, there is no algorithm in the vertical scheme yet. The only algorithm in this area is in the horizontal Space Time Space Time (a) vertical scheme (b) horizontal scheme P1Target Processors Target Processors Physical Processors Physical Processors Task Schedule Task Schedule Task Graph Task Graph Figure 1: Vertical and Horizontal Schemes. 1. Partition the graph into P equal sized sets using space domain partitioning. 2. Every PPE cooperates together to generate a schedule and each PPE maintains schedules for one or more TPEs. Figure 2: The Vertical Scheme for Parallel Scheduling. 1. Partition the graph into P equal sized sets using time domain partitioning. 2. Each PPE schedules its graph partition to generate a sub-schedule. 3. PPEs exchange information to concatenate sub-schedules. Figure 3: The Horizontal Scheme for Parallel Scheduling. scheme, which is the parallel BSA (PBSA) algorithm [1]. The BSA algorithm takes into account link contention and communication routing strategy. A PE list is constructed in a breadth-first order from the PE having the highest degree (pivot PE). This algorithm constructs a schedule incrementally by first injecting all the nodes to the pivot PE. Then, it tries to improve the start time of each node by migrating it to the adjacent PEs of the pivot PE only if the migration will improve the start time of the node. After a node is migrated to another PE, its successors are also moved with it. Then, the next PE in the PE list is selected to be the new pivot PE. This process is repeated until all the PEs in the PE list have been considered. The complexity of the BSA algorithm is O(p 2 en), where p is the number of TPEs, n the number of nodes, and e the number of edges in a graph. The PBSA algorithm parallelizes the BSA algorithm in the horizontal scheme. The nodes in the graph are sorted in a topological order and partitioned into P equal sized blocks. Each partition of the graph is then scheduled to the target system independently. The PBSA algorithm resolves the dependencies between the nodes of partitions by calculating an estimated start time of each parent node belonging to another partition, called the remote parent node (RPN). This time is estimated to be between the earliest possible start time and the latest possible start time. After all the partitions are scheduled, the independently developed schedules are concatenated. The complexity of the PBSA algorithm is O(p 2 en=P 2 ), where P is the number of PPEs. 4 The VPMCP Algorithm MCP is a list scheduling algorithm. In a list scheduling, nodes are ordered in a list according to priority. A node at the front of the list is always scheduled first. Scheduling a node depends on the nodes that were scheduled before this node. Therefore, it is basically a sequential algorithm. Its heavy dependences make parallelization of MCP very difficult. In MCP, nodes must be scheduled one by one. However, when scheduling a node, the start times of the node on different TPEs can be calculated simultaneously. This parallelism can be exploited in the vertical scheme. If multiple nodes are scheduled simultaneously, the resultant schedule may not be the same as the one produced by MCP. The scheduling length will vary, and, in general, will be longer than that produced by the sequential MCP. Exploiting more parallelism may lower scheduling quality. We will study the degree of quality degradation when increasing parallelism. In the vertical scheme, multiple nodes may be selected to be scheduled at the same time. This way, parallelism can be increased and overhead reduced. In the horizontal scheme, different partitions must be scheduled simultaneously for a parallel execution. Therefore, the resultant schedule cannot be the same as the sequential one. We call the vertical version of parallel MCP the VPMCP algorithm and the horizontal version the HPMCP algorithm. The VPMCP algorithm is described in this section and the HPMCP algorithm will be presented in the next section. Before describing the VPMCP algorithm, we present a simple parallel version of the MCP algorithm. This version schedules one node each time so that it produces the same schedule as the sequential MCP algorithm. Each PPE maintains schedules for one or more TPEs. Therefore, it is a vertical scheme. We call this algorithm VPMCP1, which is shown in Figure 4. The nodes are first sorted by the ALAP time and cyclicly divided into P partitions. That is, the nodes in places of the sorted list are assigned to PPE i. The nodes are scheduled one by one. Each node is broadcast to all PPEs along with its parent information including the scheduled TPE number and time. Then, the start times of each node on different TPEs can be calculated in parallel. The node is scheduled to the TPE that allows the earliest start time. Consequently, if a PPE has any node that is a child of the newly scheduled node, the corresponding parent information of the node is updated. 1. (a) Compute the ALAP time of each node and sort the node list in an increasing ALAP order. Ties are broken randomly. (b) Divide the node list with cyclic partitioning into equal sized partitions, and each partition is assigned to a PPE. 2. (a) The PPE that has the first node in the list broadcasts the node, along with its parent information, to all PPEs. (b) Each PPE obtains a start time for the node on each of its TPEs. The earliest start time is obtained by parallel reduction of minimum. (c) The node is scheduled to the TPE that allows the earliest start time. (d) The parent information of children of the scheduled node is updated. Delete the node from the list and repeat this step until the list is empty. Figure 4: The VPMCP1 Algorithm. The VPMCP1 algorithm parallelizes the MCP algorithm directly. It produces exactly the same schedules as MCP. However, since each time only one node is scheduled, parallelism is limited and granularity is too fine. To solve this problem, a number of nodes could be scheduled simultaneously to increase granularity and to reduce communication. When some nodes are scheduled simultaneously, they may conflict with each other. Conflict may result in degradation of scheduling quality. The following lemma states the condition that allows some nodes to be scheduled in parallel without reducing scheduling quality. Lemma 1: When a node is scheduled to its earliest start time without conflicting with its former nodes, it is scheduled to the same place as it would be scheduled by sequential MCP. Proof: In the MCP scheduling sequence, a node obtains its earliest start time after all of its former nodes in the list have been scheduled. When a set of nodes are scheduled in parallel, each node obtains its earliest start time independently. A node may obtain its earliest start time which is earlier than the one in MCP scheduling when some of its former nodes in the set have not been scheduled. In this case, it must conflict with one of its former nodes. Therefore, if a node is scheduled a place that does not conflict with its former nodes, it obtains the same earliest start time and is scheduled to the same place as it would be in the MCP scheduling sequence. 2 With this lemma, a set of nodes can obtain their earliest start time simultaneously and be scheduled accordingly. When a node conflicts with its former nodes, then this node and the rest of the nodes will not be scheduled before they obtain their new earliest start times. In this way, more than one node can be scheduled each time. However, many nodes may have the same earliest start time in the same TPE. Therefore, there could be many conflicts. In most cases, only one or two nodes can be scheduled each time. To increase the number of nodes to be scheduled in parallel, we may allow a conflict node to be scheduled to its sub-optimal place. Therefore, when a node is found to be in conflict with its former nodes, it will be scheduled to the next non-conflict place. With this strategy, p nodes can be scheduled each time, where p is the number of TPEs. We use this strategy in our vertical version of parallel MCP, which is called the VPMCP algorithm. The details of this algorithm is shown in Figure 5. Besides the sorted node list, a ready list is constructed and sorted by ALAP times. A set of nodes selected from the ready list are broadcast to all PPEs. The start time of each node is calculated independently. Then, the start times are made available to every PPE by another parallel concatenation. Some nodes may compete for the same time slot in a TPE. This conflict is resolved by the smallest-ALAP-time-first rule. A node that does not get the best time slot will try its second best place, and so on, until it is scheduled to a non-conflict place. The time for calculation of the ALAP time and sorting is O(e+n log n). The parallel scheduling step is of O(n 2 =P ). Therefore, the complexity of the VPMCP algorithm is O(e+n log n+n 2 =P ), where n is the number of nodes, e the number of edges, and P the number of PPEs. The number of communications is 2n for VPMCP1 and 2n=p for VPMCP, where p is the number of TPEs. The VPMCP algorithm is compared to VPMCP1 in Table I. The workload for testing consists of random graphs of various sizes. We use the same random graph generator in [1]. Three values of communication-computation-ratio (CCR) were selected to be 0.1, 1, and 10. The weights on the nodes and edges were generated randomly such that the average value of CCR corresponded to 0.1, 1, or 10. This set of graphs will be used in the subsequent experiment. In this section, we use a set of graphs, each of which has 2,000 nodes. Each graph is scheduled to four TPEs. In the vertical scheme, the number of PPEs cannot be larger than the number of TPEs because 1. (a) Compute the ALAP time of each node and sort the node list in an increasing ALAP order. Ties are broken randomly. (b) Divide the node list with cyclic partitioning into equal sized partitions and each partition is assigned to a PPE. Initialize a ready list consisting of all nodes with no parent. Sort the list in an increasing ALAP order. 2. (a) The first p nodes in the ready list are broadcast to all PPEs by using the parallel concatenation operation, along with their parent information. If there are less than p nodes in the ready list, broadcast the entire ready list. (b) Each PPE obtains a start time for each node on each of its TPEs. The start times of each node are made available to every PPE by parallel concatenation. (c) A node is scheduled to the TPE that allows the earliest start time. If more than one node competes for the same time slot in a TPE, the node with smaller ALAP time gets the time slot. The node that does not get the time slot is then scheduled to the time slot that allows the second earliest start time, ans so on. (d) The parent information is updated for the children of the scheduled node. Delete these nodes from the ready list and update the ready list by adding nodes freed by these nodes. Repeat this step until the ready list is empty. Figure 5: The VPMCP Algorithm. Table I: Comparison of Vertical Strategies Number Scheduling length Running time (second) CCR of PPEs VPMCP1 VPMCP VPMCP1 VPMCP each TPE must be maintained in a single PPE. It can be seen from the table that VPMCP1 produces a better scheduling quality. However, its heavy communication results in low speedup or no speedup. VPMCP reduces running times and still provides an acceptable scheduling quality. The scheduling lengths are between 0.3% and 1.2% longer than that produced by VPMCP1. 5 The HPMCP Algorithm In a horizontal scheme, different partitions must be scheduled simultaneously for a parallel exe- cution. We call a horizontal version of parallel MCP the HPMCP algorithm, which is shown in Figure 6. 1. (a) Compute the ALAP time of each node and sort the node list in an increasing ALAP order. Ties are broken randomly. (b) Partition the node list into equal sized blocks and each partition is assigned to a PPE. 2. Each PPE applies the MCP algorithm to its partition to produce a sub-schedule. Edges between a node and its RPNs are ignored. 3. Concatenate each pair of adjacent sub-schedules. Walk through the schedule to determine the actual start time of each node. Figure The HPMCP Algorithm. In the HPMCP algorithm, the nodes are first sorted by the ALAP time. Therefore, the node list is in a topological order and is then partitioned into P equal sized blocks to be assigned to P PPEs. In this way, the graph is partitioned horizontally. When the graph is partitioned, each PPE will schedule its partition to produce its sub-schedule. Then, these sub-schedules will be concatenated to form the final schedule. Three problems are to be addressed for scheduling and concatenation: information estimation, concatenation permuta- tion, and post-insertion. Information estimation The major problem in the horizontal scheme is how to resolve the dependences between par- titions. In general, a latter partition depends on its former partitions. To schedule partitions in parallel, each PPE needs schedule information of its former partitions. Since it is impossible to obtain such information before the schedules of former partitions have been produced, an estimation is necessary. Although the latter PPE does not know the exact schedules of its former partitions, an estimation can help a node to determine its earliest start time in the latter partition. In the PBSA algorithm [1], the start time of each RPN (remote parent node) is estimated. It is done by calculating two parameters: the earliest possible start time (EPST), and the latest possible start time (LPST). The EPST of a node is the largest sum of computation times from the start point to the node, excluding the node itself. The LPST of a node is the sum of computation times of all nodes scheduled before the node, excluding the node itself. The estimated start time (EST) of an RPN is defined as ffEPST is equal to 1 if the RPN is on the critical path. Otherwise, it is equal to the length of the longest path from the start point through the RPN to the end point, divided by the length of the critical path. Given the estimated start time of an RPN, it is still necessary to estimate to which TPE the RPN is scheduled. If the RPN is a critical path node, then it is assumed that it will be scheduled to the same TPE as the highest level critical path node in the local partition. Otherwise, a TPE is randomly picked to be the one to which the RPN is scheduled. We call this estimation the PBSA estimation. This estimation is not necessarily accurate or better than a simpler estimation used in HPMCP. In HPMCP, we simply ignore all dependences between partitions. Therefore, all entry nodes in its partition can start at the same time. Furthermore, we assume all schedules of the former PE end at the same time. We call this estimation the HPMCP estimation. Table II: Comparison of Estimation Algorithms Number Scheduling length Running time (second) CCR of PPEs HPMCP est. PBSA est. HPMCP est. PBSA est. Now, we will compare the two approaches of estimation. In this section, the number of nodes in a graph is 2,000 and each graph is scheduled to four TPEs. The comparison is shown in Table II. The column "HPMCP est." shows the performance of HPMCP. The column "PBSA est." shows the performance of HPMCP with PBSA estimation. The scheduling lengths and running times are compared. The running time of PBSA estimation is longer. Notice that more PPEs produce longer scheduling lengths. That shows the trade-off between scheduling quality and parallelism. On the other hand, superlinear speedup was observed due to graph partitioning. The scheduling length produced by PBSA estimation is always longer than that produced by HPMCP estimation. It implies that a more complex estimation algorithm cannot promise good scheduling and a simpler algorithm may be better. However, that is not to say that we should use the simplest one in the future. It is still possible to find a good estimation to improve performance. The simple estimation used in HPMCP sets a baseline for future estimation algorithms. Concatenation permutation After each PPE produces its sub-schedule, the final schedule is constructed by concatenating these sub-schedules. Because there is no accurate information of former sub-schedules, it is not easy to determine the optimal permutation of TPEs between adjacent sub-schedules, that is, to determine which latter TPE should be concatenated to which former TPE. A hueristics is necessary. In the PBSA algorithm, a TPE with the earliest node is concatenated to the TPE of the former sub-schedule that allows the earliest execution. Then, other TPEs are concatenated to the TPEs in the former sub-schedule in a breadth-first order [1]. In the HPMCP algorithm, we assume that the start time of each node within its partition is the same. Therefore, the above algorithm cannot be applied. We simply do not perform permutation of TPEs in HPMCP. That is, a TPE in the latter sub-schedule is concatenated to the same TPE in the former sub- schedule. An alternative hueristics can be described as follows: each PPE finds out within its sub-schedule which TPE has most critical-path nodes and permutes this TPE with TPE 0. With this permutation, as many critical path nodes as possible are scheduled to the same TPE, and the critical path length could be reduced. This permutation algorithm is compared to the non- permutation algorithm in Table III. The time spent on the permutation step causes this algorithm to be slower since extra time is spent on determine weather a node is on the critical path. In terms of the scheduling length, the permutation algorithm makes four of the test cases better than the non-permutation algorithm, and two cases worse. This permutation algorithm does not improve performance much. Therefore, no permutation is performed in the HPMCP algorithm. Post-insertion Finally, we walk through the entire concatenated schedule to determine the actual start time of each node. Some refinement can be performed in this step. In a horizontal scheme, the latter PPE is not able to insert nodes to former sub-schedules due to lack of their information. This leads to Table III: Comparison of Permutation Algorithms Number Scheduling length Running time (second) CCR of PPEs No permut. Permut. No permut. Permut. some performance loss. It can be partially corrected at the concatenation time by inserting the nodes of a latter sub-schedule into its former sub-schedules. Improvement of this post-insertion algorithm is shown in Table III. Compared to non-insertion, the post-insertion algorithm reduces scheduling length in eight test cases and increases it in four cases. Overall, this post-insertion algorithm can improve scheduling quality. However, it spends much more time for post-insertion. In the following, we do not perform this post-insertion. The time for calculation of the ALAP time and sorting is O(e n). The second step of parallel scheduling is of O(n 2 =P 2 ), and the third step spends O(e+n 2 =p 2 ) time for post-insertion and O(e) time for non-insertion. Therefore, the complexity of the HPMCP algorithm without post-insertion is O(e is the number of nodes and e the number of edges in a graph, p is the number of TPEs and P the number of PPEs. 6 Performance The VPMCP and HPMCP algorithms were implemented on Intel Paragon. We present its performance with three measures: scheduling length, running time, and speedup. Table IV: Comparison of Post-insertion Algorithms Number Scheduling length Running time (second) CCR of PPEs No insert. Insert. No insert. Insert. First, performance of the VPMCP algorithm is to be presented. In Tables V and VI, graphs of 1,000, 2,000, 3,000, and 4,000 nodes are scheduled to four TPEs. The scheduling length provides a measure of scheduling quality. The results shown in Table V are the scheduling lengths and the ratios of the scheduling lengths produced by the VPMCP algorithm to the scheduling lengths produced by the MCP algorithm. The ratio was obtained by running the VPMCP algorithm on 2 and 4 PPEs on Paragon and taking the ratios of the scheduling lengths produced by it to those of MCP running on one PPE. As one can see from this table, there was almost no effect of graph size on scheduling quality. In most cases, the scheduling lengths of VPMCP are not more than 1% longer than that produced by MCP. Running time and speedup of the VPMCP algorithm are shown in Table VI. Speedup is defined by is the sequential execution time of the optimal sequential algorithm and T P is the parallel execution time. The running times of VPMCP on more than one PPE are compared with MCP running time on one PPE. The MCP running time on a single processor is a sequential version without parallelization overhead. The low speedup of VPMCP is caused by its large number of communications. The next experiment is to study VPMCP performance of different numbers of TPEs. Tables VII and VIII show the scheduling lengths and running times of graphs of 4,000 nodes for 2, 4, Table V: The scheduling lengths produced by VPMCP and their ratios to those of MCP Graph size (number of nodes) CCR Number 1000 2000 3000 4000 of PPEs length ratio length ratio length ratio length ratio Table VI: Running time (in second) and speedup of VPMCP Graph size (number of nodes) CCR Number 1000 2000 3000 4000 of PPEs time S time S time S time S 8, and 16 TPEs. The difference between the scheduling lengths of VPMCP and MCP are within 1% in most cases. As can be noticed from this table, when the number of TPEs increases, the scheduling lengths decrease and the ratio only increases slightly. Therefore, scheduling quality scales up quit well. Higher speedups are obtained with more PPEs. Next, we study performance of the HPMCP algorithm. The number of TPEs is four in Tables IX and X. In Table IX, the ratio was obtained by running the HPMCP algorithm on 2, 4, 8, and 16 PPEs on Paragon and taking the ratios of the scheduling lengths produced by it to those of MCP running on one PPE. The deterioration in performance of HPMCP is due to estimation of the start time of RPNs and concatenation. Out of the 48 test cases shown in the table, there is only one case in which HPMCP performed more than 10% worse than MCP, 31 Table VII: The scheduling lengths and ratios for different number of TPEs produced by VPMCP Number of TPEs CCR Number 2 of PPEs length ratio length ratio length ratio length ratio Table VIII: VPMCP running time (in second) and speedup for different number of TPEs Number of TPEs CCR Number 2 of PPEs time S time S time S time S of them are within 1%, and 16 of them are between 1% to 10%. It is of interest that in two cases, HPMCP produced even better results than MCP. Since MCP is a heuristic algorithm, this is possible. Sometimes, a parallel version could produce a better result than its corresponding sequential one. Running time and speedup of the HPMCP algorithm are shown in Table X. There are some superlinear speedup cases in the table. That is because HPMCP has lower complexity than MCP. The complexity of MCP is O(n 2 ). The complexity of HPMCP on P PPEs is O(e+n log n+n 2 =P 2 ). Therefore, speedup is bounded by P 2 instead of P . Speedup on 16 PPEs is not as good as expected, because the graph size is not large enough and the relative overhead is large. Tables XI and XII show the HPMCP scheduling lengths and running times of graphs of 4,000 nodes for 2, 4, 8, and 16 TPEs. The speedups decrease with the number of TPEs. That is caused by increasing dependences between TPEs. Now, we compare performance of VPMCP and HPMCP. Performance shown in Figures 7 and 8 is for graphs of 4,000 nodes. The number of TPEs is the same as the number of PPEs. Figure 7 shows the percentage of the scheduling length over that produced by MCP. There are some negative numbers which indicate that scheduling lengths are shorter than that produced by MCP. For 2 or 4 PPEs, HPMCP produces shorter scheduling lengths. However, for more PPEs, VPMCP produces better scheduling quality than that produced by HPMCP. In general, VPMCP provides a more stable scheduling quality. Figure 8 compares the speedups of VPMCP and HPMCP algorithms. HPMCP is faster than VPMCP with a higher speedup. After scheduling, the nodes are not in the PPEs where they are to be executed in the horizontal scheme. A major communication step is necessary to move nodes. However, when the number of PPEs is equal to the number of TPEs, the vertical scheme can avoid this communication step because the nodes reside in the PPEs where they are to be executed. It becomes more important when the scheduling algorithms are used at runtime. Next, we compare two algorithms in the horizontal scheme, HPMCP and PBSA. The PBSA algorithm takes into account link contention and communication routing strategy, but HPMCP does not consider these factors. Therefore, the edge weights in PBSA vary with different topologies, whereas in HPMCP they are constant. For comparison purposes, we have implemented a simplified version of PBSA, which assumes that the edge weights are constant. PBSA is much slower than HPMCP, because its complexity is much higher. The complexity of PBSA is O(p 2 en=P 2 ) and that of HPMCP is O(e about 50 to 120 times faster than PBSA for this set of graphs. Then, we compare the scheduling lengths produced by HPMCP and PBSA. The results are shown in Table XIII. In this table, the scheduling lengths produced by the sequential MCP and BSA algorithms running on a single processor are also compared. Table IX: The scheduling lengths produced by HPMCP and their ratios to those of MCP Graph size (number of nodes) CCR Number 1000 2000 3000 4000 of PPEs length ratio length ratio length ratio length ratio Table X: Running time (in second) and speedup of HPMCP Graph size (number of nodes) Table XI: The scheduling lengths and ratios for different number of TPEs produced by HPMCP Number of TPEs CCR Number 2 of PPEs length ratio length ratio length ratio length ratio Table XII: HPMCP Running time (in second) and speedup for different number of TPEs Number of TPEs 4.0% 4.5% 5.0% Number of PPEs 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% Number of PPEs -3.0% -2.0% -1.0% 1.0% 2.0% 3.0% 4.0% Number of PPEs Figure 7: Comparison of scheduling quality produced by VPMCP and HPMCP Number of PPEs Number of PPEs Number of PPEs Figure 8: Comparison of speedups of VPMCP and HPMCP When CCR is 0.1 or 1, the scheduling lengths produced by MCP are slightly shorter than those produced by BSA. When CCR is 10, MCP is much better than BSA. The parallel versions of the two algorithms perform very differently. When the number of PPEs increases, the scheduling lengths produced by HPMCP increases only slightly, but that of PBSA increases significantly. Figure 9 compares HPMCP and PBSA by the sum of scheduling lengths of four graphs of 1,000, 2,000, 3,000, and 4,000 nodes for different CCRs and different number of PPEs. Table XIII: The scheduling lengths produced by HPMCP and PBSA Graph size (number of nodes) CCR Number 1000 2000 3000 4000 of PPEs HPMCP PBSA HPMCP PBSA HPMCP PBSA HPMCP PBSA 7 Concluding Remarks Parallel scheduling is faster and is able to schedule large macro dataflow graphs. Parallel scheduling is a new approach and is still under development. Many open problems need to be solved. High-quality parallel scheduling algorithms with low complexity will be developed. It can be achieved by parallelizing the existing sequential scheduling algorithms or by designing new parallel scheduling algorithms. We have developed the VPMCP and HPMCP algorithms by parallelizing the sequential MCP algorithm. Performance of this approach has been studied. Both VPMCP and HPMCP algorithms are much faster then PBSA. They produce high-quality scheduling in terms of the scheduling length. 30,000.0 Scheduling length Number of PPEs PBSA 30,000.0 Scheduling length Number of PPEs PBSA 30,000.0 50,000.0 Scheduling length Number of PPEs PBSA Figure 9: Comparison of scheduling lengths produced by HPMCP and PBSA Acknowledgments We are very grateful to Yu-Kwong Kwok and Ishfaq Ahmad for providing their PBSA program and random graph generator for testing. --R A parallel approach to multiprocessor scheduling. Performance comparison of algorithms for static scheduling of DAGs to multiprocessors. The LAST algorithm: A heuristics-based static task allocation algorithm Applications and performance analysis of a compile-time optimization approach for list scheduling algorithms on distributed memory multiprocessors Scheduling parallel program tasks onto arbitrary target machines. Task Scheduling in Parallel and Distributed Systems. Computers and Intractability: A Guide to the Theory of NP-Completeness Scheduling precedence graphs in systems with interprocessor communication times. A comparison of multiprocessor scheduling heuristics. Duplication scheduling heuristics (dsh): A new precedence task scheduler for parallel processor systems. Partitioning and Scheduling Parallel Programs for Multiprocessors. A compile-time scheduling heuristic for interconnection-constrained heterogeneous processor architectures A programming aid for message-passing systems DSC: Scheduling parallel tasks on an unbounded number of processors. --TR --CTR Sukanya Suranauwarat , Hideo Taniguchi, The design, implementation and initial evaluation of an advanced knowledge-based process scheduler, ACM SIGOPS Operating Systems Review, v.35 n.4, p.61-81, October 2001

parallel scheduling algorithm;static scheduling;modified critical-path algorithm;macro dataflow graph

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A Simple Algorithm for Nearest Neighbor Search in High Dimensions.

AbstractThe problem of finding the closest point in high-dimensional spaces is common in pattern recognition. Unfortunately, the complexity of most existing search algorithms, such as k-d tree and R-tree, grows exponentially with dimension, making them impractical for dimensionality above 15. In nearly all applications, the closest point is of interest only if it lies within a user-specified distance $\epsilon.$ We present a simple and practical algorithm to efficiently search for the nearest neighbor within Euclidean distance $\epsilon.$ The use of projection search combined with a novel data structure dramatically improves performance in high dimensions. A complexity analysis is presented which helps to automatically determine $\epsilon$ in structured problems. A comprehensive set of benchmarks clearly shows the superiority of the proposed algorithm for a variety of structured and unstructured search problems. Object recognition is demonstrated as an example application. The simplicity of the algorithm makes it possible to construct an inexpensive hardware search engine which can be 100 times faster than its software equivalent. A C++ implementation of our algorithm is available upon request to [email protected]/CAVE/.

Introduction Searching for nearest neighbors continues to prove itself as an important problem in many fields of science and engineering. The nearest neighbor problem in multiple dimensions is stated as follows: given a set of n points and a novel query point Q in a d-dimensional space, "Find a point in the set such that its distance from Q is lesser than, or equal to, the distance of Q from any other point in the set" [ 21 ] . A variety of search algorithms have been advanced since Knuth first stated this (post-office) problem. Why then, do we need a new algorithm? The answer is that existing techniques perform very poorly in high dimensional spaces. The complexity of most techniques grows exponentially with the dimensionality, d. By high dimensional, we mean when, say d ? 25. Such high dimensionality occurs commonly in applications that use eigenspace based appearance matching, such as real-time object recognition [ tracking and inspection [ 26 ] , and feature detection . Moreover, these techniques require that nearest neighbor search be performed using the Euclidean distance (or L 2 ) norm. This can be a hard problem, especially when dimensionality is high. High dimensionality is also observed in visual correspondence problems such as motion estimation in MPEG coding (d ? estimation in binocular stereo (d=25-81), and optical flow computation in structure from motion (also d=25-81). In this paper, we propose a simple algorithm to efficiently search for the nearest neighbor within distance ffl in high dimensions. We shall see that the complexity of the proposed algorithm, for small ffl, grows very slowly with d. Our algorithm is successful because it does not tackle the nearest neighbor problem as originally stated; it only finds points within distance ffl from the novel point. This property is sufficient in most pattern recognition problems (and for the problems stated above), because a "match" is declared with high confidence only when a novel point is sufficiently close to a training point. Occasionally, it is not possible to assume that ffl is known, so we suggest a method to automatically choose ffl. We now briefly outline the proposed algorithm. Our algorithm is based on the projection search paradigm first used by Friedman Friedman's simple technique works as follows. In the preprocessing step, d dimensional training points are ordered in d different ways by individually sorting each of their coordi- nates. Each of the d sorted coordinate arrays can be thought of as a 1-D axis with the entire d dimensional space collapsed (or projected) onto it. Given a novel point Q, the nearest neighbor is found as follows. A small offset ffl is subtracted from and added to each of Q's coordinates to obtain two values. Two binary searches are performed on each of the sorted arrays to locate the positions of both the values. An axis with the minimum number of points in between the positions is chosen. Finally, points in between the positions on the chosen axis are exhaustively searched to obtain the closest point. The complexity of this technique is roughly O(ndffl) and is clearly inefficient in high d. This simple projection search was improved upon by Yunck utilizes a precomputed data structure which maintains a mapping from the sorted to the unsorted (original) coordinate arrays. In addition to this mapping, an indicator array of n elements is used. Each element of the indicator array, henceforth called an indicator, corresponds to a point. At the beginning of a search, all indicators are initialized to the number '1'. As before, a small offset ffl is subtracted from and added to each of the novel point Q's coordinates to obtain two values. Two binary searches are performed on each of the d sorted arrays to locate the positions of both the values. The mapping from sorted to unsorted arrays is used to find the points corresponding to the coordinates in between these values. Indicators corresponding to these points are (binary) shifted to the left by one bit and the entire process repeated for each of the d dimensions. At the end, points whose indicators have the value must lie within an 2ffl hypercube. An exhaustive search can now be performed on the hypercube points to find the nearest neighbor. With the above data structure, Yunck was able to find points within the hypercube using primarily integer operations. However, the total number of machine operations required (in- teger and floating point) to find points within the hypercube are similar to that of Friedman's algorithm (roughly O(ndffl)). Due to this and the fact that most modern CPUs do not significantly penalize floating point operations, the improvement is only slight (benchmarked in a later section). We propose a data structure that significantly reduces the total number of machine operations required to locate points within the hypercube to roughly O(nffl )). Moreover, this data structure facilitates a very simple hardware implementation which can result in a further increase in performance by two orders of magnitude. Previous Work Search algorithms can be divided into the following broad categories: (a) Exhaustive search, (b) hashing and indexing, (c) static space partitioning, (d) dynamic space partitioning, and randomized algorithms. The algorithm described in this paper falls in category (d). The algorithms can be further categorized into those that work in vector spaces and those that work in metric spaces. Categories (b)-(d) fall in the former, while category (a) falls in the later. Metric space search techniques are used when it is possible to somehow compute a distance measure between sample "points" or pieces of data but the space in which the points reside lacks an explicit coordinate structure. In this paper, we focus only on vector space techniques. For a detailed discussion on searching in metric spaces, refer to [ Exhaustive search, as the term implies, involves computing the distance of the novel point from each and every point in the set and finding the point with the minimum distance. This approach is clearly inefficient and its complexity is O(nd). Hashing and indexing are the fastest search techniques and run in constant time. However, the space required to store an index table increases exponentially with d. Hence, hybrid schemes of hashing from a high dimensional space to a low (1 or 2) dimensional space and then indexing in this low dimensional space have been proposed. Such a dimensionality reduction is called geometric hashing . The problem is that, with increasing dimensionality, it becomes difficult to construct a hash function that distributes data uniformly across the entire hash table (index). An added drawback arises from the fact that hashing inherently partitions space into bins. If two points in adjacent bins are closer to each other than a third point within the same bin. A search algorithm that uses a hash table, or an index, will not correctly find the point in the adjacent bin. Hence, hashing and indexing are only really effective when the novel point is exactly equal to one of the database points. Space partitioning techniques have led to a few elegant solutions to multi-dimensional search problems. A method of particular theoretical significance divides the search space into polygons. A Voronoi polygon is a geometrical construct obtained by intersecting perpendicular bisectors of adjacent points. In a 2-D search space, Voronoi polygons allow the nearest neighbor to be found in O(log 2 n) operations, where, n is the number of points in the database. Unfortunately, the cost of constructing and storing Voronoi diagrams grows exponentially with the number of dimensions. Details can be found in [ 3 Another algorithm of interest is the 1-D binary search generalized to d dimensions [ 11 ] . This runs in O(log 2 n) time but requires storage O(n 4 ), which makes it impractical for n ? 100. Perhaps the most widely used algorithm for searching in multiple dimensions is a static space partitioning technique based on a k dimensional binary search tree, called the k-d tree . The k-d tree is a data structure which partitions space using hyperplanes placed perpendicular to the coordinate axes. The partitions are arranged hierarchically to form a tree. In its simplest form, a k-d tree is constructed as follows. A point in the database is chosen to be the root node. Points lying on one side of a hyperplane passing through the root node are added to the left child and the points on the other side are added to the right child. This process is applied recursively on the left and right children until a small number of points remain. The resulting tree of hierarchically arranged hyperplanes induces a partition of space into hyper-rectangular regions, termed buckets, each containing a small number of points. The k-d tree can be used to search for the nearest neighbor as follows. The k coordinates of a novel point are used to descend the tree to find the bucket which contains it. An exhaustive search is performed to determine the closest point within that bucket. The size of a "query" hypersphere is set to the distance of this closest point. Information stored at the parent nodes is used to determine if this hypersphere intersects with any other buckets. If it does, then that bucket is exhaustively searched and the size of the hypersphere is revised if necessary. For fixed d, and under certain assumptions about the underlying data, the k-d tree requires O(nlog 2 n) operations to construct and O(log 2 n) operations to search [ k-d trees are extremely versatile and efficient to use in low dimensions. However, the performance degrades exponentially 1 with increasing dimensionality. This is because, in high dimensions, the query hypersphere tends to intersect many adjacent buckets, leading to a dramatic increase in the number of points examined. k-d trees are dynamic data structures which means that data can be added or deleted at a small cost. The impact of adding or deleting data on the search performance is however quite unpredictable and is related to the amount of imbalance the new data causes in the tree. High imbalance 1 Although this appears contradictory to the previous statement, the claim of O(log 2 n) complexity is made assuming fixed d and varying n [ . The exact relationship between d and complexity has not yet been established, but it has been observed by us and many others that it is roughly exponential. generally means slower searches. A number of improvements to the basic algorithm have been suggested. Friedman recommends that the partitioning hyperplane be chosen such that it passes through the median point and is placed perpendicular to the coordinate axis along whose direction the spread of the points is maximum [ 15 ] . Sproull suggests using a truncated distance computation to increase efficiency in high dimensions [ 36 ] . Variants of the k-d tree have been used to address specific search problems An R-tree is also a space partitioning structure, but unlike k-d trees, the partitioning element is not a hyperplane but a hyper-rectangular region [ This hierarchical rectangular structure is useful in applications such as searching by image content [ needs to locate the closest manifold (or cluster) to a novel manifold (or cluster). An R-tree also addresses some of the problems involved in implementing k-d trees in large disk based databases. The R-tree is also a dynamic data structure, but unlike the k-d tree, the search performance is not affected by addition or deletion of data. A number of variants of R-Trees improve on the basic technique, such as packed R-trees [ 34 Although R-trees are useful in implementing sophisticated queries and managing large databases, the performance of nearest neighbor point searches in high dimensions is very similar to that of k-d trees; complexity grows exponentially with d. Other static space partitioning techniques have been proposed such as branch and bound none of which significantly improve performance for high dimensions. Clarkson describes a randomized algorithm which finds the closest point in d dimensional space in O(log 2 n) operations using a RPO (randomized post . However, the time taken to construct the RPO tree is O(n dd=2e(1+ffl) ) and the space required to store it is also O(n dd=2e(1+ffl) ). This makes it impractical when the number of points n is large or or if d ? 3. 3 The Algorithm 3.1 Searching by Slicing We illustrate the proposed high dimensional search algorithm using a simple example in 3-D space, shown in Figure 1. We call the set of points in which we wish to search for the closest point as the point set. Then, our goal is to find the point in the point set that is closest to 2e 2e 2e 2e x-e x+e z-e z+e Y Z Figure 1: The proposed algorithm efficiently finds points inside a cube of size 2ffl around the novel query point Q. The closest point is then found by performing an exhaustive search within the cube using the Euclidean distance metric. a novel query point Q(x; and within a distance ffl. Our approach is to first find all the points that lie inside a cube (see Figure 1) of side 2ffl centered at Q. Since ffl is typically small, the number of points inside the cube is also small. The closest point can then be found by performing an exhaustive search on these points. If there are no points inside the cube, we know that there are no points within ffl. The points within the cube can be found as follows. First, we find the points that are sandwiched between a pair of parallel planes X 1 and X 2 (see Figure 1) and add them to a list, which we call the candidate list. The planes are perpendicular to the first axis of the coordinate frame and are located on either side of point Q at a distance of ffl. Next, we trim the candidate list by discarding points that are not also sandwiched between the parallel pair of planes Y 1 and Y 2 , that are perpendicular to X 1 and X 2 , again located on either side of Q at a distance ffl. This procedure is repeated for planes Z 1 and Z 2 , at the end of which, the candidate list contains only points within the cube of size 2ffl centered on Q. Since the number of points in the final trimmed list is typically small, the cost of the exhaustive search is negligible. The major computational cost in our technique is therefore in constructing and trimming the candidate list. 3.2 Data Structure Candidate list construction and trimming can done in a variety of ways. Here, we propose a method that uses a simple pre-constructed data structure along with 1-D binary searches to efficiently find points sandwiched between a pair of parallel hyperplanes. The data structure is constructed from the raw point set and is depicted in Figure 2. It is assumed that the point set is static and hence, for a given point set, the data structure needs to be constructed only once. The point set is stored as a collection of d 1-D arrays, where the j th array contains the j th coordinate of the points. Thus, in the point set, coordinates of a point lie along the same row. This is illustrated by the dotted lines in Figure 2. Now suppose that novel point Q has coordinates . Recall that in order to construct the candidate list, we need to find points in the point set that lie between a pair of parallel hyperplanes separated by a distance 2ffl, perpendicular to the first coordinate axis, and centered at that is, we need to locate points whose first coordinate lies between the limits ffl. This can be done with the help of two binary searches, one for each limit, if the coordinate array were sorted beforehand. To this end, we sort each of the d coordinate arrays in the point set independently to obtain the ordered set. Unfortunately, sorting raw coordinates does not leave us with any information regarding which points in the arrays of the ordered set correspond to any given point in the point set, and vice versa. For this purpose, we maintain two maps. The backward map maps a coordinate in the ordered set to the corresponding coordinate in the point set and, conversely, the forward map maps a point in the point set to a point in the ordered set. Notice that the maps are simple integer arrays; if P [d][n] is the point set, O[d][n] is the ordered set, F [d][n] and B[d][n] are the forward and backward maps, respectively, then Using the backward map, we find the corresponding points in the point set (shown as dark shaded areas) and add the appropriate points to the candidate list. With this, the construction of the candidate list is complete. Next, we trim the candidate list by iterating POINT SET ORDERED SET Dimensions Points Forward Map Forward Map Backward Map Input -e +e -e -e +e +e ,., Figure 2: Data structures used for constructing and trimming the candidate list. The point set corresponds to the raw list of data points, while in the ordered set each coordinate is sorted. The forward and backward maps enable efficient correspondence between the point and ordered sets. on as follows. In iteration k, we check every point in the candidate list, by using the forward map, to see if its k th coordinate lies within the limits Each of these limits are also obtained by binary search. Points with k th coordinates that lie outside this range (shown in light grey) are discarded from the list. At the end of the final iteration, points remaining on the candidate list are the ones which lie inside a hypercube of side 2ffl centered at Q. In our discussion, we proposed constructing the candidate list using the first dimension, and then performing list trimming using dimensions 2; 3; d, in that order. We wish to emphasize that these operations can be done in any order and still yield the desired result. In the next section, we shall see that it is possible to determine an optimal ordering such that the cost of constructing and trimming the list is minimized. It is important to note that the only operations used in trimming the list are integer comparisons and memory lookups. Moreover, by using the proposed data structure, we have limited the use of floating point operations to just the binary searches needed to find the row indices corresponding to the hyperplanes. This feature is critical to the efficiency of the proposed algorithm, when compared with competing ones. It not only facilitates a simple software implementation, but also permits the implementation of a hardware search engine. As previously stated, the algorithm needs to be supplied with an "appropriate" ffl prior to search. This is possible for a large class of problems (in pattern recognition, for instance) where a match can be declared only if the novel point Q is sufficiently close to a database point. It is reasonable to assume that ffl is given a priori, however, the choice of ffl can prove problematic if this is not the case. One solution is to set ffl large, but this might seriously impact performance. On the other hand, a small ffl could result in the hypercube being empty. How do we determine an optimal ffl for a given problem? How exactly does ffl affect the performance of the algorithm? We seek answers to these questions in the following section. 4 Complexity In this section, we attempt to analyze the computational complexity of data structure stor- age, construction and nearest neighbor search. As we saw in the previous section, constructing the data structure is essentially sorting d arrays of size n. This can be done in O(dn log 2 n) time. The only additional storage necessary is to hold the forward and bacward maps. This requires space O(nd). For nearest neighbor search, the major computational cost is in the process of candidate list construction and trimming. The number of points initially added to the candidate list depends not only on ffl, but also on the distribution of data in the point set and the location of the novel point Q. Hence, to facilitate analysis, we structure the problem by assuming widely used distributions for the point set. The following notation is used. Random variables are denoted by uppercase letters, for instance, Q. Vectors are in bold, such as, q. Suffixes are used to denote individual elements of vectors, for instance, Q k is the k th element of vector Q. Probability density is written as and as f Q (q) if Q is continuous. 2e Axis c Figure 3: The projection of the point set and the novel point onto one of the dimensions of the search space. The number of points inside bin B is given by the binomial distribution. Figure 3 shows the novel point Q and a set of n points in 2-D space drawn from a known distribution. Recall that the candidate list is initialized with points sandwiched between a hyperplane pair in the first dimension, or more generally, in the c th dimension. This corresponds to the points inside bin B in Figure 3, where the entire point set and Q are projected to the c th coordinate axis. The boundaries of bin B are where the hyperplanes intersect the axis c, at c be the number of points in bin B. In order to determine the average number of points added to the candidate list, we must compute c to be the distance between Q c and any point on the candidate list. The distribution of Z c may be calculated from the the distribution of the point set. Define P c to be the probability that any projected point in the point set is within distance ffl from Q c ; that is, It is now possible to write an expression for the density of M c in terms of P c . Irrespective of the distribution of the points, M c is binomially distributed 2 This is equivalent to the elementary probability problem: given that a success (a point is within bin B) can occur with probability P c , the number of successes that occur in n independent trials (points) is binomially distributed. From the above expression, the average number of points in bin B, E[M c j Q c ], is easily determined to be Note that E[M c j Q c ] is itself a random variable that depends on c and the location of Q. If the distribution of Q is known, the expected number of points in the bin can be computed as we perform one lookup in the backward map for every point between a hyperplane pair, and this is the main computational effort, equation (3) directly estimates the cost of candidate list construction. Next, we derive an expression for the total number of points remaining on the candidate list as we trim through the dimensions in the sequence c 1 . Recall that in the iteration k, we perform a forward map lookup for every point in the candidate list and see if it lies between the c k th hyperplane pair. How many points on the candidate list lie between this hyperplane pair? Once again, equation (3) can be used, this time replacing n with the number of points on the candidate list rather than the entire point set. We assume that the point set is independently distributed. Hence, if N k is the total number of points on the candidate list before the iteration k, Y Define N to be the total cost of constructing and trimming the candidate list. For each trim, we need to perform one forward map lookup and two integer comparisons. Hence, if we assign one cost unit to each of these operations, an expression for N can be written with the aid of equation (4) as Y which, on the average is Y Equation (6) suggests that if the distributions f Q (q) and f Z (z) are known, we can compute the average cost E[N in terms of ffl. In the next section, we shall examine two cases of particular interest: (a) Z is uniformly distributed, and (b) Z is normally distributed. Note that we have left out the cost of exhaustive search on points within the final hypercube. The reason is that the cost of an exhaustive search is dependant on the distance metric used. This cost is however very small and can be neglected in most cases when n AE d. If it needs to be considered, it can be added to equation (6). We end this section by making an observation. We had mentioned earlier that it is of advantage to examine the dimensions in a specific order. What is this order? By expanding the summation and product and by factoring terms, equation (5) can be rewritten as It is immediate that the value of N is minimum when P c 1 . In other words, should be chosen such that the numbers of sandwiched points between hyperplane pairs are in ascending order. This can be easily ensured by simply sorting the numbers of sandwiched points. Note that there are only d such numbers, which can be obtained in time O(d) by simply taking the difference of the indices to the ordered set returned by each pair of binarysearchs. Further, the cost of sorting these numbers is O(d log 2 d) by heapsort these costs are negligible in any problem of reasonable dimensionality. 4.1 Uniformly Distributed Point Set We now look at the specific case of a point set that is uniformly distributed. If X is a point in the point set, we assume an independent and uniform distribution with extent l on each of it's coordinates as 1=l if \Gammal=2 - x - l=2 1.5e+062.5e+06Cost e d=5 e5000001.5e+062.5e+06Cost n=50000 n=150000 (a) (b) Figure 4: The average cost of the algorithm is independent of d and grows only linearly for small ffl. The point set in both cases is assumed to be uniformly distributed with extent l = 1. (a) The point set contains 100,000 points in 5-D, 10-D, 15-D, 20-D and 25-D spaces. (b) The point set is 15-D and contains 50000, 75000, 100000, 125000 and 150000 points. Using equation (8) and the fact that Z , an expression for the density of Z c can be written as f Zc jQc P c can now be written as \Gammaffl f Zc jQc (z)dz \Gammaffll dz l Substituting equation (10) in equation (6) and considering the upper bound (worst case), we get l l l l l l l By neglecting constants, we write d=5 Cost e n=50000 Cost (a) (b) Figure 5: The average cost of the algorithm is independent of d and grows only linearly for small ffl. The point set in both cases is assumed to be normally distributed with variance oe = 1. (a) The point set contains 100,000 points in 5-D, 10-D, 15-D, 20-D and 25-D spaces (b) The point set is 15-D and contains 50000, 75000, 100000, 125000 and 150000 points For small ffl, we observe that ffl d - 0, because of which cost is independent of d: In Figure 4, equation (11) is plotted against ffl for different d (Figure 4(a)) and different Figure 1. Observe that as long as ffl ! :25, the cost varies little with d and is linearly proportional to n. This also means that keeping ffl small is crucial to the performance of the algorithm. As we shall see later, ffl can in fact be kept small for many problems. Hence, even though the cost of our algorithm grows linearly with n, ffl is small enough that in many real problems, it is better to pay this price of linearity, rather than an exponential dependence on d. 4.2 Normally Distributed Point Set Next, we look at the case when the point set is normally distributed. If X is a point in the point set, we assume an independent and normal distribution with variance oe on each of it's coordinates: f Xc (x) =p 2-oe exp As before, using Z , an expression for the density of Z c can be obtained to get f Zc jQc 2-oe P c can then be written as \Gammaffl f Zc jQc (z)dz oe oe p! This expression can be substituted into equation (6) and evaluated numerically to estimate cost for a given Q. Figure 5 shows the cost as a function of ffl for As with uniform distribution, we observe that when ffl ! 1, the cost is nearly independent of d and grows linearly with n. In a variety of pattern classification problems, data take the form of individual Gaussian clusters or mixtures of Gaussian clusters. In such cases, the above results can serve as the basis for complexity analysis. 5 Determining ffl It is apparent from the analysis in the preceding section that the cost of the proposed algorithm depends critically on ffl. Setting ffl too high results in a huge increase in cost with d, while setting ffl too small may result in an empty candidate list. Although the freedom to choose ffl may be attractive in some applications, it may prove non-intuitive and hard in others. In such cases, can we automatically determine ffl so that the closest point can be found with high certainty? If the distribution of the point set is known, we can. We first review well known facts about L p norms. Figure 6 illustrates these norms for a few selected values of p. All points on these surfaces are equidistant (in the sense of the respective norm) from the central point. More formally, the L p distance between two vectors a and b is defined as These distance metrics are also known as Minkowski-p metrics. So how are these relevant to determining ffl? The L 2 norm occurs most frequently in pattern recognition problems. Unfortunately, candidate list trimming in our algorithm does not find points within L 2 , but Figure An illustration of various norms, also known as Minkowski p-metrics. All points on these surfaces are equidistant from the central point. The L1 metric bounds L p for all p. within L1 (i.e. the hypercube). Since L1 bounds L 2 , one can naively perform an exhaustive search inside L1 . However, as seen in figure 7(a), this does not always correctly find the closest point. Notice that P 2 is closer to Q than P 1 , although an exhaustive search within the cube will incorrectly identify to be the closest. There is a simple solution to this problem. When performing an exhaustive search, impose an additional constraint that only points within an L 2 radius ffl should be considered (see figure 7(b)). This, however, increases the possibility that the hypersphere is empty. In the above example, for instance, P 1 will be discarded and we would not be able to find any point. Clearly then, we need to consider this fact in our automatic method of determining ffl which we describe next. We propose two methods to automatically determine ffl. The first computes the radius of the smallest hypersphere that will contain at least one point with some (specified) probability. ffl is set to this radius and the algorithm proceeds to find all points within a circumscribing hypercube of side 2ffl. This method is however not efficient in very high dimensions; the reason being as follows. As we increase dimensionality, the difference between the hypersphere and hypercube volumes becomes so great that the hypercube "corners" contain far more points than the inscribed hypersphere. Consequently, the extra effort necessary to perform L 2 distance computations on these corner points is eventually wasted. So rather than find the circumscribing hypercube, in our second method, we simply find the length of a side of the smallest hypercube that will contain at least one point with some (specified) probability. ffl can then be set to half the length of this side. This leads to the problem we described earlier that, when searching some points outside a hypercube can be closer in the L 2 sense than points inside. We shall now describe both the methods in detail and see how we can remedy e r 2e e (a) (b) Figure 7: An exhaustive search within a hypercube may yield an incorrect result. (a) P 2 is closer to , but just an exhaustive search within the cube will incorrectly identify as the closest point. (b) This can be remedied by imposing the constraint that the exhaustive search should consider only points within an L 2 distance ffl from Q (given that the length of a side of the hypercube is 2ffl). this problem. 5.1 Smallest Hypersphere Method Let us now see how to analytically compute the minimum size of a hypersphere given that we want to be able guarantee that it is non empty with probability p. Let the radius of such a hypersphere be ffl hs . Let M be the total number of points within this hypersphere. Let Q be the novel point and define kZk to be the L 2 distance between Q and any point in the point set. Once again, M is binomially distributed with the density Now, the probability p that there is at least one point in the hypersphere is simply e hs r (a) (b) Figure 8: ffl can be computed using two methods: (a) By finding the radius of the smallest hyper-sphere that will contain at least one point with high probability. A search is performed by setting ffl to this radius and constraining the exhaustive search within ffl. (b) By finding the size of the smallest hypercube that will contain at least one point with high probability. When searching, ffl is set to half the length of a side. Additional searches have to be performed in the areas marked in bold. The above equation suggests that if we know Q, the density f Z jQ (z), and the probability p, we can solve for ffl hs . For example, consider the case when the point set is uniformly distributed with density given by equation (9). The cumulative distribution function of kZk is the uniform distribution integrated within a hypersphere; which is simply it's volume. Thus, l d d\Gamma(d=2) Substituting the above in equation 19 and solving for ffl hs , we get l d d\Gamma(d=2) Using equation (21), ffl hs is plotted against probability for two cases. In figure 9(a), d is fixed to different values between 5 to 25 with n is fixed to 100000, and in figure 9(b), n is fixed to different values between 50000 to 150000 with d fixed to 5. Both the figures illustrate an important property, which is that large changes in the probability p result in very small Probability of Success0.20.61Epsilon d=5 Probability of Success0.050.15 Epsilon n=150000 n=50000 (a) (b) Figure 9: The radius ffl necessary to find a point inside a hypersphere varies very little with probability. This means that ffl can be set to the knee where probability is close to unity. The point set in both cases is uniformly distributed with extent l = 1. (a) The point set contains 100000 points in 5, 10, 15, 20 and 25 dimensional space. (b) The point is 5-D and contains 50000, 75000, 100000, 125000 and 150000 points. changes in ffl hs . This suggests that ffl hs can be set to the right hand "knee" of both the curves where probability is very close to unity. In other words, it is easy to guarantee that at least one point is within the hypersphere. A search can now be performed by setting the length of a side of the circumscribing hypercube to 2ffl hs and by imposing an additional constraint during exhaustive search that only points within an L 2 distance ffl hs be considered. 5.2 Smallest Hypercube Method As before, we attempt to analytically compute the size of the smallest hypercube given that we want to be able guarantee that it is non empty with probability p. Let M be the number of points within a hypercube of size 2ffl hc . Define Z c to be the distance between the c th coordinate of a point set point and the novel point Q. Once again, M is binomially distributed with the density Y !k/ d Y kC A :(22) Now, the probability p that there is at least one point in the hypercube is simply Probability of Success0.10.30.5 Epsilon d=5 Probability of Success0.050.150.250.35 Epsilon n=150000 n=50000 (a) (b) Figure 10: The value of ffl necessary to find a point inside a hypercube varies very little with probability. This means that ffl can be set to the knee where probability is close to unity. The point set in both cases is uniformly distributed with extent l = 1. (a) The point set contains 100000 points in 5, 10, 15, 20 and 25 dimensional space. (b) The point set is 5-D and contains 50000, 75000, 100000, 125000 and 150000 points. d Y !n Again, above equation suggests that if we know Q, the density f Zc jQc (z), and the probability p, we can solve for ffl hc . If for the specific case that the point set is uniformly distributed, an expression for ffl hc can be obtained in closed form as follows. Let the density of the uniform distribution be given by equation (9). Using equation (10) we get, d Y l Substituting the above in equation (23) and solving for ffl hc , we get li Using equation (25), ffl hc is plotted against probability for two cases. In figure 10(a), d is fixed to different values between 5 to 25 with n is fixed to 100000, and in figure 10(b), n is fixed to different values between 50000 to 150000 with d fixed to 5. These are similar to the graphs obtained in the case of a hypersphere and again, ffl hc can be set to the right hand "knee" of both the curves where probability is very close to unity. Notice that the value of ffl hc required for the hypercube is much smaller than that required for the hypersphere, especially in high d. This is precisely the reason why we prefer the second (smallest hypercube) method. Recall that it is not sufficient to simply search for the closest point within a hypercube because a point outside can be closer than a point inside. To remedy this problem, we suggest the following technique. First, an exhaustive search is performed to compute the distance to the closest point within the hypercube. Call this distance r. In figure 8(b), the closest point within the hypercube is at a distance of r from Q. Clearly, if a closer point exists, it can only be within a hypersphere of radius r. Since parts of this hypersphere lie outside the original hypercube, we also search in the hyper-rectangular regions shown in bold (by performing additional list trimmings). When performing an exhaustive search in each of these hyper-rectangles, we impose the constraint that a point is considered only if it is less than distance r from Q. In figure 8(b), P 2 is present in one such hyper-rectangular region and happens to be closer to Q than P 1 . Although this method is more complicated, it gives excellent performance in sparsely populated high dimensional spaces (such as a high dimensional uniform distribution). To conclude, we wish to emphasize that both the hypercube and hypersphere methods can be used interchangeably and both are guaranteed to find the closest point within ffl. However, the choice of which one of these methods to use should depend on the dimensionality of the space and the local density of points. In densely populated low dimensional spaces, the hypersphere method performs quite well and searching the hyper-rectangular regions is not worth the additional overhead. In sparsely populated high dimensional spaces, the effort needed to exhaustively search the huge circumscribing hypercube is far more than the overhead of searching the hyper-rectangular regions. It is, however, difficult to analytically predict which one of these methods suits a particular class of data. Hence, we encourage the reader to implement both the methods and use the one which performs the best. Finally, although the above discussion is relevant only for the L 2 norm, an equivalent analysis can be easily performed for any other norm. 6 Benchmarks We have performed an extensive set of benchmarks on the proposed algorithm. We looked at two representative classes of search problems that may benefit from the algorithm. In the first class, the data has statistical structure. This is the case, for instance, when points are uniformly or normally distributed. The second class of problems are statistically un- structured, for instance, when points lie on a high dimensional multivariate manifold, and it is difficult to say anything about their distribution. In this section, we will present results for benchmarks performed on statistically structured data. For benchmarks on statistically unstructured data, we refer the reader to section 7. We tested two commonly occurring distributions, normal and uniform. The proposed algorithm was compared with the k-d tree and exhaustive search algorithms. Other algorithms were not included in this benchmark because they did not yield comparable performance. For the first set of benchmarks, two normally distributed point sets containing 30,000 and 100,000 points with variance 1.0 were used. To test the per search execution time, another set of points, which we shall call the test set, was constructed. The test set contained 10,000 points, also normally distributed with variance 1.0. For each algorithm, the execution time was calculated by averaging over the total time required to perform a nearest neighbor search on each of the 10,000 points in the test set. To determine ffl, we used the 'smallest hypercube' method described in section 5.2. Since the point set is normally distributed, we cannot use a closed form solution for ffl. However, it can be numerically computed as follows. Substituting equation (16) into equation (23), we get d Y oe oe p!!n By setting p (the probability that there is at least one point in the hypercube) to .99 and oe (the variance) to 1.0, we computed ffl for each search point Q using the fast and simple bisection technqiue [ Figures 11(a) and 11(b) show the average execution time per search when the point set contains 30,000 and 100,000 points respectively. These execution times include the time taken for search, computation of ffl using equation (26), and the time taken for the few (1%) additional 3 searches necessary when a point was not found within the hypercube. Although 3 When a point was not found within the hypercube, we incremented ffl by 0.1 and searched again. This Time (secs.) Dimensions Proposed algorithm k-d tree Exhaustive Time (secs.) Dimensions Proposed algorithm k-d tree Exhaustive search (a) (b)0.050.150.250.35 Time (secs.) Dimensions Proposed algorithm k-d tree Exhaustive Time (secs.) Dimensions Proposed algorithm k-d tree Exhaustive search (c) (d) Figure 11: The average execution time of the proposed algorithm is benchmarked for statistically structured problems. (a) The point set is normally distributed with variance 1.0 and contains 30,000 points. (b) The point set is normally distributed with variance 1.0 and contains 100,000 points. The proposed algorithm is clearly faster in high d. (c) The point set is uniformly distributed with extent 1.0 and contains 30,000 points. (d) The point set is uniformly distributed with extent 1.0 and contains 100,000 points. The proposed algorithm does not perform as well for uniform distributions due to the extreme sparseness of the point set in high d. ffl varies for each Q, values of ffl for a few sample points are as follows. For the values of ffl at the point corresponding to respectively. At the point the values of ffl were 0:24; 0:61; 0:86; 1:04; 1:17 corresponding to respectively. For the values of ffl at the point corresponding to respectively. At the point the values of ffl were corresponding to that the proposed algorithm is faster than the k-d tree algorithm for all d in Figure 11(a). In Figure 11(b), the proposed algorithm is faster for d ? 12. Also notice that the k-d tree algorithm actually runs slower than exhaustive search for d ? 15. The reason for this observation is as follows. In high dimensions, the space is so sparsely populated that the radius of the query hypersphere is very large. Consequently, the hypersphere intersects almost all the buckets and thus a large number of points are examined. This, along with the additional overhead of traversing the tree structure makes it very inefficient to search the sparse high dimensional space. For the second set of benchmarks, we used uniformly distributed point sets containing 30,000 and 100,000 points with extent 1.0. The test set contained 10,000 points, also uniformly distributed with extent 1.0. The execution time per search was calculated by averaging over the total time required to perform a closest point search on each of the 10,000 points in the test set. As before, to determine ffl, the 'smallest hypercube' method described in section 5.2 was used. Recall that for uniformly distributed point sets, ffl can be computed in the closed form using equation (25). Figures 11(c) and 11(d) show execution times when the point set contains 30,000 and 100,000 points respectively. For the values of ffl were corresponding to tively. For the values of ffl were corresponding to respectively. For uniform distribution, the proposed algorithm does not perform as well, although, it does appear to be slightly faster than the k-d tree and exhaustive search algorithms. The reason is that the high dimensional space is very sparsely populated and hence requires ffl to be quite large. As a result, the algorithm ends up examining almost all points, thereby approaching exhaustive search. process was repeated till a point was found. 7 An Example Application: Appearance Matching We now demonstrate two applications where a fast and efficient high dimensional search technique is desirable. The first, real time object recognition, requires the closest point to be found among 36,000 points in a 35-D space. In the second, the closest point is required to be found from points lying on a multivariate high dimensional manifold. Both these problems are examples of statistically unstructured data. Let us briefly review the object recognition technique of Murase and Nayar [ 24 ] . Object recognition is performed in two phases: 1) appearance learning phase, and 2) appearance recognition phase. In the learning phase, images of each of the hundred objects in all poses are captured. These images are used to compute a high dimensional subspace, called the eigenspace. The images are projected to eigenspace to obtain discrete high dimensional points. A smooth curve is then interpolated through points that belong to the same object. In this way, for each object, we get a curve (or a univariate manifold) parameterized by it's pose. Once we have the manifolds, the second phase, object recognition, is easy. An image of an object is projected to eigenspace to obtain a single point. The manifold closest to this point identifies the object. The closest point on the manifold identifies the pose. Note that the manifold is continuous, so in order to find the closest point on the manifold, we need to finely sample it to obtain discrete closely spaced points. For our benchmark, we used the Columbia Object Image Library [ along with the SLAM software package [ 28 ] to compute 100 univariate manifolds in a 35-D eigenspace. These manifolds correspond to appearance models of the 100 objects (20 of the 100 objects shown in Figure 12(a)). Each of the 100 manifolds were sampled at 360 equally spaced points to obtain 36,000 discrete points in 35-D space. It was impossible to manually capture the large number of object images that would be needed for a large test set. Hence, we automatically generated a test set of 100,000 points by sampling the manifolds at random locations. This is roughly equivalent to capturing actual images, but, without image sensor noise, lens blurring, and perspective projection effects. It is important to simulate these effects because they cause the projected point to shift away from the manifold and hence, substantially affect the performance of nearest neighbor search algorithms 4 . 4 For instance, in the k-d tree, a large query hypersphere would result in a large increase in the number of adjacent buckets that may have to be searched. Algorithm Time (secs.) Proposed Algorithm .0025 k-d tree .0045 Exhaustive Search .1533 Projection Search .2924 (b) Figure 12: The proposed algorithm was used to recognize and estimate pose of hundred objects using the Columbia Object Image Library. (a) Twenty of the hundred objects are shown. The point set consisted of 36,000 points (360 for each object) in 35-D eigenspace. (b) The average execution time per search is compared with other algorithms. Unfortunately, it is very difficult to relate image noise, perspective projection and other distortion effects to the location of points in eigenspace. Hence, we used a simple model where we add uniformly distributed noise with extent 5 .01 to each of the coordinates of points in the test set. We found that this approximates real-world data. We determined that setting gave us good recognition accuracy. Figure 12(b) shows the time taken per search by the different algorithms. The search time was calculated by averaging the total time taken to perform 100,000 closest point searches using points in the test set. It can be seen that the proposed algorithm outperforms all the other techniques. ffl was set to a predetermined value such that a point was found within the hypersphere all the time. For object recognition, it is useful to search for the closest point within ffl because this provides us with a means to reject points that are "far" from the manifold (most likely from objects not in the database). Next, we examine another case when data is statistically unstructured. Here, the closest point is required to be found from points lying on a single smooth multivariate high dimensional manifold. Such a manifold appears frequently in appearance matching problems such as visual tracking [ 26 ] , visual inspection [ 26 ] , and parametric feature detection [ 25 ] . As with object recognition, the manifold is a representation of visual appearance. Given a novel appearance (point), matching involves finding a point on the manifold closest to that point. Given that the manifold is continuous, to pose appearance matching as a nearest neighbor problem, as before, we sample the manifold densely to obtain discrete closely spaced points. The trivariate manifold we used in our benchmarks was obtained from a visual tracking experiment conducted by Nayar et al. [ 26 ] . In the first benchmark, the manifold was sampled to obtain 31,752 discrete points. In the second benchmark, it was sampled to obtain 107,163 points. In both cases, a test set of 10,000 randomly sampled manifold points was used. As explained previously, noise (with extent .01) was added to each coordinate in the test set. The execution time per search was averaged over this test set of 10,000 points. For this point set, it was determined that gave good recognition accuracy. Figure 13(a) shows the algorithm to be more than two orders of magnitude faster than the other algorithms. Notice the exponential behaviour of the R-tree algorithm. Also notice that Yunck's algorithm is 5 The extent of the eigenspace is from -1.0 to +1.0. The maximum noise amplitude is hence about 0.5% of the extent of eigenspace. Time (secs.) Dimensions Proposed algorithm R-tree Exhaustive search Projection search Yunck search (a)0.0050.0150.0250.035 Time (secs.) Dimensions Proposed algorithm k-d tree0.0050.0150.0250.035 Time (secs.) Dimensions Proposed algorithm k-d tree (b) (c) Figure 13: The average execution time of the proposed algorithm is benchmarked for an unstructured problem. The point set is constructed by sampling a high dimensional trivariate manifold. (a) The manifold is sampled to obtain 31,752 points. The proposed algorithm is more than two orders of magnitude faster than the other algorithms. (b) The manifold is sampled as before to obtain 31,752 points. (c) The manifold is sampled to obtain 107,163 points. The k-d tree algorithm is slightly faster in low dimension but degrades rapidly with increase in dimension. only slightly faster than Friedman's; the difference is due to use of integer operations. We could only benchmark Yunck's algorithm till due to use of a 32-bit word in the indicator array. In Figure 13(b), it can be seen that the proposed algorithm is faster than the k-d tree for all d, while in Figure 13(c), the proposed algorithm is faster for all d ? 21. 8 Hardware Architecture A major advantage of our algorithm is its simplicity. Recall that the main computations performed by the algorithm are simple integer map lookups (backward and forward maps) and two integer comparisons (to see if a point lies within hyperplane boundaries). Conse- quently, it is possible to implement the algorithm in hardware using off-the-shelf, inexpensive components. This is hard to envision in the case of any competitive techniques such as k-d trees or R-trees, given the difficulties involved in constructing parallel stack machines. The proposed architecture is shown in Figure 14. A Field Programmable Gate Array acts as an algorithm state machine controller and performs I/O with the CPU. The Dynamic RAMs (DRAMs) hold the forward and backward maps which are downloaded from the CPU during initialization. The CPU initiates a search by performing a binary search to obtain the hyperplane boundaries. These are then passed on to the search engine and held in the Static RAMs (SRAMs). The FPGA then independently begins the candidate list construction and trimming. A candidate is looked up in the backward map and each of the forward maps. The integer comparator returns a true if the candidate is within range, else it is discarded. After trimming all the candidate points by going through the dimensions, the final point list (in the form of point set indices) is returned to the CPU for exhaustive search and/or further processing. Note that although we have described an architecture with a single comparator, any number of them can be added and run in parallel with a near linear performance scaling in the number of comparators. While the search engine is trimming the candidate list, the CPU is of course free to carry out other tasks in parallel. We have begun implementation of the proposed architecture. The result is intended to be a small low-cost SCSI based module that can be plugged in to any standard workstation or PC. We estimate the module to result in a 100 fold speedup over an optimized software implementation. ALGORITHM CONTROL UNIT BACKWARD MAP FORWARD MAP A A CPU COMPARATOR WITHIN LIMIT FLAG LOWER LIMIT UPPER LIMIT CONTROL BUS FPGA/PLD Figure 14: Architecture for an inexpensive hardware search engine that is based on the proposed algorithm. 9 Discussion 9.1 k Nearest Neighbor Search In section 5, we saw that it is possible to determine the minimum value of ffl necessary to ensure that at least one point is found within a hypercube or hypersphere with high probability. It is possible to extend this notion to ensure that at least k points are found with high certainty. Recall that the probability that there exists at least one point in a hypersphere of radius ffl is given by equation (19). Now define p k to be the probability that there are at least k points within the hypersphere. We can then write p k as The above expression can now be substituted in equation (18) and, given p k , numerically solved for ffl hs . Similarly, it can be substituted in equation (22) to compute the minimum value of ffl hc for a hypercube. 9.2 Dynamic Point Insertion and Deletion Currently, the algorithm uses d floating point arrays to store the ordered set, and 2d integer arrays to store the backward and forward maps. As a result, it is not possible to efficiently insert or delete points in the search space. This limitation can be easily overcome if the ordered set is not stored as an array but as a set of d binary search trees (BST) (each BST corresponds to an array of the ordered set). Similarly, the d forward maps have to be replaced with a single linked list. The backward maps can be done away with completely as the indices can be made to reside within a node of the BST. Although BSTs would allow efficient insertion and deletion, nearest neighbor searches would no longer be as efficient as with integer arrays. Also, in order to get maximum efficiency, the BSTs would have to be well balanced (see [ 19 ] for a discussion on balancing techniques). 9.3 Searching with Partial Data Many times, it is required to search for the nearest neighbor in the absence of complete data. For instance, consider an application which requires features to be extracted from an image and then matched against other features in a feature space. Now, if it is not possible to extract all features, then the matching has to be done partially. It is trivial to adapt our algorithm to such a situation: while trimming the list, you need to only look at the dimensions for which you have data. This is hard to envision in the case of k-d trees for example, because the space has been partitioned by hyperplanes in particular dimensions. So, when traversing the tree to locate the bucket that contains the query point, it is not possible to choose a traversal direction at a node if data corresponding to the partitioning dimension at that node is missing from the query point. Acknowledgements We wish to thank Simon Baker and Dinkar Bhat for their detailed comments, criticisms and suggestions which have helped greatly in improving the paper. This research was conducted at the Center for Research on Intelligent Systems at the Department of Computer Science, Columbia University. It was supported in parts by ARPA Contract DACA-76-92-C-007, DOD/ONR MURI Grant N00014-95-1-0601, and a NSF National Young Investigator Award. --R The Design and Analysis of Computer Algorithms. Nearest neighbor searching and applications. diagrams - a survey of a fundamental geometric data struc- ture Multidimensional binary search trees used for associative searching. Multidimensional binary search trees in database applications. Data structures for range searching. Optimal expected-time algorithms for closest point problems Multidimensional indexing for recognizing visual shapes. A randomized algorithm for closest-point queries Multidimensional searching problems. Algorithms in Combinatorial Geometry. Fast nearest-neighbor search in dissimilarity spaces An algorithm for finding nearest neigh- bors An algorithm for finding best matches in logarithmic expected time. A branch and bound algorithm for computing k-nearest neighbors An effective way to represent quadtrees. A dynamic index structure for spatial searching. Fundamentals of Data Structures in. On the complexity of d-dimensional voronoi diagrams Sorting and Searching A new version of the nearest-neighbor approximating and eliminating search algorithm (aesa) with linear preprocessing time and memory requirements Visual learning and recognition of 3d objects from appear- ance Parametric feature detection. Slam: A software library for appearance matching. Digital Pictures Similarity searching in large image databases. Computational Geometry: An Introduction. Numerical Recipes in C. Direct spatial search on pictorial databases using packed r-trees Refinements to nearest-neighbor searching in k-dimensional trees Reducing the overhead of the aesa metric-space nearest neighbour searching algorithm Data structures and algorithms for nearest neighbor search in general metric spaces. A technique to identify nearest neighbors. --TR --CTR James McNames, A Fast Nearest-Neighbor Algorithm Based on a Principal Axis Search Tree, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.23 n.9, p.964-976, September 2001 David W. Patterson , Mykola Galushka , Niall Rooney, Characterisation of a Novel Indexing Technique for Case-Based Reasoning, Artificial Intelligence Review, v.23 n.4, p.359-393, June 2005 Philip Quick , David Capson, Subspace position measurement in the presence of occlusion, Pattern Recognition Letters, v.23 n.14, p.1721-1733, December 2002 T. Freeman , Thouis R. Jones , Egon C Pasztor, Example-Based Super-Resolution, IEEE Computer Graphics and Applications, v.22 n.2, p.56-65, March 2002 Bin Zhang , Sargur N. Srihari, Fast k-Nearest Neighbor Classification Using Cluster-Based Trees, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.26 n.4, p.525-528, April 2004 Jim Z. C. Lai , Yi-Ching Liaw , Julie Liu, Fast k-nearest-neighbor search based on projection and triangular inequality, Pattern Recognition, v.40 n.2, p.351-359, February, 2007 Lin Liang , Ce Liu , Ying-Qing Xu , Baining Guo , Heung-Yeung Shum, Real-time texture synthesis by patch-based sampling, ACM Transactions on Graphics (TOG), v.20 n.3, p.127-150, July 2001 Fast texture synthesis using tree-structured vector quantization, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.479-488, July 2000 Yong-Sheng Chen , Yi-Ping Hung , Ting-Fang Yen , Chiou-Shann Fuh, Fast and versatile algorithm for nearest neighbor search based on a lower bound tree, Pattern Recognition, v.40 n.2, p.360-375, February, 2007 Mineichi Kudo , Naoto Masuyama , Jun Toyama , Masaru Shimbo, Simple termination conditions for k-nearest neighbor method, Pattern Recognition Letters, v.24 n.9-10, p.1203-1213, 01 June John T. Favata, Offline General Handwritten Word Recognition Using an Approximate BEAM Matching Algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.23 n.9, p.1009-1021, September 2001 Edgar Chvez , Gonzalo Navarro, A compact space decomposition for effective metric indexing, Pattern Recognition Letters, v.26 n.9, p.1363-1376, 1 July 2005 Aaron Hertzmann , Steven M. Seitz, Example-Based Photometric Stereo: Shape Reconstruction with General, Varying BRDFs, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.8, p.1254-1264, August 2005 O. Stahlhut, Extending natural textures with multi-scale synthesis, Graphical Models, v.67 n.6, p.496-517, November 2005 Gonzalo Navarro, Searching in metric spaces by spatial approximation, The VLDB Journal The International Journal on Very Large Data Bases, v.11 n.1, p.28-46, August 2002 Edgar Chvez , Jos L. 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object recognition;benchmarks;nearest neighbor;searching by slicing;visual correspondence;pattern classification;hardware architecture

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Design and Evaluation of a Window-Consistent Replication Service.

AbstractReal-time applications typically operate under strict timing and dependability constraints. Although traditional data replication protocols provide fault tolerance, real-time guarantees require bounded overhead for managing this redundancy. This paper presents the design and evaluation of a window-consistent primary-backup replication service that provides timely availability of the repository by relaxing the consistency of the replicated data. The service guarantees controlled inconsistency by scheduling update transmissions from the primary to the backup(s); this ensures that client applications interact with a window-consistent repository when a backup must supplant a failed primary. Experiments on our prototype implementation, on a network of Intel-based PCs running RT-Mach, show that the service handles a range of client loads while maintaining bounds on temporal inconsistency.

Introduction Many embedded real-time applications, such as automated manufacturing and process control, require timely access to a fault-tolerant data repository. Fault-tolerant systems typically employ some form of redundancy to insulate applications from failures. Time redundancy protects applications by repeating computation or communication operations, while space redundancy masks failures by replicating physical resources. The time-space tradeoffs employed in most systems may prove inappropriate for achieving fault tolerance in a real-time environment. In particular, when time is scarce and the overhead for managing redundancy is too high, alternative approaches must balance the trade-off between timing predictability and fault tolerance. For example, consider the process-control system shown in Figure 1(a). A digital controller supports monitoring, control, and actuation of the plant (external world). The controller software executes a The work reported in this paper was supported in part by the National Science Foundation under Grant MIP-9203895. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the view of the NSF. Controller Plant (External World) sensors actuators repository in-memory Controller Backup Primary Controller sensors actuators Plant (External World) replicated in-memory repository (a) Digital controller interacting with a plant (b) Primary-backup control system Figure 1: Computer control system tight loop, sampling sensors, calculating new values, and sending signals to external devices under its control. It also maintains an in-memory data repository which is updated frequently during each iteration of the control loop. The data repository must be replicated on a backup controller to meet the strict timing constraint on system recovery when the primary controller fails, as shown in Figure 1(b). In the event of a primary failure, the system must switch to the backup node within a few hundred milliseconds. Since there can be hundreds of updates to the data repository during each iteration of the control loop, it is impractical (and perhaps impossible) to update the backup synchronously each time the primary repository changes. An alternative solution exploits the data semantics in a process-control system by allowing the backup to maintain a less current copy of the data that resides on the primary. The application may have distinct tolerances for the staleness of different data objects. With sufficiently recent data, the backup can safely supplant a failed primary; the backup can then reconstruct a consistent system state by extrapolating from previous values and new sensor readings. However, the system must ensure that the distance between the primary and the backup data is bounded within a predefined time window. Data objects may have distinct tolerances in how far the backup can lag behind before the object state becomes stale. The challenge is to bound the distance between the primary and the backup such that consistency is not compromised, while minimizing the overhead in exchanging messages between the primary and its backup. This paper presents the design and implementation of a data replication service that combines fault-tolerant protocols, real-time scheduling, and temporal consistency semantics to accommodate such system requirements [24, 29]. A client application registers a data object with the service by declaring the consistency requirements for the data, in terms of a time window. The primary selectively transmits to the backup, as opposed to sending an update every time an object changes, to bound both resource utilization and data inconsistency. The primary ensures that each backup site maintains a version of the object that was valid on the primary within the preceding time window by scheduling these update messages. The next section discusses related work on fault-tolerant protocols and relaxed consistency se- mantics, with an emphasis on supporting real-time applications. Section 3 describes the proposed window-consistent primary-backup architecture and replication protocols for maintaining controlled inconsistency within the service. This replication model introduces a number of interesting issues in scheduling, fault detection, and system recovery. Section 4 considers real-time scheduling algorithms for creating and maintaining a window-consistent backup, while Section 5 presents techniques for fault detection and recovery for primary, backup, and communication failures. In Section 6, we present and evaluate an implementation of the window-consistent replication service on a network of Intel-based PCs running RT-Mach [32]. Section 7 concludes the paper by highlighting the limitations of this work and discussing future research directions. Related Work 2.1 Replication Models A common approach to building fault-tolerant distributed systems is to replicate servers that fail independently. In active (state-machine) replication schemes [6, 30], a collection of identical servers maintains copies of the system state. Client write operations are applied atomically to all of the replicas so that after detecting a server failure, the remaining servers can continue the service. Passive (primary-backup) replication [2, 9], on the other hand, distinguishes one replica as the primary server, which handles all client requests. A write operation at the primary invokes the transmission of an update message to the backup servers. If the primary fails, a failover occurs and one of the backups becomes the new primary. In recent years, several fault-tolerant distributed systems have employed state-machine [7, 11, 26] or primary-backup [4, 5, 9] replication. In general, passive replication schemes have longer recovery delays since a backup must invoke an explicit recovery algorithm to replace a failed primary. On the other hand, active replication typically incurs more overhead in responding to client requests since the service must execute an agreement protocol to ensure atomic ordered delivery of messages to all replicas. In both replication models, each client write operation generates communication within the service to maintain agreement amongst the replicas. This artificially ties the rate of write operations to the communication capacity in the service, limiting system throughput while ensuring consistent data. Past work on server replication has focused, in most cases, on improving throughput and latency for client requests. For example, Figure 2(a) shows the basic primary-backup model, where a client write operation at the primary P triggers a synchronous update to the backup B [4]. The service can improve C2(a) Blocking (b) Efficient blocking (c) Non-blocking Figure 2: Primary-backup models response time by allowing the backup B to acknowledge the client C [2], as shown in Figure 2(b). Finally, the primary can further reduce write latency by replying to C immediately after sending an update message to B, without waiting for an acknowledgement [8], as shown in Figure 2(c). Similar performance optimizations apply to the state-machine replication model. Although these techniques significantly improve average performance, they do not guarantee bounded worst-case delay, since they do not limit communication within the service. Synchronization of redundant servers poses additional challenges in real-time environments, where applications operate under strict timing and dependability constraints; server replication for hard real-time systems is under investigation in several recent experimental projects [15, 16, 33]. Synchronization overheads, communication delay, and interaction with the external environment complicate the design of replication protocols for real-time applications. These overheads must be quantified precisely for the system to satisfy real-time constraints. 2.2 Consistency Semantics A replication service can bound these overheads by relaxing the data consistency requirements in the repository. For a large class of real-time applications, the system can recover from a server failure even though the servers may not have maintained identical copies of the replicated state. This facilitates alternative approaches that trade atomic or causal consistency amongst the replicas for less expensive replication protocols. Enforcing a weaker correctness criterion has been studied extensively for different purposes and application areas. In particular, a number of researchers have observed that serializability is too strict a correctness criterion for real-time databases. Relaxed correctness criteria higher concurrency by permitting a limited amount of inconsistency in how a transaction views the database state [12, 17, 18, 20, 28]. Similarly, imprecise computation guarantees timely completion of an application by relaxing the accuracy requirements of the computation [22]. This is particularly useful in applications that use discrete samples of continuous-time variables, since these values can be approximated when there is not sufficient time to compute an exact value. Weak consistency can also improve performance in non-real-time applications. For instance, the quasi-copy model permits some inconsistency between the central data and its cached copies at remote sites [1]. This gives the scheduler more flexibility in propagating updates to the cached copies. In the same spirit, window-consistent replication allows computations that may otherwise be disallowed by existing active or passive protocols that require atomic updates to a collection of replicas. 3 Window-Consistent Replication The window-consistent replication service consists of a primary and one or more backups, with the data on the primary shadowed at each backup site. These servers store objects which change over time, in response to client interaction with the primary. In the absence of failures, the primary satisfies all client requests and supplies a data-consistent repository. However, if the primary crashes, a window- consistent backup performs a failover to become the new primary. Hence, service availability hinges on the existence of a window-consistent backup to supplant a failed primary. 3.1 System Model Unlike the primary-backup protocols in Figure 2, the window-consistent replication model decouples client read and write operations from communication within the service. As shown in Figure 3, the primary object manager (OM) handles client data requests, while sending messages to the backups at the behest of the update scheduler (US). Since read and write operations do not trigger transmissions to the backup sites, client response time depends only on local operations at the primary. This allows the primary to handle a high rate of client requests while independently sending update messages to the backup sites. Although these update transmissions must accommodate the temporal consistency requirements of the objects, the primary cannot compromise the client application's processing demands. Hence, the primary must match the update rate with the available processing and network bandwidth by selectively transmitting messages to the backups. The primary executes an admission control algorithm as part of object creation, to ensure that the US can schedule sufficient update transmissions for any new objects. Unlike client reads and writes, object creation and deletion requires complete agreement between the primary and all the backups in the replication service. 3.2 Consistency Semantics The primary US schedules transmissions to the backups to ensure that each replica has a sufficiently recent version of each object. Timestamps - P versions of object O i at the primary and backup sites, respectively. At time t the primary P has a copy of O i written by the client application at time - P while a backup B stores a, possibly older, version originally written on P at time - B may have an older version of O i than P , the copy on B must be "recent Scheduler Update Object Manager Scheduler Update Object Manager Primary Backup update ack ack create/delete create/delete read/write Figure 3: Window-consistent primary-backup architecture enough." If O i has window ffi i , a window-consistent backup must believe in data that was valid on P within the last Definition 1: At time t, a backup copy of object O i has window-inconsistency i is the maximum time such that t 0 (t). Object O i is window-consistent if and only if window-consistent if and only if all of its objects are window-consistent. In other words, B has a window-consistent copy of object O i at time t if and only if For example, in Figure 4, P performs several write operations on O i , on behalf of client requests, but selectively transmits update messages to B. At time t the primary has the most recent version of the object, written by the client at time d. The backup has a copy first recorded on the primary at time b; the primary stopped believing this version at time c. Thus, - P d, B has a window-consistent version of O i at time t. The backup object has inconsistency which is less than its window-consistency requirement ffi i . A small value of allows the client to operate with a more recent copy of the object if the backup must supplant a failed primary. The i represents an object's temporal inconsistency within the replication service, as seen by an "omniscient" observer. Since the backup site does not always have up-to-date knowledge of client operations, the backup has a more conversative view of temporal consistency, as discussed in Section 5.2. The client may also require bounds on the staleness of the backup's object, relative to the primary's copy, to construct a valid system state when a failover occurs. In particular, if the client reads O i at time t on P , it receives the version that it wrote units ago. On the other hand, if B supplants a failed primary, the client would read the version that it wrote time units ago. This version is - P than that on the primary; in Figure 4, this "client view" has inconsistency d \Gamma b. a d d Btbackup view omniscient view client view client update message Figure 4: Window-consistency semantics Definition 2: At time t, object O i has recovery inconsistency - P Two components contribute to this recovery inconsistency: client write patterns and the temporal inconsistency within the service. Window-consistent replication bounds the latter, allowing the client to bound recovery inconsistency based on its access patterns. For example, suppose consecutive client writes occur at most w i time units apart; typically, w i is smaller than since the primary sends only selective updates to the backup sites. The window-consistency bound t \Gamma t 0 ensures that the backup's copy of the object was written on the primary no earlier than time t, window consistency guarantees that - P 4 Real-Time Update Scheduling This section describes how the primary can use existing real-time task scheduling algorithms to coordinate update transmissions to the backups. In the absence of link (performance or crash) failures [10], we assume a bound ' on the end-to-end communication latency within the service. For example, a real-time channel [14, 23] with the desired bound could be established between the primary and the backups. Several other approaches to providing bounds on communication latency are discussed in [3]. If a client operation modifies O i , the primary must send an update for the object within the next otherwise, the backups may not receive a sufficiently recent version of O i before the time-window elapses. In order to bound the temporal inconsistency within the service, it suffices that the primary send O i to the backups at least once every units. While bounding the temporal inconsistency, the primary may send additional updates to the backups if sufficient processing and network capacity are available; these extra transmissions increase the service's resilience to lost update messages and the average "goodness" of the replicated data. In addition to sending update transmissions to the backups, the primary must allow efficient integration of new backups into the replication service. Limited processing and network capacity necessitate a trade-off between timely integration of a new backup and keeping existing backups window-consistent. The primary should minimize the time to integrate a new replica, especially when there are no other window-consistent backups, since a subsequent primary crash would result in a server failure. The primary constructs a schedule that sends each object to the backup exactly once, and allows the primary to smoothly transition to the update transmission schedule. While several task models can accommodate the requirements of window-consistent scheduling and backup integration, we initially consider the periodic task model [19, 21]. 4.1 Periodic Scheduling of Updates The transmissions of updates can be cast as "tasks" that run periodically with deadlines derived from the objects' window-consistency requirements. The primary coordinates transmissions to the backups by scheduling an update "task" with period p i and service time e i for each object O consistency, this permits a maximum period ')=2. The end of a period serves as both the deadline for one invocation of the task and the arrival time for the subsequent invocation. The scheduler always runs the ready task with the highest priority, preempting execution if a higher-priority task arrives. For example, rate-monotonic scheduling statically assigns higher priority to tasks with shorter periods [19, 21], while earliest-due-date scheduling favors tasks with earlier deadlines [21]. The scheduling algorithm, coupled with the object parameters e i and ffi i , determines a schedulability criterion based on the total processor and network utilization. The schedulability criterion governs object admission into the replication service. The primary rejects an object registration request (specifying e i and cannot schedule sufficient updates for the new object without jeopardizing the window consistency of existing objects, i.e., it does not have sufficient processing and network resources to accommodate the object's window-consistency requirements. The scheduling algorithm maintains window consistency for all objects as long as the the collection of tasks does not exceed a certain bound on resource utilization (e.g., 0:69 for rate-monotonic and 1 for earliest-due-date) [21]. 4.2 Compressing the Periodic Schedule While the periodic model can guarantee sufficient updates for each object, the schedule updates O i only once per period p i , even if computation and network resources permit more frequent transmissions. This restriction arises because the periodic model assumes that a task becomes ready to run only at period boundaries. However, the primary can transmit the current version of an object at any time. The scheduler can capitalize on this "readiness" of tasks to improve both resource utilization and the window consistency on the backups by compressing the periodic schedule. Consider two objects O 1 (with 1), as shown in Figure 5; the unshaded boxes denote transmission of O 1 , while the shaded boxes signify transmission of O 2 . The scheduler must send an update requiring 1 unit of processing time once every 3 time units 1 The size of O i determines the time e i required for each update transmission. In order to accommodate preemptive scheduling and objects of various sizes, the primary can send an update message as one or more fixed-length packets. (a) Periodic schedule (b) Compressed periodic schedule Figure 5: Compression (p 1 (unshaded box) and an update requiring 2 units of processing time once every 5 time units (shaded box). The schedule repeats after each major cycle of length 15. Each time unit corresponds to a tick which is the granularity of resource allocation for processing and transmission of a packet. For this example, both the rate-monotonic and earliest-due-date algorithms generate the schedule shown in Figure 5(a). While each update is sent as required in the major cycle of length 15, the schedule has 4 units of slack time. The replication service can capitalize on this slack time to improve the average temporal consistency of the backup objects. In particular, the periodic schedule in Figure 5(a) can provide the order of task executions without restricting the time the tasks become active. If no tasks are ready to run, the scheduler can advance to the earliest pending task and activate that task by advancing the logical time to the start of the next period for that object. With the compressed schedule the primary still transmits an update for each O i at least once per period p i but can send more frequent update messages when time allows. As shown in Figure 5(b), compressing the slack time allows the schedule to start over at time 11. In the worst case, the compressed schedule degrades to the periodic schedule with the associated guarantees. 4.3 Integrating a New Backup To minimize the time the service operates without a window-consistent backup, the primary P needs an efficient mechanism to integrate a new or invalid backup B. P must send the new backup B a copy of each object and then transition to the normal periodic schedule, as shown in Figure 6. Although B may not have window-consistent objects during the execution of the integration schedule, each object must become consistent and remain consistent until its first update in the normal periodic schedule. As a result, B must receive a copy of O i within the "period" p i before the periodic schedule begins; this ensures that B can afford to wait until the next p i interval to start receiving periodic update messages for O i . In order to integrate the new backup, then, the primary must execute an integration schedule that would allow it to transition to the periodic schedule while maintaining window consistency. Referring to Figure 6, a window-consistent transition requires D prior post prior j is the time elapsed from the last transmission of O j to the end of the integration schedule, while D post j is the time from the start of the periodic schedule until the first transmission of O j . This O O i O i O j O k prior post transition integration schedule periodic schedule Figure Integrating a new backup repository ensures window consistency for each object, even across the schedule transition. Since the periodic task model provides D post suffices to ensure that D post A simple schedule for integration is to send objects to the new backup using the normal periodic schedule already being used for update transmissions to the existing replicas. This incurs a worst-case delay of 2 to integrate the new backup into the service. However, if the service has no window-consistent backup sites, the primary should minimize the time required to integrate a new replica. In particular, an efficient integration schedule should transmit each object exactly once before transitioning to the normal periodic schedule. The primary may adapt the normal periodic schedule into an efficient integration schedule by removing duplicate object transmissions. In particular, the primary can transmit the objects in order of their last update transmissions before the end of a major cycle in the normal schedule. For example, for the schedule shown in Figure 5(a), the integration schedule is [O because the last transmission for O 1 is at time 10 (12). A transition from the integration schedule to the normal schedule sustains window consistency on the newly integrated backup since the normal schedule guarantees window consistency across major cycles. Since the integration schedule is derived from the periodic schedule, it follows that D prior post The normal schedule order can be determined when objects are created or during the first major cycle of the normal schedule. Since the schedule transmits each object only once, the integration delay is is the number of registered objects. Although this approach is efficient for static object sets, dynamic creation and deletion of objects introduces more complexity. Since the transmission order in the normal schedule depends on the object set, the primary must recompute the integration schedule whenever a new object enters the service. The cost of constructing an integration schedule, especially for dynamic object sets, can be reduced by sending the objects to B in reverse period order, such that the objects with larger periods are sent before those with smaller periods. For object O j , this ensures that only objects with smaller or equivalent periods can follow O j in the integration schedule; these same objects can precede O j in the periodic schedule. This guarantees that the integration schedule transmits O j no more than p units before the start of the periodic schedule, ensuring a window-consistent transition. For example, in Figure 6 p i - . In the sends update to B B receives (i; O; -; s) P receives (i; -; s) select object i if (s ? t xmit last last / s last last / s send Figure 7: Update protocols periodic schedule objects O i with are transmitted at least once within time D post but exactly once within time D prior it follows that D prior post After object creations or deletions, the primary can construct the new integration schedule by sorting the new set of periods. The primary minimizes the time it operates without a window-consistent backup by transmitting each object exactly once before transitioning to the normal periodic schedule. 5 Fault Detection and Recovery Although real-time scheduling of update messages can maintain window-consistent replicas, processor and communication failures potentially disrupt system operation. We assume that servers may suffer crash failures and the communication subsystem may suffer omission or performance failures; when a site fails, the remaining replicas must recover in a timely manner to continue the data-repository service. The primary attempts to minimize the time it operates without a window-consistent backup, since a subsequent primary crash would cause a service failure. Similarly, the backup tries to detect a primary crash and initiate failover before any backup objects become window-inconsistent. Although the primary and backup cannot have complete knowledge of the global system state, the message exchange between servers provides a measure of recent service activity. 5.1 Update Protocols Figure 7 shows how the primary and backup sites exchange object data and estimate global system state. We assume that the servers communicate only by exchanging messages. Since these messages include temporal information, P and B cannot effectively reason about each other unless server clocks are synchronized within a known maximum bound. A clock synchronization algorithm can use the transmit times for the update and acknowledgement messages to bound clock skew in the service. Using the update protocols, P and B each approximate global state by maintaining the most recent information received from the other site. Before transmitting an update message at time t, the primary records the version timestamp - xmit for the selected object O i . Since - B i , this information gives P an optimistic view of the backup's window consistency. The primary's message to the backup contains the object data, along with the version timestamp and the transmission time. B uses the transmission time to detect out-of- order message arrivals by maintaining t xmit i , the time of the most recent transmission of O i that has been successfully received; the sites store monotonically non-decreasing version timestamps, without requiring reliable or in-order message delivery in the service. Upon receiving a newer transmission of O i , the backup updates the object's data, the version timestamp - B discussed in Section 5.2, the backup uses t xmit i to reason about its own window consistency. To diagnose a crashed primary, B also maintains t last , the transmission time of the last message received from P regarding any object; that is, t last g. Similarly, P tracks the transmission times of B's messages to diagnose possible crash failures. Hence, the backup's acknowledgement message to P includes the transmission time t, as well as - B i , the most recent version timestamp for O i on B. Using this information, the primary determines - ack i , the most recent version of O i that B has successfully acknowledged. Since - B i , this variable gives P a pessimistic measure of the backup's window consistency; as discussed in Section 5.3, the primary uses - ack i to select policies for scheduling update transmissions to the backup. 5.2 Backup Recovery From Primary Failures A backup site must estimate its own window consistency and the status of the primary to successfully supplant a crashed primary. While B may be unaware of recent client interaction with P for each object, B does know the time t xmit i of object O i . Although P may continue to believe version - xmit even after transmitting the update message, B conservatively estimates that the client wrote a new version of O i just after P transmitted the object at time t xmit In particular, Definition 3: At time t, the backup copy of object O i has estimated inconsistency the backup knows that O i is window-consistent if Figure 4 shows an example of this "backup view" of window consistency. Using this consistency metric, the backup must balance the possibility of becoming window- inconsistent with the likehood of falsely diagnosing a primary crash. If B believes that all of its objects are still window-consistent, B need not trigger a failover until further delay would endanger the consistency of a backup object; in particular, the backup conservatively estimates that its copy of O i could become window-inconsistent by time t xmit in the absence of further update messages from P . However, to reduce the likelihood of false failure detection, failover should only occur if B has not received any messages from P for some minimum time fi. In this adaptive failure detection mechanism, B diagnoses a primary crash at time if and only if t crash - t last After failover, the new primary site invokes the client application and begins interacting with the external environment. For a period of time, the new P operates with some partially inconsistent data but gradually constructs a consistent system state from these old values and new sensor readings. The new P later integrates a fresh backup to enhance future service availability. diagnoses a primary crash through missed update messages, lost or delayed messages could still trigger false failure detection, resulting in multiple active primary sites. When the system has multiple backups, the replicas can vote to select a single, valid primary. However, when the service has only two sites, communication failures can cause each site to assume the other has failed. In this situation, a third-party "witness" [27] can select the primary site. This witness does not act as a primary or backup server, but casts the deciding vote in failure diagnosis. In a real-time control system, the actuator devices could implicitly serve as this witness; if a new server starts issuing commands to the actuators, the devices could ignore subsequent instructions from the previous primary site. 5.3 Primary Recovery From Backup Failures Service availability also depends on timely recovery from backup failures. Since the data-repository service continues whenever a valid primary exists, the primary can temporarily tolerate backup crashes or communication failures without endangering the client application. Ultimately, though, P should minimize the portion of time it operates without a window-consistent backup, since a subsequent primary crash would cause a service failure. The primary should diagnose possible backup crashes and efficiently integrate new backup sites. If P believes that an operational backup has become window- inconsistent, due to lost update messages or transient overload conditions, the primary should quickly refresh the inconsistent objects. As in Section 5.2, timeout mechanisms can detect possible server failures. The primary assumes that the backup has crashed if P has not received any acknowledgement messages in the last ff time units (i.e., last - ff). After detecting a backup crash, P can integrate a fresh backup site into the system while continuing to satisfy client read and write requests. If the P mistakenly diagnoses a backup crash, the system must operate with one less replica while the primary integrates a new backup this new backup does not become window-consistent until the integration schedule completes, as described in Section 4.3. However, if the backup has actually failed, a large timeout value increases the failure diagnosis latency, which also increases the time the system operates without sufficient backup sites. Hence, P must carefully select ff to maximize the backups' chance of recovering from a subsequent primary failure. Even if the backup site does not crash, delayed or lost update messages can compromise the window consistency of backup objects, making B ineligible to replace a crashed primary. Using - ack and - xmit can estimate the consistency of backup objects and select the appropriate policy for scheduling update transmissions. The primary may choose to reintegrate an inconsistent backup, even when last ! ff, rather than wait for a later update message to restore the objects' window Probability of message loss0.401.20Average maximum distance w=300 msec (no compression) w=700 msec (no compression) w=300 msec (compression) w=700 msec (compression) Probability of message loss0.020.060.10 Probability that backup is inconsistent w=300 msec (no compression) w=700 msec (no compression) w=300 msec (compression) w=700 msec (compression) (a) Average maximum distance (b) Probability(backup inconsistent) Figure 8: Window consistency : The graphs show the performance of the service as a function of the client write rate, message loss, and schedule compression. Although object inconsistency increases with message loss, compressing the periodic schedule reduces the effects of communication failures. Inconsistency increases as the client writes more frequently, since the primary changes it object soon after transmitting an update message to the backup. consistency. Suppose the primary thinks that B's copy of O i is window-inconsistent. Under periodic update scheduling, P may not send another update message for this object until some time 2p later. If this object has a large window ffi i , the primary can reestablish the backup's window consistently more quickly by executing the integration schedule, which requires time P is the service time for object O i , as described in Section 4.1. Still, the primary cannot accurately determine if the backup object O i is inconsistent, since lost or delayed acknowledgement messages can result in an overly pessimistic value for - ack . The primary should not be overly aggressive in diagnosing inconsistent backup objects, since reintegration temporarily prohibits the backup from replacing a failed primary. Instead, P should ideally "retransmit" the offending object, without violating the window consistency of the other objects in the service. For example, P can schedule a special "retransmission" window for transmitting objects that have not received acknowledgement messages for past updates; when this "retransmission object" is selected for service, P transmits an update message for one of the existing objects, based on the values of - ack i . This improves the likelihood of having window-consistent backup sites, even in the presence of communication failures. 6 Implementation and Evaluation 6.1 Prototype Implementation We have developed a prototype implementation of the window-consistent replication service to demonstrate and evaluate the proposed service model. The implementation consists of a primary and a backup server, with the client application running on the primary node as shown in Figure 3. The primary implements rate-monotonic scheduling of update transmissions, with an option to enable schedule compression. Tick scheduling allocates the processor for different activities, such as handling client requests, sending update messages, and processing acknowledgements from the backup. At the start of each tick, the primary transmits an update message to the backup for one of the objects, as determined by the scheduling algorithm. Any client read/write requests and update acknowledgements are processed next, with priority given to client requests. Each server is currently an Intel-based PC running the Real-Time Mach [25, 32] operating system 2 . The sites communicate over an Ethernet through UDP datagrams using the Socket++ library [31], with extensions to the UNIX select call for priority-based access to the active sockets. At initialization, sockets are registered at the appropriate priority such that the socket for receiving client requests has a higher priority over that for receiving update acknowledgements from the backup. A tick period of 100 ms was chosen to minimize the intrusion from other runnable system processes 3 . To further minimize interference, experiments were conducted with lightly-loaded machines on the same Ethernet segment; we did not observe any significant fluctuations in network or processor load during the experiments. The primary and backup sites maintain in-memory logs of events at run-time to efficiently collect performance data with minimal intrusion. Estimates of the clock skew between the primary and the backup, derived from actual measurements of round-trip latency, are used to adjust the occurrence times of events to calculate the distance between objects on the primary and backup sites. The prototype evaluation considers three main consistency metrics representing window consistency and the backup and client views. These performability metrics are influenced by several parameters, including client write rate, communication failures, and schedule compression. The experiments vary the client write rate by changing the time w i between successive client writes to an object. We inject communication failures by randomly dropping update messages; this captures the effect of transient network load as well as lost update acknowledgements. The invariants in our evaluation are the tick period (100 ms), the objects' window size (ffi and the number of objects given the tick period and ffi i , N is determined by the schedulability criterion of the rate-monotonic scheduling algorithm. All objects have the same update transmission time of one tick, with the object size chosen such that the time to process and transmit the object is reasonably small earlier experiments on Sun workstations running Solaris 1.1 show similar results [24]. 3 The 100 ms tick period has the same granularity as the process scheduling quantum to limit the interference from other jobs running on the machine. However, smaller tick periods are desirable in order to allow objects to specify tighter windows (the window size is expressed in number of ticks) and respond to client requests in a timely manner. compared to the tick size; the extra time within each tick period is used to process client requests and update acknowledgements. Experiments ran for 45 minutes for each data point. 6.2 Omniscient View (Window Consistency) The window-consistency metric captures the actual temporal inconsistency between the primary and the backup sites, and serves as a reference point for the performance of the replication service. Figure 8(a) shows the average maximum distance between the primary and the backup as a function of the probability of message loss for three different client write periods, with and without schedule compression. This measures the inconsistency of each backup object just before receiving an update, averaged over all versions and all objects, reflecting the "goodness" of the replicated data. Figure 8(b) shows the probability of an inconsistent backup as a function message loss; this "fault-tolerance" metric measures the likelihood that the backup has one or more inconsistent objects. In these experiments, the client writes each object once every tick (w once every 3 ticks (w once every 7 ticks (w The probability of message loss varies from 0% to 10%; experiments with higher message loss rates reveal similar trends. Message loss increases the distance between the primary and the backup, as well as the likelihood of an inconsistent backup. However, the influence of message loss is not as pronounced due to conservative object admission in the current implementation. This occurs because, on average, the periodic model schedules updates twice as often as necessary, in order to guarantee the required worst-case spacing between update transmissions. Message loss should have more influence in other scheduling models which permit higher resource utilization, as discussed in Section 7. Higher client write rates also tend to increase the backup's inconsistency; as the client writes more frequently, the primary's copy of the object changes soon after sending an update message, resulting in staler data at the backup site. Schedule compression is very effective in improving both performance variables. The average maximum distance between the primary and backup under no message loss (the y-intercept) reduces by about 30% for high client rates in Figure 8(a); similar reductions are seen for all message loss probabilities. This occurs because schedule compression successfully utilizes idle ticks in the schedule generated by the rate-monotonic scheduling algorithm; the utilization thus increases to 100% and the primary sends approximately 30% more object updates to the backup. Compression plays a relatively more important role in reducing the likelihood of an inconsistent backup, as can be seen from Figure 8(b). Also, compression reduces the impact of communication failures, since the extra update transmissions effectively mask lost messages. 6.3 Backup View (Estimated Consistency) Although Figure 8 provides a system-wide view of window consistency, the backup site has limited knowledge of the primary state. The backup's view estimate of the actual window consistency, as shown in Figure 9. The backup site uses this metric to evaluate its Probability of message loss0.401.20Average maximum distance w=300 msec (no compression) w=700 msec (no compression) w=300 msec (compression) w=700 msec (compression) Probability of message loss0.020.060.10 Probability that backup is inconsistent w=300 msec (no compression) w=700 msec (no compression) w=300 msec (compression) w=700 msec (compression) (a) Average maximum distance (b) Probability(backup inconsistent) Figure 9: Backup view The plots show system performance from the backup's conservative viewpoint, as a function of the client write rate, message loss, and schedule compression. As in Figure temporal consistency improves under schedule compression but worsens under increasing message loss. The backup's view is impervious to the client write rate. own window consistency to detect a crashed primary and effect a failover. As in Figure 8 message loss increases the average maximum distance (Figure 9(a)) and the likelihood of an inconsistent backup Figure 9(b)). Schedule compression also has similar benefits for the backup's estimate of window consistency. However, unlike Figure 8, the client write rate does not influence the backup's view of its window consistency. The backup (pessimistically) assumes that the client writes an object on the primary immediately after the primary transmits an update message for that object to the backup. For this reason, the backup's estimate of the average maximum distance between the primary and the backup is always worse than that derived from the omniscient view. It follows that this estimate is more accurate for high client write rates, as can be seen by comparing Figures 8(a) and 9(a); for high client rates relative to the window, are virtually identical. The window-consistent replication model is designed to operate with high client write rates, relative to communication within the service, so the backup typically has an accurate view of its temporal consistency. 6.4 Client View (Recovery Consistency) The client view (- P measures the inconsistency between the primary and backup versions on object reads; better recovery consistency provides a more accurate system state after failover. Since the client can read at an arbitrary time, Figure 10 shows the time average of recovery inconsistency, averaged across all objects, with and without compression. We attribute the minor fluctuations in the graphs to noise in the measurements. The distance metric is not sensitive to the client write rate, since frequent client writes increase Probability of message loss0.200.601.00 Average distance w=300 msec (no compression) w=700 msec (no compression) w=300 msec (compression) w=700 msec (compression) Figure 10: Client view - P This graph presents the time average of recovery inconsistency, as a function of the client write rate, message loss, and schedule compression. Compressing the update schedule improves consistency by generating more frequent update transmission, while message loss worsens read consistency. The metric is largely independent of the client write rate. both - P when the client writes more often, the primary copy changes frequently (i.e., i (t) is close to t), but the backup also receives more recent versions of the data (i.e., - B i (t) is close to Moderate message loss does not have a significant influence on read inconsistency, especially under schedule compression. As expected, schedule compression improves the read inconsistency seen by the client significantly (- 30%). It is, therefore, an effective technique for improving the "goodness" of the replicated data. 7 Conclusion and Future Work Window consistency offers a framework for designing replication protocols with predictable timing behavior. By decoupling communication within the service from the handling of client requests, a replication protocol can handle a higher rate of read and write operations and provide more timely response to clients. Scheduling the selective communication within the service provides bounds on the degree of inconsistency between servers. While our prototype implementation has successfully demonstrated the utility of the window-consistent replication model, more extensive evaluation is needed to validate the ideas identified in this paper. We have recently added support for fault- detection, failover, and integration of new backups. Further experiments on the current platform will ascertain the usefulness of processor capacity reserves [25] and other RT-Mach features in implementing the window-consistent replication service. The present work extends into several fruitful areas of Object admission/scheduling: We are studying techniques to maximize the number of admitted objects and improve objects' window consistency by optimizing object admission and update scheduling. For the window-consistent replication service, the periodic task model is overly conservative in accepting object registration requests; that is, it may either limit the number of objects that are accepted or it may accept only those objects with relatively large windows. This occurs because, on average, the periodic model schedules updates twice as often as necessary, in order to guarantee the required worst-case spacing between update transmissions. We are exploring other scheduling algorithms, such as the distance-constrained task model [13] which assigns task priorities based on separation constraints, in terms of their implementation complexity and ability to accommodate dynamic creation/deletion of objects. We are also considering techniques to maximize the "goodness" of the replicated data. As one possible approach, we are exploring ways to incorporate the client write rate in object admission and scheduling. An alternate approach is to optimize the object window size itself by proportionally shrinking object windows such that the system remains schedulable; this should improve each object's worst-case temporal inconsistency. The selection of object window sizes can be cast as an instance of the linear programming optimization problem. Schedule compression can still be used to improve the utilization of the remaining available resources. Inter-object window consistency: We are extending our window-consistent replication model to incorporate temporal consistency constraints between objects. Our goal is to bound consistency in a replicated set of related objects; new algorithms may be necessary for real-time update scheduling of such object sets. This is related to the problem of ensuring temporally consistent objects in a real-time database system; however, our goal is to bound consistency in a replicated set of related objects. Alternative replication models: Although the current prototype implements a primary-backup architecture with a single backup site, we are studying the additional issues involved in supporting multiple backups. In addition, we are also exploring window consistency in the state-machine replication. This would enable us to investigate the applicability of window consistency to alternative replication models. Acknowledgements The authors wish to thank Sreekanth Brahmamdam and Hock-Siong Ang for their help in running experiments and post-processing the collected data, and the reviewers for their helpful comments. --R "Data caching issues in an information retrieval system," "A principle for resilient sharing of distributed resources," "Real-time communication in packet-switched networks," "A NonStop kernel," "A highly available network file server," "Reliable communication in the presence of failures," "The process group approach to reliable distributed computing," "Tradeoffs in implementing primary-backup protocols," "Tradeoffs in implementing primary-backup protocols," "Understanding fault tolerant distributed systems," "Fault-tolerance in the advanced automation system," "Partial computation in real-time database systems," "Scheduling distance-constrained real-time tasks," "Real-time communication in multi-hop networks," "Dis- tributed fault-tolerant real-time systems: The MARS approach," "TTP - a protocol for fault-tolerant real-time systems," "Triggered real time databases with consistency constraints," "Ssp: a semantics-based protocol for real-time data access," "The rate monotonic scheduling algorithm: Exact characterization and average case behavior," "A model of hard real-time transaction systems," "Scheduling algorithms for multiprogramming in a hard real-time environment," "Imprecise computations," "Structuring communication software for quality-of- service guarantees," "Design and evaluation of a window-consistent replication service," "Processor capacity reserves: Operating system support for multimedia applications," "Consul: A communication substrate for fault-tolerant distributed programs," "Using volatile witnesses to extend the applicability of available copy protocols," "Replica control in distributed systems: An asynchronous approach," "Window-consistent replication for real-time applications," "Implementing fault-tolerant services using the state machine approach: A tutorial," "Real-Time Toward a predictable real-time system," "The extra performance architecture (XPA)," --TR --CTR Hengming Zou , Farnam Jahanian, A Real-Time Primary-Backup Replication Service, IEEE Transactions on Parallel and Distributed Systems, v.10 n.6, p.533-548, June 1999

temporal consistency;real-time systems;replication protocols;scheduling;fault tolerance

266408

Decomposition of timed decision tables and its use in presynthesis optimizations.

Presynthesis optimizations transform a behavioral HDL description into an optimized HDL description that results in improved synthesis results. We introduce the decomposition of timed decision tables (TDT), a tabular model of system behavior. The TDT decomposition is based on the kernel extraction algorithm. By experimenting using named benchmarks, we demonstrate how TDT decomposition can be used in presynthesis optimizations.

Introduction Presynthesis optimizations have been introduced in [1] as source-level transformations that produce "better" HDL descriptions. For instance, these transformations are used to reduce control-flow redundancies and make synthesis result relatively insensitive to the HDL coding-style. They are also used to reduce resource requirements in the synthesized circuits by increasing component sharing at the behavior-level [2]. The TDT representation consists of a main table holding a set of rules, which is similar to the specification in a FSMD [3], an auxiliary table which specifies concurrencies, data dependencies, and serialization relations among data-path computations, or actions, and a delay table which specifies the execution delay of each action. The rule section of the model is based on the notions of condition and action. A condition may be the presence of an input, or an input value, or the outcome of a test condition. A conjunction of several conditions defines a rule. A decision table is a collection of rules that map condition conjunctions into sets of actions. Actions include logic, arithmetic, input-output(IO), and message-passing operations. We associate an execution delay with each action. Actions are grouped into action sets, or compound actions. With each action set, we associate a concurrency type of serial, parallel, or data-parallel [4]. Condition Stub Condition Entries Action Stub Action Entries Figure 1: Basic structure of TDTs. The structure of the rule section is shown in Figure 1. It consists of four quadrants. Condition stub is the set of conditions used in building the TDT. Condition entries indicate possible conjunctions of conditions as rules. Action stub is the list of actions that may apply to a certain rule. Action entries indicate the mapping from rules to actions. A rule is a column in the entry part of the table, which consists of two halves, one in the condition entry quadrant, called decision part of the rule, one in the action entry quadrant, called action part of the rule. In additional to the set of rules specified in a main table (the rule section), the TDT representation includes two auxiliary tables to hold additional information. Information specified in the auxiliary tables include the execution delay of each action, serialization, data dependency, and concurrency type between each pair of actions. Example 1.1. Consider the following TDT: a 1;1 a 1;2 a 2;1 a 2;2 a 3;1 a 3;2 a 1;1 s% a 1;2 a 2;1 d % a 2;2 a 3;1 p a 3;2 delay a a a a a a When actions a 1;1 and a 1;2 are selected for execution. Since action a 1;2 is specified as a successor of a 1;1 , action a 1;1 is executed with a one cycle delay followed by the execution of a 1;2 . Symbols 'd' and 'p' indicate actions that are data-parallel (i.e. parallel modulo data dependencies) and parallel actions respectively. An arrow '%' at row a 1;1 and column a 1;2 indicates that a 1;1 appears before a 1;2 . In contrast, an arrow '.' at row a 1;1 and column a 1;2 indicates that a 1;1 appears after a 1;2 . 2 The execution of a TDT consists of two steps: (1) select a rule to apply, (2) execute the action sets that the selected rule maps to. More than two action sets may be selected for execution. The order in which to execute those action sets are determined by the concurrency types, serialization relations, and data dependencies specified among those action sets [4], indicated by 's', `d', and 'p' in the table above. An action in a TDT may be another TDT. This is referred to as a call to the TDT contained as an action in the other TDT, which corresponds to the hierarchy specified in HDL descriptions. Consider the following example. Example 1.2. Consider the following calling hierarchy: a Here when c the action needs to be invoked is the call to TDT 2 , forces evaluation of condition c 2 resulting in actions a 2 or a 3 being executed. No additional information such as concurrency types needs to be specified between action a 1 and TDT 2 since they lie on different control paths. For the same reason, we omit the auxiliary table for TDT 2 . 2 Procedure/function calling hierarchy in input HDL descriptions results in a corresponding TDT hierarchy. TDTs in a calling hierarchy are typically merged to increase the scope of presynthesis optimizations. In the process of presynthesis optimizations, merging flattens the calling hierarchy specified in original HDL descriptions. In this paper we present TDT decomposition which is the reverse of the merging process. By first flattening the calling hierarchy and then extracting the commonalities, we may find a more efficient behavior representation which leads to improved synthesis results. This allows us to restructure HDL code. This code structuring is similar to the heuristic optimizations in multilevel logic synthesis. In this paper, we introduce code-restructuring in addition to other presynthesis optimization techniques such as column/row reduction and action sharing that have been presented earlier [1, 4, 2]. The rest of this paper is organized as follows. In the next section, we introduce the notion of TDT decomposition and relate it to the problem of kernel extraction in an algebraic form of TDT. Section 3 presents an algorithm for TDT decomposition based on kernel extraction. Section 4 shows the implementation details of the algorithm and presents the experimental results. Finally, we conclude in Section 5 and presents our future plan. TDT decomposition is the process of replacing a flattened TDT with a hierarchical TDT that represents an equivalent behavior. As we mentioned earlier, decomposition is the reverse process of merging and together with merging, it allows us to produces HDL descriptions that are optimized for subsequent synthesis tasks and are relatively insensitive to coding styles. Since this decomposition uses procedure calling abstraction, arbitrary partitions of the table (condition/action) matrices are not useful. To understand the TDT structural requirements consider the example below. Example 2.1. Consider the following TDT. Notice the common patterns in condition rows in c 6 and c 7 , and action rows in a 6 , a 7 , and a 8 . 2 Above in Example 2.1 is a flattened TDT. The first three columns have identical condition entries in c 1 and c 2 , and identical action entries in a 1 and a 2 . These columns differ in rows corresponding to conditions fc 4 , c 5 g and actions fa 3 , a 4 , a 5 g, which appear only in the first three columns. This may result, for example, from merging a sub-TDT consisting of only conditions fc 4 , c 5 g and actions fa 3 , a 4 , a 5 g. Note the common pattern in the flattened TDT may result from merging a procedure which is called twice from the main program. Or it may simply correspond to commonality in the original HDL description. Whatever the cause, we can extract the common part and make it into a separate sub-TDT and execute it as an action from the main TDT. Figure 2 shows a hierarchy of TDTs which specify the same behavior as the TDT in Example 2.1 under conditions explained later. The equivalence can be verified by merging the hierarchy of TDTs [4]. Note that the conditions and actions are partitioned among these TDTs, i.e, no conditions and actions are repeated amongs the TDTs. a 9 1 a a a 5 a a a 8 Figure 2: One possible decomposition of the TDT in Example 2.1. It is not always possible to decompose a given TDT into a hierarchical TDT as shown in Figure above. Neither is it always valid to merge the TDT hierarchy into flattened TDT [4]. These two transformations are valid only when the specified concurrency types, data dependencies, and serializations are preserved. In this particular example, we assume that the order of execution of all actions follows the order in which they appear in the condition stub. For the transformations to be valid, we also require that: ffl Actions a 1 and a 2 do not modify any values used in the evaluation of conditions c 4 and c 5 . ffl Actions a 1 and a 2 do not modify any values used in the evaluation of conditions c 6 and c 7 . Suppose we are given a hierarchical TDT as shown in Figure 2 to start with. After a merging phase, we get the flattened TDT as shown in Example 2.1. In the decomposition phase, we can choose to factor only TDT 3 because it is called more than once. Then the overall effect of merging followed by TDT decomposition is equivalent to in-line expansion of the procedure corresponding to This will not lead to any obvious improvement in hardware synthesis. However, it reduces execution delay if the description is implemented as a software component because of the overhead associated with software procedure calls. The commonality in the flattened TDT may not result from multiple calls to a procedure as indicated by TDT 3 in Figure 2. It could also be a result of commonality in the input HDL specification. If this is the case, extraction will lead to a size reduction in the synthesized circuit. The structural requirements for TDT decomposition can be efficiently captured by a two-level algebraic representation of TDTs [2]. This representation only captures the control dependencies in action sets and hence is strictly a sub set of TDT information. As we mentioned earlier, TDTs are based on the notion of conditions and actions. For each condition variable c, we define a positive condition literal, denoted as l c , which corresponds to an 'Y' value in a condition entry. We also define a negative condition literal, denoted as l - c , which corresponds to an 'N' value in a condition entry. A pair of positive and negative condition literals are related only in that they corresponds to the same condition variable in the TDT. We define a '\Delta' operator between two action literals and two conditions literals which represents a conjunction operation. This operation is both commutative and associative. A TDT is a set of rules, each of which consists of a condition part which determines when the rule is selected, and an action part which lists the actions to be executed once a rule is selected for execution. The condition part of a rule is represented as Y l c i l c i where ncond is the number of conditions in the TDT and ce(i) is the condition entry value at the ith condition row for this rule. The action part of a rule is represented as Y l a i where nact is the number of actions in the TDT and ae(i) is the action entry value at the ith action row for this rule. A rule is a tuple, denoted by As will become clear later, for the purpose of TDT decomposition a rule can be expressed as a product of corresponding action and condition literals. We call such a product a cube. For a given TDT, T , we define an algebraic expression, E T , that consists of disjunction of cubes corresponding to rules in T . For simplicity, we can drop the '\Delta' operator and `:' denotation and use 'c' or `a' instead of l c and l a in the algebraic expressions of TDTs. However, note in particular that 'c' and `-c' are short-hand notations for 'l c ' and 'l - c ' and they do not follow Boolean laws such as These symbols follow only algebraic laws for symbolic computation. For treatment of this algebra, the reader is referred to [4]. Example 2.2. Here is the algebraic expression for the TDT in Example 2.1. a 6 a 7 a 8 c 3 a 1 a 9 a 6 a 7 a 8 c 3 a 1 a 2 a 7 a 8 c 3 a 1 a 2 a 8 Note that there is no specification on delay, concurrency type, serialization relation, and data dependency. Also notice that 'c', `-c', and 'a' are short-hand notations for `l c ', 'l - c ', `l a ' respectively. 2 2.1 Kernel Extraction During TDT decomposition, it is important to keep an action literal or condition literal within one sub-TDT, that is, the decomposed TDTs must partition the condition and action literals. To capture this, we introduce the notion of support and TDT support. Definition 2.1 The support of an expression is the set of literals that appear in the expression. Definition 2.2 The TDT-support of an expression E T is the set of action literals and positive condition literals corresponding to all literals in the support of the expression E T . Example 2.3. Expression c 1 - c 2 c 3 - c 6 -c 7 a 2 a 8 is a cube. Its support is fc g. Its TDT support is fc g. 2 We consider TDT decomposition into sub-TDTs that have only disjoint TDT-supports. TDT decomposition uses algebraic division of TDT-expressions by using divisors to identify sub TDTs. We define the algebraic division as folllows: Definition 2.3 Let ff dividend remainder g be algebraic expressions. We say that f divider is an algebraic divisor of f divider when we have f dividend the TDT-support of f divisor and the TDT-support of f quotient are disjoint, and f divisor \Delta f quotient is non-empty. An algebraic divisor is called a factor when the remainder is void. An expression is said to be cannot be factored by a cube. Definition 2.4 A kernel of an expression is a cube-free quotient of the expression divided by a cube, which is called the co-kernel of the expression. Example 2.4. Rewrite the algebraic form of TDTExample 2:1 as follows. c 7 a 8 ) The expression c 4 a 3 a 4 a 5 c 5 a 3 is cube-free. Therefore it is a kernel of TDTExample 2:1 . The corresponding co-kernel is c 1 c 2 a 1 a 2 . Similarly, c 6 c 7 a 6 a 7 a 8 a 8 is also a kernel of TDTExample 2:1 , which has two corresponding co-kernels: c 1 - c 2 c 3 a 2 and - c 1 a 1 a 2 . 2 3 Algorithm for TDT Decomposition In this section, we present an algorithm for TDT decomposition. The core of the algorithm is similar to the process of multi-level logic optimization. Therefore we first discuss how to compute algebraic kernels from TDT-expressions before we show the complete algorithm which calls the kernel computing core and addresses some important issues such as preserving data-dependencies between actions through TDT decomposition. 3.1 Algorithms for Kernel Extraction A naive way to compute the kernels of an expression is to divide it by the cubes corresponding to the power set of its support set. The quotients that are not cube free are weeded out, and the others are saved in the kernel set [5]. This procedure can be improved in two ways: (1) by introducing a recursive procedure that exploits the property that a kernel of a kernel of an expression is also the kernel of this expression, (2) by reducing the search by exploiting the commutativity of the '\Delta' operator. Algorithm 3.1 shows a method adapted from a kernel extraction algorithm due to Brayton and McMullen [6], which takes into account of the above two properties to reduce computational complexity. Algorithm 3.1 A Recursive Procedure Used in Kernel Extraction INPUT: a TDT expression e, a recursion index j; OUTPUT: the set of kernels of TDT expression e; extractKernelR(e, to n do if (j getCubeSet(e; l i )j - 2) then largest cube set containing l i s.t. getCubeSet(e; if (l k 62 C8k ! i) then endfor In the above algorithm, getCubeSet(e; C) returns the set of cubes of e whose support includes C. We order the literals so that condition literals appear before action literals. We use n as the index of the last condition literal since a co-kernel containing only action literals does not correspond a valid TDT decomposition. Notice that l c and l - c are two different literals as we explained earlier. The algorithm is applicable to cube-free expressions. Thus, either the function e is cube-free or it is made so by dividing it by its largest cube factor, determined by the intersection of the support sets of all its cubes. Example 3.1. After running Algorithm 3.1 on the algebraic expression of TDT 2:1 we get the following set of kernels: c 4 -c 5 a 3 c 7 a 7 a 8 c 5 a 3 ; c 7 a 8 ; Note that k 6 has a cube with no action literals. This indicates a TDT rule with no action selected for execution if k 6 leads to a valid TDT decomposition. However, k 6 will be eliminated from the kernel set as we explained later. 2 3.2 TDT Decomposition Now we present a TDT decomposition algorithm which is based on the kernel extraction algorithm presented earlier. The decomposition algorithm works as follows. First, the algebraic expression of a TDT is constructed. Then a set of kernels are extracted from the algebraic expression. The are eventually used to reconstruct a TDT representation in hierarchical form. Not all the algebraic kernels may be useful in TDT decomposition since the algebraic expression carries only a subset of the TDT information. We use a set of filtering procedures to delete from the kernel sets kernels which corresponds to invalid TDT transformations or transformations producing models that results in inferior synthesis results. Algorithm 3.2 TDT Decomposition INPUT: a flattened TDT tdt; OUTPUT: a hierarchical TDT with root tdt f return tdt 0 The procedure constructAlgebraicExpression() builds the algebraic expression of tdt following Algorithm 3.2. The function expression() builds an expression out of a set of sets according to the data structure we choose for the two-level algrebraic expression for TDTs. The complexity of the algorithm is O(AR+ CR) where A is the number of action in tdt, R is the number of rules in tdt, and C is the number of conditions in tdt. The symbol 'OE' in the algorithm denotes an empty set. Algorithm 3.3 Constructing Algebraic Expressions of TDTs construct a positive condition literal l c i construct a negative condition literal l - c i endfor do construct an action literal l a i endfor R /\GammaOE; // empty set do r /\GammaOE; if (ce(i; if (ce(i; endfor return Procedure extractKernel(sop) calls the recursive procedure extractKernelR(sop; 1) to get a set of kernels of sop, the algebraic expression of tdt. Some kernels appear only once in the algebraic expression of a TDT. These kernels would not help in reducing the resource requirement and therefore they are trimmed from K using procedure trimKernel1(). Algorithm 3.4 below shows the details of trimKernel1(). The function co \Gamma Kernels(k; e) returns the set of co-kernels of kernel k for expression e. The number of co-kernels corresponds to the number of times sub-TDT that corresponds to a certain kernel is called in the hierarchy of TDTs. Algorithm 3.4 Removing Kernels Which Correspond to Single Occurrence of a Pattern in the TDT Matrices foreach k 2 K do if (j co-Kernels(k; endforeach Example 3.2. Look at the kernels in Example 3.1. The kernel k will be trimmed off by trimKernel1() since it has only one co-kernel. 2 Since information such as data dependency are not captured in algebraic form of TDTs, the kernels in K may not corresponds to a decomposition which preserves data-dependencies specified in the original TDT. These kernels are trimmed using procedure trimKernel2(). Algorithm 3.5 Removing Kernels Which Corresponds to an Invalid TDT Transformation e, tdt) f foreach k 2 K do flag /\Gamma0; foreach do foreach action literal l a of q do if action a modifies any condition corresponding to a condition literal of k then foreach action literal l ff in k do if (l a is specified to appear before l ff ) then flag endforeach endforeach endforeach if (flag == endforeach The worst case complexity of this algorithm is O(AR+CR) since the program checks no more than once on each condition/action literal corresponding to a condition entry or action entry of tdt. Example 3.3. Suppose in Example 2.1, a 2 modifies c 6 and the result of a 2 is also used a 6 . Because a 2 modifies c 6 , in the hierarchical TDT we need to specify that c 6 comes after TDT 2 to preserve the behavior. However, this violates the data dependency specification between a 2 and a 6 . Therefore, under the condition given here, kernel k c 7 a 8 will be removed by trimKernel2(). 2 An expression may be a kernel of itself with a co-kernel of '1' if it is kernel free. However this kernel is not useful for TDT decomposition. We use a procedure trimSelf() to delete an expression from its kernel set that will used fro TDT decomposition. Also, as we mentioned earlier, a kernel of an expression's kernel is a kernel of this expression. However, in this paper, we limit our discussion on TDT decomposition involving only two levels of calling hierarchies. For this reason, after removing an expression itself from its kernel sets, we also delete "smaller" kernels which are also kernels of other kernels of this expression. Algorithm 3.6 Other Kernel Trimming Routines foreach k 2 K do compute q and r s.t. if TDTsupport(k) and TDTsupport(r) are not disjoint then endforeach g foreach k 2 K do foreach q 2 K different from k do if q is a kernel of k then endforeach endforeach Example 3.4. Look at the kernels of E TDT2:1 . Kernels k 5 will be eliminated by trimKernel3() since a 5 and a 8 are also used in other cubes. For the same reason, k 6 and k 7 are also eliminated. 2 Finally, we reconstruct a hierarchical TDT representation using the remaining algebraic kernels of the TDT expression. The algorithm is outlined below. It consists two procedures: reconstruct TDT with Kernel(), and constructTDT () which is called by the other procedure to build a TDT out of an algebraic expression. Again, the worse case complexity of the algorithm is O(CR +AR). Algorithm 3.7 Construct a Hierarchical TDT Using Kernels INPUT: a flattened TDT tdt, its algebraic expression exp, a set of kernels K of exp; OUTPUT: a new hierarchical TDT; re Construct TDT with Kernels(tdt, K, exp) f foreach k 2 K do generate a new action literal l t for t; compute q and r s.t. endforeach return constructTDT(tdt, e); form condition stub using those conditions of tdt each of which has at least a corresponding condition literal in e; form condition matrix according to the condition literals appearing in each cube of e; form action stub using those conditions of tdt each of which has at least a corresponding action literal in e and those "new" action literals corresponding to extracted sub-TDTs; form action matrix according to the action literals appearing in each cube of e; using the above components; return Example 3.5. Assume expression c 6 c 7 a 6 a 7 a 8 a 8 is the only kernel left after trimming procedures are performed on the kernel set K of the algebraic expression of TDTExample 2:1 . A hierarchical TDT as shown in below will be constructed after running reconstruct TDT with Kernels(). a a a a 9 1 a a a 4 Implementation and Experimental Results To show the effect of using TDT decomposition in presynthesis optimizations, we have incorporated our decomposition algorithm in PUMPKIN, the TDT-based presynthesis optimization tool [4]. Figure 3 shows the flow diagram of the process presynthesis optimizations. The ellipse titled "kernel extraction" in Figure 3 show where the TDT decomposition algorithm fits in the global picture of presynthesis optimization using TDT. Assertions parser merger merged TDT optimizer optimized TDT code generator optimized HDL input HDL (a) Assertions user specification (b) merged TDT column reduction row reduction kernel extraction optimized TDT action sharing Figure 3: Flow diagram for presynthesis optimizations: (a) the whole picture, (b) details of the optimizer. Our experimental methodology is as follows. The HDL description is compiled into TDT mod- els, run through the optimizations, and finally output as a HardwareC description. This output is provided to the Olympus High-Level Synthesis System [7] for hardware synthesis under minimum area objectives. We use Olympus synthesis results to compare the effect of optimizations on hardware size on HDL descriptions. Hardware synthesis was performed for the target technology of LSI Logic 10K library of gates. Results are compared for final circuits sizes, in term of number of cells. In addition to the merging algorithms, the column and row optimization algorithms originally implemented in PUMPKIN [1], we have added another optimization step of TDT decomposition. To evaluate the effectiveness of this step, we turn off column reduction, row reduction, and action sharing pahses and run PUMPKIN with several high-level synthesis benchmark designs. Table 1: Synthesis Results: cell counts before and after TDT decomposition is carried out. design module circuit size (cells) \Delta% before after daio phase decoder 1252 1232 2 receiver 440 355 19 comm DMA xmit 992 770 22 exec unit 864 587 cruiser State 356 308 14 Table 1 shows the results of TDT decomposition on examples designs. The design 'daio' refers to the HardwareC design of a Digital Audio Input-Output chip (DAIO) [8]. The design 'comm' refers to the HardwareC design of an Ethernet controller [9]. The design 'cruiser' refers to the HardareC design of a vehicle controller. The description 'State' is the vehicle speed regulation module. All designs can be found in the high-level synthesis benchmark suite [7]. The percentage of circuit size reduction is computed for each description and listed in the last column of Table 1. Note that this improvement depends on the amount of commonality existing in the input behavioral descriptions. 5 Conclusion and Future Work In this paper, we have introduced TDT decomposition as a complementary procedure to TDT merging. We have presented a TDT decomposition algorithm based on kernel extraction on an algebraic form of TDTs. Combining TDT decomposition and merging, we can restructure HDL descriptions to obtain descriptions that lead to either improved synthesis results or more efficient compiled code. Our experiment on named benchmarks shows a size reduction in the synthesized circuits after code restructuring. Sequential Decomposition (SD) has been proposed in [10] to map a procedure to a separate hardware component which is typically specified with a process in most HDLs. Using SD, a procedure can be mapped on an off-shelf component with fixed communication protocol while a complement protocol can be constructed accordingly on the rest (synthesizable part) of the system. Therefore as a future plan of the research presented in this paper, we plan to combine SD and TDT decomposition to obtain a novel system partitioning scheme which works on tabular representations. We will investigate the possible advantages/disadvantages of this approach over other partitioning approaches. --R "HDL Optimization Using Timed Decision Tables," "Limited exception modeling and its use in presynthesis optimizations," Specification and Design of Embedded Systems. "System modeling and presynthesis using timed decision tables," Synthesis and Optimization of Digital Circuits. "The decomposition and factorization of boolean expressions," "The Olympus Synthesis System for Digital Design," "Design of a digital input output chip," "Decomposition of sequential behavior using interface specification and complementation," --TR High level synthesis of ASICs under timing and synchronization constraints Specification and design of embedded systems HDL optimization using timed decision tables Limited exception modeling and its use in presynthesis optimizations Synthesis and Optimization of Digital Circuits The Olympus Synthesis System --CTR Jian Li , Rajesh K. Gupta, HDL code restructuring using timed decision tables, Proceedings of the 6th international workshop on Hardware/software codesign, p.131-135, March 15-18, 1998, Seattle, Washington, United States J. Li , R. K. Gupta, An algorithm to determine mutually exclusive operations in behavioral descriptions, Proceedings of the conference on Design, automation and test in Europe, p.457-465, February 23-26, 1998, Le Palais des Congrs de Paris, France Sumit Gupta , Rajesh Kumar Gupta , Nikil D. Dutt , Alexandru Nicolau, Coordinated parallelizing compiler optimizations and high-level synthesis, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.9 n.4, p.441-470, October 2004

benchmarks;presynthesis optimizations;timed decision table decomposition;decision tables;system behavior model;TDT decomposition;behavioral HDL description;circuit synthesis;optimized HDL description;kernel extraction algorithm

266421

Effects of delay models on peak power estimation of VLSI sequential circuits.

Previous work has shown that maximum switching density at a given node is extremely sensitive to a slight change in the delay at that node. However, when estimating the peak power for the entire circuit, the powers estimated must not be as sensitive to a slight variation or inaccuracy in the assumed gate delays because computing the exact gate delays for every gate in the circuit during simulation is expensive. Thus, we would like to use the simplest delay model possible to reduce the execution time for estimating power, while making sure that it provides an accurate estimate, i.e., that the peak powers estimated will not vary due to a variation in the gate delays. Results for four delay models are reported for the ISCAS85 combinational benchmark circuits, ISCAS89 sequential benchmark circuits, and several synthesized circuits.

Introduction The continuing decrease in feature size and increase in chip density in recent years give rise to concerns about excessive power dissipation in VLSI chips. As pointed out in [1], large instantaneous power dissipation can cause overheating (lo- cal hot spots), and the failure rate for components roughly doubles for every increase in operating temperature. Knowledge of peak power dissipation can help to determine the thermal and electrical limits of a design. The power dissipated in CMOS logic circuits is a complex function of the gate delays, clock frequency, process param- eters, circuit topology and structure, and the input vectors applied. Once the processing and structural parameters have been fixed, the measure of power dissipation is dominated by the switching activity (toggle counts) of the circuit. It has been shown in [2] and [3] that, due to uneven circuit delay paths, multiple switching events at internal nodes can result, and power estimation can be extremely sensitive to different gate delays. Both [2] and [3] computed the upper bound of maximum transition (or switching) density of individual internal nodes of a combinational circuit via propagation of uncertainty waveforms across the circuit. However, these measures cannot be used to compute a tight upper bound of the overall power dissipation of the entire circuit. Although maximum switching density of a given internal node can be extremely sensitive to the delay model used, it is unclear whether peak power dissipation of the entire circuit is also equally sensitive to the delay model. Since This research was conducted at the University of Illinois and was supported in part by the Semiconductor Research Corporation under contract SRC 96-DP-109, in part by DARPA under contract DABT63-95-C-0069, and by Hewlett-Packard under an equipment grant. glitches and hazards are not taken into account in a zero-delay framework, the power dissipation measures from the zero-delay model may differ greatly from the actual powers. Will the peak power be vastly different among various non-zero delay models (where glitches and hazards are accounted for) in sequential circuits? Moreover, the delays used for internal gates in most simulators are mere estimates of the actual gate delays. Can we still have confidence in the peak power estimated using these delay measures considering that the actual delays for the gates in the circuit may be different? Since computing the exact gate delays for every gate in the circuit during simulation is expensive, does there exist a simple delay model such that the execution time for estimating power is reduced and an accurate peak power estimate can be obtained (i.e., the peak powers estimated will not vary due to a variation in the gate delays)? Several approaches to measuring maximum power in CMOS VLSI circuits have been reported [4-10]. Unlike average power estimations [11-17], where signal switching probabilities are sufficient to compute the average power, peak power is associated with a specific starting circuit state S and a specific sequence of vectors Seq that produce the power. Two issues are addressed in this paper. First, given a (S i , that generates peak power under delay model DM i , is it possible to obtain another tuple (S generates equal or higher dissipation under a different delay model DM j ? Second, will the (S i , peak power under delay model DM i also produce near peak power under a different delay model DM j ? Four different delay models are studied in this work: zero delay, unit delay, type-1 variable delay, and type-2 variable delay. The three delay models used in [9] are the same as the first three delay models of this work. Measures of peak power dissipation are estimated for all four delay models over various time periods. Genetic algorithms (GA's) are chosen as the optimization tool for this problem as in [10]. GA's are used to find the vector sequences which most accurately estimate the peak power under various delay models as well as over various time durations. The estimates obtained for each delay model are compared with the estimates from randomly-generated sequences, and the results for combinational circuits are compared with the automatic test generation (ATG) based technique [9] as well. The GA-based estimates will be shown to achieve much higher peak power dissipation in short execution times when compared with the random simulation. The remainder of the paper is organized as follows. Section 2 explains the delay models, and Section 3 discusses the peak power measures over various time periods. The GA framework for power estimation is described in Section 4. Experimental results on the effects of various delay models are discussed in Section 5, and Section 6 concludes the paper. Delay Models Zero delay, unit delay, and two types of variable delay models are studied here. The zero delay model assumes that no delay is associated with any gate; no glitches or hazards will occur under this model. The unit delay model assigns identical delays to every gate in the circuit independent of gate size and numbers of fanins and fanouts. The type-1 variable delay model assigns the delay of a given gate to be proportional to the number of fanouts at the output of the gate. This model is more accurate than the unit delay model; however, fanouts that feed bigger gates are not taken into ac- count, and inaccuracies may result. The fourth model is a different variable delay model which is based on the number of fanouts as well as the sizes of successor gates. The gate delay data for various types and sizes of gates are obtained from a VLSI library. The difference between the type-1 and variable delay models for a typical gate is illustrated in Figure 1. From the figure, the output capacitance of gate Type 1 variable: G3 Delay for gate G1 is 2 units. Delay for driving 2-input Type 2 variable: Delay for G1 is Delay for driving 4-input units. Figure 1: Variable Delay Models. G1 is estimated to be 2 (the number of fanouts) in the type- delay model, while the delay calculated using the variable delay model is proportional to the delay associated with driving the successor gates G2 and G3, or simply 3. Since the type-1 variable delay model does not consider the size of the succeeding gates, the delay calculations may be less accurate. 3 Peak Power Measures Three types of peak power are used in the context of sequential circuits for comparing the effects of delay models. An automatic procedure was implemented for various delay models that obtains these measures and generates the actual input vectors that attain them, as described in [10]. They are Peak Single-Cycle Power, Peak n-Cycle Power, and Peak Sustainable Power, covering time durations of one clock cy- cle, several consecutive clock cycles, and an infinite number of cycles, respectively. The unit of power used throughout the paper is energy per clock cycle and will simply be referred to as power. In a typical sequential circuit, the switching activity is largely controlled by the state vectors and less influenced by input vectors, because the number of flip-flops far outweighs the number of primary inputs. In all three cases, the power dissipated in the combinational portion of the sequential circuit can be computed as dd 2 \Theta cycle period \Theta for all gates g [toggles(g) \Theta C(g)]; where the summation is performed over all gates g, and toggles(g) is the number of times gate g has switched from 0 to 1 or vice versa within a given clock cycle; C(g) represents the output capacitance of gate g. In this work, we made the assumption that the output capacitance for each gate is equal to the number of fanouts for all four delay models; however, assigned gate output capacitances can be handled by our optimization technique as well. Peak single-cycle switching activity generally occurs when the greatest number of capacitive nodes are toggled between two consecutive vectors. For combinational circuits, the task is to search for a pair of vectors (V1 , V2 ) that generates the most gate transitions. For sequential circuits, on the other hand, the activity depends on the initial state as well as the primary input vectors. The estimate for peak power dissipation can be used as a lower-bound for worst-case power dissipation in the circuit in any given time frame. Our goal in this work is to find and compare such bounds for the peak power dissipation of a circuit under different delay assumptions Peak n-cycle switching activity is a measure of the peak average power dissipation over a contiguous sequence of n vectors. This measure serves as an upper-bound to peak sustainable power, which is a measure of the peak average power that can be sustained indefinitely. Both measures are considered only for sequential circuits. The n-cycle power dissipation varies with the sequence length n. When n is equal to 2, the power dissipation is the same as the peak single-cycle power dissipation, and as n increases, the average power is expected to decrease if the peak single-cycle power dissipation cannot be sustained over the n vectors in the sequence. We computed peak sustainable power by finding the synchronizing sequence for the circuit that produces greatest power [10]. Unlike the approach proposed in [7], where symbolic transition counts based on binary decision diagrams (BDD's) were used to compute maximum power cycles, no state transition diagrams are needed in our approach. Ba- sically, the method used in [7] was to find the maximal average length cycle in the state transition graph, where the edge weights in the STG indicate the power dissipation in the combinational portion of the circuit between two adjacent states. However, the huge sizes of STG's and BDD's in large circuits make the approach infeasible and impracti- cal. Our approach restricts the search for peak power loops to be a synchronizing sequence, thereby limiting the search to a subset of all loops, and the resulting peak power may not be a very tight lower bound. However, our experiments have shown that this approach still yields peaks higher than extensive random search [10]. 4 GA Framework for Power Estimation The GA framework used in this work is similar to the simple GA described by Goldberg [18]. The GA contains a population of strings, also called chromosomes or individuals, in which each individual represents a sequence of vectors. Peak n-cycle power estimation requires a search for the (n tuple (S1 , V1 , ., Vn+1) that maximizes power dissipation. This (n + 2)-tuple is encoded as a single binary string, as illustrated in Figure 2. The population size used is a function of the string length, which depends on the number of primary inputs, the number of flip-flops, and the vector sequence length n. Larger populations are needed to accommodate longer vector sequences in order to maintain diversity. Figure 2: Encoding of an Individual. The population size is set equal to 32 \Theta sequence length when the number of primary inputs is less than 16 and 128 \Theta sequence length when the number of primary inputs is greater than or equal to 16. Each individual has an associated fitness, which measures the quality of the vector sequence in terms of switching activity, indicated by the total number of capacitive nodes toggled by the individual. The delay model and the amount of capacitive-node switching are taken into account during the evolutionary processes of the GA via the fitness function. The fitness function is a simple counting function that measures the power dissipated in terms of amount of capacitive-node switching in a given time period. The population is first initialized with random strings. A variable-delay logic simulator is then used to compute the fitness of each individual. The evolutionary processes of se- lection, crossover, and mutation are used to generate an entirely new population from the existing population. This process continues for 32 generations, and the best individual in any generation is chosen as the solution. We use tournament selection without replacement and uniform crossover, and mutation is done by simply flipping a bit. In tournament selection without replacement, two individuals are randomly chosen and removed from the population, and the best is selected; the two individuals are not replaced into the original population until all other individuals have also been removed. Thus, it takes two passes through the parent population to completely fill the new population. In uniform crossover, bits from the two parents are swapped with probability 0.5 at each string position in generating the two offspring. A crossover probability of 1 is used; i.e., the two parents are always crossed in generating the two offspring. Since the majority of time spent by the GA is in the fitness evaluation, parallelism among the individuals can be exploited. Parallel-pattern simulation [19] is used to speed up the process in which candidate sequences from the population are simulated simultaneously, with values bit-packed into 32-bit words. 5 Experimental Results Peak powers for four different delay models were estimated for ISCAS85 combinational benchmark circuits, ISCAS89 sequential benchmark circuits, and several synthesized circuits. Functions of the synthesized circuits are as follows: am2910 is a 12-bit microprogram sequencer [20]; mult16 is a 16-bit two's complement shift-and-add multiplier; div16 is a 16-bit divider using repeated subtraction; and proc16 is a 16-bit microprocessor with 373 flip-flops. Table 1 lists the total number of gates and capacitive nodes for all circuits. The total number of capacitive nodes in a circuit is computed as the total number of gate inputs. All computations were performed on an HP 9000 J200 with 256 MB RAM. The GA-based power estimates are compared against the best estimates obtained from randomly generated vector se- quences, and the powers are expressed in peak switching frequency per node (PSF), which is the average frequency of peak switching activity of the nodes (ratio of the Table 1: Numbers of Capacitive Nodes in Circuits Circuit Gates Cap Circuit Gates Cap Nodes Nodes c7552 3828 6252 s5378 3043 4440 s526 224 472 number of transitions on all nodes to the total number of capacitive nodes) in the circuit. Notice that for the non-zero delay models, PSF's greater than 1.0 are possible due to repeated switching on the internal nodes within one clock cycle. We have previously shown that estimates made by our GA-based technique are indeed good estimates of peak power, and that if random-based methods were to achieve similar levels of peak power estimates, execution times are several orders of magnitude higher than for the GA-based technique [10]. Peak power estimation results will now be presented for the various delay models, and correlations between the delay models will be discussed. 5.1 Effects of the four delay models For the ISCAS85 combinational circuits, Table 2 compares the GA-based results against the randomly-generated sequences as well as those reported in [9]. For each circuit, results for all four delay models are reported. The estimates obtained from the best of randomly-generated vector-pairs, an ATG-based approach [9], and the GA-based technique are shown. The ATG-based approach used in [9] attempts to optimize the total number of nodes switching in the expanded combinational circuit; its results were compared against the best of only 10,000 random simulations. In our work, on the other hand, the number of random simulations depends on the GA population size, which is a function of the number of primary inputs in the circuit. Typically, the number of random simulations exceeds 64,000. The GA-based estimates are the highest for all of the circuits and for all delay models, except for one zero-delay estimate in c1355, where a 1.1% lower power was obtained. The average improvements made by the GA-based technique over the random simulations are 10.8%, 27.4%, 38.5%, and 35.5%, for the four delay models, respectively. Since [9] estimated power on the expanded circuits, backtrace using zero-delay techniques was used. Consequently, only hazards were captured; zero-width pulses were not taken into account in the non-zero delay models. For this reason, our GA-based estimates are higher than all the estimates obtained in [9]. Furthermore, results from [9] suggested that peak power estimated from the unit-delay model consistently gave higher values than both zero-delay and type-1 variable-delay estimates. We do not see such consistency from our results in Table 2. The estimated powers from random simulations do not show such a trend either. Nevertheless, it is observed that peak powers Table 2: Peak Single-Cycle Power for ISCAS85 Combinational Circuits Circuit Zero Unit Type-1 Variable Type-2 Variable Rnd [9] GA Rnd [9] GA Rnd [9] GA Rnd [9] GA c432 0.636 0.536 0.805 1.985 1.985 2.362 2.557 1.181 3.399 2.286 - 2.522 Impr 10.8% 27.4% 38.5% 35.5% Impr: Average improvements over the best random estimates for some circuits are very sensitive to the underlying delay model. For instance, in circuit c3540, the peak power estimates are 0.600, 2.684, 3.555, and 4.678 for zero, unit, type-1 variable, and type-2 variable delay models. The results for sequential circuits using the four delay models are shown in Table 3. The length of the sequence in each case is set to 10 for N-cycle and sustainable power estimates. Results for best of the random simulations are not shown; however, the average improvements made by the GA-based estimates are shown at the bottom of the table. The number of random simulations is also the same as the number of simulations required in the GA-based technique. For the small circuits, the number of simulations is around 64,000. The GA-based estimates surpass the best random estimates for all circuits for all three peak power measures under all four delay models. The average improvements of the GA-based estimates for each delay model are shown in the bottom row of Table 3. Up to 32.5% improvement is obtained on the average for peak N-cycle powers. Across the four delay models, estimates made from the zero-delay model consistently gave significantly lower power, since glitches and hazards were not accounted for. However, there is not a clear trend as to which of the other three delay models will consistently give peak or near peak power esti- mates. In circuits such as s641, s713, and am2910, the peak powers are very sensitive to the delay model. Over 100% increase in power dissipation can result when a different delay model is used. For some other circuits, such as s400, s444, s526, and s1238, the peak powers estimated are quite insensitive to the underlying delay model; less than 5% difference in the estimates is observed from the three delay models. The execution times needed for the zero-delay estimates are typically smaller, since few events need to be evaluated, while execution times for the other three delay models are comparable. Table 4 shows the execution times for the GA optimization technique for the unit-delay model. The execution times are directly proportional to the number of events generated in the circuit during the course of estimation. For this reason, peak n-cycle power estimates do not take 10 times as much computation as peak single-cycle power for since the amount of activity across the 10 cycles is not 10 times that of the peak single cycle. For circuits in which peak power estimates differ significantly among various delay models, the computation costs will also differ significantly according to the number of activated events. The execution times required for the random simulations are very close to the GA-based technique since identical numbers of simulations are performed. 5.2 Cross-effects of delay models Although peak powers are sensitive to the underlying delay model, it would be interesting to study if the vectors optimized under one delay model will produce peak or near-peak power under the other delay models. If we can find a delay model in which derived sequences consistently generate near peak powers under the other delay models, we can conclude that, although peak power estimation is sensitive to the underlying delay model, there exists a delay model by which optimized sequences generate high powers regardless of the delay model used. Experiments were therefore carried out to study these cross-effects of delay models. Table 5 shows the results for the combinational circuits. For each circuit, the previously optimized peak power measures for various delay models (the same as those in Table 2) are shown in the Opt columns. The power produced from applying vectors optimized for a given delay model on the other three delay models are listed in the adjacent (Eff) columns. For example, in circuit c2670 under the zero-delay model (upper-left quadrant of the ta- ble), the original peak single-cycle switching frequency per node (PSF) for the unit-delay model was 2.251 (Opt); how- ever, when vectors optimized for the zero-delay model are simulated using the unit-delay model, the measured PSF is 1.150 (Eff). The Eff measures that exceed the optimized (Opt) measures are italicized. At the bottom of the table, Dev shows the average amount that the Eff values deviated from the average Opt values. For instance, when the optimized vectors on the unit-delay model are simulated using the zero-delay assumption, an average deviation of 18.9% results from the GA-optimized zero-delay powers. On the average, the Eff peak powers deviated from the optimized values for all delay models as indicated by the Dev values. However, the vectors optimized for the zero-delay model produced significantly lower power when they were simulated in the non-zero delay environments; over 60% drops were observed from the GA-optimized powers, as indicated by the Dev metric. The vectors that were optimized on the remaining three delay models, on the other hand, deviated less significantly when simulated using other non-zero delay models, all less than 10% deviations. For sequential circuits, we will first look at the vectors optimized under the zero-delay assumption. Table 6 shows the results. The Opt and Eff values are defined the same way as before. For example, in circuit s382, the peak single-cycle switching frequency per node (PSF) under unit delay was however, when the vector sequence optimized for the zero-delay model is simulated using the unit-delay model, the Table 3: GA-Based Power Estimates for ISCAS89 Sequential Circuits Circuit Single Cycle N-Cycle Sustainable Z U V1 V2 Z U V1 V2 Z U V1 V2 Impr 11.9% 12.3% 20.4% 22.3% 17.4% 32.5% 26.7% 32.0% 12.3% 23.8% 24.4% 30.2% Z: Zero U: Unit V1: Type-1 Variable V2: Type-2 Variable Impr: Average improvements over the best random estimates Table 4: Execution Times for the GA-Based Technique (seconds) Circuit Single N- Sustain Circuit Single N- Sustain Circuit Single N- Sustain cycle cycle able cycle cycle able cycle cycle able measured PSF is 0.952. A similar format is used for the n-cycle and sustainable powers. Average deviation values Dev are also displayed at the bottom of the table. On average, the Eff peak powers deviated significantly from the powers optimized by the GA for both unit and type-1 variable delay models, as indicated in the Dev row. When examining the results for each circuit individually, none of the single-cycle vectors derived under the zero-delay model exceed the previously optimized peak powers for unit and type-1 variable delay models. However, several occasions of slightly higher peak n-cycle and sustainable powers have been obtained by the zero-delay-optimized vectors because of the difficulty in finding the optimum in the huge search space for the (n+2)-tuple needed for the peak n-cycle and sustainable powers. Nevertheless, the number of these occasions is quite small. For the small circuits s298 to s526, the results show that the zero-delay-optimized vectors produced peak or near-peak powers for the unit and type-1 variable delay models as well. One plausible explanation for this phenomenon is that the peak switching frequencies (PSF's) for these circuits are small, typically less than 1.2, indicating that most nodes in the circuit do not toggle multiple times in a single clock cycle. For the other circuits, especially for circuits where high PSF's are obtained, the vectors obtained which generated high peak powers under zero-delay assumptions do not provide peak powers under non-zero delay assumptions. The last four synthesized circuits, am2910, mult16, div16, and proc16, showed widened gaps. The effects on type-2 variable delay are similar to those on type-1 variable delay. Similarly, vectors optimized under the non-zero-delay models were simulated using other delay models, and the results are shown in Table 7 for type-2 variable delay. The trends for the unit-delay and type-1 variable delay models are similar to those seen for the type-2 variable delay model. The results are not compared with the zero-delay model here, since their trends are similar to those for the combinational circuits, where large deviations exist. When the optimized powers are great, i.e., PSF's are greater than 2, a greater deviation is observed. Such cases can be seen in circuits s641, s713, am2910, and div16. For instance, the optimized peak single-cycle unit-delay power for s713 is 2.815, but the power produced by applying the vectors optimized for the type-2 variable delay model is only 1.273. On the contrary, less significant deviation is observed in circuits for which smaller PSF's are obtained. For the n-cycle and sustainable pow- ers, although deviations still exist when compared with the GA-optimized vectors (shown in Dev), they are small devi- ations. Furthermore, deviations between type-1 and type-2 delay models are smaller when compared with the unit-delay model, suggesting that the two variable delay models are more correlated. 6 Conclusions When estimating peak power under one delay model, it is crucial to have confidence that the estimate will not vary significantly when the actual delays in the circuit differ from Table 5: Effects of Various Delay Models for ISCAS85 Combinational Circuits Effects of Zero-delay Vectors On. Effects of Unit-delay Vectors On. Circuit Unit Type-1 Var Type-2 Var Zero Type-1 Var Type-2 Var Opt Eff Opt Eff Opt Eff Opt Eff Opt Eff Opt Eff c432 2.362 2.035 3.399 2.181 2.522 2.192 0.636 0.455 3.399 2.904 2.522 3.032 c1355 3.260 1.000 2.707 1.011 2.676 1.042 0.533 0.441 2.707 2.391 2.676 3.143 c2670 2.251 1.150 2.750 1.294 2.825 1.320 0.623 0.572 2.750 2.364 2.825 2.355 c7552 2.821 1.619 2.833 1.757 3.238 1.788 0.602 0.537 2.833 3.089 3.238 3.012 Effects of Type-1 Var-delay Vectors On. Effects of Type-2 Var-delay Vectors On. Circuit Zero Unit Type-2 Var Zero Unit Type-1 Var Opt Eff Opt Eff Opt Eff Opt Eff Opt Eff Opt Eff c432 0.636 0.496 2.362 2.373 2.522 3.399 0.636 0.411 2.362 1.776 3.399 2.155 Dev: Average deviation of Eff from Opt. the delay model assumed. Peak power estimation under four different delay models for single-cycle, n-cycle, and sustainable power dissipation has been presented. For most circuits, vector sequences optimized under the zero-delay assumption do not produce peak or near-peak powers when they are simulated under a non-zero delay model. Similarly, the vectors optimized under non-zero delay models do not produce peak or near-peak zero-delay powers. However, vector sequences optimized under non-zero delay models provide good measures for other non-zero delay models, with only a slight deviation, for most combinational circuits. For sequential circuits, small deviations are observed when the optimized peak powers are small, but when the optimized peak powers are large, i.e., nodes switch multiple times in a single cycle on the average, the estimated peak powers will be sensitive to the underlying delay model. --R "Shrinking devices put the squeeze on system packag- ing," "Extreme delay sensitivity and the worst-case switching activity in VLSI circuits," "Estimation of maximum transition counts at internal nodes in CMOS VLSI circuits," "Estimation of power dissipation in CMOS combinational circuits using boolean function manipulation," "Worst case voltage drops in power and ground busses of CMOS VLSI circuits," "Resolving signal correlations for estimating maximum currents in CMOS combinational circuits," "Computing the maximum power cycles of a sequential circuit," "Maximum power estimation for sequential circuits using a test generation based technique," "Maximizing the weighted switching activity in combinational CMOS circuits under the variable delay model," "K2: An estimator for peak sustainable power of VLSI circuits," "Estimation of average switching activity in combinational and sequential circuits," "A survey of power estimation techniques in VLSI circuits," "Statistical estimation of sequential circuit activity," "Accurate power estimation of CMOS sequential circuits," "Switching activity analysis using boolean approximation method," "Power estimation methods for sequential logic circuits," "Power estimation in sequential circuits," Genetic Algorithms in Search Digital Systems Testing and Testable Design. "The AM2910, a complete 12-bit microprogram sequence controller," --TR Estimation of average switching activity in combinational and sequential circuits Resolving signal correlations for estimating maximum currents in CMOS combinational circuits Worst case voltage drops in power and ground buses of CMOS VLSI circuits A survey of power estimation techniques in VLSI circuits Power estimation methods for sequential logic circuits Computing the maximum power cycles of a sequential circuit Extreme delay sensitivity and the worst-case switching activity in VLSI circuits Power estimation in sequential circuits Switching activity analysis using Boolean approximation method Statistical estimation of sequential circuit activity Estimation of maximum transition counts at internal nodes in CMOS VLSI circuits Accurate power estimation of CMOS sequential circuits K2 Genetic Algorithms in Search, Optimization and Machine Learning Maximizing the weighted switching activity in combinational CMOS circuits under the variable delay model --CTR Chia-Chien Weng , Ching-Shang Yang , Shi-Yu Huang, RT-level vector selection for realistic peak power simulation, Proceedings of the 17th great lakes symposium on Great lakes symposium on VLSI, March 11-13, 2007, Stresa-Lago Maggiore, Italy Hratch Mangassarian , Andreas Veneris , Sean Safarpour , Farid N. Najm , Magdy S. Abadir, Maximum circuit activity estimation using pseudo-boolean satisfiability, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France Nicola Nicolici , Bashir M. Al-Hashimi, Scan latch partitioning into multiple scan chains for power minimization in full scan sequential circuits, Proceedings of the conference on Design, automation and test in Europe, p.715-722, March 27-30, 2000, Paris, France V. R. Devanathan , C. P. Ravikumar , V. Kamakoti, Interactive presentation: On power-profiling and pattern generation for power-safe scan tests, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France Michael S. Hsiao, Peak power estimation using genetic spot optimization for large VLSI circuits, Proceedings of the conference on Design, automation and test in Europe, p.38-es, January 1999, Munich, Germany Nicola Nicolici , Bashir M. Al-Hashimi, Multiple Scan Chains for Power Minimization during Test Application in Sequential Circuits, IEEE Transactions on Computers, v.51 n.6, p.721-734, June 2002

n-cycle power;sustainable power;variable delay;peak power;genetic optimization

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Power optimization using divide-and-conquer techniques for minimization of the number of operations.

We develop an approach to minimizing power consumption of portable wireless DSP applications using a set of compilation and architectural techniques. The key technical innovation is a novel divide-and-conquer compilation technique to minimize the number of operations for general DSP computations. Our technique optimizes not only a significantly wider set of computations than the previously published techniques, but also outperforms (or performs at least as well as other techniques) on all examples. Along the architectural dimension, we investigate coordinated impact of compilation techniques on the number of processors which provide optimal trade-off between cost and power. We demonstrate that proper compilation techniques can significantly reduce power with bounded hardware cost. The effectiveness of all techniques and algorithms is documented on numerous real-life designs.

INTRODUCTION 1.1 Motivation The pace of progress in integrated circuits and system design has been dictated by the push from application trends and the pull from technology improvements. The goal and role of designers and design tool developers has been to develop design methodologies, architectures, and synthesis tools which connect changing worlds of applications and technologies. A preliminary version of this paper was presented at the 1997 ACM/IEEE International Conference on Computer-Aided Design, San Jose, California, November 10-13, 1997. Authors' addresses: I. Hong and M. Potkonjak, Computer Science Department, University of Cali- fornia, Los Angeles, CA 90095-1596; R. Karri, Department of Electrical & Computer Engineering, University of Massachusetts, Amherst, MA 01003. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior specific permission and/or a fee. Recently, a new class of portable applications has been forming a new market at an exceptionally high rate. The applications and products of portable wireless market are defined by their intrinsic demand for portability, flexibility, cost sensitivity and by their high digital signal processing (DSP) content [Schneiderman 1994]. Portability translates into the crucial importance of low power design, flexibility results in a need for programmable platforms implementation, and cost sensitivity narrows architectural alternatives to a uniprocessor or an architecture with a limited number of off-the-shelf standard processors. The key optimization degree of freedom for relaxing and satisfying this set of requirements comes from properties of typical portable computations. The computations are mainly linear, but rarely 100 % linear due to either need for adaptive algorithms or nonlinear quantization elements. Such computations are well suited for static compilation and intensive quantitative optimization. Two main recent relevant technological trends are reduced minimal feature size and therefore reduced voltages of deep submicron technologies, and the introduction of ultra low power technologies. In widely dominating digital CMOS technologies, the power consumption is proportional to square of supply voltage (V dd ). The most effective techniques try to reduce V dd while compensating for speed reduction using a variety of architectural and compilation techniques [Singh et al. 1995]. The main limitation of the conventional technologies with respect to power minimization is also related to V dd and threshold voltage (V t ). In traditional bulk silicon technologies both voltages are commonly limited to range above 0.7V. However, in the last few years ultra low power silicon on insulator (SOI) technologies, such as SIMOX (SOI using separation of oxygen), bond and etchback SOI (BESOI) and silicon-on-insulator-with-active-substrate (SOIAS), have reduced both V dd and V t to well below 1V [El-Kareh et al. 1995; Ipposhi et al. 1995]. There are a number of reported ICs which have values for V dd and V t in a range as low as 0.05V - 0.1V [Chandrakasan et al. 1996]. Our goal in this paper is to develop a system of synthesis and compilation methods and tools for realization of portable applications. Technically restated, the primary goal is to develop techniques which efficiently and effectively compile typical DSP wireless applications on single and multiple programmable processors assuming both traditional bulk silicon and the newer SOI technologies. Furthermore, we study achievable power-cost trade-offs when parallelism is traded for power reduction on programmable platforms. 1.2 Design Methodology: What is New? Our design methodology can be briefly described as follows. Given throughput, power consumption and cost requirements for a computation, our goal is to find cost-effective, power-efficient solutions on single or multiple programmable processor platforms. The first step is to find power-efficient solutions for a single processor implementation by applying the new technique described in Section 4. The second step is to continue to add processors until the reduction in average power consumption is not enough to justify the cost of an additional processor. This step generates cost-effective and power efficient solutions. This straightforward design methodology produces implementations with low cost and low power consumption for the given design requirements. The main technical innovation of the research presented in this paper is the first approach for the minimization of the number of operations in arbitrary computa- tions. The approach optimizes not only significantly wider set of computations than the other previously published techniques [Parhi and Messerschmitt 1991; Srivastava and Potkonjak 1996], but also outperforms or performs at least as well as other techniques on all examples. The novel divide-and-conquer compilation procedure combines and coordinates power and enabling effects of several transformations (using a well organized ordering of transformations) to minimize the number of operations in each logical partition. To the best of our knowledge this is the first approach for minimization of the number of operations which in an optimization intensive way treats general computations. The second technical highlight is the quantitative analysis of cost vs power trade-off on multiple programmable processor implementation platforms. We derive a condition under which the optimization of the cost-power product using parallelization is beneficial. 1.3 Paper Organization The rest of the paper is organized in the following way. First, in the next sec- tion, we summarize the relevant background material. In Section 3 we review the related work on power estimation and optimization as well as on program optimization using transformations, and in particular the minimization of the number of operations. Sections 4 and 5 are the technical core of this paper and present a novel approach for minimization of the number of operations for general DSP computations and explore compiler and technology impact on power-cost trade-offs of multiple processors-based low power application specific systems. We then present comprehensive experimental results and their analysis in Section 6 followed by conclusions in Section 7. 2. PRELIMINARIES Before we delve into technical details of the new approach, we outline the relevant preliminaries in this section. In particular, we describe application and computation abstractions, selected implementation platform at the technology and architectural level, and power estimation related background material. 2.1 Computational Model We selected as our computational model synchronous data flow (SDF) [Lee and Messerschmitt 1987; Lee and Parks 1995]. Synchronous data flow (SDF) is a special case of data flow in which the number of data samples produced or consumed by each node on each invocation is specified a priori. Nodes can be scheduled statically at compile time onto programmable processors. We restrict our attention to homogeneous SDF (HSDF), where each node consumes and produces exactly one sample on every execution. The HSDF model is well suited for specification of single task computations in numerous application domains such as digital signal processing, video and image processing, broadband and wireless communications, control, information and coding theory, and multimedia. The syntax of a targeted computation is defined as a hierarchical control-data flow graph (CDFG) [Rabaey et al. 1991]. The CDFG represents the computation as a flow graph, with nodes, data edges, and control edges. The semantics underlying the syntax of the CDFG format, as we already stated, is that of the synchronous data flow computation model. The only relevant speed metric is throughput, the rate at which the implementation is capable of accepting and processing the input samples from two consecutive iterations. We opted for throughput as the selected speed metric since in essentially all DSP and communication wireless computations latency is not a limiting factor, where latency is defined to be the delay between the arrival of a set of input samples and the production of the corresponding output as defined by the specification. 2.2 Hardware Model The basic building block of the targeted hardware platform is a single programmable processor. We assume that all types of operations take one clock cycle for their execution, as it is the case in many modern DSP processors. The adaptation of the software and algorithms to other hardware timing models is straightforward. In the case of a multi-processor, we make the following additional simplifying assumptions: (i) all processors are homogeneous and (ii) inter-processor communication does not cost any time and hardware. This assumption is reasonable because multiple processors can be placed on single integrated circuit due to increased integration, although it would be more realistic to assume additional hardware and delay penalty for using multiple processors. 2.3 Power and Timing Models in Conventional and Ultra Low Power Technology It is well known that there are three principal components of power consumption in CMOS integrated circuits: switching power, short-circuit power, and leakage power. The switching power is given by P switching dd f clock , where ff is the probability that the power consuming switching activity, i.e. transition from 0 to 1, occurs, CL is the loading capacitance, V dd is the supply voltage, and f clock is the system clock frequency. ffC L is defined to be effective switched capacitance. In CMOS technology, switching power dominates power consumption. The short-circuit power consumption occurs when both NMOS and CMOS transistors are "ON" at the same time while the leakage power consumption results from reverse biased diode conduction and subthreshold operation. We assume that effective switched capacitance increases linearly with the number of processors and supply voltage can not be lowered below threshold voltage V t , for which we use several different values between 0.06V and 1.1V for both conventional and ultra low power technology. It is also known that reduced voltage operation comes at the cost of reduced throughput [Chandrakasan et al. 1992]. The clock speed T follows the following is a constant [Chandrakasan et al. 1992]. The maximum rate at which a circuit is clocked monotonically decreases as the voltage is reduced. As the supply voltage is reduced close to V t , the rate of clock speed reduction becomes higher. 2.4 Architecture-level Power Models for Single and Multiple Programmable Processors The power model used in this research is built on three statistically validated and experimentally established facts. The first fact is that the number of operations at the machine code-level is proportional to the number of operations at high-level language [Hoang and Rabaey 1993]. The second fact is that the power consumption in modern programmable processors such as the Fujitsu SPARClite MB86934, a 32-bit RISC microcontroller, is directly proportional to the number of operations, regardless of the mix of operations being executed [Tiwari and Lee 1995]. Tiwari and Lee [1995] report that all the operations including integer ALU instructions, floating point instructions, and load/store instructions with locked caches incur similar power consumption. Since the use of memory operands results in additional power overhead due to the possibility of cache misses, we assume that the cache locking feature is exploited as far as possible. If the cache locking feature can not be used for the target applications, the power consumption by memory traffic is likely to be reduced by the minimization of the number of operations since less operations usually imply less memory traffic. When the power consumption depends on the mix of operations being executed as in the case of the Intel 486DX2 [Tiwari et al. 1994], more detailed hardware power model may be needed. However, it is obvious that in all proposed power models for programmable processors, significant reduction in the number of operations inevitably results in lower power. The final empirical observation is related to power consumption and timing models in digital CMOS circuits presented in the previous subsection. Based on these three observations, we conclude that if the targeted implementation platform is a single programmable CMOS processor, a reduction in the number of operations is the key to power minimization. When the initial number of operations is N init , the optimized number of operations is N opt , the initial voltage is V init and the scaled voltage is V opt , the optimized power consumption relative to the initial power consumption is ( Vopt . For multiprocessors, assuming that there is no communication overhead, the optimized power consumption for n processors relative to that for a single processor is ( Vn Vn are the scaled voltages for single and n processors, respectively. 3. RELATED WORK The related work can be classified along two lines: low power and implementation optimization, and in particular minimization of the number of operations using transformations. The relevant low power topics can be further divided in three directions: power minimization techniques, power estimation techniques, and technologies for ultra low power design. The relevant compilation techniques are also grouped in three directions: transformations, ordering of transformations, and minimization of the number of operations. In the last five years power minimization has been arguably the most popular optimization goal. This is mainly due to the impact of the rapidly growing market for portable computation and communication products. Power minimization efforts across all level of design abstraction process are surveyed in [Singh et al. 1995]. It is apparent that the greatest potential for power reduction is at the highest levels (behavioral and algorithmic). Chandrakasan et al. [1992] demonstrated the effectiveness of transformations by showing an order of magnitude reduction in several DSP computationally intensive examples using a simulated annealing-based transformational script. Raghunathan and Jha [1994] and Goodby et al. [1994] also proposed methods for power minimization which explore trade-offs between voltage scaling, throughput, and power. Chatterjee and Roy [1994] targeted power reduction in fully hardwired designs by minimizing the switching activity. Chandrakasan et. al. [1994], and Tiwari et. al. [1994] did work in power minimization when programmable platforms are targeted. Numerous power modeling techniques have been proposed at all levels of abstraction in the synthesis process. As documented in [Singh et al. 1995] while there have been numerous efforts at the gate level, at the higher level of abstraction relatively few efforts have been reported. Chandrakasan et al. [1995] developed a statistical technique for power estimation from the behavioral level which takes into account all components at the layout level including interconnect. Landman and Rabaey [1996] developed activity-sensitive architectural power analysis approach for execution units in ASIC designs. Finally, in a series of papers it has been established that the power consumption of modern programmable processor is directly proportional to the number of operations, regardless of what the mix of operations being executed is [Lee et al. 1996; Tiwari et al. 1994]. Transformations have been widely used at all levels of abstraction in the synthesis process, e.g. [Dey et al. 1992]. However, there is a strong experimental evidence that they are most effective at the highest levels of abstractions, such as system and in particular behavioral synthesis. Transformations only received widespread attention in high level synthesis [Ku and Micheli 1992; Potkonjak and Rabaey 1992; Walker and Camposano 1991]. Comprehensive reviews of use of transformations in parallelizing compilers, state- of-the-art general purpose computing environments, and VLSI DSP design are given in [Banerjee et al. 1993], [Bacon et al. 1994], and [Parhi 1995] respectively. The approaches for transformation ordering can be classified in seven groups: local (peephole) optimization, static scripts, exhaustive search-based "generate and test" methods, algebraic approaches, probabilistic search techniques, bottleneck removal methods, and enabling-effect based techniques. Probably the most widely used technique for ordering transformations is local (peephole) optimization [Tanenbaum et al. 1982], where a compiler considers only a small section of code at a time in order to apply one by one iteratively and locally all available transformations. The advantages of the approach are that it is fast and simple to implement. However, performance are rarely high, and usually inferior to other approaches. Another popular technique is a static approach to transformations ordering where their order is given a priori, most often in the form of a script [Ullman 1989]. Script development is based on experience of the compiler/synthesis software developer. This method has at least three drawbacks: it is a time consuming process which involves a lot of experimentation on random examples in an ad-hoc manner, any knowledge about the relationship among transformations is only implicitly used, and the quality of the solution is often relatively low for programs/design which have different characteristics than the ones used for the development of the script. The most powerful approach to transformation ordering is enumeration-based "generate and test" [Massalin 1987]. All possible combinations of transformations are considered for a particular compilation and the best one is selected using branch- 7and-bound or dynamic programming algorithms. The drawback is the large run time, often exponential in the number of transformations. Another interesting approach is to use a mathematical theory behind the ordering of some transformations. However, this method is limited to only several linear loop transformations [Wolf and Lam 1991]. Simulated annealing, genetic programming, and other probabilistic techniques in many situations provide a good trade-off between the run time and the quality of solution when little or no information about the topology of the solution space is available. Recently, several probabilistic search techniques have been proposed for ordering of transformations in both compiler and behavioral synthesis literature. For example, backward-propagation-based neural network techniques were used for developing a probabilistic approach to the application of transformations in compilers for parallel computers [Fox and Koller 1989] and approaches which combine both simulated annealing-based probabilistic and local heuristic optimization mechanism were used to demonstrate significant reductions in area and power [Chandrakasan et al. 1995]. In behavioral and logic synthesis several bottleneck identification and elimination approaches for ordering of transformations have been proposed [Dey et al. 1992; Iqbal et al. 1993]. This line of work has been mainly addressing the throughput and latency optimization problems, where the bottlenecks can be easily identified and well quantified. Finally, the idea of enabling and disabling transformations has been recently explored in a number of compilation [Whitfield and Soffa 1990] and high level synthesis papers [Potkonjak and Rabaey 1992; Srivastava and Potkonjak 1996]. Using this idea several very powerful transformations scripts have been developed, such as one for maximally and arbitrarily fast implementation of linear computations [Potkonjak and Rabaey 1992], and joint optimization of latency and throughput for linear computations [Srivastava and Potkonjak 1994]. Also, the enabling mechanism has been used as a basis for several approaches for ordering of transformations for optimization of general computations [Huang and Rabaey 1994]. The key advantage of this class of approaches is related to intrinsic importance and power of enabling/disabling relationship between a pair of transformations. Transformations have been used for optimization of a variety of design and program metrics, such as throughput, latency, area, power, permanent and temporal fault-tolerance, and testability. Interestingly, the power of transformations is most often focused on secondary metrics such as parallelism, instead on the primary metrics such as the number of operations. In compiler domain, constant and copy propagation and common subexpression techniques are often used. It can be easily shown that the constant propagation problem is undecidable, when the computation has conditionals [Kam and Ullman 1977]. The standard procedure to address this problem is to use so called conservative algorithms. Those algorithms do not guarantee that all constants will be detected, but that each data declared constant is indeed constant over all possible executions of the program. A comprehensive survey of the most popular constant propagation algorithms can be found in [Wegman and Zadeck 1991]. Parhi and Messerschmitt [1991] presented optimal unfolding of linear computations in DSP systems. Unfolding results in simultaneous processing of consecutive iterations of a computation. Potkonjak and Rabaey [1992] addressed the minimization of the number of multiplications and additions in linear computations in their maximally fast form so that the throughput is preserved. Potkonjak et al. [1996] presented a set of techniques for minimization of the number of shifts and additions in linear computations. Sheliga and Sha [1994] presented an approach for minimization of the number of multiplications and additions in linear computations. Srivastava and Potkonjak [1996] developed an approach for the minimization of the number of operations in linear computations using unfolding and the application of the maximally fast procedure. A variant of their technique is used in "conquer" phase of our approach. Our approach is different from theirs in two respects. First, their technique can handle only very restricted computations which are linear, while our approach can optimize arbitrary computations. Second, our approach outperforms or performs at least as well as their technique for linear computations. 4. SINGLE PROGRAMMABLE PROCESSOR IMPLEMENTATION: MINIMIZING THE NUMBER OF OPERATIONS The global flow of the approach is presented in subsection 4.1. The strategy is based on divide-and-conquer optimization followed by post optimization step, merging of divided sub parts which is explained in subsection 4.2. Finally, subsection 4.3 provides a comprehensive example to illustrate the strategy. 4.1 Global Flow Of the Approach The core of the approach is presented in the pseudo-code of Figure 1. The rest of this subsection explains the global flow of the approach in more detail. Decompose a computation into strongly connected components(SCCs); Any adjacent trivial SCCs are merged into a sub part; Use pipelining to isolate the sub parts; For each sub part Minimize the number of delays using retiming; If (the sub part is linear) Apply optimal unfolding; Else Apply unfolding after the isolation of nonlinear operations; Merge linear sub parts to further optimize; Schedule merged sub parts to minimize memory usage; Fig. 1. The core of the approach to minimize the number of operations for general DSP computation The first step of the approach is to identify the computation's strongly connected components(SCCs), using the standard depth-first search-based algorithm [Tarjan 1972] which has a low order polynomial-time complexity. For any pair of operations A and B within an SCC, there exist both a path from A to B and a path from B to A. An illustrated example of this step is shown in Figure 2. The graph formed by all the SCCs is acyclic. Thus, the SCCs can be isolated from each other using pipeline delays, which enables us to optimize each SCC separately. The inserted pipeline delays are treated as inputs or outputs to the SCC. As a result, every output and state in an SCC depend only on the inputs and states of the SCC. Addition Constant Multiplication Functional Delay (State) Variable Multiplication Strongly Connected Component Fig. 2. An illustrated example of the SCC decomposition step Thus, in this sense, the SCC is isolated from the rest of the computation and it can be optimized separately. In a number of situations our technique is capable to partition a nonlinear computation into partitions which consist of only linear computations. Consider for example a computation which consists of two strongly connected components SCC 1 and SCC 2 . SCC 1 has as operations only additions and multiplications with constants. SCC 2 has as operations only max operation and additions. Obviously, since the computations has additions, multiplications with constants and max operations, it is nonlinear. However, after applying our technique of logical separation using pipeline states we have two parts which are linear. Note that this isolation is not affected by unfolding. We define an SCC with only one node as a trivial SCC. For trivial SCCs unfolding fails to reduce the number of operations. Thus, any adjacent trivial SCCs are merged together before the isolation step, to reduce the number of pipeline delays used. where X, Y , and S are the input, output, and state vectors respectively and A; B; C; and D are constant coefficient matrices. Fig. 3. State-space equations for linear computations The number of delays in each sub part is minimized using retiming in polynomial time by the Leiserson-Saxe algorithm [Leiserson and Saxe 1991]. Note that smaller number of delays will require smaller number of operations since both the next states and outputs depend on the previous states. SCCs are further classified as either linear or nonlinear. Linear computations can be represented using the (2R\Gamma1)R (2R\Gamma1)R which gives the smaller value of of multiplications for i times unfolded system of additions for i times unfolded system Fig. 4. Closed-form formula of unfolding for dense linear computation with P inputs, Q outputs, and R states. state-space equations in Figure 3. Minimization of the number of operations for linear computations is NP-complete [Sheliga and Sha 1994]. We have adopted an approach of [Srivastava and Potkonjak 1996] for the optimization of linear sub parts, which uses unfolding and the maximally fast procedure [Potkonjak and Rabaey 1992]. We note that instead of maximally fast procedure, the ratio analysis by [Sheliga and Sha 1994] can be used. [Srivastava and Potkonjak 1996] has provided the closed-form formula for the optimal unfolding factor with the assumption of dense linear computations. We provide the formula in Figure 4. For sparse linear computations, they have proposed a heuristic which continues to unfold until there is no improvement. We have made the simple heuristic more efficient with binary search, based on the unimodality property of the number of operations on unfolding factor [Srivastava and Potkonjak 1996]. iteration i iteration i+1 iteration i+2 Fig. 5. An example of isolating nonlinear operations from 2 times unfolded nonlinear sub part When a sub part is classified as nonlinear, we apply unfolding after the isolation of nonlinear operations. All nonlinear operations are isolated from the sub part so that the remaining linear sub parts can be optimized by the maximally fast procedure. All arcs from nonlinear operations to the linear sub parts are considered as inputs to the linear sub parts, and all arcs from linear sub parts to the nonlinear operations are considered as outputs from the linear sub parts. The process is illustrated in Figure 5. All arcs denoted by i are considered to be inputs and all arcs denoted by are considered to be outputs for unfolded linear sub part. We observe that if every output and state of the nonlinear sub part depend on nonlinear operations, then unfolding with the separation of nonlinear operations is ineffective in reducing the number of operations. Fig. 6. A motivational example for sub part merging Sometimes it is beneficial to decompose a computation into larger sub parts than SCCs. We consider an example given in Figure 6. Each node represents a sub part of the computation. We make the following assumptions only specifically for clarifying the presentation of this example and simplifying the example. We stress here that the assumptions are not necessary for our approach. Assume that each sub part is linear and can be represented by state-space equations in Figure 3. Also assume that every sub part is dense, which means that every output and state in a sub part are linear combinations of all inputs and states in the sub part with no 0, 1, or -1 coefficients. The number inside a node is the number of delays or states in the sub part. Assume that when there is an arc from a sub part X to a sub part Y, every output and state of Y depends on all inputs and states of X. Separately optimizing SCCs P 1 and P 2 in Figure 6 costs 211 operations from the formula in Figure 4. On the otherhand, optimizing the entire computation entails only 63.67 operations. The reason why separate optimization does not perform well in this example is because there are too many intermediate outputs from SCC P 1 to This observation leads us to an approach of merging sub parts for further reducing the number of operations. Since it is worthwhile to explain the sub part merging problem in detail, the next subsection is devoted to the explanation of the problem and our heuristic approaches. Since the sub parts of a computation are unfolded separately by different unfolding factors, we need to address the problem of scheduling the sub parts. They should be scheduled so that memory requirements for code and data are minimized. We observe that the unfolded sub parts can be represented by multi-rate synchronous dataflow graph [Lee and Messerschmitt 1987] and the work of [Bhattacharyya et al. 1993] can be directly used. Note that the approach is in particular useful for such architectures that require high locality and regularity in computation because it improves both locality and regularity of computation by decomposing into sub parts and using the maximally fast procedure. Locality in a computation relates to the degree to which a computation has natural clusters of operations while regularity in a computation refers to the repeated occurrence of the computational patterns such as a multiplication followed by an addition [Guerra et al. 1994; Mehra and Rabaey 1996]. 4.2 Subpart Merging Initially, we only consider merging of linear SCCs. When two SCCs are merged, the resulting sub part does not form an SCC. Thus, in general, we must consider merging of any adjacent arbitrary sub parts. Suppose we consider merging of sub parts i and j. The gain GAIN(i; j) of merging sub parts i and j can be computed as follows: is the number of operations for sub part i and COST (i; j) is the number of operations for the merged sub part of i and j. To compute the gain, COST (i; j) must be computed, which requires constant coefficient matrices A; B; C; and D for only the merged sub part of i and j. It is easy to construct the matrices using the depth-first search [Tarjan 1972]. Fig. 7. i times unfolded state-space equations \Gamma 1c, or d which gives the smaller value of i opt ( states in state group j outputs in output group j inputs that output group j depends on inputs that state group j depends on states that output group j depends on states that state group j depends on Fig. 8. Closed-form formula for unfolding; If two outputs depend on the same set of inputs and states, they are in the same group, and the same is true for states. The i times unfolded system can be represented by the state-space equations in Figure 7. From the equations, the total number of operations can be computed for i times unfolded sub part as follows. Let denote the number of multiplications and the number of additions for i times unfolded system, respectively. The resulting number of operations is N( ;i)+N(+;i) because i times unfolded system uses a batch of samples to generate a batch of output samples. We continue to unfold until no improvement is achieved. If there are no coefficients of 1 or \Gamma1 in the matrices A, B, C, and D, then the closed-form formula for the optimal unfolding factor i opt and for the number of operations for times unfolded system are provided in Figure 8. While (there is improvement) For all possible merging candidates, Compute the gain; Merge the pair with the highest gain; Fig. 9. Pseudo-code of a greedy heuristic for sub part merging Generate a starting solution S. Set the best solution S Determine a starting temperature T . While not yet frozen, While not yet at equilibrium for the current temperature, Choose a random neighbor S 0 of the current solution. Else Generate a random number r uniformly from [0, 1]. Update the temperature T . Return the best solution S . Fig. 10. Pseudo-code for simulated annealing algorithm for sub part merging Now, we can evaluate possible merging candidates. We propose two heuristic algorithms for sub part merging. The first heuristic is based on greedy optimization approach. The pseudo-code is provided in Figure 9. The algorithm is simple. Until there is no improvement, merge the pair of sub parts which produces the highest gain. The other heuristic algorithm is based on a general combinatorial optimization technique known as simulated annealing [Kirkpatrick et al. 1983]. The pseudo-code is provided in Figure 10. The actual implementation details are presented for each of the following areas: the cost function, the neighbor solution generation, the temperature update function, the equilibrium criterion and the frozen criterion. Firstly, the number of operations for the entire given computation has been used as the cost function. Secondly, the neighbor solution is generated by the merging of two adjacent sub parts. Thirdly, the temperature is updated by the function old . For the temperature T ? 200:0, ff is chosen to be 0.1 so that in high temperature regime where every new state has very high chance of acceptance, the temperature reduction occurs very rapidly. For is set to 0.95 so that the optimization process explores this promising region more slowly. For T 1:0, ff is set to 0.8 so that T is quickly reduced to converge to a local minimum. The initial temperature is set to 4,000,000. Fourthly, the equilibrium criterion is specified by the number of iterations of the inner loop. The number of Fig. 11. An explanatory example iterations of the inner loop is set to 20 times of the number of sub parts. Lastly, the frozen criterion is given by the temperature. If the temperature falls below 0.1, the simulated annealing algorithm stops. Both heuristics performed equally well on all the examples and the run times for both are very small because the examples have a few sub parts. We have used both greedy and simulated annealing based heuristics for generating experimental results and they produced exactly the same results. Fig. 12. A simple example of # operations calculation 4.3 Explanatory Example: Putting It All Together We illustrate the key ideas of our approach for minimizing the number of operations by considering the computation of Figure 11. We use the same assumptions made for the example in Figure 6. The number of operations per input sample is initially 2081 (We illustrate how the number of operations is calculated in a maximally fast way [Potkonjak and Rabaey 1992] using a simple linear computation with 1 input X , 1 output Y , and which is described in Figure 12). Using the technique of [Srivastava and Potkonjak 1996] which unfolds the entire computation, the number can be reduced to 725 with an unfolding factor of 12. Our approach optimizes each sub part separately. This separate optimization is enabled by isolating the sub parts using pipeline delays. Figure 13 shows the computation after the isolation step. Since every sub part is linear, unfolding is performed to optimize the number of operations for each sub part. The sub parts cost 120.75, 53.91, 114.86, 129.75, and 103.0 operations per input sample with unfolding factors 3, 10, 6, 7, and 2, respectively. The total number of operations per input sample for the entire computation is 522.27. We now apply SCC merging to further reduce the number of operations. We first consider the greedy heuristic. The heuristic Fig. 13. A motivational example after the isolation step considers merging of adjacent sub parts. Initially, the possible merging candidate are which produce the gains of -51.48, - 112.06, -52.38, 122.87, and -114.92, respectively. SCC P 3 and SCC P 4 are merged with an unfolding factor of 22. In the next iteration, there are now 4 sub parts and 4 candidate pairs for merging all of which yield negative gains. So, the heuristic stops at this point. The total number of operations per input sample has further decreased to 399.4. Simulated annealing heuristic produced the same solution for this example. The approach has reduced the number of operations by a factor of 1.82 from the previous technique of [Srivastava and Potkonjak 1996], while it has achieved the reduction by a factor of 5.2 from the initial number of operations. For single processor implementation, since both the technique of [Srivastava and Potkonjak 1996] and our new method yield higher throughput than the original, the supply voltage can be lowered up to the extent that the extra throughput is compensated by the loss in circuit speed due to reduced voltage. If the initial voltage is 3.3V, then our technique reduces power consumption by a factor of 26.0 with the supply voltage of 1.48V while the technique of [Srivastava and Potkonjak 1996] reduces it by a factor of 10.0 with the supply voltage of 1.77V. The scheduling of the unfolded sub parts is performed to generate the minimum code and data memory schedule. The schedule period is the least common multiple of (the unfolding factor+1)'s which is 3036. Let P 3;4 denote the merged sub part of While a simple minded schedule (759P 1 , 276P 2 , 132P 3;4 , 1012P 5 ) to minimize the code size ignoring loop overheads generates 9108 units of data memory requirement, a schedule (759P 1 , 4(69P 2 , 33P 3;4 , 253P 5 which minimizes the data memory requirement among the schedules minimizing the code size generates 4554 units of data memory requirement. 5. MULTIPLE PROGRAMMABLE PROCESSORS IMPLEMENTATION When multiple programmable processors are used, potentially more savings in power consumption can be obtained. We summarize the assumptions made in Section 2: (i) processors are homogeneous, (ii) inter-processor communication does not cost any time and hardware, (iii) effective switched capacitance increases linearly with the number of processors, (iv) both addition and multiplication take one clock cycle, and (v) supply voltage can not be lowered below threshold voltage V t , for which we use several different values between 0.06V and 1.1V. Based on these assumptions, using k processors increases the throughput k times when there is enough parallelism in the computation, while the effective switched capacitance increases k times as well. In all the real-life examples considered, sufficient parallelism actually existed for the numbers of processors that we used. R )e , if k R 2 Fig. 14. Closed-form condition for sufficient parallelism when using k processors for a dense linear computation with R states We observe that the next states, i.e., the feedback loops can be computed in parallel. Note that the maximally fast procedure by [Potkonjak and Rabaey 1992] evaluates a linear computation by first doing the constant-variable multiplications in parallel, and then organizing the additions as a maximally balanced binary tree. Since all the next states are computed in a maximally fast procedure, in the bottom of the binary computation tree there exists more paral- lelism. All other operations not in the feedback loops can be computed in parallel because they can be separated by pipeline delays. As the number of processors becomes larger, the number of operations outside the feedback loops gets larger to result in more parallelism. For dense linear computations, we provide the closed-form condition for sufficient parallelism when using k processors in Figure 14. We note that although the formulae were derived for the worst case scenario, the required number of operations outside the feedback loops is small for the range of the number of processors that we have tried in the experiment. There exist more operations outside feedback loops than are required for full parallelism in all the real-life examples we have considered. Now one can reduce the voltage so that the clock frequency of all k processors is reduced by a factor of k. The average power consumption of k processors is reduced from that of a single processor by a factor of ( V1 is a scaled supply voltage for k processor implementation, and V k satisfies the equation et al. 1992]. From this observation it is always beneficial to use more processors in terms of power consumption with the following two limitations: (i) the amount of parallelism available limits the improvement in throughput and the critical path of the computation is the maximum achievable throughput and (ii) when supply voltage approaches close to threshold voltage, the improvement in power consumption becomes so small that the cost of adding a processor is not justified. With this in mind, we want to find the number of processors which minimizes power consumption cost-effectively in both standard CMOS technology and ultra low power technology. Since the cost of programmable processors is high, and especially the cost of processors on ultra low power platforms such as SOI is very high [El-Kareh et al. 1995; Ipposhi et al. 1995], the guidances for cost-effective design are important. We need a measure to differentiate between cost-effective and cost-ineffective solutions. We propose a PN product, where P is the power consumption normalized to that of optimized single processor implementation and N is the number of processors number of processors 1.1 5.0 5.0 0.90 0.93 0.98 1.04 1.10 1.17 1.24 1.30 1.37 4.0 1.00 1.08 1.17 1.28 1.38 1.49 1.59 1.70 1.80 3.0 1.14 1.33 1.51 1.70 1.88 2.06 2.24 2.42 2.60 2.0 1.42 1.82 2.22 2.60 2.98 3.35 3.71 4.08 4.44 0.7 3.3 3.3 0.89 0.91 0.95 1.00 1.06 1.12 1.19 1.25 1.31 2.0 1.12 1.28 1.45 1.62 1.79 1.96 2.12 2.28 2.45 1.0 1.63 2.22 2.80 3.38 3.94 4.50 5.05 5.60 6.15 0.3 1.3 1.3 0.92 0.96 1.02 1.08 1.16 1.23 1.30 1.38 1.45 0.7 1.24 1.49 1.75 2.00 2.24 2.48 2.72 2.95 3.19 Table I. The values of PN products with respect to the number of processors for various combinations of the initial voltage V init , the scaled voltage for single processor V 1 , and the threshold used. The smaller the PN product is the more cost-effective the solution is. If PN is smaller than 1.0, using N processors has decreased the power consumption by a factor of more than N. It depends on the power consumption requirement and the cost budget for the implementation how many processors the implementation should use. Table I provides the values of PN products with respect to the number of processors used for various combinations of the initial voltage V init , the scaled voltage for single processor V 1 , and the threshold voltage V t . V init is the initial voltage for the implementation before optimization. We note that PN products monotonically increase with respect to the number of processors. Init. New IF From RP From IF From RP From Design Ops [Sri96] Method [Sri96] [Sri96] Init. Ops Init. Ops dist 48 47.3 36.4 1.30 23.0 1.32 24.2 chemical modem 213 213 148.83 1.43 30.1 1.43 30.1 GE controller 180 180 105.26 1.71 41.5 1.71 41.5 APCM receiver 2238 N/A 1444.19 N/A N/A 1.55 35.4 Audio Filter 1 154 N/A 76.0 N/A N/A 2.03 50.7 Audio Filter 2 228 N/A 92.0 N/A N/A 2.48 59.7 Filter 1 296 N/A 157.14 N/A N/A 1.88 46.8 Filter 2 398 N/A 184.5 N/A N/A 2.16 53.7 Table II. Minimizing the number of operations for real-life examples; IF - Improvement Factor, Reduction Percentage, N/A - Not Applicable, [Sri96] - [Srivastava and Potkonjak 1996] From the Table I, we observe that cost effective solutions usually use a few processors in all the cases considered on both the standard CMOS and ultra low power platforms. We also observe that if the voltage reduction is high for single Vnew PRF dist 5.0 1.1 3.76 2.33 3.3 0.7 2.70 1.96 1.3 0.3 1.10 1.84 chemical 5.0 1.1 3.61 2.65 3.3 0.7 2.61 2.21 1.3 0.3 1.07 2.04 DAC 5.0 1.1 3.81 2.72 3.3 0.7 2.50 2.75 1.3 0.3 1.00 2.69 modem 5.0 1.1 4.02 2.21 3.3 0.7 2.65 2.23 1.3 0.3 1.05 2.19 GE controller 5.0 1.1 3.65 3.21 3.3 0.7 2.39 3.25 1.3 0.3 0.96 3.16 Table III. Minimizing power consumption on single programmable processor for linear examples; Reduction Factor processor case, then there is not much room to further reduce power consumption by using more processors. Based on those observations, we have developed our strategy for the multiple processor implementation. The first step is to minimize power consumption for single processor implementation using the proposed technique in Section 4. The second step is to increase the number of processors until the PN product is below the given maximum value. The maximum value is determined based on the power consumption requirement and the cost budget for the implementation. The strategy produces solutions with only a few processors, in many cases single processor for all the real-life examples because our method for the minimization of the number of operations significantly reduces the number of operations and in turn the supply voltage for the single processor implementation, adding more processors does not usually reduce power consumption cost-effectively. Our method achieves cost-effective solutions with very low power penalty compared to the solutions which only optimize power consumption without considering hardware cost. 6. EXPERIMENTAL RESULTS Our set of benchmark designs include all the benchmark examples used in [Sri- vastava and Potkonjak 1996] as well as the following typical portable DSP, video, communication, and control applications: DAC - 4 stage NEC digital to analog converter (DAC) for audio signals; modem - 2 stage NEC modem; GE controller - 5-state GE linear controller; APCM receiver - Motorola's adaptive pulse code Vnew PRF APCM receiver 5.0 1.1 3.85 2.62 3.3 0.7 2.53 2.64 1.3 0.3 1.01 2.58 Audio Filter 1 5.0 1.1 3.34 4.54 3.3 0.7 2.03 4.43 1.3 0.3 0.88 4.45 Audio Filter 2 5.0 1.1 3.03 6.76 3.3 0.7 1.85 6.54 1.3 0.3 0.8 6.58 Filter 1 5.0 1.1 3.45 3.97 3.3 0.7 2.26 4.03 1.3 0.3 0.91 3.90 Filter 2 5.0 1.1 3.24 5.15 3.3 0.7 2.12 5.24 1.3 0.3 0.85 5.04 VSTOL 5.0 1.1 3.43 4.10 3.3 0.7 2.25 4.15 1.3 0.3 0.90 4.02 Table IV. Minimizing power consumption on single programmable processor for nonlinear ex- amples; PRF - Power Reduction Factor modulation receiver; Audio Filter 1 - analog to digital converter (ADC) followed by 14 order cascade IIR filter; Audio Filter 2 - ADC followed by two ADCs followed by 10-order two dimensional (2D) IIR two ADCs followed by 12-order 2D IIR filter; and VSTOL - VSTOL robust observer structure aircraft speed controller. DAC, modem, and GE controller are linear computations and the rest are nonlinear computations. The benchmark examples from [Srivastava and Potkonjak 1996] are all linear, which include ellip, iir5, wdf5, iir6, iir10, iir12, steam, dist, and chemical. Table II presents the experimental results of our technique for minimizing the number of operations for real-life examples. The fifth and seventh columns of Table II provide the improvement factors of our method from that of [Srivastava and Potkonjak 1996] and from the initial number of operations, respectively. Our method has achieved the same number of operations as that of [Srivastava and Potkonjak 1996] for ellip, iir5, wdf5, iir6, iir10, iir12, and steam while it has reduced the number of operations by 23 and 10.3 % for dist and chemical, respectively. All the examples from [Srivastava and Potkonjak 1996] are single-input single-output linear computations, except dist and chemical which are two-inputs single-output linear computations. Since the SISO linear computations are very small, Vnew PRF N PRF N PRF N PRF dist 5.0 1.1 3.76 2.33 1 2.33 4 7.53 6 9.47 3.3 0.7 2.70 1.96 2 4.04 5 8.11 8 10.54 1.3 0.3 1.10 1.84 2 3.71 4 6.31 7 8.74 chemical 5.0 1.1 3.61 2.65 1 2.65 3 6.87 5 9.39 3.3 0.7 2.61 2.21 2 4.49 5 8.86 7 10.69 1.3 0.3 1.07 2.04 1 2.04 4 6.83 6 8.67 3.3 0.7 2.50 2.75 1 2.75 4 9.22 6 11.71 1.3 0.3 1.00 2.69 1 2.69 3 7.05 5 9.67 modem 5.0 1.1 4.02 2.21 2 4.45 4 7.56 7 10.45 3.3 0.7 2.65 2.23 2 4.56 5 9.07 7 10.97 1.3 0.3 1.05 2.19 1 2.19 4 7.22 6 9.13 GE 5.0 1.1 3.65 3.21 1 3.21 3 8.39 5 11.49 controller 3.3 0.7 2.39 3.25 1 3.25 4 10.49 6 13.20 1.3 0.3 0.96 3.16 1 3.16 3 8.05 5 10.92 Table V. Minimizing power consumption on multiple processors for linear examples; PN T - threshold PN product, N - # of processors, PRF - Power Reduction Factor there is no room for improvement from [Srivastava and Potkonjak 1996]. Our method has reduced the number of operations by an average factor of 1.77 (average 43.5 %) for the examples that previous techniques are either ineffective or inap- plicable. Tables III and IV present the experimental results of our technique for minimizing power consumption on single programmable processor of real-life examples on various technologies. Our method results in power consumption reduction by an average factor of 3.58. For multiple processor implementations, Tables V and VI summarize the experimental results of our technique for minimizing power consumption. We define threshold PN product PN T to be the value of PN product, where we should stop increasing the number of processors. When PN i.e., the power reduction by the addition of a processor must be greater than 2 to be cost effective, in almost all cases single processor solution is optimum. When PN T gets larger, the number of processors used increases, but the solutions still use only a few processors which result in an order of magnitude reduction in power consumption. All the results clearly indicate the effectiveness of our new method. 7. CONCLUSION We introduced an approach for power minimization using a set of compilation and architectural techniques. The key technical innovation is a compilation technique for minimization of the number of operations which synergistically uses several Vnew PRF N PRF N PRF N PRF APCM 5.0 1.1 3.85 2.62 1 2.62 4 8.64 6 10.92 receiver 3.3 0.7 2.53 2.64 2 5.29 4 8.95 7 12.33 1.3 0.3 1.01 2.58 1 2.58 3 6.81 6 10.32 Audio 5.0 1.1 3.34 4.54 1 4.54 3 11.12 4 13.22 Filter 1 3.3 0.7 2.03 4.43 1 4.43 2 7.99 4 12.38 1.3 0.3 0.88 4.45 1 4.45 2 8.07 4 12.56 Audio 5.0 1.1 3.03 6.76 1 6.76 2 11.88 4 18.04 Filter 2 3.3 0.7 1.85 6.54 1 6.54 2 11.26 3 14.45 1.3 0.3 0.8 6.58 1 6.58 2 11.38 3 14.64 Filter 1 3.3 0.7 2.26 4.03 1 4.03 3 10.32 5 14.04 1.3 0.3 0.91 3.90 1 3.90 3 9.55 4 11.34 Filter 2 3.3 0.7 2.12 5.24 1 5.24 3 12.82 4 15.22 1.3 0.3 0.85 5.04 1 5.04 2 8.99 4 13.79 3.3 0.7 2.25 4.15 1 4.15 3 10.60 5 14.40 1.3 0.3 0.90 4.02 1 4.02 3 9.76 4 11.58 Table VI. Minimizing power consumption on multiple processors for nonlinear examples; PN T threshold PN product, N - # of processors, PRF - Power Reduction Factor transformations within a divide and conquer optimization framework. The new approach not only deals with arbitrary computations, but also outperforms previous techniques for limited computation types. Furthermore, we investigated coordinated impact of compilation techniques and new ultra low power technologies on the number processors which provide optimal trade-off of cost and power. The experimental results on a number of real-life designs clearly indicates the effectiveness of all proposed techniques and algorithms. --R Compiler transformations for high performance computing. Automatic program parallelization. A scheduling framework for minimizing memory requirements of multirate signal processing algorithms expressed as dataflow graphs. Optimizing power using transformations. Energy efficient programmable computation. Design considerations and tools for low-voltage digital system design Synthesis of low power DSP circuits using activity metrics. Performance optimization of sequential circuits by eliminating retiming bottlenecks. Silicon on insulator - an emerging high-leverage technology Code generation by a generalized neural network. Microarchitectural synthesis of performance-constrained Scheduling of DSP programs onto multiprocessors for maximum throughput. Maximizing the throughput of high performance DSP applications using behavioral transformations. An advanced 0.5 mu m CMOS/SOI technology for practical ultrahigh-speed and low-power circuits Critical path minimization using retiming and algebraic speedup. Monotone data flow analysis frameworks. Optimization by simulated annealing. Synchronous dataflow. Dataflow process networks. Power analysis and minimization techniques for embedded dsp software. Retiming synchronous circuitry. A look at the smallest program. Exploiting regularity for low-power design Journal of VLSI Signal Processing Static rate-optimal scheduling of iterative data-flow programs via optimum unfolding Maximally fast and arbitrarily fast implementation of linear computations. Multiple constant multi- plications: efficient and versatile framework and algorithms for exploring common subexpression elimination Fast prototyping of data path intensive architectures. Behavioral synthesis for low power. Personal Communications. Global node reduction of linear systems using ratio analysis. Power conscious cad tools and methodologies: A perspective. Transforming linear systems for joint latency and throughput optimization. Power optimization in programmable processors and ASIC implementations of linear systems: Transformation-based approach Using peephole optimization on intermediate code. Depth first search and linear graph algorithms. Power analysis of a 32-bit embedded microcontroller In Asia and South Pacific Design Automation Conference Power analysis of embedded software: a first step towards software power minimization. Database and Knowledge-Base Systems A Survey of High-level Synthesis Systems ACM Transactions on Programming Languages An approach to ordering optimizing transformations. In ACM Symposium on Principles and Practice of Parallel Programming A loop transformation theory and an algorithm to maximize parallelism. --TR Static Rate-Optimal Scheduling of Iterative Data-Flow Programs Via Optimum Unfolding Power analysis of embedded software Power optimization in programmable processors and ASIC implementations of linear systems Global node reduction of linear systems using ratio analysis Maximally fast and arbitrarily fast implementation of linear computations Fast Prototyping of Datapath-Intensive Architectures --CTR Johnson Kin , Chunho Lee , William H. Mangione-Smith , Miodrag Potkonjak, Power efficient mediaprocessors: design space exploration, Proceedings of the 36th ACM/IEEE conference on Design automation, p.321-326, June 21-25, 1999, New Orleans, Louisiana, United States Luca Benini , Giovanni De Micheli, System-level power optimization: techniques and tools, Proceedings of the 1999 international symposium on Low power electronics and design, p.288-293, August 16-17, 1999, San Diego, California, United States Luca Benini , Giovanni de Micheli, System-level power optimization: techniques and tools, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.5 n.2, p.115-192, April 2000

power consumption;data flow graphs;portable wireless DSP applications;architectural techniques;DSP computations;compilation;divide-and-conquer compilation

266458

Approximate timing analysis of combinational circuits under the XBD0 model.

This paper is concerned with approximate delay computation algorithms for combinational circuits. As a result of intensive research in the early 90's efficient tools exist which can analyze circuits of thousands of gates in a few minutes or even in seconds for many cases. However, the computation time of these tools is not so predictable since the internal engine of the analysis is either a SAT solver or a modified ATPG algorithm, both of which are just heuristic algorithms for an NP-complete problem. Although they are highly tuned for CAD applications, there exists a class of problem instances which exhibits the worst-case exponential CPU time behavior. In the context of timing analysis, circuits with a high amount of reconvergence, e.g. C6288 of the ISCAS benchmark suite, are known to be difficult to analyze under sophisticated delay models even with state-of-the-art techniques. For example [McGeer93] could not complete the analysis of C6288 under the mapped delay model. To make timing analysis of such corner case circuits feasible we propose an approximate computation scheme to the timing analysis problem as an extension to the exact analysis method proposed in [McGeer93]. Sensitization conditions are conservatively approximated in a selective fashion so that the size of SAT problems solved during analysis is controlled. Experimental results show that the approximation technique is effective in reducing the total analysis time without losing accuracy for the case where the exact approach takes much time or cannot complete.

Introduction During late 80's and early 90's significant progress [2, 8] was made in the theory of exact gate-level timing analysis. In this, false paths are correctly identified so that exact delays can be computed. As the theory progressed, the efficiency and size limitation of actual implementations of timing analysis tools were dramatically improved [3, 8]. Although state-of-the-art implementations can handle circuits composed of thousands of gates under mapped delay models, it is evident that the current size limitation is far from satisfactory for analyzing industrial-strength circuits. Furthermore, even if they can handle large circuits, the computation time is often prohibitively large especially when delay models are elaborate. To alleviate this problem several researchers have proposed approximate timing analysis algorithms. The goal is to compute a conservative yet accurate enough approximation of true delays in less computation time to make analysis of large circuits tractable. Huang et al. [4, 6] proposed, as part of optimization techniques used in exact analysis, a simple approximation heuristic, in which a complex timed Boolean calculus expression at an internal node is simplified to a new independent variable arriving at the latest This work was supported by SRC-97-DC-324. time referred to in the original expression. This simplification is applied only when the number of terms in the Boolean calculus expression exceeds a certain limit, to control the computational complexity. Accuracy loss comes from the fact that the original functional relationship is completely lost by the substitution. They also investigated a more powerful approximation technique in [5], in which each timed Boolean calculus formula is under- and over- approximated by sum of literals and products of literals respectively so that each sensitizability check, which is a satisfiability problem in the exact analysis, can be performed conservatively in polynomial time. Since this approximation is fairly aggressive to guarantee the polynomial time complexity, estimated delays do not seem accurate enough to be useful. Unfortunately their results, shown in [5], are not clear about the accuracy of approximate delays. They merely showed ratios of internal nodeswhose delays match the exact delays at the nodes. No result was shown on the accuracy of circuit delays. More recently Yalcin et al. [11] proposed an approximation technique, which utilizes user's knowledge about primary inputs. They categorize each primary input either as data or control and label all the internal nodes either data or control using a certain rule. The sensitization condition at each node is then simplified conservatively so that it becomes independent of the data variables. The intuition behind this is that the delay of a circuit is most likely determined by control signals while data signals have only minor effects in the final delay. [11] shows experimentally that a dramatic speed-up is possible without losing much accuracy for unit-delay timing analysis based on static sensitization. Unfortunately this sensitization criterion is known to underapproximate true delays, i.e. it is not a safe criterion, which defeats the whole purpose of timing analysis. More recently they confirmed that a similar speed-up and accuracy can be achieved for a correct sensitization criterion (the floating mode) under the unit-delay model [9]. Although an application of the same technique to more sophisticated delay models is theoretically possible, it is not clear whether their algorithm can handle large circuits under those delay models. Moreover, their CPU times for exact analysis are much worse than state-of-the- art implementations available, which cancels some of the speed-up since their speed-up is reported relative to this slower algorithm 1 . In this paper we apply their idea of using data/control separation to a state-of-the-art timing analysis technique [8] to design an approximate algorithm. The sensitization criterion here is the XBD0 model [8], which is one of the well-accepted delay models shown to be correct and accurate. In addition a novel technique to control the complexity of the analysis is proposed. The combination of these two ideas leads to a new approximation scheme, which for 1 One of the reasons why their exact algorithm is slower is that they try to represent in BDD all the input minterms that activate the longest sensitizable delay while most of the state-of-the-art techniques determine the delay without representing those input minterms explicitly. some extreme cases shows a speed-up of 70x, while maintaining accuracy within the noise range. This paper is organized as follows. Section 2 summarizes false path analysis, which forms a basis of this work. We specially focus on the technique proposed in [8]. Section 3 proposes two approximation schemes and discusses how they can be selectively applied to trade off accuracy and speed-up. Experimental results are given in Section 4. Section 5 concludes the paper. Preliminaries In this section, we review sensitization theory for the false path problem. Specifically, the theory developed in [8] is detailed below since the analysis following this section is heavily based on this particular theory. 2.1 Functional Delay Analysis Functional delay analysis, or false path analysis, seeksto determine when all the primary output signals of a Boolean network become stable at their final values given maximum delays of each gate and arrival times at the primary inputs. Since some paths may never be sensitized, the stable time computed by functional delay analysis can be earlier than the time computed by topological delay analysis, thereby capturing the timing characteristic of the network more accurately. Those paths along which signals never propagate are called false paths. The extended bounded delay-0 model [8], the XBD0 model, is the delay model most commonly used in false path analysis. It is the underlying model for the floating mode analysis [1] and viability analysis [7]. Under the XBD0 model, each gate in a network has a maximum positive delay and a minimum delay which is zero. Sensitization analysis is done under the assumption that each gate can take any delay between its maximum value and zero. The core idea of [8] is to characterize recursively the set of all input vectors that make the signal value of a primary output stable to a constant by a given required time. Once these sets are identified both for constants 0 and 1, one can compare these against the on-set and the off-set of the primary output respectively to see if the output is indeed stable for all input vectors by the required time. The overall scenario of computing true delay is to start by setting the required time to the longest topological delay minus gradually decrease it until some input vector cannotmake the output stable by the required time. The next to the last required time gives an approximation to the true arrival time at the output. This process of guessing the next required time can be sped up and refined by making use of a binary search. Let us illustrate how we can compute these sets. Let n and dn be a node (gate) in a Boolean network N and the maximum delay of the node n respectively 2 . Let - t n;v be the characteristic function of the set of input minterms under which the output of the node becomes stable to a constant v 2 f0; 1g by time t. Let fn be the local functionality of the node n in terms of immediate fanins of n. For ease of explanation, let is a two-input AND gate. It is clear from the functionality of the AND gate that to set n to a constant 1 by time t, both of the fanins of are required to be stable at 1 by time This is equivalent to Note that the two - functions for the fanins are AND'ed to take the intersection of the two sets. Similarly, to set n to a constant 0 2 It is possible to differentiate rise delays from fall delays. In this paper, however, we do not distinguish between them to simplify exposition. by time t, at least one of the fanins must be stabilized to 0 by time Here the two - functions are OR'ed to take the union of the two conditions. It is easy to see that the above computations can be generalized to the case where the local functionality of n is given as an arbitrary function in terms of its fanins as follows. Y Y n are the sets of all primes of fn and fn respec- tively. One can easily verify that the recursive formulations for the AND gate shown above are captured in this general formulation by noticing . The terminal cases are given when the node n is a primary input x. where arr(x) denotes the arrival time of x. The above formulas simply say that a primary input is stable only after its given arrival time. The key observation of this formulation is that characteristic functions can be computed recursively. Once characteristic functions for constants 0 and 1 are computed at a primary output, two comparisons are made: one for the characteristic function for 1 against the on-set of the output, and the other for the characteristic function for 0 against the off-set of the output. Each comparison is done by creating a Boolean network which computes the difference between two functions and using a SAT solver to checkwhether the output of the network is satisfiable. The Boolean network is called a -network. 2.2 Optimal Construction of -Networks To argue the approximation algorithms presented in this paper, further details on the construction of -networks need to beunderstood. We have mentioned that a -network is constructed recursively from a primary output. In [8] further optimization to reduce the size of -networks is discussed. Given a required time at a primary output, assume that a backward required-time propagation of N is done to primary inputs so that the list of all required times at each internal node is computed. The propagation is done so that all the potential required times are computed at each node instead of the earliest required time. If the -network is constructed naively, for each internal node in N , a distinct node is to be created for each required time in the list. This, however, is not necessary since it is possible that different required times exhibit the same stability behavior, in which case having a single node in the -network for the required times is enough. To detect such a case a forward arrival-time propagation from primary inputs to primary outputs is performed to compute the list of all potential arrival times at each node. Note that each potential arrival time corresponds to the topological delay of a path from a primary input to the internal node. Therefore the stability of the node can only change at those times. In other words between two adjacent potential arrival times, one cannot see any change in the stability. Consider an internal node n 2 N . Let and the sorted list of required times and that of arrival times respectively at node n. Consider - function A be the maximum arrival time such that a . Since there is no event happeningbetween time a j and r i , n;v . Matchings from required times to arrival times are performed in this fashion to identify the subset of A that is required to compute the final - function. This optimization avoids creating redundant nodes in the - network thereby reducing the size of the - network without losing any accuracy in analysis. Only those arrival times which have a match with required times yield nodes in the - network. Another type of optimization suggested in [8] is to generate the list of arrival times more carefully. For each potential arrival time, equivalence between the corresponding - function and the on-set or the off-set (whichever suitable) is checked by a satisfiability call and a new node is created in - network only if the two functions are different. Otherwise, the original function or its complement is used as it is. Although this requires additional CPU time spent on satisfiability calls, it is experimentally confirmed that the size reduction of the final - network is so significant that the the total run-time decreases in most cases. 3 Approximation Algorithms 3.1 Limitation of the Exact Algorithm Although the exact algorithm proposed in [8] can handle many circuits of thousands of gates, it still has a size limitation. If a large network is given and timing analysis is requested under a detailed delay model like the technology mapped delay model, it is likely that the algorithm runs practically forever 3 . Even if timing analysis is tractable, the computation time can be too large to be practical. As seen in the previous section, the exact timing analysis consists of repeated SAT solver calls. More precisely, for each time tested at a primary output, a -network is constructed such that the network computes the difference between the on-set (off-set) of the primary output and the set of input vectors which make the primary output stable to value 1 (0) by the given time. If the output never becomes 1 for any input assignment, i.e. it is not satisfiable, we know that the output becomes stable completely by the time tested. To test whether this condition holds, a SAT formula which is satisfiable only if the output is satisfiable is created directly from the - network, and a SAT solver is called on it. The size of the SAT formula is roughly proportional to the size of the - network. The main difficulty in the analysis of large networks is that due to a potentially large size of the - networks, the size of SAT formulas generated can be too large for a SAT solver to solve even after the optimization discussed in the previous section has been applied 4 . In the following we discuss how to control the size of - networks without losing much accuracy. 3.2 Reducing the Size of - Networks for Effective Approximation The main reason why - networks become large in the exact approach is that - functions at many distinct arrival times must be computed for internal nodes. This size increase occurs when there are many distinct path delays to internal nodes due to the reconvergence of the circuit. Therefore our goal is to control the number of distinct arrival times considered at each internal node. More specifically we only create a small number of - functions at each internal node. This strategy avoids the creation of huge - networks thereby controlling the size of SAT formulas generated. Although this idea certainly helps reduce the size of - networks, it must be done carefully so that the correctness of the analysis is 3 The algorithm is CPU intensive rather than memory intensive since the core part of the algorithm is SAT. Theoretically it is not necessarily true that a smaller SAT formula is easier to solve. However we have observed that the size of SAT formulas is well correlated with the time the solver takes. guaranteed. We must never underapproximate true delays since otherwise the timing analysis could miss timing violations when used in the context of timing verification. Overapproximation is acceptable as long as reasonable accuracy is maintained. We guarantee this property by selectively underapproximating stability of signals. This underapproximation in turn overapproximates instability of signals thereby guaranteeing that estimated delays are never underapproximated. The key idea on approximation is to modify the mapping from required times to arrival times discussed in Section 2.2 so that only a small set of arrival times forms the image of the mapping. Given the sorted set of required times and the sorted set of arrival times at an internal node n, the mapping f : R 7! A used in the exact analysis is defined as ae A such that a i - r if r - a 1 Since the stability of the signal at the node increases monotonically as time elapses by the definition of - functions, it is safe to change the mapping so that it maps a required time to a time earlier than the time defined in the above. This corresponds to underapproximation of the signal stability. Thus, by modifying the mapping under this constraint so that only a small set of arrival times is required, one can control the number of nodes to be introduced in the - network without violating the correctness of the analysis. Depending on how the original mapping in the exact analysis is changed several conservative approximation schemes can be devised. Two such approximation schemes are described next. 3.2.1 Topological Approximation The most aggressive approximation, which we call topological ap- proximation, is to map required times either to the topological arrival time (aq 5 ) or to \Gamma1. More formally, the mapping f T is defined as follows. ae It is easy to see that f T is a conservative approximation of f . Since no need to create a new node for the - function in the - network 6 . Instead the node function or its complement of the original network can be used for the - function. For the other arrival time \Gamma1, - \Gamma1 1g. Therefore it is sufficient to have a constant zero node in the - network and use it for all the cases where the zero function is needed. Since neither of the arrival times needs any additional node in the - network, this approximation never increases the size of the - network. If this reduction is applied at all nodes, the analysis simply becomes pure topological analysis. Therefore, this approximation makes sense only if it is selectively invoked on some subset of nodes. A selection strategy is described later. 3.2.2 Semi-Topological Approximation Thesecondapproximationscheme, called semi-topological approx- imation, is slightly milder than the first in terms of the power of simplifying - networks. In this, required times are mapped to two arrival times again, but the times chosen are different. The times to be picked are 1) the arrival time, say ae , matched with r 1 in the exact mapping f and 2) the topological arrival time aq , which is the same as in the first approximation. The first approximation and this one are different only if ae 6= \Gamma1, in which case the second one 5 To be precise, aq can be earlier than the topological arrival time if an intermediate satisfiability call has already verified that by time aq the signal is stabilized completely. 6 Notice that the - network always includes the original circuit. gives a more accurate approximation. To be precise, the definition of the new mapping function f S is as follows. ae ae if r ! aq aq otherwise If ae 6= \Gamma1, the - function for time ae is now computed explic- itly, and the corresponding node is added to the - network. Similar extensions which give tighter approximations are possible by allowing more arrival times to remain after the mapping. A set of various approximations gives a tradeoff between compactness of - networks and accuracy of analysis. 3.3 Control/Data Dichotomy in Approximation Strategie In [11] Yalcin et al. proposed to use designer's knowledge on control-data separation of primary inputs for effective approximate timing analysis. They applied this idea to speed up their timing analysis technique using conditional delays [10] by simplifying signal propagation conditions of data variables. We adapt their idea, of using this knowledge, to the XBD0 analysis to develop a selection strategy of various approximation schemes. 3.3.1 Labeling Data/Control Types Given data/control types of all primary inputs, each internal node is labeled data or control based on the following procedure. All the nodes in the network are visited from primary inputs to primary outputs in a topological order. At each node the types of its fanins are examined. If all of them are data, the node is labeled data; otherwise it is labeled control. Hence nodes labeled data are pure data variables with no dependencyon control variables, while those labeled control are all the other variables with some dependency on control variables. This labeling policy is different from the one used in [11], where a node is labeled data if at least one of its fanins is labeled data. In their labeling, nodes labeled data are variables with some dependency on data whereas nodes labeled control are pure control variables. The difference between the two labelings is whether pure data variables or pure control variables are distinguished. Our labeling will lead to tighter approximations. 3.3.2 Applying Different Approximations based on Types Once all the nodes are labeled, different approximation schemes are applied at nodes based on their types. The strategy is as follows. If a node is a control variable, the semi-topological approximation f S is applied while if a node is a data variable, the topological approximation f T is applied. The intuition is to use a tighter approximation for control variables to preserve accuracy while performing maximum simplification for data variables assuming they have less impact on delays than control variables. 3.3.3 Extracting Control Circuitry for Further Ap- proximation If the approximation so far is not powerful enough to make analysis tractable, further approximation is possible by extracting only the control-intensive portion of the circuit and performing timing analysis on the subcircuit. The extraction of the control portion is done by stripping off all pure data nodes from the original network under analysis. Note that any circuit can be decomposed into a cascade circuit where the nodes in the driving circuit are labeled as data and those in the driven circuit control by the definition of data variables. Therefore, the primary inputs of the subcircuit are the boundary variables which separate the subcircuit from the pure data portion. We assume conservatively that delays of the pure data portion of the circuit are the same as topological delays, which gives arrival times at the primary inputs of the extracted circuit. Analysis is then performed on this subcircuit as if it were the circuit given. Notice that this has a similar flavor to the approximation proposed in [4]. The difference between this approximation and the previous method is that the subcircuit has a new set of primary inputs, which are assumed independent. However, it is possible that in the original circuit only a certain subset of signal combinations appears at the boundary variables. Since this approximation assumes that all signal combinations can show up, the analysis becomes pessimistic 7 . For example, if a signal combination which does not appear on the cut makes a long path sensitizable, it can make delay estimation unnecessarily pessimistic. Although this method is more conservative than the one without subcircuit extraction, it reduces the size of a circuit to be analyzed much more significantly than the other one. 4 Experimental Results We implemented the new approximation scheme on top of the implementation of [8] under SIS environment. To evaluate the effectiveness of the approximation, we focused on timing analysis of mapped ISCAS combinational circuits, which is generally much more time-consuming than analysis basedon simpler delay models. In Table 1 8 the results on three circuits whose exact analysis takes more than 20 secondson a DEC Alpha Server 7000/610 are shown 9 . Each circuit is technology-mapped first with the option specified in the second column using the lib2.genlib library. The delay of the circuit is then analyzed using three techniques. The first one (exact) is the exact method presented in [8]. The remaining two are approximate methods; the second, called approx(1), is the technique in Section 3.3.2 and the third, called approx(2), is the one in Section 3.3.3 which involves subcircuit extraction. Control/Data specification for the primary inputs of these circuits are the same as those in [11] 10 . For each of the three analyses, estimated delay and CPU time are shown in the last two columns. One can observe that accuracy is preserved in the three examples in both of the approximation methods while CPU time is reduced significantly. Table summarizes a similar experiment for C6288, an integer multiplier, which is known to be difficult for exact timing analysis due to a huge amount of reconvergence. Since all the primary inputs are data variables, the approximate techniques proposed are degenerated into topological analysis. To avoid this inaccuracy all the primary inputs were set to control. Note that this sets all intermediate nodes to control. We then applied the first approximate method under this labeling. Although the approximation is not so powerful as the original algorithms, this at least enables us to reduce the size of - networks without giving up accuracy completely. Since there is no data variable in the network, only approx(1) was tried. Significant time saving was achieved with only a slight overapproximation in terms of analysis quality. The exact analysis is not only more CPU-time intensive but also much more memory-intensive than the approximate analysis. In fact we could not completeany of the three exact analyses within 150MB of memory. They ran out of memory in a couple of minutes. These exact analyses were possible after 7 If the set of all possible signal combinations at the boundaryvariables can be represented compactly, one can safely avoid this pessimism by multiplying the additional constraint to the SAT formula generated. 8 Timing analysis was done in the linear search mode [8] where the decrement time step is 0.1 and the error tolerance is 0.01. 9 If exact analysis is already efficient, approximation cannot make significant improvement in CPU time; in fact the overall performance can be degraded due to additional tasks involved in approximation. precisely, C1908(1) and C3540(1) in [11] were used. circuit tech.map #gates topological delay type of approx. estimated delay CPU time exact 34.77 29.1 exact 35.76 41.2 exact 35.66 727.0 Table 1: Exact analysis vs. Approximate analysis (CPU time in seconds on DEC AlphaServer 7000/610) circuit tech.map #gates topological delay type of approx. estimated delay CPU times exact 123.87 7850.2 exact 119.16 18956.2 exact 112.92 15610.5 Table 2: Exact analysis vs. Approximate analysis on C6288 (CPU time in seconds on DEC AlphaServer 7000/610) the memory limit was expanded to 1GB. The last example needs an additional explanation. In this example the estimated delay by the approximate algorithm is smaller than that by the exact algorithm although in Section 3 we claimed that the approximation algorithm never underapproximates exact delay. The reason for this is that the SAT solver is not perfect. Given a very hard SAT problem, the solver may not be able to determine the result under a given resource, in which case the solver simply returns Unknown. This is conservatively interpreted as being satisfiable in the timing analysis. In this particular example the SAT solver returned Unknown during the exact timing analysis, which resulted in an overapproximation of the estimated delay, while in the approximate analysis the SAT solver never aborted because of the simplification of - networks and gave a better overapproximation. This example shows that the approximate analysis gives not only computational efficiency but also better accuracy in some cases. To compare the exact and the approximate methods further, we examined the total CPU time of the exact analysis to see how it can be broken down. For the first example of C6288 the exact analysis took 714.7 seconds to conclude that any path of length 123.93 is false, which is about four times longer for the approximate analysis to conclude that the delay of the circuit is 123.94. The situation is much worse in the second example, where the exact analysis took seconds to conclude that any path of length 119.21 is false while the approximate method took only about 1.4% of this time to finish off the entire analysis. Conclusions We have proposed new approximation algorithms as an extension to the XBD0 timing analysis [8]. The core idea of the algorithms is to control the size of sensitization networks to prevent the size of SAT formulas to be solved from getting large. The use of knowledge on data/control separation of primary inputs originally proposed in [11] was adapted to choose an appropriate approximation at each node. We showed experimentally that the technique helps simplify the analysis while maintaining accuracy well within the accuracy of the delay model. Acknowledgments Hakan Yalcin kindly offered detailed data on ISCAS benchmark circuits. --R Path sensitization in critical path problem. Computation of floating mode delay in combinational circuits: Theory and algorithms. Computation of floating mode delay in combinational circuits: Practice and implementation. A new approach to solving false path problem in timing analysis. A polynomial-time heuristic approach to approximate a solution to the false path problem Timed boolean calculus and its applications in timing analysis. Integrating Functional and Temporal Domains in Logic Design. Delay models and exact timing Private communication Hierarchical timing analysis using conditional delays. An approximate timing analysis method for datapath circuits. --TR A polynomial-time heuristic approach to approximate a solution to the false path problem Hierarchical timing analysis using conditional delays An approximate timing analysis method for datapath circuits Integrating Functional and Temporal Domains in Logic Design --CTR David Blaauw , Rajendran Panda , Abhijit Das, Removing user specified false paths from timing graphs, Proceedings of the 37th conference on Design automation, p.270-273, June 05-09, 2000, Los Angeles, California, United States Hakan Yalcin , Mohammad Mortazavi , Robert Palermo , Cyrus Bamji , Karem Sakallah, Functional timing analysis for IP characterization, Proceedings of the 36th ACM/IEEE conference on Design automation, p.731-736, June 21-25, 1999, New Orleans, Louisiana, United States Mark C. Hansen , Hakan Yalcin , John P. Hayes, Unveiling the ISCAS-85 Benchmarks: A Case Study in Reverse Engineering, IEEE Design & Test, v.16 n.3, p.72-80, July 1999 David Blaauw , Vladimir Zolotov , Savithri Sundareswaran , Chanhee Oh , Rajendran Panda, Slope propagation in static timing analysis, Proceedings of the 2000 IEEE/ACM international conference on Computer-aided design, November 05-09, 2000, San Jose, California

false path;delay computation;timing analysis

266472

Sequential optimisation without state space exploration.

We propose an algorithm for area optimization of sequential circuits through redundancy removal. The algorithm finds compatible redundancies by implying values over nets in the circuit. The potentially exponential cost of state space traversal is avoided and the redundancies found can all be removed at once. The optimized circuit is a safe delayed replacement of the original circuit. The algorithm computes a set of compatible sequential redundancies and simplifies the circuit by propagating them through the circuit. We demonstrate the efficacy of the algorithm even for large circuits through experimental results on benchmark circuits.

Introduction Sequential optimisation seeks to replace a given sequential circuit with another one optimised with respect to some criterion area, performance or power, in a way such that the environment of the circuit cannot detect the replacement. In this work, we deal with the problem of optimising sequential circuits for area. We present an algorithm which computes sequential redundancies in the circuit by propagating implications over its nets. The redundancies we compute are compatible in the sense that they form a set that can be removed simultaneously. Our algorithm works for large circuits and scales better than those algorithms that depend on state space exploration. The starting point of our work is [1], in which a method was described to identify sequential redundancies without exploring the state space. The basic algorithm is that for any net, two cases are considered: the net value is 0 and the net value is 1. For each case, constants as well as unobservability conditions are learnt on other nets. If some other net is either set to the same constant for both cases, or to a constant in one case and is unobservable in the other, it is identified as redundant. For example, consider the trivial circuit shown in Figure 1. For the value the net n2 is unobservable and for the value the net n2 is 1. Thus net n2 is stuck-at-1 redundant. However, the redundancies found by the method in [1] are not compatible in the sense that they remain redundant even in the University of California at Berkeley, Berkeley, CA 94720 Cadence Berkeley Labs, Berkeley, CA 94704 # University of Texas at Austin, Austin, Figure 1: Example of incompatible redundancies presence of each other. For instance, the redundancy identification algorithm will declare both the inputs n 1 and n 2 as stuck-at-1 redundant. However, for logic optimisation, it is incorrect to replace both the nets by a constant 1. The straightforward application of Iyer's method to redundancy removal is to identify one redundancy by their implication procedure, remove the redundancy and iterate until con- vergence. Our goal to learn all compatible implications in the circuit in one step and use the compatibility of these implications to remove all the redundancies simultaneously (in this sense our method for finding compatible unobservabilities is related to the work in [2, 3] for computing compatible ODC's (observability don't cares). This is our first contribution. Sec- ondly, we generalise the implication procedure by combining it with recursive learning [4] to enhance the capability of the redundancy identification procedure. Recursive learning lets us perform case split on unjustified gates so that it is possible to learn more implications at the expense of computation time. Consider the circuit in Figure 2. Setting net a to 0 implies that net f is 0. If we set a to 1, a1 becomes 1, but the AND-gate connected to remains unjustified. If we perform recursive learning for the two justifications: the former case, net f becomes 0, and for the latter case, f becomes unobservable because e is 1. Thus, for all the possible cases, either f is 0 or it is unobservable. Hence f is declared stuck-at-0 redundant. Recursive learning helps identify these kinds of new redundancies. We present data which shows that we are able to gain significant optimisations on large benchmark circuits using these two new improvements. In fact, for some circuits, we find that recursive learning not only gives us more optimisation, it is even faster since a previous recursive learning step makes the circuit simpler for a later stage. We do not assume designated initial states for circuits. For sequential optimisation, we use the notion of c-delay replacement [1, 5]. This notion guarantees that every possible input-output behaviour that can be observed in the new circuit after it has been clocked for c cycles after power-up, must have been present in the old circuit. In contrast to the work in [5, 6], the synthesis method presented here does not require state space a e d c f Figure 2: Example of recursive learning a d c o1 a d c Figure 3: A circuit and its graph traversal, and can therefore be applied to large sequential cir- cuits. Recursive learning has been used earlier for optimi- sation, as described in [7], but their method is applied only to combinational circuits and they do not use unobservability conditions. Another procedure to do redundancy removal is described in [8], but as [9] shows, their notion of replacement is not compositional and may also identify redundancies which destroy the initialisability of the circuit. We have therefore chosen to use the notion of safe delayed replacement which preserves responses to all initializing sequences. We are interested in compositionality because we would like a notion of replacement that is valid without making any assumptions about the environment of the circuit. This is why our replacement notion is safer than that used in [10] which identifies sequential redundancies by preserving weak synchronizing se- quences. Their work implicitly assumes that the environment of the circuit has total control so that it can supply the arbitrary sequence that the redundancy identification tool has in mind. Our approach does not pose any such restrictions. The rest of the paper is organised as follows. In Section 2, we present our algorithm to compute compatible redundancies on combinational and sequential circuits. In Section 3, we present experimental results on some large circuits from the ISCAS benchmark set. In Section 4, we conclude with some directions for future work. Redundancy Removal We present an algorithm for sequential circuits that have been mapped using edge-triggered latches, inverters and 2-input gates; note that any combinational implementation can be mapped to a circuit containing only inverters and 2-input gates. We use the notion of circuit graph for explaining our algorithm. A circuit graph is a labelled directed graph whose vertices correspond to primary inputs, primary outputs, logic gates and latches, and edges correspond to wires between the elements of the circuit. The label of a vertex identifies the type of ele- Figure 4: Rules for implying constants ment it represents (e.g. two-input gates, inverters or latches). We refer to an edge in the circuit graph as a net. Figure 3 shows an example of a circuit graph. 2.1 Combinational redundancies We explain our algorithm and prove its correctness for combinational circuits and later extend it to sequential circuits. Consider a circuit graph of a circuit, where V is the set of vertices and E is the set of nets. An assumption A on the subset P ' E is a labelling of the nets in P by values from the set f0;1g. Let n 2 P be a net. We labels the net n with the value v. An assumption is denoted by an ordered tuple. The set of all possible assumptions on the set P of nets is denoted by A P . Consider the set The assumption labeling m with 0 and n with 1 is denoted by hm 7! 0;n 7! 1i and A 1i;hm 7! 1;n 7! 0i;hm 7! 1;n 7! 1ig. An assumption A 2 A P is inconsistent if it is not satisfiable for any assignments to the primary inputs of the circuits. For instance, an assumption of 0 at the input and 1 at the output of an AND gate is inconsistent. In the algorithm, values are implied at nets in E nP from an assumption on P. We imply either constants or unobservability indicators at nets. We indicate unobservability at a net by implying a symbolic value\Omega at it. Let be the set of all possible value that can be implied at any net. An implication is a label (n is a net and r 2 R. Figure 4 illustrates the rules for implying constants. Rules C1, C2, C3 and C5 are self-explanatory. Rule C4 states that for an AND gate, 0 at the output and 1 at an input implies 0 at the other input. Rule C6 states that a constant at some fanout net of a gate implies the same constant at all other fanout nets. Figure 5 illustrates the rules for implying\Omega 's. Rule O1 states that 0 at an input of an AND gate implies a\Omega at the other input. Rule O2 states that a\Omega at every fanout net of a gate implies a\Omega at every fanin net of that gate. Note that constants can be implied in both directions across a gate while\Omega propagates only backwards. We have shown rules only for inverters and AND gates but similar rules can be easily formulated for other gates as well. We use these rules to label the edges of the circuit graph. A constant (0 or 1) label on a net indicates that \Omega \Omega \Omega \Omega \Omega \Omega Figure 5: Rules for implying unobservability0a e1 c b d \Omega \Omega \Omega Figure Overwriting constants with unobservability indicator the net assumes the respective constant value under the current assumption. A\Omega label indicates that the net is not observable at any primary output. Hence, it can be freely assigned to either or 1 under the current assumption. Suppose for every assumption in A P , some net n is labelled either with constant v or with\Omega , then we can safely replace n with constant v. This is because we have shown that under every possible assump- tion, either the net takes the value v or its value does not affect the output. We can therefore conclude that net n is stuck-at-v redundant. We are concerned about the compatibility of all labellings because otherwise we run the danger of marking nets with labels so that all labels are not consistent. For example, consider the circuit in Figure 1. For the purpose of identifying redundancies, [1] would infer the implications from the assumption hn 7! 1i. Additionally, the assumption hn 7! 0i implies that hn 7! 0i implies that (notice that we use the symbol\Omega to denote compatible observability as opposed to which simply denotes observability). So, [1] would rightly claim that both n1 and are stuck-at-1 redundant in isola- tion; however, for redundancy removal it is easy to see that we cannot This is why we want to make all labelings compatible. A sufficient condition for the redundancies to be compatible is to ensure that the procedure for computing implications from an assumption returns compatible implications, i.e., every implication is valid in the presence of all other implications. It is easy to see that if the labelling of edges in the circuit graph is done by invoking the rules described above and no label is ever overwritten, then the set of learnt implications will be compat- ible. For instance, in the circuit of Figure 1, once n1 is labelled a\Omega cannot be inferred at because (n1 be overwritten with 0). But this approach is conservative and will miss some redundancies. In Figure 6, we show an example where overwriting a constant with a\Omega yields a redundancy which could not have been found otherwise. We propagate implications from assumptions on the net a. The redundancy remove /* find and remove redundancies from the circuit graph */ while (there is an unvisited net n in the circuit graph) f S := learn implications (G , hn 7! 1i) S := learn implications (G , hn 7! 0i) R := T -T for every implication set net n to constant v propagate constants and simplify learn implications propagate implications on the circuit graph given an assignment */ f forall n such that A : n 7! v f label n / v while (some rule can be invoked) f b) be the new implication if (b label n / b conflicts with a current label) return else label n / b return set of all current labels Figure 7: Combinational redundancy removal algorithm implications from ha 7! 0i are written below and those from ha 7! 1i are written above the wires. Note that while propagating implications from ha 7! 1i, a2 and d are initially labeled with 1 but after labelling c with 0, the labels at d and a2 are successively overwritten 's. Hence, a2 is found to be stuck-at-0 redundant. As a result, the OR gate can be re- moved. We prove later in this section that this overwriting does not make previously learnt implications invalid, i.e., compatibility of implications is maintained, if the only overwriting that is allowed is that of constants with unobservability indicators. Our algorithm for removing combinational redundancies is given in Figure 7. The function learn implications takes as input an assumption A on an arbitrary subset of nets and labels nets with values from f0;1; learnt through implications. Initially all nets n such that A : n 7! v is an assumption, are la- belled. Then we derive new labels by invoking the rules C1-C6 and O1-O2 and similar rules for other kinds of two input gates. Note that at all times each net has a unique label and constants can be overwritten with\Omega 's but not vice-versa. It returns the set of all final labels. The function redundancy remove takes as input a circuit graph G and calls learn implications successively with assumptions hn i 7! 0i and hn i 7! 1i on the singleton subset fn i g. The two sets of labels are used to compute all pairs n and v such that n is stuck-at-v redundant. We later show that our labelling procedure for learning implications guarantees that all such redundancies can be removed f d c d e f a c e a Figure 8: An implication graph simultaneously. These redundancies are used to simplify the network. The process is repeated until all nets have been con- sidered. Note that the function redundancy remove considers assumptions on only a single net but in general any number of nets could be used to generate assumptions. We later show results for the case when we considered assumptions on two nets, the second one corresponding to the unjustified node closest to This is an instance of recursive learning. We now formalise the notion of a valid label as one for which an implication graph exists. We will use the notion of implication graph for proving the compatibility of the set of labels generated by the algorithm. Let A be an assumption on a set P of nets. An implication graph for the label from assumption A is a directed acyclic graph G I = (V I where L I is a set of labels of the form (m = a) for some net m and some a 2 f0;1; labelling every vertex v 2V I , such that ffl Every root 1 vertex is labelled with m 7! a ffl There is exactly one leaf 2 vertex v 2V I which is labelled ffl For any vertex v 2 V I , if v is not a root node the implication labelling it can be obtained from the implications labelling its parents by invoking an inference rule. An example of an implication graph for the label from the assumption hn8 7! 1i is shown in Figure 8. A set of labels C derived from an assumption A is compatible if for every label C2C there exists an implication graph of C from A such that LC ' C . We now prove the compatibility of implications returned by our labelling procedure. At each step, the labelling procedure either labels a node for the first time or overwrites a constant with a\Omega . We prove the invariant that at any time, the current set of implications C is compatible. We must prove that if a label is overwritten with a new label, every other label must have an implication graph which does not depend on the over-written label. This claim is proved in the following lemma and is needed for all current labels to be simultaneously valid. 1 A vertex with no incoming edges A vertex with no outgoing edges Note that overwriting a 0 with a 1 (or vice-versa) implies an inconsistent assumption and the procedure exits. Lemma 2.1 Let A be a consistent assumption. If a label a) is overwritten by the label (m =\Omega ) in the current set of labels, then for all labels (n there is an implication graph such that a) is not a label of any vertex in the graph. Proof: We call net m a parent of net n if there is a node v of the circuit graph such that m is an incoming arc and n an outgoing arc of v. We also say that n is a child of m. We say m is a sibling of n if there is a node v such that both m and n are outgoing edges of v. We prove the claim by contradiction. Suppose it is false. Let the replacement of a) by (m =\Omega ) be the first instance that makes it false. Therefore, there was an implication graph for each current implication before this happened. Let be an implication that does not have a valid implication graph now. Consider any path in the old implication graph for a net n j , (n is the ith implication on the path. We consider the case where b j is a constant. Hence, all b k 's in the path are constants since a\Omega at a net can only imply a\Omega at another. The case in which =\Omega is considered later. We show that if the assumption A is consistent then it is possible to replace n in the implication graph for n j . There are three cases on the relation between Case 1: The circuit edge n i\Gamma1 is a child of n i .\Omega can be inferred at n i only if either n =\Omega is a current implication or current implication and n i 0 and n i are inputs to an AND gate. In the first case, the fact that an implication graph existed in which n i\Gamma1 was labelled with a constant is contradicted. In the second case, n i\Gamma1 is the output of an AND gate, whose two inputs are n i and n i 0 . Since (n and In either case Case 2: n i\Gamma1 and n i are siblings and (n is an application of Rule C6. If n i+1 is either the parent or a sibling of n i then can be removed from the implication graph for implication. If n i+1 is a child of n i , then\Omega can be inferred at =\Omega is a current implication or n i is a current implication and n i and n i are inputs to an AND gate. In the first case, the fact that an implication graph existed in which n i+1 was labelled with a constant is contradicted. In the second case, clearly n i+1 is labelled with 0, i.e., b otherwise the assumption A is inconsistent, and the path (n can be replaced by the path (n Note that to get a new implication graph for we need the implication graph for n but that exists and is not affected by the overwriting of the previous label of with\Omega . e a g x c d f y Figure 9: Sequential circuit C Case 3: n i\Gamma1 is a parent of n i . The reasoning is same as in Case 2. Thus we have shown that if the assumption was consistent, each vertex labelled with (n in the implication graph of a current implication (n can be replaced with some other current implication. This shows that the replacement of n by =\Omega does not falsify the claim which is a contradiction. Now we consider the case in which b j =\Omega . Then, there is a greatest k such that b k is a constant, b l is constant for all =\Omega for all k ! l - j. From the proof before, we know there exists an implication graph for n k in which not used. This yields an implication graph for in which n used. Lemma 2.2 Let A be a consistent assumption. Then the set of labels returned by the algorithm is compatible. Proof: At each step in the algorithm, either a value is implied at a net for the first time or a constant is overwritten by a\Omega . The proof of this lemma follows by induction on the number of steps of the algorithm and by using Lemma 2.1 to prove the induction step. Theorem 2.1 Let n i stuck-at-v i redundant, for all 1 be the set of redundant faults reported by the algorithm. Then the circuit obtained by setting combinationally equivalent to the original. 2.2 Sequential redundancies Now we extend the algorithm for combinational circuits described in the previous section to find sequential redundancies by propagating implications across latches. The implications may not be valid on the first clock cycle since the latches power-up nondeterministically and have a random boolean value initially. Nevertheless, we can use the notion of k- delayed replacement which requires that the modified circuit produce the same behaviour as the original only after k clock cycles have elapsed. Thus, for example, if implying constant v at a latch output from constant v at its input yields a redun- dancy, a 1-delay replacement 3 is guaranteed on the removal of that redundancy. 3 If we have latches where a reset value is guaranteed on the first cycle of operation, it is sufficient to ensure that the constant v is equal to the reset value; in this case the replacement is a 0-delay replacement. Figure 10: A sequential implication graph from assumption a for the circuit C Figure 11: An incorrect sequential implication graph from assumption a for the circuit C The notion of a label in the implication graph is modified so that it also contains an integer time offset with respect to a global symbolic time step t. The rules for learning implications are exactly the same as before with the addition of a new rule which allows us to propagate implications across latches: when we go across a latch we modify the time offset accord- ingly, e.g. if the output of a latch is labelled with 1 and offset -2, the input of the latch can be labelled with 1 and offset -3. An example of an implication graph for the circuit C in Figure 9 is shown in Figure 10. This example also shows a potential problem with learning sequential implications. Consider the circuit C in Figure 9. For the two assumptions ha t 7! 0i (a is 0 at t and t denotes the global symbolic time) and ha t 7! 1i we get two implication graphs (in Figures 10 and 11) which both imply (c This might lead us to believe that the dundancy. However, the new circuit obtained by replacing c with 0, if it powers up in state 11 (each latch at 1), remains forever in 11 with the circuit output x = 1. However, the original circuit produces no matter which state it powers up in. Thus we do not have a k-delay replacement for any k. The reason for this incorrect redundancy identification is that in order to infer (c from the assumption needed (c with 0 (i.e., for all times), c could not have been 1 at t + 1. One way of solving the above problem is to ensure that no net is labelled with different labels for different times. We will label a net with at most one label, and if a net is labelled we will associate a list of integers with this label which denotes the time offset when this label is valid. Thus, for the above example, during the implication propagation phase for the assumption never infer (a and we will not get the second implication graph in Figure 10. Labeling one net with at most one label also obviates the need for the validation step described in [1]. The algorithm replaces a net n with the constant v if for some time offset t 0 , it is either labelled with v or is unobservable for all assumptions. With each such replacement, we associate a time k as follows [1]. To validate a redundancy n stuck-at-v at time t 0 , we have a set of implication graphs, one for each assumption, that imply either n t 0 =\Omega . Let t 00 be the least time offset on any label in these implication graphs such that for some net m, m t 00 is labelled with a constant. Then We say that n is k-cycle stuck-at-v redundant. We use the following theorem to claim that the circuit obtained by replacing net n with constant v is a k-delayed safe replacement. Lemma 2.3 ([1]) Let a net n be k-cycle stuck-at-v redundant. Then the circuit obtained by setting net results in a k- delayed safe replacement of the original circuit. As in the combinational case, we allow overwriting of constants with unobservability indicators. We make sure that the label at net n at time t +a is overwritten only if the new label is\Omega and net n is not labelled at any other time offset (this is to prevent the problem shown in Figure 11). This may make our algorithm dependent on the order of application of rules, but we have not explored the various options. The proof of the following two lemmas follows by easy extensions of Lemmas 2.1 and 2.2. Lemma 2.4 Let A be a consistent assumption. If a label m a is replaced with m t =\Omega in the current set of labels, then for all labels m t there is an implication graph such that a is not a label in the graph. Lemma 2.5 Let A be a consistent assumption. Then the set of labels returned by the algorithm is compatible. Hence, the redundancies reported by the algorithm are compatible with each other and all redundancies can be removed simultaneously to get a delayed safe replacement. Theorem 2.2 Let n i k i -cycle stuck-at-v i redundant, for all 1 - be the set of redundant faults reported by the algorithm. Then, the circuit obtained by setting net K-delay safe replacement of the original. Proof: From Lemma 2.5, we know from that for all 1 redundant in the circuit obtained by setting It has been shown in [5] that for any circuits C, D and E, if C is an a-delay replacement for D and D is a b-delay replacement for E then C is (a delay replacement for E. The desired result follows easily by induction on n from this property of delay replacements. 3 Experimental Results We present some experimental results for this algorithm. We demonstrate that our approach of identifying sequential redundancies yields significant reduction in area and is better than Circuit Redundancy Removal With Recursive Learning Name red LR A1 % red LR A2 % cordic For legend see Table 2. Table 1: Experimental results for combinational redundancies the approach which removes only combinational redundancies. We also show that for most examples, recursive learning gives better results then the simple implication propagation scheme. In fact for many circuits, recursive learning could identify redundancies where the simple implication propagation scheme is unable to find any. This algorithm was implemented in SIS [11]. The circuit was first optimised using script.rugged which performs combinational optimisation on the network. The optimised circuit was mapped with a library consisting of 2-input gates and inverters. The sequential redundancy removal algorithm was run on the mapped circuit. The propagation of implications was allowed to propagate 15 time steps forward and 15 time-steps backward from the global symbolic time. Table 2 shows the mapped (to MCNC91 library) area of the circuits obtained by running script.ruggedand that obtained by starting from that result and applying redundancy removal algorithm. For very large circuits (s15850 and larger), BDD operations during the full simplify step in script.ruggedwere not per- formed. We report results for those circuits on which our algorithm was able to find redundancies. As mentioned earlier, our algorithm starts with an assumption on the nets and implies values on other nets of the circuit. We implemented two flavors of selection of assumptions. In the first case a conflicting assignment was assumed on one net and values were implied on other nets. The second case was similar to the first except that once the implications could not propagate for an assumption on a net, we performed a na-ve Circuit Attributes Redundancy Removal With Recursive Learning Name PI PO L A red C LR A1 % time red C LR A2 % time s953 43 26 183 3775 28 7035 8.4 66.9 92 733 70 6317 17.8 32.1 43 9380 10.0 493.7 s38417* 28 106 1464 33055 591 887 42 31943 3.4 1139.4 1129 9245 97 29718 10.1 1763.7 * full simplify not run. All times reported on an Alpha 21164 300MHz dual processor with 2G of memory. PI number of primary inputs PO number of primary outputs L number of latches A Mapped area after script.rugged A1 Mapped area after redundancy removal A2 Mapped area after redundancy removal with recursive learning red number of redundancies removed LR Number of latches removed C Upper bound on c, where the new circuit is a c-delay replacement time CPU time % Percentage area reduction Table 2: Experimental results for sequential redundancies version of case splitting only on the net which was closest to the original net from which the implications were propagated and implications common in the two cases were also added in the set of implications learnt for the original net. 4 This enabled us to propagate implications over a larger set of nets in the network and hence to discover more redundancies at the expense of CPU time. Table 2 indicates the area reduction obtained both by simple propagation and by performing this recursive learning. We find that even for this na-ve recursive learning we get reduction in area in most of the circuits over that obtained without case split. For instance, for S5378 we were able to obtain 37.5% area reduction with recursive learning as against 19.6% without it. For most of the medium sized circuits we were not able to obtain any reduction in area without recursive learning. For large circuits also we were able to obtain approximately 5-10% area reduction. S35952 was an exception where we did not obtain any more reduction in area. Except for this circuit the CPU time for recursive learning was less than twice the CPU time for redundancy removal without it. This suggests that more sophisticated recursive learning 4 If a node is unjustified during forward propagation of implications then case-split is performed by setting the output net to 0 and 1. If the node is unjustified during backward propagation case split is achieved by setting one of the two inputs to the input controlling value (0 for (N)AND gate and 1 for (N)OR gate) at a time and propagating the implications backward. based techniques could yield larger area reduction without prohibitive overhead in terms of CPU time. Since our algorithm also identified combinational redundan- cies, we wanted to quantify how many of the redundancies were purely combinational. To verify this we ran our algorithm on the circuits for combinational redundancy removal only. Table 1 shows the area reduction due to combinational redundancies only with and without recursive learning. In most cases, the number of redundancies identified in Table 2 is significantly larger than the set of combinational redundancies identified by our algorithm. Only for S35952 and S953 did the combinational redundancy removal result in approximately the same area reduction as the sequential redundancy case. For the example circuits presented here we were able to achieve 0-37% area reduction. In a number of cases the algorithm was able to remove a significant number of latches. In all cases, the new circuit is a C-delay safe replacement of the original circuit. The C reported in Table 2 is actually an upper bound. For most of the delay replaced circuits C ! 10000. However most practical circuits operate at speeds exceeding 100 MHz in present technology. C ! 10000 for a circuit would require the user to wait for at most 100 -s before useful operation can begin. This is not a severe restriction. We are unable to compare sequential redundancy removal results with the previous work of Entrena and Cheng [8] because as we noted earlier, their notion of sequential replace- ment, which is based on the conservative 0,1,X-valued simula- tion, is not compositional (unlike the notion of delay replacement that we use). 4 Future Work Our redundancy removal algorithm does not find the complete set of redundancies. We can extend this scheme in several ways to identify larger sets. For instance, instead of analyzing two assumptions due to a case split on a single net we could case split on multiple nets and intersect the implications learnt on this larger set of assumptions. One such method is to incrementally select those which are at the frontier where the first phase of implications died out. Additionally, if we split on multiple nets it is possible to detect pairs of nets such that if one is replaced with another the circuit functionality does not change. With our current approach, because we split on a single net, one of the nets in this pair is always a 1 or a 0, which means that we are only identifying stuck-at-constant redundancies For this algorithm we map a given circuit using a library of two input gates and inverters. A different approach would be to use the original circuit and propagate the implications forward and backward by building the BDD's for the node function in terms of it's immediate fanins. We intend to compare the running times and area reduction numbers of our approach with such a BDD based approach. In addition, BDD based approaches may allow us to do redundancy removal for multi-valued logic circuits as well in a relatively inexpensive way. We can extend the notion of redundancy for multi-valued circuits to identify cases where a net can take only a subset of its allowed values. Then latches of this kind can be encoded using fewer bits. Acknowledgements We had very useful discussions with Mahesh Iyer and Miron Abramovici during the course of this work. The comments by the referees also helped to improved the paper. --R "Identifying Sequential Redundancies Without Search," "The Transduction Method - Design of Logic Networks Based on Permissible Functions," Don't Cares in Multi-Level Network Optimiza- tion "Recursive Learning: A New Implication Technique for Efficient Solution to CAD Problems - Test, Verification and Optimization," "Ex- ploiting Power-up Delay for Sequential Optimization," "Latch Redundancy Removal without Global Reset," "LOT: Logic Optimization with Testability - New Transformations using Recursive Learning," "Sequential Logic Optimization by Redundancy Addition and Removal," On Redundancy and Untestability in Sequential Circuits. "On Removing Redundancies from Synchronous Sequential Circuits with Synchronizing Sequences," "SIS: A System for Sequential Circuit Synthesis," --TR The Transduction Method-Design of Logic Networks Based on Permissible Functions Don''t cares in multi-level network optimization Exploiting power-up delay for sequential optimization On Removing Redundancies from Synchronous Sequential Circuits with Synchronizing Sequences On redundancy and untestability in sequential circuits sequential redundancies without search Sequential logic optimization by redundancy addition and removal Latch Redundancy Removal Without Global Reset --CTR Vigyan Singhal , Carl Pixley , Adnan Aziz , Shaz Qadeer , Robert Brayton, Sequential optimization in the absence of global reset, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.8 n.2, p.222-251, April

sequential optimization;recursive learning;sequential circuits;safe delay replacement;compatible unobservability

266476

Decomposition and technology mapping of speed-independent circuits using Boolean relations.

Presents a new technique for the decomposition and technology mapping of speed-independent circuits. An initial circuit implementation is obtained in the form of a netlist of complex gates, which may not be available in the design library. The proposed method iteratively performs Boolean decomposition of each such gate F into a two-input combinational or sequential gate G, which is available in the library, and two gates H/sub 1/ and H/sub 2/, which are simpler than F, while preserving the original behavior and speed-independence of the circuit. To extract functions for H/sub 1/ and H/sub 2/, the method uses Boolean relations, as opposed to the less powerful algebraic factorization approach used in previous methods. After logic decomposition, overall library matching and optimization is carried out. Logic resynthesis, performed after speed-independent signal insertion for H/sub 1/ and H/sub 2/, allows for the sharing of decomposed logic. Overall, this method is more general than existing techniques based on restricted decomposition architectures, and thereby leads to better results in technology mapping.

Introduction Speed-independent circuits, originating from D.E. Muller's work [12], are hazard-free under the unbounded gate delay model . With recent progress in developing efficient analysis and synthesis techniques, supported by CAD tools, this sub-class has moved closer to practice, bearing in mind the advantages of speed- independent designs, such as their greater temporal robustness and self-checking properties. The basic ideas about synthesis of speed-independent circuits from event-based models, such as Signal Transition Graphs (STGs) and Change Diagrams, are described e.g. in [4, 9, 6]. They provide general conditions for logic implementability of specifications into complex gates . The latter are allowed to have an arbitrary fanin and include internal feedback (their Boolean functions being self-dependent). To achieve greater practicality, synthesis of speed-independent circuits has to rely on more realistic assumptions about implementation logic. Thus, more recent work has been focused on the development of logic decomposition techniques. It falls into two categories. One of them includes attempts to achieve logic decomposition through the use of standard architectures (e.g. the standard-C architecture mentioned below). The other group comprises work targeting the decomposition of complex gates directly, by finding a behavior-preserving interconnection of simpler gates. In both cases, the major functional issue, in addition to logic simplification, is that the decomposed logic must not violate the original speed-independent specification. This criterion makes the entire body of research in logic decomposition and technology mapping for speed-independent circuits quite specific compared to their synchronous counterparts. This work has been partially supported by ACiD-WG (ESPRIT 21949), UK EPSRC project ASTI GR/L24038 and CICYT TIC 95-0419 Two examples of the first category [1, 8] present initial attempts to move from complex gates to a more structured implementation. The basic circuit architecture includes C elements (acting as latches) and combinational logic, responsible for the computation of the excitation functions for the latches. This logic is assumed to consist of AND gates with potentially unbounded fain and unlimited input inversions and bounded fanin OR gates. Necessary and sufficient conditions for implementability of circuits in such an architecture (called the standard-C architecture), have been formulated in[8, 1]. They are called Monotonic Cover (MC) requirements. The intuitive objective of the MC conditions is to make the first level (AND) gates work in a one-hot fashion with acknowledgment through one of the C- elements. Following this approach, various methods for speed-independent decomposition and technology mapping into implementable libraries have been developed,e.g. in [14] and [7]. The former method only decomposes existing gates (e.g., a 3-input AND into two 2-input ANDs), without any further search of the implementation space. The latter method extends the decomposition to more complex (algebraic) divisors, but does not tackle the limitation inherent in the initial MC architecture. The best representative of the second category appears to be the work of S. Burns [3]. It provides general conditions for speed-independent decomposition of complex (sequential) elements into two sequential elements (or a sequential and a combinational element). Notably, these conditions are analyzed using the original (unexpanded) behavioral model, thus improving the efficiency of the method. This work is, in our opinion, a big step in the right direction, but addresses mainly correctness issues. It does not describe how to use the efficient correctness checks in an optimization loop, and does not allow the sharing of a decomposed gate by different signal networks. The latter issues were successfully resolved in but only within a standard architecture approach. In [15, 13] methods for technology mapping of fundamental mode and speed-independent circuits using complex gates were presented. These methods however only identify when a set of simple logic gates can be implemented as a complex gate, but cannot perform a speed-independent decomposition of a signal function in case it does not fit into a single gate. In fact, a BDD-based implementation of the latter is used as a post-optimization step after our proposed decomposition technique. In our present work we are considering a more general framework which allows use of arbitrary gates and latches available in the library to decompose a complex gate function, as shown in Figure 1. In that respect, we are effectively making progress towards the more flexible second approach. The basic idea of this new method is as follows. An initial complex gate is characterized by its function F . The result of decomposition is a library component designated by G and a set of (possibly still complex) gates labeled H . The latter are decomposed recursively until all elements are found in the library and optimized to achieve the lowest possible cost. We thus by and large put no restrictions on the implementation architecture in this work. However, as will be seen further, for the sake of practical efficiency, our implemented procedure deals only with the 2-input gates and/or latches to act as G-elements in the decomposition. The second important change of this work compared to [7] is that the new method is based on a full scale Boolean decomposition rather than just on algebraic factorization. This allows us to widen the scope of implementable solutions and improve on area cost (future work will tackle performance-oriented decomposition). Our second goal in generalizing the C-element based decomposition has been to allow the designer to use more conventional types of latches, e.g. D-latches and SR-latches, instead of C-elements that may not exist in conventional standard-cell libraries. Furthermore, as our experimental results show (see Section 6), in many cases the use of standard latches instead of C-elements helps improving the circuit implementations considerably. The power of this new method can be appreciated by looking at the example hazard.g taken from a set of asynchronous benchmarks. The original STG specification and its state graph are shown in Figure 2,a and b. The initial implementation using the "standard C-architecture" and its decomposition using two input gates by the method described in [7] are shown in Figure 2,c and d. Our new method produces a much cheaper solution with just two D-latches, shown in Figure 2,e. Despite the apparent triviality (for an experienced human designer!) of this solution, none of the previously existing automated tools has been able to obtain it. Also note that the D-latches are used in a speed-independent fashion, G F Hn y y Figure 1: General framework for speed-independent decomposition and are thus free from meta-stability and hazard problems 1 . a,d - inputs outputs d a z c c d c d D R c z R R c R c C z (a) c- z- a- a- d- (b) a- a- d- d- z- c-1000 a- a z z a c a d a d c c z a z a z c z d (c) (d) Figure 2: An example of Signal Transition Graph (a), State Graph (b) and their implementation (c)(d)(e) (benchmark hazard.g) The paper is organized as follows. Section 2 introduces the main theoretical concepts and notation. Section 3 presents an overview of the method. Section 4 describes the major aspects of our Boolean relation-based decomposition technique in more detail. Section 5 briefly describes its algorithmic imple- mentation. Experimental results are presented in Section 6, which is followed by conclusions and ideas about further work. Background In this section we introduce theoretical concepts and notation required for our decomposition method. Firstly, we define State Graphs , which are used for logic synthesis of speed-independent circuits. The State Graph itself may of course be generated by a more compact, user-oriented model, such as the Signal Transition Graph. The State Graph provides the logic synthesis procedure with all information necessary for deriving Boolean functions for complex gates. Secondly, the State Graph is used for a property-preserving transformation, called signal insertion. The latter is performed when a complex gate is decomposed into smaller gates, and the thus obtained new signals must be guaranteed to be speed-independent (hazard-free in input/output mode using the unbounded gate delay model). 1 For example, all transitions on the input must be acknowledged by the output before the clock can fall and close the latch. E.g. there is no problem with setup and hold times as long as the propagation time from D to Q is larger than both setup and hold times, which is generally the case. 2.1 State Graphs and Logic Implementability A State Graph (SG) is a labeled directed graph whose nodes are called states . Each arc of an SG is labeled with an event , that is a rising (a+) or falling (a\Gamma) transition of a signal a in the specified circuit. We also allow notation a if we are not specific about the direction of the signal transition. Each state is labeled with a vector of signal values. An SG is consistent if its state labeling v is such that: in every transition sequence from the initial state, rising and falling transitions alternate for each signal. Figure 2,b shows the SG for the Signal Transition Graph in Figure 2,a, which is consistent. We write s a there is an arc from state s (to state s 0 ) labeled with a. The set of all signals whose transitions label SG arcs are partitioned into a (possibly empty) set of inputs, which come from the environment, and a set of outputs or state signals that must be implemented. In addition to consistency, the following two properties of an SG are needed for their implementability in a speed-independent logic circuit. The first property is speed-independence. It consists of three parts: determinism, commutativity and output-persistence. An SG is called deterministic if for each state s and each label a there can be at most one state s 0 such that s a An SG is called commutative if whenever two transitions can be executed from some state in any order, then their execution always leads to the same state, regardless of the order. An event a is called persistent in state s if it is enabled at s and remains enabled in any other state reachable from s by firing another event b . An SG is called output-persistent if its output signal events are persistent in all states and no output signal event can disable input events. Any transformation (e.g., insertion of new signals for decomposition), if performed at the SG level, may affect all three properties. The second requirement, Complete State Coding (CSC), becomes necessary and sufficient for the existence of a logic circuit implementation. A consistent SG satisfies the CSC property if for every pair of states s; s 0 such that the set of output events enabled in both states is the same. (The SG in Figure 2,b is output-persistent and has CSC.) CSC does not however restrict the type of logic function implementing each signal. It requires that each signal is cast into a single atomic gate. The complexity of such a gate can however go beyond that provided in a concrete library or technology. The concepts of excitation regions and quiescent regions are essential for transformation of SGs, in particular for inserting new signals into them. A set of states is called an excitation region (ER) for event a (denoted by ER(a it is the set of states such that s 2 ER(a ) , s a !. The quiescent region (QR) (denoted by QR(a )) of a transition a , with excitation region ER(a ), is the set of states in which a is stable and keeps the same value, i.e. for ER(a+) (ER(a\Gamma) ), a is equal to 1(0) in QR(a+) (QR(a\Gamma)). Examples of ER and QR are shown in Figure 2,b. 2.2 Property-preserving event insertion Our decomposition method is essentially behavioral - the extraction of new signals at the structural (logic) level must be matched by an insertion of their transitions at the behavioral (SG) level. Event insertion is an operation on an SG which selects a subset of states, splits each of them into two states and creates, on the basis of these new states, an excitation region for a new event. Figure 3 shows the chosen insertion scheme, analogous to that used by most authors in the area [16]. We shall say that an inserted signal a is acknowledged by a signal b, if b is one of the signals delayed by the insertion of a, (the same terminology will be used for the corresponding transitions). For example, d acknowledges x in Figure 3. State signal insertion must preserve the speed-independence of the original specification. The events corresponding to an inserted signal x are denoted x , x+, x\Gamma, or, if no confusion occurs, simply by x. Let A be a deterministic, commutative SG and let A 0 be the SG obtained from A by inserting event x. We say that an insertion state set ER(x) in A is a speed-independence preserving set (SIP-set) iff: (1) for each event a in A, if a is persistent in A, then it remains persistent in A 0 , and (2) A 0 is deterministic and commutative. The formal conditions for the set of states r to be a SIP-set can be given in terms of intersections of r with the so-called state diamonds of SG [5]. These conditions are illustrated by Figure 4, where all possible cases of illegal intersections of r with state diamonds are shown. The first (rather (a) d c b a (b) d x Figure 3: Event insertion scheme: (a) before insertion, (b) after insertion inefficient) method for finding SIP-sets based on a reduction to the satisfiability problem was proposed in [16]. An efficient method based on the theory of regions has been described in [5]. d r a d a s3 d d a r d a r d a) b) c) Figure 4: Possible violations of SIP conditions Assume that the set of states S in an SG is partitioned into two subsets which are to be encoded by means of an additional signal. This new signal can be added either in order to satisfy the CSC condition, or to break up a complex gate into a set of smaller gates. In the latter case, a new signal represents the output of the intermediate gate added to the circuit. Let r and r denote the blocks of such a partition. For implementing such a partition we need to insert transitions of the new signals in the border states between r and r. The input border of a partition block r, denoted by IB(r), is informally a subset of states of r by which r is entered. We call IB(r) well-formed if there are no arcs leading from states in r \Gamma IB(r) to states in IB(r). If a new signal is inserted using an input border, which is not well-formed, then the consistency property is violated. Therefore, if an input border is not well-formed, its well-formed speed-independent preserving closure is constructed, as described by an algorithm presented in [7]. The insertion of a new signal can be formalized with the notion of I-partition ([16] used a similar definition). Given an SG, A, with a set of states S, an I-partition is a partition of S into four blocks: QR(x\Gamma)g. QR(x\Gamma)(QR(x+)) defines the states in which x will have the stable value 0 (1). ER(x+) (ER(x\Gamma)) defines the excitation region of x in the new SG A 0 . To distinguish between the sets of states for the excitation (quiescent) regions of the inserted signal x in an original SG A and the new SG A 0 we will refer to them as ERA (x ) and ER A 0 respectively. If the insertion of x preserves consistency and persistency, then the only transitions crossing boundaries of the blocks are the following: QR(x\Gamma) ! ERA Example 2.1 Figure 5 shows three different cases of the insertion of a new signal x into the SG for the hazard.g example. The insertion using ERA (x+) and ERA (x\Gamma) of Figure 5,a does not preserve speed-independence as the SIP set conditions are violated for ERA (x+) (violation type in Figure 4,b). When signal x is inserted by the excitation regions in Figure 5,b then its positive switching is acknowledged by transitions a\Gamma, d+, while its negative switching by transition z \Gamma. The corresponding excitation regions satisfy the SIP conditions and the new SG A 0 , obtained after insertion of signal x, is shown in Figure 5,b. Note that the acknowledgment of x+ by transitions a\Gamma, d+ results in delaying some input signal transitions in A 0 until x+ fires. This changes the original I/O interface for SG A, because it requires the environment to look at a new signal before it can change a and d. This is generally incorrect (unless we are also separately finding an implementation for the environment or we are working under appropriate timing assumptions), and hence this insertion is rejected. a,d - inputs outputs a+ a- a- d- d- z- c-1000 a- A A a- a- d- d- c- a- a- a- a- d- d- z- c-1000 a- (a) A A A' x- x- z- (b) A' a- a- d- d- z- c-1000 a- A A (c) Figure 5: Different cases of signal insertion for benchmark hazard.g: violating the SIP-condition (a), changing the I/O interface (b), correct insertion (c) The excitation regions ERA (x+) and ERA (x\Gamma) shown in Figure 5,c are SIP sets. They are well-formed and comply with the original I/O interface because positive and negative transitions of signal x are acknowledged only by output signal z. This insertion scheme is valid. 2.3 Basic definitions about Boolean Functions and Relations An important part of our decomposition method is finding appropriate candidates for characterization (by means of Boolean covers) of the sets of states ERA (x+) and ERA (x\Gamma) for the inserted signal x. For this, we need to reference here several important concepts about Boolean functions and relations [11]. An incompletely specified (scalar) Boolean function is a functional mapping and '\Gamma' is a don't care value. The subsets of domain B n in which F holds the 0, 1 and don't care value are respectively called the OFF-set , ON-set and DC-set . F is completely specified if its DC-set is empty. We shall further always assume that F is a completely specified Boolean function unless said otherwise specifically. be a Boolean function of n Boolean variables. The set is called the support of the function F . In this paper we shall mostly be using the notion of true support , which is defined as follows. A point (i.e. binary vector of values) in the domain B n of a function F is called a minterm. A variable x 2 X is essential for function F (or F is dependent on x) if there exist at least two minterms v1; v2 different only in the value of x, such that F (v1) 6= F (v2). The set of essential variables for a Boolean function F is called the true support of F and is denoted by sup(F ). It is clear that for an arbitrary Boolean function its support may not be the same as the true support. E.g., for a c the true support of F (X) is sup(F a subset of X . Let F (X) be a Boolean function F (X) with support g. The cofactor of F (X) with respect to x i defined as F x i respectively). The well-known Shannon expansion of a Boolean function F (X) is based on its cofactors: . The Boolean difference, or Boolean derivative, of F (X) with respect to x defined as ffiF do 1: foreach non-input signal x do solutions(x):=;; 2: foreach gate G 2 flatches, and2, or2g do endfor 3: best H(x) := Best SIP candidate from solutions(x); endfor 4: if foreach x, best H(x) is implementable or foreach x, best H(x) is empty then exit loop; 5: Let H be the most complex best H(x); Insert new signal z implementing H and derive new SG; forever 7: Library matching; Figure Algorithm for logic decomposition and technology mapping. A function F or F x i under ordering In the former case it is called positive unate in x i , in the latter case negative unate in x i . A function that is not unate in x i is called binate in x i . A function is (positive/negative) unate if it is (positive/negative) unate in all support variables. Otherwise it is binate. For example, the function positive unate in variable a because F a For an incompletely specified function F (X) with a DC-set, let us define the DC function FDC : We will say that a function e F is an implementation of F if Boolean relation is a relation between Boolean spaces [2, 11]; it can be seen as a generalization of a Boolean function, where a point in the domain B n can be associated with several points in the codomain. More formally, a Boolean relation R is R ' B n \Theta f0; 1g m . Sometimes, we shall also use the "\Gamma" symbol as a shorthand in denoting elements in the codomain vector, e.g. 10 and 00 will be represented as one vector \Gamma0. Boolean relations play an important role in multi-level logic synthesis [11], and we shall use them in our decomposition method. Consider a set of Boolean functions with the same domain. Let R ' be a Boolean relation with the same domain as functions from H. We will say that H is compatible with R if for every point v in the domain of R the vector of values (v; H 1 is an element of R. An example of compatible functions will be given in Section 4. 3 Overview of the method In this section we describe our proposed method for sequential decomposition of speed-independent circuits aimed at technology mapping. It consists of three main steps: 1. Synthesis via decomposition based on Boolean relations; 2. Signal insertion and generation of a new SG; 3. Library matching The first two steps are iterated until all functions are decomposed into implementable gates or no further progress can be made. Each time a new signal is inserted (step 2), resynthesis is performed for all output signals (step 1). Finally, step 3 collapses decomposed gates and matches them with library gates. The pseudo-code for the technology mapping algorithm is given in Figure 6. By using a speed-independent initial SG specification, a complex gate implementation of the Boolean function for each SG signal is guaranteed to be speed-independent. Unfortunately this gate may be too large to be implemented in a semi-custom library or even in full custom CMOS, e.g. because it requires too many stacked transistors. The goal of the proposed method is to break this gate starting from its output by using sequential (if its function is self-dependent, i.e. it has internal feedback) or combinational gates. Given a vector X of SG signals and given one non-input signal y 2 X (in general the function F (X) for y may be self-dependent), we try to decompose the function F (X) into (line 2 of algorithm in Figure 6): ffl a combinational or sequential gate with function G(Z; y), where Z is a vector of newly introduced signals, ffl a vector of combinational 2 functions H(X) for signals Z, so that G(H(X)) implements F (X). Moreover, we require the newly introduced signals to be speed- independent (line 3). We are careful not to introduce any unnecessary fanouts due to non-local ac- knowledgment, since they would hinder successive area recovery by gate merging (when allowed by the library). The problem of representing the flexibility in the choice of the H functions has been explored, in the context of combinational logic minimization, by [19] among others. Here we extend its formulation to cover also sequential gates (in Sections 4.1 and 4.3). This is essential in order to overcome the limitations of previous methods for speed-independent circuit synthesis that were based on a specific architecture. Now we are able to use a broad range of sequential elements, like set and reset dominant SR latches, transparent D latches, and so on. We believe that overcoming this limitation of previous methods (that could only use C elements and dual-rail SR-latches) is one of the major strengths of this work. Apart from dramatically improving some experimental results, it allows one to use a "generic" standard-cell library (that generally includes SR and D latches, but not C elements) without the need to design and characterize any new asynchronous-specific gates. The algorithm proceeds as follows. We start from an SG and derive a logic function for all its non-input signals (line 1). We then perform an implementability check for each such function as a library gate. The largest non-implementable function is selected for decomposition. In order to limit the search space, we currently try as candidates for G (line 2): ffl all the sequential elements in the library (assumed to have two inputs at most, again in order to limit the search space), ffl two-input AND, OR gates with all possible input inversions. The flexibility in the choice of functions represented as a Boolean relation, that represents the solution space of F described in Section 4.1. The set of function pairs compatible with the Boolean relation is then checked for speed- independence (line 3), as described in Section 2.2. This additional requirement has forced us to implement a new Boolean relation minimizer, that returns all compatible functions , as outlined in Section 5.1. If both are not speed-independent, the pair is immediately rejected. Then, both H 1 and H 2 are checked for approximate (as discussed above) implementability in the library, in increasing order of estimated cost. We have two cases: 1. both are speed-independent and implementable: in this case the decomposition is accepted, 2. otherwise, the most complex implementable H i is selected, and the other one is merged with G. 2 The restriction that H(X) be combinational will be partially lifted in Section4.3. The latter is a heuristic technique aimed at keeping the decomposition balanced. Note that at this stage we can also implement H 1 or H 2 as a sequential gate if the sufficient conditions described in Section 4.3 are met. The procedure is iterated as long as there is progress or until everything has been decomposed (line 4). Each time a new function H i is selected to be implemented as a new signal, it is inserted into the SG (line and resynthesis is performed in the next iteration. The incompleteness of the method is essentially due to the greedy heuristic search that accepts the smallest implementable or non-implementable but speed-independent solution. We believe that an exhaustive enumeration with backtracking would be complete even for non-autonomous circuits, by a relatively straightforward extension of the results in [17]. At the end, we perform a Boolean matching step ([10]) to recover area and delay (line 7). This step can merge together the simple 2-input combinational gates that we have (conservatively) used in the decomposition into a larger library gate. It is guaranteed not to introduce any hazards if the matched gates are atomic. 4 Logic decomposition using Boolean relations 4.1 Specifying permissible decompositions with BRs In this paper we apply BRs to the following problem. Given an incompletely specified Boolean function F (X) for signal y, y 2 X, decompose it into two-levels such that G(H(X); y) implements F (X) and functions G and H have a simpler implementation than F (any such H will be called permissible). Note that the first-level function (X)g is a multi-output logic function, specifying the behavior of internal nodes of the decomposition, The final goal is a function decomposition to a form that is easily mappable to a given library. Hence only functions available in the library are selected as candidates for G. Then at each step of decomposition a small mappable piece (function G) is cut from the potentially complex and unmappable function F . For a selected G all permissible implementations of function H are specified with a BR and then via minimization of BRs a few best compatible functions are obtained. All of them are verified for speed- independence by checking SIP-sets. The one which is speed-independent and has the best estimated cost is selected. Since the support of function F can include the output variable y, it can specify sequential behavior. In the most general case we perform two-level sequential decomposition such that both function G and function H can be sequential, i.e., contain their own output variables in the supports. The second level of the decomposition is made sequential by selecting a latch from the library as a candidate gate, G. The technique for deriving a sequential solution for the first level H is described in Section 4.3. We next show by example how all permissible implementations of decomposition can be expressed with BRs. Example 4.1 Consider the STG in Figure 7,a, whose SG appears in Figure 8,a. Signals a, c and d are inputs and y is an output. A possible implementation of the logic function for y is F (a; c; d; us decompose this function using as G a reset-dominant Rs-latch represented by the equation Figure 7,b). At the first step we specify the permissible implementations for the first level functions by using the BR specified in the table in Figure 8,b. Consider, for example, vector a; c; d; It is easy to check that F(0; 0; 0; Hence, for vector 0000 the table specifies that (R; implementation of R and S must keep for this input vector either 1 at R or 0 at S, since these are the necessary and 3 For simplicity we consider the decomposition problem for a single-output binary function F , although generalization for the multi-output and multi-valued functions is straightforward. y d D Rs c dc+yc Rs a Rs Rs d- a- c- a- y y G R c d y a) c) d) e) R y d c a S R c a d Figure 7: Sequential decomposition for function d) Region C-element D-latch Rs Sr AND OR QR(y\Gamma) f0\Gamma; \Gamma0g f0\Gamma; \Gamma0g f1\Gamma; \Gamma0g 0\Gamma f0\Gamma; \Gamma0g 00 unreachable \Gamma\Gamma \Gamma\Gamma \Gamma\Gamma \Gamma\Gamma \Gamma\Gamma \Gamma\Gamma Table 1: Boolean relations for different gates sufficient conditions for the Rs-latch to keep the value 0 at the output y, as required by the specification. On the other hand, only one solution possible for the input vector 1100 which corresponds to setting the output of the Rs-latch to 1. The Boolean relation solver will find, among others, the two solutions illustrated in Figure 7,c,d: (1) a and (2) acd. Any of these solutions can be chosen depending on the cost function. Table specifies compatible values of BRs for different types of gates: a C-element, a D-latch, a reset-dominant Rs-latch, a set-dominant Sr-latch, a two input AND gate and a two input states of an SG are partitioned into four subsets, ER(y+); QR(y+);ER(y \Gamma); and QR(y \Gamma), with respect to signal y with function F (X) for which decomposition is performed. All states that are not reachable in the SG form a DC-set for the BR. E.g., for each state, s, from ER(y+) only one compatible solution, 11, is allowed for input functions H of a C-element. This is because the output of a C-element in all states, s 2 ER(y+) is at 0 and F these conditions the combination 11 is the only possible input combination that implies 1 at the output of a C-element. On the other hand, for each state s 2 QR(y+), the output hence it is enough to keep at least one input of a C-element in 1. This is expressed by values f1\Gamma; \Gamma1g in the second line of the table. Similarly all other compatible values are derived. 4.2 Functional representation of Boolean relations Given an SG satisfying CSC requirement, each output signal y 2 X is associated with a unique incompletely specified function F (X), whose DC-set represents the set of unreachable states. F (X) can be represented by three completely specified functions, denoted ON(y)(X), OFF (y)(X) and DC(y)(X) representing the ON-, OFF-, and DC-set of F (X), such that they are pairwise disjoint and their union is a tautology. c- c- d- a- c- c- d+ a- a- a- ON(y) a,c,d,y (a) acdy F R S (b) Figure 8: (a) State graph, (b) Decomposition of signal y by an RS latch Let a generic n-input gate be represented by a Boolean equation are the inputs of the gate, and q is its output 4 . The gate is sequential if q belongs to the true support of G(Z; q). We now give the characteristic function of the Boolean relation for the implementation of F (X) with gate G. This characteristic function represents all permissible implementations of z allow F to be decomposed by G. Given characteristic function (1), the corresponding table describing Boolean relation can be derived using cofactors. For each minterm m with support in X , the cofactor BR(y)m gives the characteristic function of all compatible values for z (see example below). Finding a decomposition of F with gate G is reduced to finding a set of n functions Example 4.2 (Example 4.1 continued.) The SG shown in Figure 8.a corresponds to the STG in Figure 7. Let us consider how the implementation of signal y with a reset-dominant Rs latch can be expressed using the characteristic function of BR. Recall that the table shown in Figure 8.b represents the function F (a; c; d; and the permissible values for the inputs R and S of the Rs latch. The ON-, OFF-, and DC-sets of function F (a; c; d; y) are defined by the following completely specified functions: 4 In the context of Boolean equations representing gates we shall liberally use the "=" sign to denote "assignment", rather than mathematical equality. Hence q in the left-hand side of this equation stands for the next value of signal q while the one in the right-hand side corresponds to its previous value. The set of permissible implementations for R and S is characterized by the following characteristic function of the BR specified in the table. It can be obtained using equation 1 by substituting expressions for ON(y); OFF (y); DC(y), and the function of an Rs-latch, R(S BR(y)(a; c; d; (R This function has value 1 for all combinations represented in the table and value 0 for all combinations that are not in the table (e.g., for (a; c; d; For example, the set of compatible values for given by the cofactor which correspond to the terms 1\Gamma and \Gamma0 given for the Boolean relation for that minterm. Two possible solutions for the equation BR(y)(a; c; d; corresponding to Figure 7,c,d are: 4.3 Two-level sequential decomposition Accurate estimation of the cost of each solution produced by the Boolean relation minimizer is essential in order to ensure the quality of the final result. The minimizer itself can only handle combinational logic, but often (as shown below) the best solution can be obtained by replacing a combinational gate with a sequential one. This section discusses some heuristic techniques that can be used to identify when such a replacement is possible without altering the asynchronous circuit behavior, and without undergoing the cost of a full-blown sequential optimization step. Let us consider our example again. Example 4.3 (Example 4.1 continued.) Let us assume that the considered library contains three-input AND, OR gates and Rs-, Sr- and D-latches. Implementation (1) of signal y by an Rs-latch with inputs R=cd and S=acd matches the library and requires two AND gates (one with two and one with three inputs) and one Rs-latch. The implementation (2) of y by an Rs-latch with inputs R=cd+ y c and S=a would be rejected, as it requires a complex AND-OR gate which is not in the library. However, when input y in the function cd replaced by signal R, the output behavior of R will not change, i.e. function R=cd+ y c can be safely replaced by R=cd+Rc. The latter equation corresponds to the function of a D-latch and gives the valid implementation shown in Figure 7,e. Our technique to improve the precision of the cost estimation step, by partially considering sequential gates, is as follows: 1. Produce permissible functions z via the minimization of Boolean relations (z 1 and z 2 are always combinational as z 1 ; z 2 62 X). 2. Estimate the complexity of H 1 and matches the library then Complexity = cost of the gate else Complexity = literal count 3. Estimate the possible simplification of H 1 and H 2 due to adding signals z 1 and z 2 to their supports, i.e. estimate the complexity of the new pair fH 0 2 g of permissible functions z 4. Choose the best complexity between H 1 Let us consider the task of determining H 0 2 as in step 3. Let A be an SG encoded by variables from set V and let z = H(X; y), such that X ' V; y 2 V , be an equation for the new variable z which is to be inserted in A. The resulting SG is denoted A 0 =Ins(A; z=H(X; y)) (sometimes we will simply A 0 =Ins(A; z) or A 0 =Ins(A; z one signal is inserted). A solution for Step 3 of the above procedure can be obtained by minimizing functions for signals z 1 and z 2 in an SG A However this is rather inefficient because the creation of SG A 0 is computationally expensive. Hence instead of looking for an exact estimation of complexity for signals z 1 and z 2 we will rely on a heuristic solution, following the ideas on input resubstitution presented in Example 4.3. For computational efficiency, the formal conditions on input resubstitution should be formulated in terms of an original SG A rather than in terms of the SG A 0 obtained after the insertion of new signals 5 . Lemma 4.1 Let Boolean function H(X; y) implement the inserted signal z and be positive (negative) unate in y. Let H 0 (X; z) be the function obtained from H(X; y) by replacing each literal y (or y) by literal z. The SGs A 0 =Ins(A; z=H(X; y)) and A 00 =Ins(A; z=H 0 (X; z)) are isomorphic iff the following condition is satisfied: where S is the characteristic function describing the set of states (ERA (z+) [ ERA (z \Gamma)) in A. Informally Lemma 4.1 states that resubstitution of input y by z is permissible if in all states where the value of function H(X; y) depends on y, the inserted signal z has a stable value. Example 4.4 (Example 4.1 continued.) Let input R of the RS-latch be implemented as cd Figure 7,d). The ON-set of function H=cd shown by the dashed line in Figure 8,a. The input border of H is the set of states by which its ON-set is entered in the original SG A, i.e. By similar consideration we have that f0100g. These input borders satisfy the SIP conditions and hence IB(H) can be taken as ERA (R+), while ERA (R\Gamma) must be expanded beyond IB(H) by state 1100 for not to delay the input transition a+ (ERA (R\Gamma) = f0100; 1100g). The set of states where the value of function H essentially depends on signal y is given by the function negative unate in y and cube ac has no intersection with ERA (R+)[ERA (R\Gamma). Therefore by the condition of Lemma 4.1 literal y can be replaced by literal R, thus producing a new permissible function R=cd This result can be generalized for binate functions, as follows. Lemma 4.2 Let Boolean function H(X; y) implement the inserted signal z and be binate in y. Function H can be represented as H(X; are Boolean functions not depending on y. Let H 0 (X; are isomorphic iff the following conditions are satisfied: are characteristic Boolean functions describing sets of states ERA (z+) [ ERA (z \Gamma) and ERA (z+) in A, respectively. The proof is given in the Appendix. The conditions of Lemma 4.2 can be efficiently checked within our BDD-based framework. They require to check two tautologies involving functions defined over the states of the original SG A. This heuristic solution is a trade-off between computational efficiency and optimality. Even though the estimation is still not completely exact (the exact solution requires the creation of A 0 =Ins(A; z)), it allows us to discover and possibly use the implementation of Figure 7,e. 5 Note that this heuristic estimation covers only the cases when one of the input signals for a combinational permissible is replaced by the feedback z i from the output of H i itself. Other cases can also be investigated, but checking them would be too complex. 5 Implementation aspects The method for logic decomposition presented in the previous section has been implemented in a synthesis tool for speed-independent circuits. The main purpose of such implementation was to evaluate the potential improvements that could be obtained in the synthesis of speed-independent circuits by using a Boolean-relation-based decomposition approach. Efficiency of the current implementation was considered to be a secondary goal at this stage of the research. 5.1 Solving Boolean relations In the overall approach, it is required to solve BRs for each output signal and for each gate and latch used for decomposition. Furthermore, for each signal and for each gate, several solutions are desirable in order to increase the chances to find SIP functions. Previous approaches to solve BRs [2, 18] do not satisfy the needs of our synthesis method, since (1) they minimize the number of terms of a multiple-output function and (2) they deliver (without significant modifications to the algorithms and their implementation) only one solution for each BR. In our case we need to obtain several compatible solutions with the primary goal of minimizing the complexity of each function individually . Term sharing is not significant because two-level decomposition of a function is not speed-independent in general, and hence each minimized function must be treated as an atomic object. Sharing can be exploited, on the other hand, when re-synthesizing the circuit after insertion of each new signal. For this reason we devised a heuristic approach to solve BRs. We next briefly sketch it. Given a BR BR(y)(X; Z), each function H i for z i is individually minimized by assuming that all other functions will be defined in such a way that H(X) will be a compatible solution for BR. In general, an incompatible solution may be generated when combining all H i 's. Taking the example of Figure 8, an individual minimization of R and S could generate the solution Next, a minterm with incompatible values is selected, e.g. - d-y for which but only the compatible values 1\Gamma or \Gamma0 are acceptable. New BRs are derived by freezing different compatible values for the selected minterm. In this case, two new BRs will be produced with the values 1\Gamma and \Gamma0, respectively for the minterm - a-c - d-y. Next, each BR is again minimized individually for each output function and new minterms are frozen until a compatible solution is obtained. This approach generates a tree of BRs to be solved. This provides a way of obtaining several compatible solutions for the same BR. However, the exploration may become prohibitively expensive if the search tree is not pruned. In our implementation, a branch-and-bound-like pruning strategy has been incorporated for such purpose. Still, the time required by the BR solver dominates the computational cost of the overall method in our current implementation. Ongoing research on solving BRs for our framework is being carried out. We believe that the fact that we pursue to minimize functions individually, i.e. without caring about term sharing among different output functions, and that we only deal with 2-output decompositions, may be crucial to derive algorithms much more efficient than the existing approaches. 5.2 Selection of the best decomposition Once a set of compatible solutions has been generated for each output signal, the best candidate is selected according to the following criteria (in priority 1. At least one of the decomposed functions must be speed-independent. 2. The acknowledgment of the decomposed functions must not increase the complexity of the implementation of other signals (see section 5.3). 3. Solutions in which all decomposable functions are implementable in the library are preferred. 4. Solutions in which the complexity of the largest non-implementable function is minimized are preferred. This criterion helps to balance the complexity of the decomposed functions and derive balanced tree-like structures rather than linear ones 6 . 5. The estimated savings obtained by sharing a function for the implementation of several output signals is also considered as a second order priority criterion. Among the best candidate solutions for all output signals, the function with the largest complexity, i.e. the farthest from implementability, is selected to be implemented as a new output signal of the SG. The complexity of a function is calculated as the number of literals in factored form. In case it is a sequential function and it matches some of the latches of the gate library, the implementation cost is directly obtained from the information provided by the library. 5.3 Signal acknowledgment and insertion For each function delivered by the BR solver, an efficient SIP insertion must be found. This reduces to finding a partition fERA (x+); QRA (x+); ERA (x\Gamma); QRA (x\Gamma)g of the SG A such that ERA (x+) and ERA (x\Gamma) (that are restricted to be SIP-sets, Section 2.2) become the positive and negative ERs of the new signal x. QRA (x+) and QRA (x\Gamma) stand for the corresponding state sets where x will be stable and equal to 1 and 0, respectively. In general, each function may have several ERA (x+) and ERA (x\Gamma) sets acceptable as ERs. Each one corresponds to a signal insertion with different acknowledging outputs signals for its transitions. In our approach, we perform a heuristic exploration seeking for different ERA (x+) and ERA (x\Gamma) sets for each function. We finally select one according to the following criteria: ffl Sets that are only acknowledged by the signal that is being decomposed (i.e. local acknowledgment) are preferred. ffl If no set with local acknowledgment is found, the one with least acknowledgment cost is selected. The selection of the ERA (x+) and ERA (x\Gamma) sets is done independently. The cost of acknowledgment is estimated by considering the influence of the inserted signal x on the implementability of the other signals. The cost can be either increased or decreased depending on how ERA (x+) and ERA (x\Gamma) are selected, and is calculated by incrementally deriving the new SG after signal insertion. As an example consider the SG of Figure 5,c and the insertion of a new signal x for the function d. A valid SIP set for ERA (x+) would be the set of states f1100; 0100; 1110; 0110g, where the state f1100g is the input border for the inserted function. A valid SIP set for ERA (x\Gamma) would be the set of states f1001; 0001g. With such insertion, ERA (x+) will be acknowledged by the transition z+ and ERA (x\Gamma) by z \Gamma. However, this insertion is not unique. For the sake of simplicity, let us assume that a and d are also output signals. Then an insertion with ERA would be also valid. In that case, the transition x+ would be acknowledged by the transitions a\Gamma and d+. 5.4 Library mapping The logic decomposition of the non-input signals is completed by a technology mapping step aimed at recovering area and delay based on a technology-dependent library of gates. These reductions are achieved by collapsing small fanin gates into complex gates, provided that the gates are available in the library. The collapsing process is based on the Boolean matching techniques proposed by Mailhot et al. [10], adapted to the existence of asynchronous memory elements and combinational feedback in speed- independent circuits. The overall technology mapping process has been efficiently implemented based on the utilization of BDDs. 6 Different criteria, of course, may be used when we also consider the delay of the resulting implementation, since then keeping late arriving signals close to the output is generally useful and can require unbalanced trees. 6 Experimental results 6.1 Results in decomposition and technology mapping The method for logic decomposition presented in the previous sections has been implemented and applied to a set of benchmarks. The results are shown in Table 2. Circuit signals literals/latches CPU Area non-SI Area SI I/O old new (secs) lib 1 lib 2 best 2 inp map best chu150 3/3 14/2 10/1 converta 2/3 12/3 16/4 252 352 312 312 338 296 296 drs ebergen 2/3 20/3 6/2 4 184 160 160 160 144 144 hazard 2/2 12/2 0/2 1 144 120 120 104 104 104 nak-pa 4/6 20/4 18/2 441 256 248 248 250 344 250 nowick 3/3 16/1 16/1 170 248 248 248 232 256 232 sbuf-ram-write 5/7 22/6 20/2 696 296 296 296 360 338 338 trimos-send 3/6 36/8 14/10 2071 576 480 480 786 684 684 Total 252/52 180/37 4288 3984 3976 4180 4662 3982 Table 2: Experimental results. The columns "literals/latches" report the complexity of the circuits derived after logic decomposition into 2-input gates. The results obtained by the method presented in this paper ("new") are significantly better than those obtained by the method presented in [7] ("old"). Note that the library used for the "new" experiments was deliberately restricted to D, Sr and Rs latches (i.e. without C-elements, since they are generally not part of standard cell libraries). This improvement is mainly achieved because of two reasons: ffl The superiority of Boolean methods versus algebraic methods for logic decomposition. ffl The intensive use of different types of latches to implement sequential functions compared to the C-element-based implementation in [7]. However, the improved results obtained by using Boolean methods are paid in terms of a significant increase in terms of CPU time. This is the reason why some of the examples presented in [7] have not been decomposed. We are currently exploring ways to alleviate this problem by finding new heuristics to solve Boolean relations efficiently. 6.2 The cost of speed independence The second part of Table 2 is an attempt to evaluate the cost of implementing an asynchronous specification as a speed-independent circuit. The experiments have been done as follows. For each bench- mark, the following script has been run in SIS, using the library asynch.genlib: astg to f; source script.rugged; map. The resulting netlists could be considered a lower bound on the area of the circuit regardless of its hazardous behavior (i.e. the circuit only implements the correct function for each output signal, without regard to hazards). script.rugged is the best known general-purpose optimization script for combinational logic. The columns labeled "lib 1" and "lib 2" refer to two different libraries, one biased towards using latches instead of combinational feedback 7 , the other one without any such bias. The columns labeled SI report the results obtained by the method proposed in this paper. Two decomposition strategies have been experimented before mapping the circuit onto the library: ffl Decompose all gates into 2-input gates (2 inp). ffl Decompose only those gates that are not directly mappable into gates of the library (map). In both cases, decomposition and mapping preserve speed independence, since we do not use gates (such as MUXes) that may have a hazardous behavior when the select input changes. There is no clear evidence that performing an aggressive decomposition into 2-input gates is always the best approach for technology mapping. The insertion of multiple-fanout signals offers opportunities to share logic in the circuit, but also precludes the mapper from taking advantage of the flexibility of mapping tree-like structures. This trade-off must be better explored in forthcoming work. Looking at the best results for non-SI/SI implementations, we can conclude that preserving speed independence does not involve a significant overhead. In our experiments we have shown that the reported area is similar. Some benchmarks were even more efficiently implemented by using the SI-preserving decomposition. We impute these improvements to the efficient mapping of functions into latches by using Boolean relations. 7 Conclusions and future work In this paper we have shown a new solution to the problem of multi-level logic synthesis and technology mapping for asynchronous speed-independent circuits. The method consists of three major parts. Part 1 uses Boolean relations to compute a set of candidates for logic decomposition of the initial complex gate circuit implementation. Thus each complex gate F is iteratively split into a two-input combinational or sequential gate G available in the library and two gates H 1 and H 2 that are simpler than F , while preserving the original behavior and speed-independence of the circuit. The best candidates for H 1 and are selected for the next step, providing the lowest cost in terms of implementability and new signal insertion overhead. Part 2 of the method performs the actual insertion of new signals for H 1 and/or H 2 into the state graph specification, and re-synthesizes logic from the latter. Thus parts 1 and 2 are applied to each complex gate that cannot be mapped into the library. Finally, Part 3 does library matching to recover area and delay. This step can collapse into a larger library gate the simple 2-input combinational gates (denoted above by G) that have been (conservatively) used in decomposing complex gates. No violations of speed-independence can arise if the matched gates are atomic. This method improves significantly over previously known techniques [1, 8, 7]. This is due to the significantly larger optimization space exploited by using (1) Boolean relations for decomposition and (2) a broader class of latches 8 . Furthermore, the ability to implement sequential functions with SR and D latches significantly improves the practicality of the method. Indeed one should not completely rely, as earlier methods did, on the availability of C-elements in a conventional library. In the future we are planning to improve the Boolean relation solution algorithm, aimed at finding a set of optimal functions compatible with a Boolean relation. This is essential in order to improve the CPU times and synthesize successfully more complex specifications. --R Automatic gate-level synthesis of speed-independent circuits An exact minimizer for boolean relations. General conditions for the decomposition of state holding elements. Synthesis of Self-timed VLSI Circuits from Graph-theoretic Specifications Complete state encoding based on the theory of regions. Concurrent Hardware. Technology mapping for speed- independent circuits: decomposition and resynthesis Basic gate implementation of speed-independent circuits Algorithms for synthesis and testing of asynchronous circuits. Algorithms for technology mapping based on binary decision diagrams and on boolean operations. Synthesis and Optimization of Digital Circuits. A theory of asynchronous circuits. Structural methods for the synthesis of speed-independent circuits Decomposition methods for library binding of speed-independent asynchronous designs Automatic technology mapping for generalized fundamental mode asynchronous designs. A generalized state assignment theory for transformations on Signal Transition Graphs. Heuristic minimization of multiple-valued relations Permissible functions for multioutput components in combinational logic optimization. --TR Automatic technology mapping for generalized fundamental-mode asynchronous designs Decomposition methods for library binding of speed-independent asynchronous designs Basic gate implementation of speed-independent circuits A generalized state assignment theory for transformation on signal transition graphs Automatic gate-level synthesis of speed-independent circuits Synthesis and Optimization of Digital Circuits Algorithms for Synthesis and Testing of Asynchronous Circuits General Conditions for the Decomposition of State-Holding Elements Complete State Encoding Based on the Theory of Regions Technology Mapping for Speed-Independent Circuits Structural Methods for the Synthesis of Speed-Independent Circuits --CTR Jordi Cortadella , Michael Kishinevsky , Alex Kondratyev , Luciano Lavagno , Alexander Taubin , Alex Yakovlev, Lazy transition systems: application to timing optimization of asynchronous circuits, Proceedings of the 1998 IEEE/ACM international conference on Computer-aided design, p.324-331, November 08-12, 1998, San Jose, California, United States Michael Kishinevsky , Jordi Cortadella , Alex Kondratyev, Asynchronous interface specification, analysis and synthesis, Proceedings of the 35th annual conference on Design automation, p.2-7, June 15-19, 1998, San Francisco, California, United States

technology mapping;two-input sequential gate;complex gates;decomposed logic sharing;netlist;speed-independent circuits;two-input combinational gate;signal insertion;optimization;library matching;boolean decomposition;boolean relations;circuit CAD;logic resynthesis;design library;logic decomposition

266552

A deductive technique for diagnosis of bridging faults.

A deductive technique is presented that uses voltage testing for the diagnosis of single bridging faults between two gate input or output lines and is applicable to combinational or full-scan sequential circuits. For defects in this class of faults the method is accurate by construction while making no assumptions about the logic-level wired-AND/OR behavior. A path-trace procedure starting from failing outputs deduces potential lines associated with the bridge and eliminates certain faults. The information obtained from the path-trace from failing outputs is combined using an intersection graph to make further deductions. The intersection graph implicitly represents all candidate faults, thereby obviating the need to enumerate faults and hence allowing the exploration of the space of all faults. The above procedures are performed dynamically and a reduced intersection graph is maintained to reduce memory and simulation time. No dictionary or fault simulation is required. Results are provided for all large ISCAS89 benchmark circuits. For the largest benchmark circuit, the procedure reduces the space of all bridging faults, which is of the order of 10^9 to a few hundred faults on the average in about 30 seconds of execution time.

Introduction A bridging fault [1] between two lines A and B in a circuit occurs when the two lines are unintentionally shorted. When the lines A and B have different logic values, the gates driving the lines will be engaged in a drive fight (logic contention). Depending on the gates driving the lines A and B, their input values, and the resistance of the bridge, the bridged lines can have intermediate voltage values VA and VB (not well defined logic values of 1 or 0). This is interpreted by the logic that fans out from the bridge as shown in the shaded region in Figure 1 (a). The logic gates downstream from the bridged nodes can have variable input logic thresholds. Thus the intermediate voltage at a bridged node may be interpreted differently by different gates. This is known as the Byzantine Generals Problem [2, 3] and is illustrated in Figure (b). The voltage at the node A ( VA ) is interpreted as a faulty value (0) by gate d and a good value (1) by gate c. Thus, different branches from a single fanout stem can have different logic values. The feasibility of any diagnosis scheme can be evaluated using the parameters: accuracy, precision, storage requirements and computational complexity. Accurate simulation of bridging faults [4, 3] is computationally expensive. Thus, it may not be feasible to perform bridging fault simulation during diagnosis. Further, the space of all bridging faults is extremely large. For example, for the large ISCAS89 benchmark circuits, it is of the order of 10 9 faults. Several techniques have been proposed for the diagnosis of bridging faults in combinational circuits using voltage testing. Mill- This research was supported in part by Defense Advanced Research Projects Agency (DARPA) under contract DABT 63-96-C-0069, by the Semiconductor Research Corporation (SRC) under grant 95-DP-109, by the Office of Naval Research (ONR) under grant N00014-95-1-1049, and by an equipment grant from Hewlett-Packard. man, McCluskey and Acken [5] presented an approach to diagnose bridging faults using stuck-at dictionaries. Chess et al. [6], and Lavo et al. [7] improved on this technique. These techniques enumerate bridging faults and are hence constrained to using a reduced set of bridging faults extracted from the layout. Further, the construction and storage requirements of fault dictionaries may be prohibitive. Chakravarty and Gong [8] describe a voltage-based algorithm that uses the wired-AND (wired-OR) model and stuck-at fault dictionaries. The wired-AND and wired-OR models work only for technologies for which one logic value is always more strongly driven than the other. x x x (a) Primary Outputs (b) effect propagation Primary Inputs Threshold: 2.6 V Threshold: 2.4 VBridge a A d c A Bridging fault Threshold: 2.3 V Figure 1: Bridging Fault Effect Propagation In this paper we present a deductive technique that does not require fault dictionaries and does not explicitly simulate faults, either stuck-at or bridging. Further, no model such as wired-AND or wired-OR is assumed at the logic-level. The class of bridging faults considered are all single bridging faults between two lines in the circuit. The lines could be gate outputs, gate inputs, or primary inputs. For defects in this class of faults, the method is accurate in that the defect is guaranteed to be in the candidate list. In the fol- lowing, a failing vector and a failing output refer to a vector and a primary output that fail the test on a tester (not during simulation). The deductive technique consists of two deductive procedures. The first is a path-trace procedure that starts from failing outputs and uses the logic values obtained by the logic simulation of the good circuit for each failing vector. This is used to deduce lines potentially associated with the bridging faults. The second procedure is an intersection graph constructed from the information obtained through path-tracing from failing outputs. The path-trace and the intersection graph are constructed and processed dynamically during diagnosis. The intersection graph implicitly represents all candidate bridging faults under consideration, thereby allowing processing of the entire space of all bridging faults in an implicit man- ner. During diagnosis, a reduced version of the graph is maintained that retains all diagnostic information. This reduces memory usage and simulation time. Since the technique uses only logic simulation and does not explicitly simulate faults, it is fast. The technique outputs a list of candidate faults. If the resolution (the size of the candidate list) is adequate, the diagnosis is complete. Otherwise, either the candidate list can be simulated with a bridging fault simulator or other techniques [5, 6, 7, 8] can be used to improve the resolution. 2 The Path-Trace Procedure The path-trace procedure deduces lines in the circuit that are potentially associated with a bridging fault. A potential source of error with respect to a failing output is defined as follows. potential source of error, with respect to a failing output, is a line in the circuit from which there exists a sensitized path to that failing output on the application of the corresponding failing vector. Note that there is a distinction between potential sources of error and the actual source(s) of error associated with the defect. In the following, the actual source(s) of error are simply referred to as the source(s) of error. The path-trace procedure is similar to critical path tracing [9], and star algorithm [10]. However, there are some important differences. The above procedures were developed for single stuck-at faults. Hence, only one line in the circuit is assumed to be faulty. However, for bridging faults, due to the Byzantine Generals Problem, both lines could be sources of fault effects. Further, these effects may reconverge, leading to effects such as multiple-path sensitization as shown in Figure 1 (b). The voltages at lines A (VA ) and B (VB ) are both interpreted as faulty by gates d and e, and the fault effect reconverges at gate f . However, the assumption of a single bridging fault between two lines ensures that at most two lines in the circuit can be sources of error. The logic-value of a gate input is said to be controlling if it determines the gate's output value regardless of other input values [11]. The path-trace procedure proceeds as follows. Start from a failing output and process the lines of the circuit in a reverse topological order up to the inputs. When a gate output is reached, observe the input values. If all inputs have noncontrolling values, continue the trace from all inputs. If one or more inputs have controlling values, continue the trace from any one controlling input. When a fanout branch is reached, continue tracing from the stem. The choice in selecting the controlling value can be exploited, as will be explained later. We first consider the case of a single line being the source of error for a failing output, and then consider the case where both lines of a bridging fault are sources of error for a failing output. Consider a single line being the source of errors on a failing vector. When reconvergent fanout exists, the following situations could occur. In Figure 2 (a), the effects of an error from the stem c propagate to the output. However, if the paths have different parities, they will cancel each other when they reconverge. This is referred to as self- masking [9]. Figure 2 (b) shows an example of multiple path sensitization [9]. The bold lines indicate error propagation. The error from line c propagates through two paths before reconverging and propagating to an output. a0h f d e f e c propagation (a) error from stem c propagates (b) multiple-path sensitization from c Figure 2: Reconvergent Fanout with Single Source of Error Lemma 1 On any failing vector, the path-trace procedure includes all potential sources of error with respect to the failing out- puts, assuming there is a single source of error. Proof: When the path-trace reaches a fanout branch, it continues from the stem. Hence, if the stem were a source of error, it would be included. If a gate has multiple controlling values on its inputs, then fault effects can propagate through this gate only if there exists a stem from which errors reconverge at this gate to collectively change all the controlling values. When the path-trace reaches this gate, it will continue along one of the lines having controlling val- ues. Hence it will include the stem. Lemma 1 can be interpreted as follows. If the defect causes a single line to be faulty on some failing vector and this fault effect propagates to some failing output, then the path-trace includes all lines that are sensitized to that failing output. The path-trace procedure is conservative with respect to single sources of error. Not all lines in the path-trace may be potential sources of error. For example, line h in Figure 2 (b) is not a potential source of error but would be included in the path-trace. However, this conservative approach is necessary when both lines of a bridging fault could be sources of error with respect to some failing output. Note that for a single source of error, the potential sources of error are the same as critical lines [9] in the circuit. Next, we consider the case where both lines of a bridging fault are sources of error on some failing vector. If there exists at least one path between the lines of a bridging fault, then the bridging fault creates one or more feedback loops. Such a fault is referred to as a feedback bridging fault [11]. If no paths exist between the lines of a bridging fault, then it is called a non-feedback bridging fault. A feedback bridging fault may cause oscillations to occur if the input vector creates a sensitized path from one line of the bridging fault to the other and this path has odd inversion parity. If such oscillations are detectable by the tester, then they can be used as additional failing outputs for the path-trace pro- cedure. The following Lemma, Theorem and Corollary are applicable to both feedback and nonfeedback bridging faults. The symbol A@B is used to represent a bridging fault. Lemma 2 If a bridging fault A@B causes fault effect propagation to an output due to reconvergence of bridging fault effects from both lines of the bridging fault, then the path-trace procedure starting from that failing output will include at least one of the lines of the bridging fault. Proof: At the reconvergent gate, there exist one or more controlling input values. The path-trace continues from one of the lines with controlling input value. Thus, one of the lines of the bridging fault is covered by the path-trace. A case of Lemma 2 is illustrated in Figure 3. The output of gate e fails. Path-trace starts from this output and proceeds to the inputs. Since gate e has two controlling inputs, the trace continues from one of them. Node B, which is part of the bridging fault A@B, is covered by the path-trace. x x d e Bridging fault A EI F G Node Test Vector PO Primary Output Figure 3: Path-Trace and Node Set 2 The node set N ij is defined as the set of lines that lie on the path-trace starting from failing output PO i under the application of test-vector t j . Theorem 1 If neither line A nor line B of a bridging fault A@B is in a node set N ij , then the fault A@B could not have caused output PO i to fail under test vector t j . Proof: [By contradiction] Assume that the bridging fault A@B caused an output PO i to fail on some test-vector t j . This implies that there exists a sensitized path from A, or B, or the interaction of fault effects from both A and B, to the primary output PO i under the application of test-vector t j . If neither line A nor line B is in then due to Lemmas 1 and 2, there exists no sensitized path to PO i . This leads to a contradiction. Corollary 1 If the defect is a single bridging fault A@B, then a node set N ij must contain at least one of the lines A and B. Proof: Follows directly from Theorem 1. Note that Theorem 1 and Corollary 1 are conservative in that they make no assumptions about the resistance of the bridging fault, the gates feeding the bridging fault and their input values, and the logic input thresholds of the gates downstream from the bridging fault. The only assumption made is the presence of a single bridging fault. The information from a group of node sets can be used to make further deductions. This is performed using the concept of an intersection graph. 3 The Intersection Graph and Its Processing Given a group of node sets fN ij g the intersection graph is defined as follows. Definition 3 The intersection graph is a simple undirected graph (no loops or multiple edges) is a node setg and edge or (j 6= l)) and Figure 4 shows an intersection graph with 7 vertices. The corresponding node sets are shown within the curly brackets. The intersection graph has similar structure to the initialization graph proposed by Chakravarty and Gong [8]. However, there are important differences. The initialization graph is constructed using only structural information while the intersection graph is constructed using logic information exploited by the path-trace procedure. The initialization graph is created statically once before diagnosis and processed. However, the intersection graph is updated and processed dynamically during diagnosis. A reduction procedure maintains a reduced version of the graph without losing diagnostic in- formation. The intersection graph has interesting structural properties that are useful for performing deduction and for maintaining reduced graphs to help reduce memory requirements and simulation time. G 2v G Figure 4: Intersection Graph and Its Properties 3.1 Structural Properties Property 1 If GI has two vertices such that (v1 ; v2 . The set of vertices can be partitioned into three sets 0such that 8v 2 EI and (v 0 (v 0 Proof: Let N 1 ij and N 2 ij be the node sets corresponding to v1 and v2 . From Corollary 1, N 1 ij and N 2 ij contain at least one of the lines of the bridging fault A@B. Since (v1 ; v2 contains only one of the lines A and B (say A). This implies that N 2 contains the other line (B). Consider any arbitrary vertex v2g. From Corollary 1 it follows that the node set corresponding to v3 contains at least one of the lines A and B. Thus v3 is adjacent to at least one of v1 and v2 . This implies that one of the following three conditions holds: v3 is adjacent to v1 and not adjacent to v2 ; v3 is adjacent to v2 and not adjacent to v1 ; v3 is adjacent to both v1 and v2 . 0are the three sets obtained by Property are cliques. Proof: From Corollary 1 and Property 1 it follows that the node sets corresponding to every v i 2 V 0[ fv1g contain one and only one of A and B (say A), while the node sets corresponding to every contain the other line (B). Thus V 0 are cliques. Figure 4 illustrates these properties. The intersection graphs can be reduced while maintaining their properties. This reduces the number of vertices and edges. Further, this also reduces the number of node sets that need to be maintained and their sizes. Thus the reduction process, which is done dynamically during diagnosis, can help reduce memory and simulation time. The following which follows from Property 2, is used in the reduction process. Corollary 2 If the intersection graph is not a clique, and are the subgraphs induced by V 0 fv1g and V 0[ fv2g respectively using Property 1, then all node sets in V1 contain only one of the lines A or B of the bridge, while all the node sets in V2 contain the other line. 3.2 Intersection Graph Processing Corollary 2 is used by the procedure shown in Figure 5 to reduce the intersection graph. An irreducible intersection graph is either a complete graph or has the following characteristic. For each EI , the V 0and V 0sets are empty. An example of the reduction procedure is shown in Figure 6. The initial intersection The intersection graph Vertex corresponding to node set N ij Possible to find v1 ; v2 since GI is not a clique reduced reduced reduced reduced edges incident on v 0and v 0 Figure 5: Procedure for Reducing the Intersection Graph graph is reduced two times to obtain an irreducible graph with two disjoint vertices. The dynamic processing of GI proceeds as fol- lows. After each node set N ij is obtained, update GI . Reduce the intersection graph until an irreducible graph is obtained. After all node sets are processed, the irreducible intersection graph obtained contains the candidate bridging faults. The candidate list (C) is obtained from the irreducible graph (GIR ) using the following rules. 1. If GIR has two disconnected components, each of which has one vertex, then let N 1 ij and N 2 ij be the node sets associated with the two vertices. g. 2. If GIR has one component that is not a complete graph, then for each (v1 ; ij and N 2 ij be the node sets associated with v1 and v2 . g. 3. If GIR is a complete graph, then let g. The reduced intersection graph is a compact way to implicitly represent the space of candidate bridging faults. Further, the reduction procedure prunes the space of candidate bridging faults without losing diagnostic information. The defect is guaranteed to be in the candidate list by construction. The candidate list will include other faults which are logically equivalent or diagnostically equivalent with respect to the test set. A better test set may distinguish between some of these faults, thereby increasing the diagnostic res- olution. WhenGIR is a complete graph, only one of the lines of the bridge can be determined with certainty. This results in a partial diagnosis. The experimental results indicate that partial diagnosis does not occur often. 3.3 Implementation Issues and Complexity The major operation performed during GI processing is its reduc- tion. The basic operation needed by the reduction procedure is set intersection. Further, the node sets need to be stored for each vertex G 2G'v'v'G' (a) Initial intersection graph (b) Reduced intersection graph Figure An Example of the Intersection Graph Reduction of GI . The node sets are represented as bit-vectors with a value of 1 indicating the presence of a node and a 0 indicating the absence of one. If there are n lines in the circuit, the size of a node set is dn=8e bytes. The bit vector representation allows for efficient set intersection using the bitwise AND operator. As a result of the dynamic processing of GI , its size grows and shrinks. Hence, the data structure chosen to represent GI is a two-dimensional linked list. GI has jVI j vertices. Assuming 4 bytes each for pointers and vertex indices, the worst-case memory requirement for GI and its associated node sets is (jVI j \Theta dn=8e+8jVI j +8jVI j 2 bytes. Since jVI j is typically much smaller than n, the worst-case space complexity is O(jVI j \Theta n). The worst-case size of jVI j is n fail , where n fail is the total number of failing outputs on all failing vectors. The reduction procedure results in jVI j being much smaller than nfail , thereby reducing the memory requirements. The reduction procedure computes the V 0and V 0sets based on Corollary 2 by exploring the edges of EI . Typically, jEI j is small. For each edge in EI the reduction procedure computes V 0and Each intersection operation between the node sets of two vertices of GI reduces the number of vertices of GI by 1. Thus the maximum number of intersections possible in the procedure reduce intersection graph() is jVI 2. Thus the worst-case time complexity for the procedure reduce intersection graph() is O(jVI intersections. Here again, the reduction procedure results in small jVI j values, thereby reducing the simulation time. 4 Heuristics to Improve Resolution When the path-tracing procedure reaches a gate with multiple controlling inputs, one of them is chosen. The choice of input impacts the size of the resultant node set, its elements, and hence, impacts the diagnostic resolution. The smaller the size of the node set, the smaller is the intersection with other node sets, and the greater is the likelihood of reducing the intersection graph. Two conditions are checked to select the controlling input in such a manner that the size of the resultant node set is reduced. The first is based on fanout. When the path-trace reaches a stem, it continues from the stem unconditionally. When a controlling input is the branch of a stem, one of whose other branches has been chosen, then this input should be selected, since the stem has to be selected anyway [10]. The second condition involves checking the controllability of the line. SCOAP controllability measures are used. The most easily controllable input (check for 0-controllability if the logic value of the line is 0 and vice-versa) is likely to give the smallest node set. If the same gate is reached in two different applications of the path-trace and the same choice of controlling inputs exists, then selecting different inputs for the two runs can potentially result in a smaller intersection between the two resultant node sets. A dirty bit is set when the path-trace chooses a controlling input. This input is avoided in future invocations of the path-trace procedure. Based on the above conditions, three heuristics are defined be- low. Heuristic 1 chooses controlling input randomly. Heuristic 2 chooses controlling input by checking for fanout followed by con- trollability. Heuristic 3 chooses controlling input by checking for dirty bit followed by fanout and then controllability. The overall diagnosis procedure is shown in Figure 7. for each test vector t i outputs simulation with t i for each failing output PO j path-trace from failing output PO j Figure 7: The Diagnosis Procedure 5 Experimental Results The diagnosis procedure was implemented in C++. All experiments were performed on a SUN SPARCStation 20 with 64MB of memory for the full-scan versions of the ISCAS89 sequential benchmark circuits [12]. In practice, the failing responses used as input for the diagnosis procedure would be obtained by testing the failing circuit on a tester. For our diagnosis experiments, the failing responses were generated using the accurate bridging fault simulator E-PROOFS [4] to ensure that the diagnostic experiments were as realistic as possible. The cell libraries for the circuits were generated manually [4]. The test vectors used were compact tests generated to target stuck-at faults [13]. Ideally, diagnostic test sets for bridging faults would be the best choice. All large ISCAS89 benchmark circuits were considered. For each of the benchmark circuits, a random sample of 500 single two-line bridging faults were injected one at a time. For each one of these faults, the failing responses were obtained by performing bridging fault simulation on the given test set using E-PROOFS. Faults that do not produce any failing outputs were dropped. For the rest of the faults, the failing responses were used to perform di- agnosis. The diagnosis results are summarized in Tables 1 and 2. The average, minimum and maximum sizes of the candidate lists are shown in Table 1 for the three different heuristics. The average size of the candidate list is a few hundred faults, which is a significant reduction from the space of all faults. Further, as expected, heuristics 2 and 3 improve the diagnostic resolution over heuristic 1. The reduction can be significant. For example, for s38584, the average size of the candidate list is reduced by a factor of 4. Note that in some cases, the method uniquely identifies the fault (reso- lution of 1). The best resolutions are indicated in bold. The average sizes of the node sets and the intersection graph are shown in Figure 2. As expected, heuristic 2 does the best in terms of node set sizes. Both heuristic 2 and heuristic 3 do better than heuristic 1 in terms of the average size of the intersection graph. The average values of the execution time, number of failing outputs and percentage of faults that were partially diagnosed is given for heuristic 2. Other interesting observations can be made from Table 2. Note that the average size of the sets is very small and appears to be independent of the circuit size. Further, it is about 2-3 orders of magnitude smaller than the total number of lines in the circuit, thereby suggesting that the path-trace procedure is ef- ficient. The average size of the intersection graph (jVI j) is about a quarter of the total number of failing outputs, indicating that the graph reduction procedure is useful. As expected heuristic 2 does the best in terms of the average size of the node set and heuristic 3 does the best in terms of the average size of the intersection graph (jVI j). Note that the procedure is accurate by construction; that is, the defect is guaranteed to be in the candidate list. The distribution of the sizes of the candidate lists is shown in Figure 8 for s13207 and s38417. This trend is observed for the other circuits as well. For about 10% of the faults, the resolution is adequate (less than 20 candidates) to consider the diagnosis complete. For about 80% of the faults, the resolution is such (between that the candidate list is small enough to be accurately simulated using a bridging fault simulator as a post-processing step. In about 25% of the cases, the diagnosis is partial; that is, only one of the lines of the bridge can be determined with certainty. In such cases, and if the resolution is so large that bridging fault simulation cannot be performed, then the diagnosis procedure can be followed with other techniques [5, 6, 7, 8] using the candidate list to improve the resolu- tion. Note that these resolutions were obtained using a compacted stuck-at test set. We expect that there would be better resolution with better test sets. Table 1: Diagnostic Resolution Circuit Candidate List Size Ave. Min. Max. Ave. Min. Max. Ave. Min. Max. The diagnosis procedure requires very small execution times, as seen in column 8 of Table 2. The procedure requires only the logic simulation of failing vectors and the path-trace procedure from failing outputs. Both of these procedures are linear in the size of the circuit. Further, the graph reduction procedure is linear in the size of its vertex set (jVI j). Techniques such as those used in [5, 6, 7, 8] require either the storage of stuck-at fault dictionaries or the simulation of stuck-at faults during diagnosis. As seen in columns 12 and 13 of Table 2, the storage requirements for dictionaries can be very large, and the simulation time is about an order of magnitude larger than that required for the diagnosis procedure. This is expected since fault simulation has greater than linear complexity in the size of the circuit. Further, fault simulation without fault dropping needs to be performed. Techniques such as those used in [5, 6, 7] also need to enumerate bridging faults and are hence constrained to use a small set of realistic faults. This trade-off between resolution and complexity suggests that our diagnosis procedure, which is both space- and time-efficient, could be attempted first, and then be complemented by other procedures if greater resolution is required. Table 2: Diagnosis Results and Comparison with Techniques Using Stuck-at Fault Information Circuit Average Size of Average Average Values Stuck-at fault Node Set jV I j (Heuristic 2) Information Heu.1 Heu.2 Heu.3 Heu.1 Heu.2 Heu.3 Exec. # Fail Partial # of Storage y Exec. Time z Time s9234f 40.2 36.3 37.0 18.7 19.3 19.3 2.32 65.4 0.35 6927 47.1 M 34.61 s13207f 29.6 28.2 27.4 62.5 58.2 27.4 31.1 298.3 0.16 12311 0.51 G 131.55 s38584f 38.9 28.7 29.4 31.3 28.7 29.4 30.4 64.9 0.38 36303 1.56 G 214.51 y Full fault dictionary in matrix format z w/o fault dropping0.20.30.50.70.9 Candidate list size Normalized ratio Figure 8: Distribution of Candidate List Size 6 Conclusions and Future Work A deductive procedure for the diagnosis of bridging faults, which is accurate and experimentally shown to be both space- and time- efficient, has been described. The information obtained from a path-trace procedure from failing outputs is combined using an intersection graph, which is constructed and processed dynamically, to make the deduction. The intersection graph provides an implicit means of representing and processing the space of candidate bridging faults without using dictionaries or explicit fault simulation. The procedure assumes a single bridging fault between two lines. If the defect involves multiple faults or shorts between multiple lines, then the properties of GI may be violated. Extensions to multiple faults or shorts between multiple lines require looking for larger sized cliques (Kn , n - 3) in the graph GI . An interesting application of this work is in the area of design error location. For design errors of multiplicity 2, the diagnosis procedure can be used without any modification. Higher multiplicity errors require extensions --R "Bridging and Stuck-at Faults," "Fault Model Evolution for Diagno- sis: Accuracy vs. Precision," "Biased Voting: A Method for Simulating CMOS Bridging Faults in the Presence of Variable Gate Logic Thresholds," "E-PROOFS: A CMOS Bridging Fault Simulator," "Diagnosing CMOS Bridging Faults with Stuck-at Fault Dictionaries," "Diagnosing of Realistic Bridging Faults with Stuck-at Information," "Beyond the Byzantine Gen- erals: Unexpected Behavior and Bridging Faults Diagnosis," "An Algorithm for Diagnosing Two-Line Bridging Faults in CMOS Combinational Circuits," "SCRIPT: A Critical Path Tracing Algorithm for Synchronous Sequential Circuits," "Why is Less Information From Logic Simulation More Useful in Fault Simulation?," Digital System Testing and Testable Design. "Combinational Profiles of Sequential Benchmark Circuits," "Cost- Effective Generation of Minimal Test Sets for Stuck-at Faults in Combinational Logic Circuits," --TR An algorithm for diagnosing two-line bridging faults in combinational circuits Diagnosis of realistic bridging faults with single stuck-at information E-PROOFS Beyond the Byzantine Generals Biased Voting --CTR Srikanth Venkataraman , Scott Brady Drummonds, Poirot: Applications of a Logic Fault Diagnosis Tool, IEEE Design & Test, v.18 n.1, p.19-30, January 2001 Yu-Shen Yang , Andreas Veneris , Paul Thadikaran , Srikanth Venkataraman, Extraction Error Modeling and Automated Model Debugging in High-Performance Low Power Custom Designs, Proceedings of the conference on Design, Automation and Test in Europe, p.996-1001, March 07-11, 2005 Andreas Veneris , Jiang Brandon Liu, Incremental Design Debugging in a Logic Synthesis Environment, Journal of Electronic Testing: Theory and Applications, v.21 n.5, p.485-494, October 2005

bridging faults;deduction;diagnosis

266569

A SAT-based implication engine for efficient ATPG, equivalence checking, and optimization of netlists.

The paper presents a flexible and efficient approach to evaluating implications as well as deriving indirect implications in logic circuits. Evaluation and derivation of implications are essential in ATPG, equivalence checking, and netlist optimization. Contrary to other methods, the approach is based on a graph model of a circuit's clause description called implication graph. It combines both the flexibility of SAT-based techniques and high efficiency of structure based methods. As the proposed algorithms operate only on the implication graph, they are independent of the chosen logic. Evaluation of implications and computation of indirect implications are performed by simple and efficient graph algorithms. Experimental results for various applications relying on implication demonstrate the effectiveness of the approach.

Introduction Recently, substantial progress has been achieved in the fields of Boolean equivalence checking and optimization of netlists. Techniques for deriving indirect implications, which were originally developed for ATPG tools, play a key role in this development. Indirect implications have been successfully applied in algorithms for optimizing netlists. For this task, either a set of permissible transformations is derived [1, 2, 3] or promising transformations are applied and their permissibility is later verified by an ATPG tool [4, 5, 6]. Furthermore, they are of great importance in ATPG-based approaches to Boolean equivalence checking of both combinational and sequential circuits [7, 8, 9, 10, 11] as they help identify equivalent internal signals in the circuits to be compared. In the late 1980s, Schulz et al. incorporated computation of indirect implications into the ATPG tool SOCRATES[12]. Indirect implications are indispensable when dealing with redundant faults as they help to efficiently prune the search space of the branch- and-bound search. In order to derive more indirect implications, the originally static technique of SOCRATES, which the authors refer to as (static) learning, has been extended to dynamic learning [13, 14]. Recursive learning [7], proposed by Kunz et al. in 1992, was the first complete algorithm for determining indirect implications. As the problem of finding all indirect implications is NP-complete, only small depths of recursion are feasible. Recently, it has been shown that recursive learning can be adequately modelled by AND-OR reasoning graphs [3]. Another complete method for deriving indirect implications based on BDDs was suggested by Mukherjee et al. [15]. Very recently, Zhao et al. presented an approach that combines iterated static learning with recursive learning constrained to recursion level one [16]. It is based on set algebra and is similar to single pass deductive fault simulation. Contrary to the above methods, which work on the structural description of a circuit, other approaches use a Boolean satisfiability based model. The SAT-model allows an elegant problem formulation which can easily be adapted to various log- ics. This abstraction, however, often impedes development of efficient algorithms as structural information is lost. Larrabee included a clause based formulation of Schulz's algorithm into NEMESIS[17]. Her approach has been improved by the iterated method of TEGUS [18]. The transitive closure algorithms suggested by Chakradhar et al. rely on a relational model of binary clauses [19]. Silva et al. proposed another form of dynamic learning in GRASP [20] where indirect implications are determined by a conflict analysis during the backtracking phase of a SAT-solver. In many areas of logic synthesis and formal verification Binary Decision Diagrams (BDD) have become the most widely used data structure as they provide many advantageous properties, e.g. canonicity and high flexibility. Besides their exponential memory complexity, when used for ATPG, equivalence checking, and optimization of large netlists, BDDs suffer from the drawback that implications cannot be derived efficiently on this data structure. For a given signal assignment it can only be decided if another signal assignment is implied or not. So, finding all possible implications from a given signal assignment is expensive because theoretically all possible combinations of signal pairs have to be checked. Therefore, BDD-based approachessuch as functional learning [15] restrict their search to potential learning areas, which are identified by non BDD-based implication. Consequently, structural or hybrid approaches, i.e. BDDs combined with other methods, are predominant in ATPG, equivalence checking and optimization of netlists. Even though most of these approachesmake heavy use of implica- tions, the data structures that are used for deriving and evaluating implications are often suboptimal and inflexible. That is why we propose a flexible data structure which is specifically optimized with respect to implication. In this paper, we introduce a framework for implication based algorithms which inherits the advantages of structural as well as SAT-based approaches. Our approach combines both the flexibility and elegance of a SAT-based algorithm and the efficiency of a structural method by working on a graph model of the clause system, called implication graph. Its memory complexity is only linear in the number of modules in the circuit. Due to structural information available in the graph, fundamental problems such as justification, propagation and particularly implication are carried out efficiently on the graph. The search for indirect implications reduces to graph algorithms that can be executed very fast and are easily extended to exploit bit-parallelism. As the implication graph can automatically be generated for any arbitrary logic, all presented algorithms remain valid independent of the chosen logic. This allows rapid prototyping of implication based tools for new multi-valued logics. The remainder of this paper is organized as follows. In Sec. 2, we show how to derive the implication graph. Next, we discuss how implications are evaluated and how indirect implications can be computed in Sec. 3 and 4, respectively. In order to demonstrate the high efficiency of our approach, experimental results for various applications using the proposed implication engine are presented in Sec. 5. Sec. 6 concludes the paper. Implication graph As performing implications is one of the most prominent and time consuming tasks in ATPG, equivalence checking, and optimization of netlists, it is of utmost importance to use a data structure that is best suited. Unlike other graphical representations of clause systems, our data structure represents all information contained in both the structural netlist and the clause database. The implication graphs used in NEMESIS[17] and TRAN[19] model only binary clauses, clauses of a higher order are solely included in the clause database. Since our approach is generic in nature, any combinational circuit can automatically be compiled into its implication graph rep- resentation. Only information about a logic and its encoding as well as the truth table descriptions of supported module types have to be provided. The basic steps of compilation are given in Fig. 1. First, all supported module types are individually compiled into encoded table clauses module database circuit implication subgraph logic encoding module optimization implication graph Figure 1: Deriving the implication graph encoded truth tables. Then, these tables are optimized by a two-level logic optimizer, e.g. ESPRESSO. This step is explained in Sec. 2.1. Next, a set of clauses is extracted from the optimized ta- ble, which is shown in Sec. 2.2. As shown in Sec. 2.3, the set of clauses is transformed into an implication subgraph that is stored in the module database. Then, for every module in the circuit the appropriate generic subgraph is taken from the module database and personalized with the input and output signals of the given module. Finally, all identical nodes are merged into a single node resulting in the complete implication graph. The following sections only consider the 3-valued logic L f0;1;Xg in order to present the basic ideas of our approach. Generation of an implication graph for an arbitrary multi-valued logic, e.g. the 10-valued logic L 10 known from robust path delay ATPG, is discussed in [21]. 2.1 Encoding A signal variable x 2L 3 requires two encoding bits c x and c x for its internal representation. The complete scheme of encoding for L 3 is shown in Table 1. In order to easily detect inconsistencies, encoding interpretation x signal x is 1 signal x is unknown conflict at signal x Table 1: 3-valued logic and its encoding conflicting signal assignments are denoted by c property is expressed in the following definition: DEFINITION 1 An assignment is called non-conflicting iff c x - c holds for all signal variables x. Based on this encoding, the truth tables of all supported module types are converted into encoded tables. For example, the truth table of a 2-input AND-gate found in Table 2 is converted into the encoded table of Table 2. This encoded table can truth table a b c encoded table c a c a c b c c optimized table c a c a c b c c Table 2: AND-gate: truth table - encoded table - optimized table be interpreted as specifying the on-set as well as the off-set of two Boolean functions c c and c c . Conflicting assignmentsbelong to the don't-care-set, as they are explicitly checked for by the implication engine. Exploiting these don't-cares, functions c c and c c in the encoded table are optimized by ESPRESSO. 2.2 Clause description The characteristic function describing the AND-gate with respect to the given encoding can easily be given in its Conjunctive Normal Form (CNF) by analyzing the individual rows of the optimized table of Table 2. Every row in this table corresponds to a clause contained in the CNF. Here, the CNF comprises the three clauses :c a - c c , :c c , and :c a -:c b - c c . That is, all valid value assignments to the inputs and outputs of the AND-gate are implicitly given by the non-conflicting satisfying assignments to the characteristic equation: CNF , (:c a -c 2.3 Building the implication graph By exploiting the following equivalencies the clause description of Eq. (1) is converted into the corresponding implication graph. x-y It is sufficient to provide equivalencies for binary and ternary clauses only, as any clause system of a higher order can be decomposed into a system of binary and ternary clauses [21]. Having transformed all clauses into binary and ternary clauses, the sub-graphs shown in Fig. 2 are used for representation of these clauses. These graphs contain two types of nodes. While the first type rep- :y :y y z x Figure 2: Implication subgraph for a binary and a ternary clause resents the encoded signal values, the second one symbolizes the conjunction operation. The latter type is depicted by - or a shaded triangle. Every ternary clause has three associated -nodes that uniquely represent the ternary clause in the implication graph. Coming back to the 2-input AND-gate, its CNF-description is transformed into the implication graph shown in Fig. 3. Every bit of the encoding for a signal x is represented by a corresponding node in the implication graph, e.g. node c a (c a ) in Fig. 3 gives bit c a (c a ) of signal a. As we require non-conflicting assignments, literals x ) can be replaced by c x only nodes corresponding to non-negated encoding bits are contained in Fig. 3. So far, the implication graph only captures the logic functionality of a circuit. Since structural information is indispensable for some tasks, such as justification and propagation, we provide this information within the implication graph by marking its edges with three different tags f , b, and o. Edges that denote an implication from an input to an output signal of a module are marked with f (forward edge). Relations from output to input signals are tagged with b (backward edge). All other edges, e.g. input to input relations and indirect implications, are given tag . The tags for the 2-input AND-gate are found in Fig. 3. By means of these tags, a directed acyclic graph (DAG) can be extracted from the implication graph. If all edges but the forward edges are re- moved, we obtain a DAG that forms the base of an efficient algorithm for backtracing and justification. For a simple circuit, the three different circuit descriptions introduced above are presented in Ex. 2.1. Please observe that most clause based approaches work on a CNF in L 2 . Our approach operates on a CNF of variables encoded with respect to a given logic, here L 3 . denoting other edges have been omitted in later examples. c c c c a c c c a f f f f Figure 3: Implication graph for 2-input AND-gate 2.4 Advantages Using the proposed implication graph as a core data structure in CAD algorithms has many advantages. (1) Important tasks such as implication and justification can be carried out on the implication graph in the same manner for any arbitrary logic. The peculiarities of the chosen logic are included in the graph. Implication and derivation of indirect implications reduce to efficient graph algorithms as will be shown in Sec. 3.3 and 4.4. (2) Most SAT-based algorithms use a static order for variable assignments during their search for a satisfying assignment [17, 19]. Furthermore, these algorithms assign values to internal signals during justification. Since PODEM, it has been well known that assigning values only to primary input signals helps to reduce the search space. Obviously, primary inputs are a special property of the given instance of SAT which is not exploited by algorithms for solving arbitrary SAT problems. The algorithm of TEGUS tries to mimic PODEM by ordering the clauses in a special manner [18]. Our approach does not need such techniques, as structural information is provided by edge tags. (3) Algorithms working on the implication graph can easily exploit bit-parallelism as the status of every node can be represented by one bit only. For example, on a 64-bit machine 64 value assignments can be processed in parallel, making bit-parallel implication very efficient. Sequential circuits are often modelled as an iterative logic array (ILA). In this model the time domain is unfolded into multiple copies of the combinational logic block. These logic blocks can be compiled into the corresponding implication graphs. Using bit-parallel techniques, a 64-bit machine allows to keep 64 time-frames without increasing the size of the implication graph. 3 How to perform implications 3.1 Structure based Structure based implication is a special form of event-driven simulation. Contrary to ordinary simulation, which starts at the primary inputs, implication is started at an arbitrary signal in the circuit. Therefore, it has to proceed towards the primary outputs as well as the primary inputs such that implications are often categorized into forward and backward implications. Obviously, this technique requires many table lookups for evaluating the module functions. This becomes particularly costly for multi-valued log- ics, e.g. the ones used in path delay ATPG. 3.2 Clause based Clause based implication relies on Boolean Constraint Propagation (BCP). BCP corresponds to an iterative application of the Example 2.1 Circuit descriptions: structural - clauses - implication graph a c d e f ffl CNF for (:c d -c e -c (:c a -c d a -:c c -c ffl Implication graph for f f f f f f f f f f f f f c d c e c f c c d c e c a c c c c c a c b unit clause rule proposed by Davis et al. in 1960 [22]. In BCP, unary clauses are used to simplify other clauses until no further simplification is possible or some clause becomes unsatisfied. Implication is started by adding a unary clause, which represents the initial signal assignment, to the CNF. All unary clauses computed by BCP correspond to implications from the initial assignment as they force the corresponding signals to a certain logic value. The most time consuming task in BCP is the search for clauses that can be simplified by the unit clause rule. This search is not necessary when working on the implication graph since clauses that share common variables are connected in the graph. 3.3 Implication graph based Implication graph based implication is simple and efficient, as it only requires a partial traversal of the implication graph. Implying from a signal assignment means that first the corresponding nodes are marked in the implication graph. Then, the implication procedure traverses the implication graph obeying the following rule: Starting from an initial set S I of marked nodes, all successor nodes s j are marked ffl if node s j is a -node and all its predecessors are marked. ffl if node s j represents an encoding bit and at least one predecessor is marked . This rule is applied until no further propagation of marks is possible All nodes that have been marked represent signal values that can be implied from the initial assignment given by S I . Conflicting signal assignments are easily detected during implication, since they cause both nodes c x and c x to be marked. Let us use the circuit of Ex. 2.1 for the sake of explanation. Assigning logical value 0 to signal e corresponds to marking node c e in the implication graph. After running the implication proce- dure, the following nodes are marked: c c and c f . To finally obtain the implied signal values with respect to the given logic, the marked nodes are decoded according to the given encoding, i.e. we determine Deriving indirect implications Contrary to direct implications, detection of indirect implications requires a special analysis of the logic function of a circuit as they represent information on the circuit that is not obvious from its description. Most methods for computation of indirect implications are subject to order dependency. That is, some indirect implications can only be found if certain other indirect implications have already been discovered. In order to avoid this problem, it has been suggested to iterate their computation [18]. 4.1 Structure based The SOCRATES algorithm [12] was the first to introduce computation of indirect implications using the following tautologies: While Eq. (4) (law of contraposition) may generate a candidate for an indirect implication, Eq. (5) identifies a fix value. Indirect implications are primarily computed in a pre-processing phase. The idea is to temporarily set a given signal to a certain logic value. Then, all possible direct implications from this signal assignment are computed. For all implied signal values, it is checked if the contrapositive cannot be deduced by direct implications (learning criterion). In this case, the contrapositive is an indirect implication. As indirect implications cannot be represented within the data structure used to describe the circuit, structural algorithms have to store them in an external data structure. This adds additional complexity to structure based algorithms. 4.2 Clause based Clause based computation [17, 18] is similar to the structural algorithm of Sec. 4.1. Each free literal a contained in the CNF is temporarily set to 1. Then BCP is used to derive all possible direct implications, i.e. unary clauses. For all generated unary clauses b, it is checked if the contrapositive :b ! :a is an indirect im- plication. In this case, the corresponding clause b -:a is added to the clause database. Thereby, indirect implications enrich the data structure used for representing the circuit functionality. Once an indirect implication has been added to the clause database, it does no longer require any special attention. This is one important advantage of clause based algorithms over structure based approaches [18]. 4.3 AND-OR enumeration A different approach, known as recursive learning, has been taken by Kunz et al. [3, 7]. Indirect implications are deduced by an search [23] for all possible implications resulting from a signal assignment. This search is performed by recursively injecting and reversing signal assignments, which correspond to the different possibilities for justifying a gate, followed by deriving all direct implications. Signal values that are common to all justifications of a gate yield indirect implications. Only a simple structural algorithm for executing implications is applied. Let us illustrate the principles of the AND-OR enumeration with the circuit of Ex. 2.1 and the AND-OR tree found in Fig. 4. The level 2 level 0 initial assignment f level 1 Figure 4: AND-OR enumeration root node of the AND-OR tree reflects the initial assignment, it is of the AND-type 2 . In our example, a logical 0 is assigned to signal f . As no further signal values can be implied, OR-node is the only successor of the root node. The justifications for 0g. In order to derive an indirect implication, we have to search for implied signal values that are common to both justifications. Here, implied for both justifications. This is represented by a new OR-node level 0 of the AND-OR tree. In general, new OR-nodes in level 0 correspond to indirect implications. Further examination of gates in level 2, which have become unjustified because of setting b to 0, does not yield additional indirect implications. 4.4 Implication graph based An implication graph based method for computing indirect implications inherits all advantages of clause based techniques but eliminates the costly search process required during BCP-based implication. Moreover, our approach integrates computation of indirect implications based on the law of contraposition and AND-OR enumeration into the same framework. In general, an AND-node (marked by an arc) represents a signal assignment due to justification of an unjustified gate, whereas an OR-node denotes a signal value that can be implied from a chosen justification. Justified gates correspond to OR-leaves and unjustified gates to internal OR-nodes in the AND-OR graph [3]. 4.4.1 Reconvergence analysis The basic idea of determining indirect implications by a search for reconvergencies is shown in Fig. 5. While implication c a ! c b c a indirect direct c c x c x c a c a - Figure 5: Learning by contraposition on the implication graph is deduced by direct implication, c a forms an indirect impli- cation. The -node can only be passed if both of its predecessors are marked, i.e. it forms a reconvergent -node during implication. If we start implication at node c b , however, we cannot pass the - node, as its other predecessor c x is not marked. Applying the law of contraposition to c a ! c b , we deduce c a such that c a is implied from c b . This observation is expressed in the following lemma: assignment. A reconvergent structure c y ) in the implication graph yields an indirect implication c x only if ffl c x is a fanout node in the implication graph. ffl a node c y is marked via a -node and both predecessors of the -node have been marked by implying along disjoint paths in the implication graph. (Proof: [21]) Using Lemma 1 it can be shown that the search for reconver- gencies in the implication graph detects all indirect implications, which are found by clause and structural based approaches. implications found by BCP on the (en- coded) clause description can be identified by a search for the reconvergent structures defined in Lemma 1. (Proof: [21]) We explain the reconvergence analysis with the implication graph of Ex. 2.1. Let's assume that fanout node c b is marked. Then, the implication procedure of Sec. 3.3 is invoked. As both c d and c e have been marked, the succeeding -node and c f are marked, too. The -node has been reached via two disjoint paths in the graph (indicated by the dashed and solid line, respectively) such that the contrapositive c b forms an indirect implication. This indirect implication is included into the graph in form of the grey edge leading from node c f to node c b . Applying our graph analysis offers the following advantages: (1) The search for reconvergence regions in the implication graph reduces the set of candidate signals that may yield an indirect im- plication. Clause based methods have to temporarily assign a value to all literals contained in the CNF. (2) Reconvergence analysis is carried out very fast by an adapted version of the algorithm presented in [24]. (3) Our method does not require a learning criterion such as the approach of [12]. 4.4.2 Extended reconvergence analysis Contrary to the reconvergence analysis of Sec. 4.4.1, the extended reconvergence analysis detects conditional reconvergencies at signal nodes. As it corresponds to an AND-OR search in the implication graph, we need the following definitions: clause C= c 1 -c 2 -c n is called unjustified do not evaluate to 1 and at least one complement :c i of a literal c i is 1. Unjustified ternary clauses are found in the implication graph without effort. They are represented by -nodes that have exactly one of their two predecessors marked. unspecified literals in a that is unjustified, and let V 1 denote the assigned values. Then, the set of non-conflicting assignments is called a justification of clause C, if the value assignments in J makeC evaluate to 1. In a clause based framework a complete set of justifications J c for an unjustified clause C is easily given by J 1gg. For our approach, set J c is even simpler, as only ternary clauses can be unjustified. 3 Therefore, J c always consists of exactly two justifications. We will now explain how these two justifications can be derived in the implication graph with Fig. 6. The given ternary clause c y c x c z c x c z c y Figure Unjustified ternary clause c x -c y -c z due to assignment c z is unjustified due to an assignment of c is indicated by the two -nodes that have exactly one predecessor Here, the ternary clause can be justified by setting c z or c y to 1. If we consider that the subgraph denoting the ternary clause c x - c y - c z is a straightforward graphical representation of the following formulae c x -c x -c y -c it becomes apparent that both possible justifications in J c are found in the consequents of those implications which have the literal making the clause unjustified, i.e. c x , in their antecedent. These consequents correspond to the successors of the two -nodes. Let us now explain how the extended reconvergence analysis corresponds to an efficient AND-OR search on the implication graph with help of Fig. 7 showing the implication graph of Ex. 2.1. An initial assignment of c clause C unjustified. Next, the possible justifications J a are determined as the successors of the two - nodes a 1 and a 2 belonging to clause C a . These -nodes correspond to AND-nodes J a 1 and J a 2 in the AND-OR tree, respec- 3 If a binary clause is unjustified according to Definition 2, it reduces to a unary clause. Unary clauses represent necessary assignments (implied signal values) for the given signal assignment. tively. So as to distinguish between the consequences of the two justifications, each one is assigned a different color. Thus, node c marker (represented by dashed lines in Fig. 7) and all signals that can be implied from c are marked green. The same is done for c using a red marker (dotted lines in Fig. 7). Nodes that are assigned both colors, i.e. nodes where the markers reconverge, can be implied independent of the chosen justification. These nodes can therefore be elevated to the previous level in the AND-OR tree. In our example, only node c b is marked by both colors and we derive the indirect implication c b . Further analysis of unjustified clauses C b and C g in level 2 of the AND-OR tree does not yield additional indirect implications This example indicates that the trace of the extended reconvergence analysis is identical to the AND-OR tree generated by AND-OR enumeration if marked -nodes are converted to AND-nodes and marked signal nodes to OR-nodes. Obviously the extended reconvergence analysis is capable of determining all indirect implications given enough colors, i.e. it is complete. An efficient procedure implementing this extended reconvergence analysis is given in [21]. It takes advantage of the implication graph by encoding the colors locally at the nodes using only bit slices of a full machine word. Thus, subtrees of the AND-OR tree are stored in parallel in different bit-levels. Additionally, a bit-parallel version of the implication algorithm introduced in Sec. 3.3 is used. Our algorithm supports a depth of r levels in the AND-OR tree on a 2 r -bit architecture. On a DECAlphaStation, for example, a maximal depth of 6 levels is available. Let us briefly summarize the advantages of our (1) The implication graph model allows the full word size to be exploited by means of bit-parallel techniques. The search for indirect implications, requires efficient set operations as an OR-node may only be elevated if it is a successor of both AND-nodes belonging to an unjustified clause. These set operations are carried out effectively on the implication graph by performing local bit-operations at signal nodes such that no separate data structure is needed. Please note, that the advantage of efficient set operations remains, if we extend our algorithm to handle arbitrary depths of AND-OR enumeration, which has already been done. (2) The notion of unjustified gates necessary in [3, 7] reduces to the simple concept of unjustified ternary clauses. Due to this concept and the uniformity of our description, AND-OR enumeration can easily be performed for arbitrary logics applying the same pro- cedure. This has already been done for logic L 10 . On the contrary, higher valued logics are complicated to deal with in the structural approach of [7, 3]. (3) Detected indirect implications can be included into the graph immediately, which often facilitates the computation of other indirect implications. (4) Some indirect implications are easily computed by the law of contraposition while requiring a high depth of AND-OR search. As our approach integrates both methods into one framework, indirect implications can be identified by the best suited technique. 5 Experimental results The implication engine, presented in this paper, has been implemented in a C language library of functions that has been applied successfully to several CAD problems. Please note, that some of c d c e c f c c d c e c a c c c c c a c b level 2 level 1 level 0 J a initial assignment fc c c c c c c a 3 a 1 J a a 2 Figure 7: Extended reconvergence analysis on the implication graph the presented results have already been published in papers dealing with application specific issues. The underlying implication engine was not discussed. We have included these results in order to show the efficiency of our flexible approach. While the experiments for ATPG and netlist optimization were carried out on a DECStation3000/600, the experiments for equivalence checking were performed on a DECAlphaStation250 4=266 . ATPG and netlist optimization rely on an earlier version of our implication engine, that does not support the techniques of Sec. 4.4.2. So far, these advanced techniques have only been used for equivalence checking. Table 3 presents results for ATPG considering various fault models [25, 26, 27]. Due to the flexibility of the implication graph non-robust robust stuck-at c5315 342117 643.4 81435 5251.8 5291 1.2 c7552 277244 1499.4 86252 5746.0 7419 5.2 Table 3: Result of test pattern generation the various logics (L 3 required for the different fault models could easily be handled. Table 3 gives the number of tested faults and CPU time required for performing ATPG for non-robust and robust path delay faults as well as stuck-at faults in combinational circuits (or sequential circuits with enhanced scan design). The excellent quality of the achieved results can be seen from fur- circuit # gates # literals delay time before after before after before after [s] c1908 488 402 933 803 41.2 33.9 1364 red.: 8.1% 3.2% 18.8% - Table 4: Results of delay optimization ther tables in [25, 26, 27] where an extensive comparison to other state-of-the-art tools is made. Results for optimization of mapped netlists with respect to delay are provided in Table 4. The basic idea and the approach, that applies our implication engine to verify the permissibility of circuit transformations, is described in [6]. The number of gates, literals, and the circuit delay before and after optimization, as well as the required CPU time are given. Results for equivalence checking of netlists are presented in Table 5. It lists the total time required for equivalence checking, i.e. circuit time[s] level max total indirect implications c432 1.3 1.2 1 Table 5: Results for verifying against redundancy free circuits ATPG plus computation of indirect implications, and the time consumed by the latter in columns 2 and 3, respectively. The maximal depth of AND-OR search necessary for successful verification is also given in column 4. We provide these early results in order to show that our implication engine forms a suitable data structure for building an efficient equivalence checker. Our straightforward approach adopts the basic idea of the well-known equivalence checker HANNIBAL [28] but does not include its advanced heuris- tics, e.g. observability implications and heuristics for candidate se- lection. Nevertheless, the results shown in Table 5 are comparable to the ones reported in [28]. This indicates that our implication engine is well suited for equivalence checking. Please note, that it is easily incorporated into state-of-the-art implication based or hy- brid, i.e. BDDs combined with implications, equivalence checkers such that these approaches can benefit, too. 6 Conclusion In this paper we have proposed an efficient implication engine working on a flexible data structure called implication graph. It has been shown that indirect implications can be effectively computed by analysis of the graph. Experimental results confirm the efficiency and flexibility of our approach. In the future, our preliminary equivalence checker will be extended by deriving observability implications directly on the implication graph. Furthermore, we will investigate how a hybrid technique using BDDs and the implication graph can be advantageous for equivalence checking. Acknowledgements The authors are very grateful to Prof. Kurt J. Antreich for many valuable discussions and his advice. They like to thank Bernhard Rohfleisch and Hannes Wittmann for using the implication engine in the netlist optimization tool and developing the path delay ATPG tool, respectively. --R "Multi-level logic optimization by implication analysis," "LOT: Logic optimization with testability - new transformations using recursive learning," "And/or reasoning graphs for determining prime implicants in multi-level combinational networks," "Combinational and sequential logic optimization by redundancy addition and removal," "Perturb and simplify: Multi-level boolean network optimizer," "Logic clause analysis for delay optimization," "Recursive learning; a new implication technique for efficient solutions to cad problems - test, veri- fication, and optimization," "Advanced verification techniques based on learning," "A novel framework for logic verification in a synthesis environment," "Verilat: Verification using logic augmentation and transformations," "Aquila: An equivalence verifier for large sequential circuits," "Socrates: A highly efficient automatic test pattern generation system," "Improved deterministic test pattern generation with applications to redundancy identification," "Accelerated dynamic learning for test pattern generation in combinational circuits," "Functional learning: A new approach to learning in digital circuits," "Static logic implication with application to redundancy identification," "Test pattern generation using boolean satisfiability," "Com- binational test generation using satisfiability," "A transitive closure algorithm for test generation," "Grasp - a new search algorithm for satisfiability," "A sat-based implication engine," "A computing procedure for quantification theory," "A method of fault simulation based on stem regions," "A formal non-heuristic atpg approach," "Bit parallel test pattern generation for path delay faults," "Path delay atpg for standard scan designs," "Hannibal: An efficent tool for logic verification based on recursive learning," --TR Artificial intelligence Perturb and simplify Multi-level logic optimization by implication analysis Advanced verification techniques based on learning Logic clause analysis for delay optimization Path delay ATPG for standard scan design A formal non-heuristic ATPG approach VERILAT GRASPMYAMPERSANDmdash;a new search algorithm for satisfiability A Computing Procedure for Quantification Theory Bit parallel test pattern generation for path delay faults Static logic implication with application to redundancy identification --CTR Joo Marques-Silva , Lus Guerra e Silva, Solving Satisfiability in Combinational Circuits, IEEE Design & Test, v.20 n.04, p.16-21, January F. Lu , M. K. Iyer , G. Parthasarathy , L.-C. Wang , K.-T. Cheng , K. C. Chen, An Efficient Sequential SAT Solver With Improved Search Strategies, Proceedings of the conference on Design, Automation and Test in Europe, p.1102-1107, March 07-11, 2005 Alexander Smith , Andreas Veneris , Anastasios Viglas, Design diagnosis using Boolean satisfiability, Proceedings of the 2004 conference on Asia South Pacific design automation: electronic design and solution fair, p.218-223, January 27-30, 2004, Yokohama, Japan Paul Tafertshofer , Andreas Ganz, SAT based ATPG using fast justification and propagation in the implication graph, Proceedings of the 1999 IEEE/ACM international conference on Computer-aided design, p.139-146, November 07-11, 1999, San Jose, California, United States Sean Safarpour , Andreas Veneris , Rolf Drechsler , Joanne Lee, Managing Don't Cares in Boolean Satisfiability, Proceedings of the conference on Design, automation and test in Europe, p.10260, February 16-20, 2004 Ilia Polian , Bernd Becker, Multiple Scan Chain Design for Two-Pattern Testing, Journal of Electronic Testing: Theory and Applications, v.19 n.1, p.37-48, February Christoph Scholl , Bernd Becker, Checking equivalence for partial implementations, Proceedings of the 38th conference on Design automation, p.238-243, June 2001, Las Vegas, Nevada, United States Ilia Polian , Hideo Fujiwara, Functional constraints vs. test compression in scan-based delay testing, Proceedings of the conference on Design, automation and test in Europe: Proceedings, March 06-10, 2006, Munich, Germany Lus Guerra e Silva , L. Miguel Silveira , Joa Marques-Silva, Algorithms for solving Boolean satisfiability in combinational circuits, Proceedings of the conference on Design, automation and test in Europe, p.107-es, January 1999, Munich, Germany E. Goldberg , M. Prasad , R. Brayton, Using SAT for combinational equivalence checking, Proceedings of the conference on Design, automation and test in Europe, p.114-121, March 2001, Munich, Germany Joo Marques-Silva , Thomas Glass, Combinational equivalence checking using satisfiability and recursive learning, Proceedings of the conference on Design, automation and test in Europe, p.33-es, January 1999, Munich, Germany Ilia Polian , Hideo Fujiwara, Functional Constraints vs. Test Compression in Scan-Based Delay Testing, Journal of Electronic Testing: Theory and Applications, v.23 n.5, p.445-455, October 2007 Ilia Polian , Alejandro Czutro , Bernd Becker, Evolutionary Optimization in Code-Based Test Compression, Proceedings of the conference on Design, Automation and Test in Europe, p.1124-1129, March 07-11, 2005 Joo P. Marques-Silva , Karem A. Sakallah, Boolean satisfiability in electronic design automation, Proceedings of the 37th conference on Design automation, p.675-680, June 05-09, 2000, Los Angeles, California, United States Ilia Polian , Bernd Becker, Scalable Delay Fault BIST for Use with Low-Cost ATE, Journal of Electronic Testing: Theory and Applications, v.20 n.2, p.181-197, April 2004

efficient ATPG;structure based methods;SAT-based implication engine;logic circuits;implication evaluation;indirect implications;implication graph;equivalence checking;graph algorithms;netlist optimization;automatic testing;graph model;circuit clause description

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Java as a specification language for hardware-software systems.

The specification language is a critical component of the hardware-software co-design process since it is used for functional validation and as a starting point for hardware- software partitioning and co-synthesis. This paper pro poses the Java programming language as a specification language for hardware-software systems. Java has several characteristics that make it suitable for system specification. However, static control and dataflow analysis of Java programs is problematic because Java classes are dynamically linked. This paper provides a general solution to the problem of statically analyzing Java programs using a technique that pre-allocates most class instances and aggressively resolve memory aliasing using global analysis. The output of our analysis is a control dataflow graph for the input specification. Our results for sample designs show that the analysis can extract fine to coarse-grained concurrency for subsequent hardware-software partitioning and co-synthesis steps of the hardware-software co- design process to exploit.

Introduction Hardware-software system solutions have increased in popularity in a variety of design domains [1] because these systems provide both high performance and flexibility. Mixed hardware-software implementations have a number of benefits. Hardware components provide higher performance than can be achieved by software for certain time-critical subsystems. Hardware also provides interfaces to sensors and actuators that interact with the physical envi- ronment. On the other hand, software allows the designer to specify the system at high levels of abstraction in a flexible environment where errors - even at late stages in the design - can be rapidly corrected [2]. Software therefore contributes to decreased time-to-market and decreased system cost. Hardware-software system design can be broken down into the following main steps: system specification, parti- tioning, and co-synthesis. The first step in an automatic hardware-software co-design process is to establish a complete system specification. This specification is used to validate the desired behavior without considering implementation details. Functional validation of the system specification is critical to keep the system development time short because functional errors are easier to fix and less costly to handle earlier in the development process. Given a validated system specification, the hardware-software partitioner step divides the system into hardware, software subsystems, and necessary interfaces by analyzing the concurrency available in the specification. The partitioner maps concurrent blocks into communicating hardware and software components in order to satisfy performance and cost constraints of the design. The final co-synthesis step generates implementations of the different subsystems by generating machine code for the software subsytems and hardware configuration data for the hardware subsystems. The system specification is a critical step in the co-design methodology because it drives the functional validation step and the hardware-software partitioning process. Thus, the choice of a specification language is important. Functional validation entails exploration of the design space using simulation; hence, the specification must allow efficient execution. This requires a compile and run-time environment that efficiently maps the specification onto general-purpose processor platforms. On the other hand, the partitioning process requires a precise input specification whose concurrency can be clearly identified. Generating a precise specification requires language constructs and abstractions that directly correspond to characteristics of hardware or software. Traditionally, designers have not been able to reconcile these two objectives in one specification language, but have instead been forced to maintain multiple specifications. Obviously, maintaining multiple specifications of the design is at best tedious due to the need to keep all specifications synchronized. It is also error-prone because different specification languages tend to have different programming models and semantics. This need for multiple specifications is due to shortcomings of current specification languages used in hardware-software co-design. Hardware-software specification languages currently used by system designers can be divided into software programming languages and hardware description languages. Software languages such as C or C++ generate high-performance executable specifications of system behavior for functional validation. Software languages are traditionally based on a sequential execution model derived from the execution semantics of general purpose processors. However, software languages generally do not have support for modeling concurrency or dealing with hardware issues such as timing or events. These deficiencies can be overcome by providing the designer with library packages that emulate the missing features [15]. A more serious problem is that software languages allow the use of indirect memory referencing which is very difficult to analyze statically. This makes it difficult for static analysis to extract implicit concurrency within the specification. Hardware description languages such as Verilog [5] and VHDL [6] are optimized for specifying hardware with support for a variety of hardware characteristics such as hierarchy, fine-grained con- currency, and elaborate timing constructs. Esterel is another specification language similar to Verilog with more constructs for handling exceptions [7]. SpecCharts builds on a graphical structural hierarchy while using VHDL to specify the implementations of the various structures in the hierarchy [4]. These languages do not have high-level programming constructs, and this limits their expressiveness and makes it difficult to specify software. Fur- thermore, these languages are based on execution models that require a great deal of run-time interpretation such as event-driven semantics. This results in low-performance execution compared to software languages. This paper advocates the use of Java as a single specification language for hardware-software systems by identifying language characteristics that enable both efficient functional validation and concurrency exploration by the hardware-software partitioner. Java is a general-pur- pose, concurrent, object-oriented, platform-independent programming language [10]. Java is implementation-independent because its run-time environment is an abstract machine called the Java virtual machine (JVM) with its own instruction set called bytecodes [11]. The virtual machine uses a stack-based architecture; therefore, Java byte-codes use an operand stack to store temporary results to be used by later bytecodes. Java programs are set in an object-oriented framework and consist of multiple classes, each of which is compiled into a binary representation called the classfile format. This representation lays out all the class information including the class's data fields and methods whose code segments are compiled into bytecodes. These fields and methods can be optionally declared as static. Static fields or methods of a class are shared by all instances of that class while non-static fields or methods are duplicated for each new instance. Data types in Java are either primitive types such as integers, floats, and characters or references (pointers) to class instances and arrays [10]. Since Java classes are predominantly linked at run-time, references to class instances cannot be resolved at compile- time. This presents a challenge to static analyzers in determining data flow through data field accesses and control flow through method calls. This paper also outlines a control/dataflow analysis technique that can be used as a framework for detecting concurrency in the design. Our analysis technique provides a general solution for the problem of dynamic class allocation by aggressively pre-allocating most class instances at compile-time and performing global reference analysis. The rest of the paper is organized as follows. In Section 2 we explain why Java is well-suited for hardware-software system specification. In Section 3 we identify the problems that arise when analyzing Java programs and present a general solution for building control flow and dataflow dependence information. We apply our technique to three sample designs and analyze both explicit and implicit concurrency in these designs in Section 4. We conclude and briefly discuss future directions in Section 5. Hardware-Software Specification with Java It is desirable for the hardware-software co-design process to use a single specification language for design entry because specifications using different languages for software and hardware combine different execution mod- els. This makes these specifications difficult to simulate and to analyze. Some researchers begin with a software programming language usually C++ and extend this language with constructs to support concurrency, timing, and events by providing library packages or by adding new language constructs. Examples of these approach are Scenic [15] and V++ [17]. We take a slightly different approach. Instead of requiring the designer to specify the hardware implementation details in the specification, in our approach the designer models the complete system in an algorithmic or behavioral fashion. Software languages are well-suited for this type of modeling. Once the specification is com- plete, an automatic compilation process is used to analyze the specification to identify the coarse-grained concurrency described by the designer and uncover the finer-grained concurrency implicit in the specification. The partitioning and synthesis steps of the hardware-software co-design process use the concurrency uncovered by this analysis to create an optimized hardware-software system. The specification language used with this approach must have the ability to specify explicit concurrency and make it easy to uncover the implicit concurrency. Coarse-grained concurrency is intuitive for the designer to specify because hardware-software systems are often conceptualized as sets of concurrent behaviors [2]. Java is a multi-threaded language and can readily express this sort of concurrency. Such concurrent behaviors can be modeled by sub-classing the Thread class and overriding its run method to encode the thread behavior as shown in Figure 1. The Thread class provides methods such as suspend and resume, yield, and sleep that manipulate the thread. Syn- chronization, however, is supported at a lower level using monitors implemented in two bytecode operations that pro- Read Data loop: Generate x-array Proc A x-array Process x-array class system { static Thread procA, procB; public static void main(String argv[]) { new procAClass(); new procBClass(procA); // Launch threads system.class procAClass.class class procAClass extends Thread { boolean synchronized void setXArray(.) { . // Write x-array data synchronized arr_t getXArray(.) { . // Return x-array data public void run() { while (.not done.) { while (this.xready == true) for (int class procBClass extends Thread { procAClass procA; procBClass(Thread procA_in) { public void run() { while (.not done.) { while (this.xready == false) . // Process data procBClass.class Figure 1. Concurrency in Java vide an entry and an exit to the monitor. The sample design shown in Figure 1 maintains synchronization when reading and writing the x-array in methods getXArray and setXAr- ray which are tagged as synchronized. Fine-grained concurrency is usually either non-intuitive or cumbersome for the designer to express in the spec- ification. This implies that an automated co-design tool must be able to uncover fine-grained concurrency by analyzing the specification. The primary form of concurrency to look for is loop-level concurrency where multiple iterations of the same loop can be executed simultaneously. This form of concurrency is important to detect because algorithms generally spend most of their time within core loops. Identifying and exploiting parallel core loops can thus provide significant performance enhancements. Determining whether loop iterations are parallel requires analysis to statically determine if data dependencies exist across these loop iterations. In the run method of procA- Class shown in Figure 1, if the compute_xarray call does not depend on values generated in previous iterations of the for-loop, then all the iterations of the loop may be executed simultaneously. The major hurdle that the data dependence analysis must overcome is dealing with memory references because these references introduce a level of indirection in reading and writing physical memory locations. Compile-time analysis has to be conservative in handling such refer- ences. This conservatism is necessary to guarantee correct system behavior across transformations introduced by the partitioning step based on the results of the analysis. How- ever, this conservatism causes the analysis to generate false data dependences which are nonexistent at the system lev- el. These dependences reduce the data parallelism that the analysis detects. In the simple design shown in Figure 1, without the ability to analyze dependences within the for-loop and across the associated method call, conservative analysis would determine that the loop iterations are inter-dependent and hence can only be performed sequentially reducing the degree of data parallelism in that section of the specification by 100-fold. The advantage that Java has over a language like C++ is that Java restricts the programmer's use of memory references. In Java, memory references are strongly typed. Also, references are strictly treated as object handles and not as memory addresses. Consequently, pointer arithmetic is disallowed. This restrictive use of references enables more aggressive analysis to reduce false data dependences. A co-design specification language should provide high-performance execution to enable rapid functional val- idation. Java's execution environment uses a virtual ma- chine. The JVM provides platform-independence; however, this independence requires Java code to be executed by an interpreter which reduces execution performance compared to an identical specification modeled in C++. Although this performance degradation is at least an order of magnitude for Sun's JDK 1.0 run-time environ- ment, techniques such as just-in-time compilation are closing the performance gap to less than two-times that of C++[12][16]. This evolution in Java tools and technology has been and will be driven by Java's success in other do- mains, especially network-based applications. Moreover, the Java run-time environment makes it easy to instrument and gather profiling information which can be used to guide hardware-software partitioning. Analyzing Java Programs Control and dataflow analysis of the Java specification is required for partitioning and co-synthesis steps of the co-design process. This analysis examines the bytecodes of invoked methods to determine their relative ordering and data dependencies. These bytecodes have operand and result types that are either primitive types, or classes and ar- rays. While primitive types are always handled by value, class and array variables are handled by reference. These object (class instance) references are pointers; however, they are well-behaved compared to their C/C++ counter-parts because these references are strongly typed and cannot be manipulated. Object references point to class instances that are linked dynamically during run-time. So, prior to executing the Java program, we can only allocate the static fields and methods of the program's classes. This makes it difficult to statically analyze Java programs because if object references cannot be resolved, calls to methods of these dynamically linked objects cannot be resolved either. This makes it impossible to determine control flow. The only way to deal with this problem is to conservatively assign the method invocation to software so that the software run-time system can handle the dynamic resolution. However, this reduces the opportunities for extracting parallelism in hardware and thus leads to inferior hardware-software design. In order to avoid the problem with dynamically linked objects, the specification could be restricted to use only static fields and methods or be forced to allocate all necessary objects linearly at the beginning of the program. How- ever, this would significantly restrict the use of the language. Our solution is to attempt to pre-allocate objects during static analysis. It should be noted that this approach does not handle class instantiations within loops or recursive method invocations. Pre-allocation only partially solves the problem with dynamically allocated class instances. A class reference can point to any instance of compatible class type; there- fore, two references of compatible class types can alias. Conservative handling of reference aliasing reduces the apparent concurrency in the specification. More aggressive reference aliasing analysis requires global dataflow analysis to determine a class instance or set of instances that a reference may point to. An outline of our analysis technique is shown in Figure 2. The analysis starts with the static main method. For each method processed, local analysis is performed to determine local control and dataflow. Next, all methods invoked by the current method are recursively analyzed. Fi- nally, reference point-to values are resolved in order to determine global data dependence information. Before elaborating on the techniques used to perform the local and global analyses, we describe the target representation of the CDFG. The CDFG representation shown in Figure 3 involves two main structures. The first structure is a table of static and pre-allocated class instances. Aside from object accounting information, this table maintains a list of entries per object; each entry represents either a method or a non- Figure 2. Analysis technique outline ProcessMethod (current_method) { Perform local analysis on current_method to build local control flow information and resolve local dependencies. Pre-allocate new instantiations if not inside loops or recursion. For each method invoked { ProcessMethod (invoked_method) Resolve reference global analysis impacted by invoked_method Resolve global dependencies given complete reference analysis ProcessMethod (main) Figure 3. Target representation Method Call Graph Basic Block Control Flow Pre-allocated Entities Static Entities Graph primitive type data field. The data field entry is necessary for global analysis because data fields have a global scope during the life of their instances. Arrays are treated exactly as class instances. In fact, arrays are modeled as classes with no methods. The method entries point to portions of the second main structure in the representation. The second structure is the control dataflow information. Its nodes are bytecode basic blocks. The edges represent local control flow between basic blocks within a methods as well as global control flow across method invocations and returns. The CDFG representation models multi-threading and exceptions using special control flow edges that annotate information about the thread operation performed or the exception trapped. Thread operations in Java are implemented in methods of the Thread class. The CDFG abstracts invocations of these methods by encoding the associated operation in the control flow edge corresponding to the method call. For example, Java threads are initiated by invoking Thread class start method. When the CDFG encounters an invocation of the start method, a new control flow edge is inserted between the invocation and the start of the thread's run method. This edge also indicates that a new thread is being forked. On the other hand, Exceptions in Java use try-catch blocks where the code which may cause an exception is placed inside the try clause followed by one or more subsequent catch clauses. Catch blocks trap on a specified thrown exception and execute the corresponding handler code. The CDFG inserts special control flow edges between the block that may cause the exception and the handler block. These edges are annotated with the type of exception the handler is trap- ping. An example of how exceptions are handled in shown in Figure 4. 3.1 This step targets a particular method, identifying and sequencing its basic blocks to capture the local control flow. It also resolves local dependencies at two distinct lev- els. First, since Java bytecodes rely on an operand stack for Arithmetic Handler Code Exception Figure 4. Exception edges in CDFG try { int catch (ArithmeticException e) { // handler code. the intermediate results, the extra level of dependency indirection through the stack needs to be factored out. This is achieved using bytecode numbering. Second, dependencies through local method variables are identified using reaching definition dataflow analysis. flow is represented by the method's basic blocks and the corresponding sequenc- ing. Basic blocks are sequences of bytecodes such that only the first bytecode can be directly reached from outside the block and if the first bytecode is executed, then all the byte-codes are sequentially executed. The control flow edges simply represent the predecessor-successor ordering of all the basic blocks. Analysis. Dependencies exist between bytecodes through an extra level of indirection - the operand stack. We resolve this indirection by using "bytecode numbering." Bytecode numbering simply denotes the replacement of the stack semantics of each bytecode analyzed with physical operands that point to the bytecode that generated the required result. This is simply achieved by traversing the method's bytecodes in program order. Instead of executing the bytecode, its stack behavior is simulated using a compile-time operand stack, OpStack. If the bytecode reads data off the stack, entries are popped off OpStack, and new operands are created with the values retrieved from the stack. If the bytecode writes a result to the stack, a pointer to it is pushed onto OpStack. This process has to account for data that requires more than one stack entry such as double precision floating point and long integer results. Also, stack-manipulating bytecodes such as dup (duplicate top entry) or swap (swap top two entries) are interpreted by manipulating OpStack accordingly. Then, these bytecodes are discarded since they are no longer needed for the purposes of code functionality. An outline and an example of bytecode numbering are shown in Figure 5. Data dependencies across local variables are resolved by computing the reaching definitions for the particular method. A definition of a variable is a bytecode that may assign a value to that variable. A definition d reaches some point p if there exists a path from the position of d to p such that no other definition that overwrites d is encountered. Once all the reaching definitions are computed, it would be clear that there exists a data dependency between bytecode m and bytecode n if m defines a local variable used by n and m's definition reaches the point immediately following n. Computing the reaching definitions uses the iterative dataflow Worklist algorithm [8]. This algorithm iterates over all the basic blocks. A particular basic block propagates definitions it does not overwrite. At a join point of multiple control branches, the set of reaching definitions is the union of the individual sets. The algorithm iterates over the set of successors of all basic blocks whose output set of reaching definitions changes and converges when no more changes in these sets of reaching definitions materialize. 3.2 Global Analysis To handle data dependencies between references, global analysis generates for each reference the set of object instances to which it may point out of the set of pre-allocated instances. Once this points-to relation is determined, simple dataflow analysis techniques such as global reaching definition can compute dataflow dependencies between these references. A straightforward solution is to examine the entire control flow graph while treating method invocations as regular control flow edges. Then, iterative dataflow analysis can generate the points-to information for every refer- ence. However, this approach suffers from the problem of unrealizable paths which cause global aliasing information to propagate from one invocation site to a non-corresponding return site [9]. Figure 5. Bytecode numbering example179iload_1 ldc_w #4 ireturn ireturn Initialize symbolic operand stack, OpStack, to empty. Traverse basic blocks in reverse postorder. If current bytecode reads data from the stack, pop OpStack into the appropriate bytecode operand slot. If current bytecode writes data to the stack, push the bytecode's PC unto OpStack. push local variable onto operand stack push constant onto operand stack pop two entries, if first < second -> jump to PC= 9 pop entry and return from method with entry as return value pop entry and return from method with entry as return value OpStack Status Current Bytecode 9 iconst_m1 8 ireturn A more context-sensitive solution motivated by [9] is to generate a transfer function for each method to summarize the impact of invoking that method on globally accessible data and references. The variables that this transfer function maps are the formal method parameters that are references. In addition, this set of variables is extended to include global references used inside the method through (1) creating new instances, (2) invoking other methods that return object references, or (3) accessing class instance fields that are references. Input to this transfer function is the initial points-to values of the extended parameters set. Output generated by this transfer function is the final points-to values of the extended parameters due to the method invocation. This transfer function is a summary of the accesses (reads and writes) of the method's extended parameters generated using interval analysis [8]. These accesses are ordered according to the method's local control flow infor- mation. Accesses can be one of the following five primitive operations: read, assign, new, meet and invoke. The read primitive requires one operand which is a reference; the result is the set of potential class instances to which the reference points. The assign primitive is used to summarize an assignment whose left-hand side is an extended parameter. It requires two operands the first of which is the target ref- erence. The second is a set of potential point-to instances. The new primitive indicates the creation of a new class in- stance. This primitive returns a set composed of a single in- stance, if pre-allocation is possible (not within loop or recursion). Otherwise, it conservatively points-to the set of compatible class instances. The meet primitive is necessary to handle joining branches in the control flow. At a meet point, the alias set of some reference assigned in one or more of the meeting branches is the union of the alias sets for that reference from each of the meeting control flow edges. Finally, the invoke primitive is used to resolve change in reference alias sets due to invoking some meth- od. Effectively, this primitive causes the transfer function of the invoked method to be executed. some_method (obj a, obj b) { new obj(); if (test1) Figure 6. Transfer functions for global reference analysis TF{some_method}: assign ( a , new ( obj 4 Experimental Results Hardware-software systems are multi-process sys- tems, so partitioning and co-synthesis tools which map behavioral specifications to these systems need to make hardware-software trade-offs [13]. To make these trade-offs with the objective of maximizing the cost-performance of the mixed implementation, it is necessary to be able to identify the concurrency in the input specification. We have implemented our Java front-end analysis step as a stand-alone compilation pass that reads the design's class files and generates a corresponding CDFG representation. We tested our technique using the designs listed in Table 1. The first design, raytracer, is a simple graphical appli- cation. It renders two spheres on top of a plane with shadows and reflections due to a single, specular light source. The second application, robotarm, is a robot arm control- ler. The third design, decoder, is a digital signal processing application featuring a video decoder for H.263-encoded bitstreams [14]. The resulting control-dataflow graphs were analyzed to identify concurrency in the specification. The analysis examined concurrency at three levels: thread-level, loop- level, and bytecode-level. Thread-level concurrency is exhibited as communicating, concurrent processes which can span the control flow of several methods. Loop-level concurrency is exhibited by core loops usually confined to a single method. Bytecode-level concurrency is exhibited by bytecode operations that can proceed provided their data dependencies are satisfied irrespective of a control flow or- dering. This form of concurrency exists within basic blocks. Thread-level concurrency is explicitly expressed by the designer through Java threads. Since threads are uniquely identified in the CDFG, no work is required to uncover this form of parallelism. Loop-level concurrency requires analysis of control and dataflow information associated with inner loops to identify data dependencies spanning different loop iterations and determine if these are true dependencies, that is, dependencies between a write in some iteration of the loop and a read in a subsequent itera- tion. So, loops with independent iterations can execute these iterations concurrently as mini-threads. The coarse-grained concurrency expressed at the thread or loop level Lines of Java Classes Instances Basic Blocks raytracer 698 6 37 358 Table 1: Design characteristics can be exploited by allocating these threads to different subsystems in our target architecture. On the other hand, bytecode-level concurrency in the CDFG does not span multiple basic blocks; it exists at the bytecode level within each basic block. Its degree depends on the basic block's ``inter-bytecode'' data dependencies. This fine-grained concurrency impacts the performance improvement of a hardware implementation of the basic block. Hardware is inherently parallel; therefore, parallelism in the design is implemented without any cost overhead given enough structural resources to support the parallelism. The only limitation on the degree of parallelism is synchronization due to data dependencies. Hence the execution time of some block in hardware decreases with increased data parallelism. Table 2 presents the results of analyzing the three different forms of parallelism in our sample designs . The first column indicates the number of designer-specified threads. The second column shows the number of parallelizable loops while the third column indicates the average number of bytecodes per loop. The fourth column shows the average number of bytecodes per basic block while the fifth column assesses the average data parallelism in these basic blocks. This bytecode-level concurrency is measured as the average number of bytecodes that can execute simultaneously during a cycle of the JVM. These results show that it is possible to extract parallelism at various levels of granularity for Java programs. 5 Conclusions and Future Work The specification language is the starting point of the hardware-software co-design process. We have described key requirements of such a language. A specification language should be expressive so that design concepts can be easily modeled but should provide a representation that is relatively easy to analyze and optimize for performance. The language should also provide high-performance exe- cution. We have shown that the Java programming language satisfies these requirements. Thread-level Concurrency Loop-level Concurrency Bytecode-level Concurrency Number of threads Number of loops Avg. bytecodes per loop Avg. basic block Avg. bytecode parallelism decoder 3 28 27 7.1 2.5 Table 2: Parallelism assessment results To be able to partition and eventually co-synthesize input Java specifications, we must be able to analyze the specification. However, a major problem facing this analysis step in Java are dynamic links to class instances. To make static analysis possible, we proposed a technique that relies on aggressive reference analysis to resolve ambiguity in global control and dataflow. This technique generates a control dataflow graph representation for the specification. Our results show that using this technique it is possible to extract concurrency which can be exploited from the Java specification. In the future, our analysis technique will serve as a front-end to a co-design tool which maps the Java system specification to a target architecture composed of one or more microprocessors tightly coupled to programmable hardware resources. Acknowledgments This work was sponsored by ARPA under grant no. MIP DABT 63-95-C-0049. --R "Hardware-Software Co- Design," "Specification and Design of Embedded Hardware-Software Systems," Specification and Design of Embedded Systems. The Verilog Hardware Description Language. IEEE Inc. "The Esterel Synchronous Programming Language: Design, Semantics, Implementation," Compilers Principles "Efficient Context-Sensitive Pointer Analysis for C Programs," The Java Language Specification. The Java Virtual Machine Specification. "Java Performance Advancing Rapidly," "Multiple-Process Behavioral Synthesis for Mixed Hardware-Software Systems," Enhanced H. 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Nayak , M. Haldar , A. Choudhary , P. Banerjee, Precision and error analysis of MATLAB applications during automated hardware synthesis for FPGAs, Proceedings of the conference on Design, automation and test in Europe, p.722-728, March 2001, Munich, Germany Annette Bunker , Ganesh Gopalakrishnan , Sally A. Mckee, Formal hardware specification languages for protocol compliance verification, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.9 n.1, p.1-32, January 2004

specification languages;hardware-software co-design

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Delay bounded buffered tree construction for timing driven floorplanning.

As devices and lines shrink into the deep submicron range, the propagation delay of signals can be effectively improved by repowering the signals using intermediate buffers placed within the routing trees. Almost no existing timing driven floorplanning and placement approaches consider the option of buffer insertion. As such, they may exclude solutions, particularly early in the design process, with smaller overall area and better routability. In this paper, we propose a new methodology in which buffered trees are used to estimate wire delay during floorplanning. Instead of treating delay as one of the objectives, as done by the majority of previous work, we formulate the problem in terms of Delay Bounded Buffered Trees (DBB-tree) and propose an efficient algorithm to construct a DBB spanning tree for use during floorplanning. Experimental results show that the algorithm is very effective. Using buffer insertion at the floorplanning stage yields significantly better solutions in terms of both chip area and total wire length.

Introduction In high speed design, long on-chip interconnects can be modeled as distributed delay lines, where the delay of the lines can often be reduced by wire sizing or intermediate buffer insertion. Simple wire sizing is one degree of freedom available to the designer, but often it is ineffective due to area, routability, and capacitance considerations. On the other hand, driver sizing and buffer insertion are powerful tools for reducing delay, given reasonable power constraints. Intermediate buffers can effectively decouple a large load off of a critical path or divide a long wire into smaller segments, each of which has less line resistance and makes the path delay more linear with overall length. As the devices and lines shrink into deep submicron, it is more effective, in terms of power, area, and routability, to insert intermediate buffers than to rely solely on wire sizing. Because floorplanning and placement have a significant impact on critical path delay, research in the area has focused on timing driven approaches. Almost no existing floorplanning and placement techniques consider the option of buffer insertion, particularly early in the design cycle. Typically, only wire length or Elmore delay is used for delay calcula- tion. This practice is too restrictive as evidenced by the reliance industry has placed on intermediate buffering as a means for achieving aggressive cycle times. It is commonplace for production chips to contain tens of thousands of buffers. This paper attempts to leverage the additional freedom gained by inserting buffers during floorplanning and placement. The resulting formulation provides an additional degree of freedom not present in past approaches and typically leads to solutions with smaller area and increased routability. To incorporate buffer insertion into early planning stage, we propose a new methodology of floorplanning and placement using buffered trees to estimate the wiring delay. We formulate the Delay Bounded Buffered Tree (DBB-tree) problem as follows: Given a net with delay bounds on the critical sinks that are associated with critical paths, construct a tree with intermediate buffers inserted to minimize both the total wiring length and the number of buffers, while satisfying the delay bounds. We propose an efficient algorithm based on the Elmore delay model to construct DBB spanning trees for use during floorplanning and placement. The experimental results of the DBB spanning tree show that using buffer insertion at the floorplanning stage yields significantly better solutions in terms of both chip area and total wiring length. The remainder of the paper is organized as follows. Section 2 reviews the related works on interconnect optimization and intermediate buffer insertion, and introduces the idea of our DBB spanning tree algorithm. Section 3 describes the DBB algorithm in detail. The experimental results of DBB spanning tree algorithm applied for signal nets and for general floorplanning are given in Section 4, followed by conclusions in Section 5. Related Works and Overview of DBB-tree Algorithm 2.1 Elmore Delay Model As VLSI design reaches deep submicron, interconnect delay models have evolved from the simplistic lumped RC model to the sophisticated high-order moment-matching delay model [1]. The Elmore delay model [2] provides a simple closed-form expression with greatly improved accuracy for delay compared to the lumped RC model. Elmore is the most commonly used delay model in recent works of interconnect design. For each wire segment modeled as a - \Gammatype circuit, given the interconnect tree T , the Elmore delay from the source s 0 to sink s i can be expressed as follows: rl u;v ( cl u;v where R 0 is the driver resistance at the source and C 0 is the total capacitance charged by the driver. Path(0; i) denotes the path from s 0 to s i and wire e(u; v) connecting s v to its parent s u . Given a uniform wire width, r and c denote the unit resistance and unit capacitance respectively. The wire resistance rl u;v and wire capacitance cl u;v are proportional to the wire length l u;v . Let C v denote the total capacitance of a subtree rooted at s v , which is charged through wire e(u; v). The first term of -(0; i) is linear with the total wire length of T , while the second term has quadratic dependence on the length of the path from the source to s i . 2.2 Topology Optimization for Interconnect From the previous discussion of Elmore delay, we can conclude that for interconnect topology optimization, two major concerns are the total wire length and the path length from the driver to the critical sinks. The early work of Cohoon Randall [3] and Cong et al. [4] observed the existence of conflicting min-cost and min-radius (the longest source-to-sink path length of the tree) objectives in performance-driven routing [5]. A number of algorithms have been proposed to make the trade-offs between the total wiring length and the radius of the Steiner or spanning tree [6, 7, 8, 9]. Cong et al. 2.3 Buffered Tree Construction 3 proposed the "Bounded Radius, Bounded Cost" (BRBC) spanning tree algorithm which uses the shallow-light approach. BRBC constructs a routing tree with total wire length no greater than (1 times that of a minimum spanning tree and radius no greater than times that of a shortest path tree where ffl - 0. Alpert et al. [10] proposed AHHK trees as a direct trade-off between Prim's MST algorithm and Dijkstra's shortest path tree algorithm. They used a parameter 0 - c - 1 to adjust the preference between tree length and path length. For deep submicron design, path length is no longer an accurate estimate of path delay. Several attempts have been made to directly optimize Elmore delay taking into account different loading capacitances of the sinks. With exponential timing complexity, the branch and the bound algorithms proposed by Boese et al. [11, 12] provide the optimal and near-optimal solutions that minimize the delay from the source to an identified critical sink or a set of critical sinks of Steiner tree. For a set of critical sinks, it minimizes a linear combination of the sink delays. However it is very difficult to choose the proper weights, or the criticality, for this linear combination. Hong et al. [13] proposed a modified Dreyfus-Wagner Steiner tree algorithm for minimizing the maximal source-to-sink delay, The maximal source-to-sink delay is not necessarily interesting when the corresponding sink is off the critical path. Also, there may be more than one critical sink in the same net associated with multiple critical paths. Prasitjutrakul and Kubitz [14] proposed an algorithm for maximizing the minimal delay slack, where the delay slack is defined as the difference between the real delay and the given delay bound at a sink. 2.3 Buffered Tree Construction Intermediate buffer insertion creates another degree of freedom for interconnect optimiza- tion. Early works on fanout optimization problem focused on the construction of buffered trees during logic synthesis [15, 16, 17] without taking into account the wiring effect. Re- cently, layout driven fanout optimization have been proposed [18, 19]. For a given Steiner tree, a polynomial time dynamic programming algorithm was proposed in [20] for the delay-optimal buffer insertion problem. Using dynamic programming, Lillis et al. [21] integrated wire sizing and power minimization with the tree construction under a more accurate delay model taking signal slew into account. Inspired by the same dynamic programming algorithm, Okamoto and Cong [22] proposed a simultaneous Steiner tree construction and buffer insertion algorithm. Later the work was extended to include wire sizing [23]. In the formulation of the problem [22, 23], the main objective is to maximize the required arrival time at the root of the tree, which is defined as the minimum among the differences between the arrival time of the sinks and the delay from the root to the sinks. To achieve optimal delay, multiple buffers may be necessary for a single edge. An early work of S. Dhar and M. Franklin [24] developed the optimal solution for the size, number and position of buffers driving a uniform line that minimizes the delay of the line. The work further considered the area occupied by the buffers as a constraint. Recently C. Alpert and A. Devgan [25] calculated the optimal number of equally spaced buffers on a uniform wire to minimize the Elmore delay of the wire. 2.4 Delay Minimized vs. Delay Bounded Since timing driven floorplanning and placement are usually iterated with static timing analysis tools, the critical path information is often available and the timing requirement for critical sinks converges as the design and layout progresses. It is sufficient to have bounded delay rather than minimized delay. On the other hand, the minimization of total wire length is of interest since total wire length contributes to circuit area and routing congestion. In addition, total wire capacitance contributes a significant factor to the switching power. The reduction of wire length reduces circuit area and improves routability, also reduces power consumption, which are important factors for manufacturing cost and fabrication yield [1]. In this paper, instead of minimizing the source to sink delays, we will present an algorithm that constructs buffered spanning trees to minimize the total wire length subject to timing constraints. Zhu [26] proposed the "Delay Bounded Minimum Steiner Tree" (DBMST) algorithm to construct a low cost Steiner tree with bounded delay at critical sinks. The DBMST algorithm consists of two phases: (1) initialization of Steiner tree subject to timing constraints and (2) iterative refinement of the topology to reduce the wiring length while satisfying the delay bounds associated with critical sinks. Since the Elmore delays at sinks are very sensitive to topology and they have to be recomputed every time the topology is changed, DBMST algorithm searches all possible topological updates exhaustively at each iteration and so it is very time consuming. 2.5 Overview of DBB-tree Algorithm In this paper, we formulate the new Delay Bounded Buffered tree (DBB-tree) problem as follows: Given a signal net and delay bounds associated with critical sinks, construct a routing tree with intermediate buffers inserted to minimize the total wiring length and the number of buffers while satisfying the delay bounds. Based on Elmore delay, we develop an efficient algorithm for DBB spanning tree construction. The DBB-tree algorithm consists of three phases: (1) Calculate the minimum Elmore delay for each critical sink to allow immediate exclusion of floorplanning/placement solutions that are clearly infeasible from a timing perspective; (2) Construct a buffered spanning tree to minimize the total wire length subject to the bounded delay; (3) Based on the topology obtained in (2), delete unnecessary buffers without violating timing constraints to minimize the total number of buffers. The overall time complexity of DBB-tree algorithm is O(kn 2 ), where k is the maximum number of buffers inserted on a single edge, and n the number of sinks in the net. Our DBB-tree algorithm makes the following three major contributions: ffl Treating the delay bounds provided by static timing analysis tools as constraints rather than formulating the delay into the optimization objectives. ffl Constructing a spanning tree and placing intermediate buffers simultaneously. The algorithm is very effective to minimize both wire length and the number of buffers. ffl Allowing more than one buffer to be inserted on each single edge and calculating the precise buffer positions for the optimal solution. In contrast, most previous work assumes at most one buffer is inserted for each edge and the buffer location is fixed. 3 Description of DBB-tree Algorithm For floorplanning purpose, we assume uniform wire width. In the DBB-tree algorithm presented here, we consider only non-inverting buffers. However, the algorithm can be easily extended to handle inverting buffers. Given a signal net the source and s sinks. The geometric location for each terminal of S is determined by floorplanning. Let ~ denote the vector describing the parameters of non-inverting buffers, in which t b , r b and c b are the internal delay, resistance and capacitance of each buffer respectively. Before presenting the detailed DBB-tree algorithm, we first state some theoretical results developed by Alpert and Devgan [25] which will be used to calculate the number and position of identical buffers placed on a single edge to minimize the edge delay in DBB-tree algorithm: Theorem 1 Given a uniform line e(0; i) connecting sink s i to source s 0 , and the parameter vector ~ B, the number of buffers placed on the wire to obtain the minimum Elmore delay of Figure 1: Given a uniform line e(0; i) connecting sink s i to source s 0 , -(0; i) buffers are placed on the wire in such way that the wire delay is minimized: the first buffer is ff - away from source s 0 , the distance between two adjacent buffers equals to ffi - and the last buffer is fi - away from sink s i . e is given by: s where R 0 is the driver output resistance at source s 0 and c i the loading capacitance at sink s i . Given - buffers inserted on e(0; i), the optimal placement of buffers which obtains the minimum wire delay is places the buffers at equal spacing from each other. Let ff - be the distance from the source to the first buffer, ffi - the distance between two adjacent buffers, and fi - the distance from the last buffer to sink s i . They can be derived as follows: r c r c The minimized wire delay with - buffers is given by: cl 0;i (R c r buffers instead of - buffers are placed on wire e the wire delay will be increased by: \Delta- (0; 3.1 Lower Bound of Elmore Delay for Critical Sinks 7 Figure 2: If we place a buffer right after s 0 as in (a), the total capacitance driven by the driver at source is reduced to c b and the first term of -(0; i) equals to R 0 c b . The second term, the propagation delay of the path from source to s i , can be minimized by directly connecting s i to the source and placing -(0; i) buffers on the wire as in (b). Combining (a) and (b), we calculate the lower bound of Elmore delay for s i . By replacing R 0 with 0, Equations 2 - 5 can be applied to the wire connecting any two sinks in routing tree T . Based on the theoretical results discussed above, we will present the detailed DBB-tree algorithm in the following section. 3.1 Lower Bound of Elmore Delay for Critical Sinks The first phase of DBB-tree algorithm calculates the lower bound of Elmore delay for each sink s i . It may not be possible to achieve this delay simultaneously for all sinks, but no achievable delay will exceed it. The floorplanning is timing infeasible if there exists s i in S such that the lower bound - (0; i) is greater than the given delay bound D i The first term in Eq.1, R 0 C 0 , can be reduced to R 0 c b by placing a buffer right after s 0 as shown in Fig. 2 (a). And the second term, the propagation delay of the path from source to s i , can be minimized by directly connecting source to s i and placing buffers as shown in Fig. 2 (b). Formally, the lower bound of Elmore delay for s i can be given by: If for all sinks in S, the lower bound of Elmore delay is less than the given delay bound, then the algorithm continues to phases 2 and 3, otherwise the timing constraints are too Figure 3: For particular sink s is the last buffered edge on the path from the source to s v and the last buffer on edge through the resistance between the buffer and s v , defined as driving resistance of T v , denoted by R(T v ). Since there is no buffer between s u and s v , the driver of T v also drives T i for are the intermediate sinks from s u to s v . After adding the new edge e(v; w), the loading capacitance of T v is increased by \DeltaC v , the Elmore delay of sinks in will be increased by R(T i )\DeltaC v . On the other hand, due to the buffers on edge not affect on the delay of sinks which are not in T u . Therefore the timing constraints of T will be satisfied if and only if the timing constraints of T u are satisfied. tight for the given floorplanning and the solution is excluded. 3.2 DBB Spanning Tree Construction The second phase of DBB-tree algorithm constructs a buffered spanning tree to minimize the total wire length subject to the timing constraints. Similar with Prim's MST algorithm, it starts with the trivial tree: g. Iteratively edge e(v; w) with -(v; w) buffers is added into T , where s are chosen such that l v;w is minimized and timing constraints are satisfied. T grows incrementally until it spans all terminals of S, or there is no edge e(v; w) that can be added without violating the timing constraints. In the later case, the floorplanning is considered to be timing infeasible and the solution is excluded. For the incremental construction of the DBB-tree, the key issue is how to quickly evaluate the timing constraints each time a new edge is added, i.e. whether or not the 3.2 DBB Spanning Tree Construction 9 delay bound at each critical sink is satisfied. For particular edge e(v; w) where s the number and the precise positions of buffers inserted on the edge which minimize the edge delay can be calculated according to Equations 2 and 3. Let T v denote the subtree rooted at s v , after adding edge e(v; w) into T , the loading capacitance of T v , is increased by \DeltaC cl denote the last buffered edge on the path from the source to s v as shown in Fig. 3, the last buffer on edge . If there is no buffer from the source to s v , the source drives T v . According to Elmore delay, T v is driven through the resistance between the driver and s v , defined as driving resistance of T v , denoted by R(T v ). Given s v\Gamma1 is the parent of s v , R(T v ) can be calculated as follows: Since there is no buffer on the path from s u to s v , the driver of T v also drives T i for are the intermediate sinks from s u to s v as shown in Fig. 3. Let T denote the set of sinks in subtree T i but not in T i+1 . Due to the increased loading capacitance \DeltaC v of T v , the Elmore delay of sinks in T On the other hand, due to the buffers on edge the increased loading capacitance of T v will not affect on the delay of sinks which are not in T u . We define the delay slack of a sink s 2 T as: and the delay slack of T i to be: the timing constraints will be satisfied for the sinks in T if and only if the following condition holds: By introducing the loading capacitance slack of each subtree Eq. 12 can be rewritten as: Let oe (v) denote the minimum slack of loading capacitance among the subtrees T i for oe the condition in Eq. 14 can be simply rewritten as: oe (v) - \DeltaC By keeping track of oe (v), this condition can be checked in constant time. The Elmore delay of s w can be calculated from the Elmore delay of s where - (v; w) is calculated from Eq. 4 and the timing bound at s w can also be checked in constant time. From above analysis, we can conclude that the necessary and sufficient condition for satisfying the timing constraints of T after adding the new edge e(v; w) is: oe (v) - \DeltaC v and Dw -(0; w); (18) and this condition can be checked in constant time. At each iterative step of DBB-tree construction, s can be selected in linear time such that l v;w is minimum and the timing constraints are satisfied. After adding the new edge e(v; w), a two-pass traversal of T is sufficient to update the delay slack and loading capacitance slack of each subtree in T : (1) traverse T bottom up and calculate the delay slack and loading capacitance slack of each subtree T i according to Equations 11 and 13; (2) traverse T top down and calculate oe (i) from oe (i \Gamma 1), given s i\Gamma1 is the parent of oe Since each new edge can be added into T in linear time, the overall DBB spanning tree can be constructed in O(n 2 ) time for net S with n sinks. 3.3 Buffer Deletion In phase 2, one or more buffers are inserted on each edge to minimize wire delay. Some of the buffers may not be necessary for meeting the delay bound. The third phase of the 3.3 Buffer Deletion 11 Figure 4: In case of -(v; shown in (a), edge e(v; w) becomes unbuffered edge after deleting the buffer, the load capacitance of subtree T v is increased by: \DeltaC cl v;w buffers are re-inserted on e(v; w), as shown DBB-tree algorithm deletes buffers from the spanning tree obtained in the second phase to reduce the total number of buffers. In general the buffers closest to the source can unload the critical path the most. The algorithm traverses T bottom up and deletes one buffer at a time without violating timing constraints. The deletion continues until all the buffers left in T are necessary, that is, the timing constraints would not be satisfied if one more buffer is deleted. For particular edge e(v; w) with - ? 0 buffers, if one buffer is deleted from e(v; w), this wire delay will be increased by \Delta- (v; w) according to Eq. 5, and buffers will be re-inserted: In case of shown in Fig. 4 (a), wire e(v; w) becomes unbuffered edge after deleting the buffer, the load capacitance of subtree T v is increased by: \DeltaC cl v;w buffers are re-inserted on edge e(v; w), as shown in Fig. 4 (b): \DeltaC Similar to phase 2, let denote the last buffered edge from the source to s v . The delay of the sinks in subtree T u will be increased due to the increased loading capacitance of T v . In addition, the delay of sinks in subtree Tw will be further increased due to the increased edge delay of e(v; w). Based on the analysis in phase 2, a buffer can be deleted without causing timing violation if and only if following condition holds: oe (v) - \DeltaC v and -(Tw Table 1: Experimental Parameters of DBB-tree Algorithm on Signal Nets Output Resistance of Driver R 0 500\Omega \Gamma1000\Omega Unit Wire Resistance c 0:12\Omega =-m Unit Wire Capacitance r 0:15fF=-m Output Resistance of Buffer r b 500\Omega Loading Capacitance of Buffer c b 0:05pF Intrinsic Delay of Buffer t b 0:1ns Loading Capacitance of Sink c i 0:05pF \Gamma 0:15pF Therefore the timing constraints of T can be evaluated in constant time for deleting a buffer from edge e(v; w). The buffer can be found by searching at most n \Gamma 1 edges. After deleting a buffer, the delay slack and loading capacitance slack of subtrees in T are incrementally updated in O(n) time as in phase 2. So one buffer will be deleted in linear time. There are at most kn buffers in T where k is the maximum number of buffers on single edge, the timing complexity of buffer deletion is O(kn 2 ) which dominates the overall DBB-tree algorithm. Following experimental results show that the buffer deletion effectively minimizes the total number of buffers and it can delete more than 90% of the buffers inserted in the previous phase. 4 Experimental Results In the first part of the experiments, we implemented the DBB spanning tree algorithm on a Sun SPARC 20 workstation under the C/UNIX environment. The algorithm was tested on signal nets with 2; 5; 10; 25; 50 and 100 pins. For each net size, 100 nets were randomly generated on a 10mm \Theta 10mm routing region, and we report the average results. The driver output resistance at the source and the loading capacitances of sinks are randomly chosen from the respectively. The parameters used in the experiments are based on [22], which are summarized in Table 1. The average results of the DBB spanning tree construction are shown in Table 2. The delay bounds of critical sinks for each net size are randomly chosen from the interval titled "Delay Bounds". The average wire length and number of buffers for DBB spanning tree are reported in this table. The average CPU time consumed per net shows that DBB spanning tree algorithm is fast enough that can be applied during the stochastic optimization. Table 2: Experimental Results of DBB Spanning Trees on Signal Nets Pins(#) Delay Bounds(ns) Wire Length(mm) Buffers(#) CPU (sec:) To evaluate the DBB spanning trees generated by the experiments, we constructed both minimum spanning tree (MST) and shortest path tree (SPT) for the same signal nets using the same parameters. The comparison of the average results is shown in Table 3. "DBB/MST" and "DBB/SPT" is the average length ratio of DBB-tree to MST and DBB-tree to SPT respectively. The column "% sinks meeting bound" gives the average percentage of critical sinks which satisfy the delay bounds. For the nets with small number of terminals, the length of DBB-tree is very close to MST. As the number of terminals in the nets increases, the length of DBB-tree to MST is increased, but only 9% through 0% critical sinks can meet the bound in MST for 25-pin through 100-pin nets. It can be concluded that it is very difficult to satisfy the timing constraints using MST especially for the large nets. On the other hand, the length ratio of DBB-tree to SPT is decreased from 1:0 down to 0:24, and SPT is also not ideal to meet the delay bounds for the large nets. The DBB-tree approach can achieve the short wire length with 100% critical sinks meeting the delay bounds. In Table 4, the average number of buffers inserted in DBB spanning trees are listed and the result is very reasonable considering the number of terminals in the net. To evaluate the buffer deletion algorithm, we compare the average number of buffers inserted in DBB spanning tree before and after buffer deletion. The percentage of buffers reduced by the third phase of DBB-tree algorithm is as high as 79% through 93%. The results presented in Table 4 demonstrate that the third phase of the algorithm is quite effective at removing any unnecessary buffers estimated during phase 2 and the DBB-tree algorithm will not lead to unrealistic, impractical results. In the second part of the experiments, we apply DBB-tree to evaluate the wiring delay of floorplanning solutions considered by the Genetic Simulated Annealing method [27]. Table Table 3: Comparison of DBB-tree, MST and SPT of Signal Nets. Pins (#) Legnth (mm) % sinks meeting bound DBB MST DBB/MST SPT DBB/SPT DBB MST SPT Table 4: Average Number of Buffers Before vs. After Buffer Deletion. Pins(#) w/o Deletion with Deletion Reduced (%) Table 5: Four Examples of Floorplanning Applying DBB-tree Algorithm. Blocks Block size Aspect ratio Nets Net size Delay bound CPU (#) (mm) of blocks (#pins/net) (ns) (min:) Table Achieved Floorplanning Solutions by Using DBB-tree, MST and SPT Approaches. Blocks sinks meeting bound (#) DBB MST SPT DBB MST SPT DBB MST SPT 100 213.57 274.77 274.02 6039.93 7037.06 16339.61 100 90.82 95.61 5 presents four examples which includes 10, 25, 50 and 100 rectangular blocks, respectively. The sizes (widths and heights) and aspect ratios of blocks are randomly chosen within a nominal range. Netlists are also randomly generated for the four examples. The technology parameters are consistent with those shown in Table 1. To compare with the traditional approaches which do not consider buffer insertion during the floorplanning, we also apply MST and SPT methods to evaluate the floorplanning solution in the same examples. Based on the same stochastic search strategy, the floorplanning solutions achieved by the three methods are shown in Table 6. Similarly, the column "% sinks meeting bound" measures the percentage of critical sinks which satisfy the tim- Table 7: The Improvement by Considering Buffer Insertion in Floorplanning Stage. Blocks Area Improvement(%) Wire Length Improvement(%) Buffers(#) (#) DBB vs. MST DBB vs. SPT DBB vs. MST DBB vs. SPT in DBB Figure 5: Floorplanning of 50 blocks with 150 nets sized from 2-pin to 25-pin. SPT is applied to evaluated the wiring delay. Achieved chip area is 124:38mm 2 and total wire length 2696:10mm with 97:7% critical sinks meeting the delay bounds. ing bounds. Table 7 calculates the improvement of both chip area and total wire length by using DBB-tree method. For the examples, the area can be improved up to 31% over MST and 22% over SPT, respectively. On the other hand, the total wire length can be improved up to 19% over MST and 63% over SPT, respectively. This substantial improvement demonstrates that using buffer insertion at the floorplanning stage yields significantly better solutions in terms of both chip area and total wire length. In addition, the total number of buffers estimated by the DBB-tree approach are also shown in this table. Figures 5 and 6 show the floorplanning solution with 50 blocks by using SPT and DBB-tree algorithm, respectively. In addition, Fig. 6 also displays the buffers estimated by DBB- tree approach. It should be noted that future research is needed to extend the approach to distribute buffers into the empty space between macros subject to timing constraints. However, the area of such buffers is typically a small fraction of a given macro area and can be typically accommodated. 5 Conclusion In this paper, we propose a new methodology of floorplanning and placement where intermediate buffer insertion is used as another degree of freedom in the delay calculation. An efficient algorithm to construct Delay Bounded Buffered(DBB) spanning trees has been de- veloped. One of the key reasons this approach is effective is that we treat the delay bounds as constraints rather than formulating the delay into the optimization objectives as is done Figure Floorplanning of the same example in Fig. 5. DBB-tree is applied to evaluate the wiring delay. Achieved chip area is 112:59mm 2 and total wire length 1455:47mm with 100% critical sinks meeting the delay bounds. The area and total wire length are improved by 9:48% and 46:02% respectively. The dots shown in the figure represent the buffers estimated by DBB-tree. in most of the previous work. In fact, our problem formulation is more realistic for the path based timing driven layout design. The timing constraints of a floorplan are evaluated many times during our stochastic optimization process. The efficient DBB spanning tree algorithm made our buffered tree based floorplanning and placement highly effective and practically applicable to industrial problems. --R "The transient response of damped linear networks with particular regard to wide-band amplifiers," "Critical net routing," "A new class of iterative steiner tree heuristics with good perfor- mance," "A direct combination of the prim and dijkstra constructions for improved performance-driven global routing," "Performance-Driven interconnect design based on distributed RC delay model," "Performance oriented rectilinear steiner trees," "Bounded-diameter spanning tree and related problems," "Prim-Dijkstra tradeoffs for improved performance-Driven routing tree design," "Rectilinear steiner trees with minimum elmore delay," "High-Performance routing trees with identified critical sinks," "Performance-Driven steiner tree algorithms for global routing," "A timing-Driven global router for custom chip design," "A heuristic algorithm for the fanout problem," "Performance oriented technology mapping," "The fanout problem: From theory to practice," "A methodology and algorithms for post-Placement delay optimization," "Routability-Driven fanout optimization," "Buffer placement in distributed RC-tree networks for minimal elmore delay," "Optimal and efficient buffer insertion and wire sizing," "Interconnect layout optimization by simultaneous steiner tree construction and buffer insertion," "Buffered steiner tree construction with wire sizing for interconnect layout optimization," "Optimum buffer circuits for driving long uniform lines," "Wire segmenting for improved buffer insertion," Chip and Package Co-Synthesis of Clock Networks "Genetic simulated annealing and application to non-slicing floorplan design," --TR Bounded diameter minimum spanning trees and related problems The fanout problem: from theory to practice Performance-oriented technology mapping A heuristic algorithm for the fanout problem Performance oriented rectilinear Steiner trees Performance-driven Steiner tree algorithm for global routing High-performance routing trees with identified critical sinks Routability-driven fanout optimization Performance-driven interconnect design based on distributed RC delay model A methodology and algorithms for post-placement delay optimization Rectilinear Steiner trees with minimum Elmore delay Buffered Steiner tree construction with wire sizing for interconnect layout optimization Wire segmenting for improved buffer insertion Performance-Driven Global Routing for Cell Based ICs Critical Net Routing Chip and package cosynthesis of clock networks --CTR Weiping Shi , Zhuo Li, An O(nlogn) time algorithm for optimal buffer insertion, Proceedings of the 40th conference on Design automation, June 02-06, 2003, Anaheim, CA, USA Yuantao Peng , Xun Liu, Power macromodeling of global interconnects considering practical repeater insertion, Proceedings of the 14th ACM Great Lakes symposium on VLSI, April 26-28, 2004, Boston, MA, USA Xun Liu , Yuantao Peng , Marios C. Papaefthymiou, Practical repeater insertion for low power: what repeater library do we need?, Proceedings of the 41st annual conference on Design automation, June 07-11, 2004, San Diego, CA, USA Ruiming Chen , Hai Zhou, Efficient algorithms for buffer insertion in general circuits based on network flow, Proceedings of the 2005 IEEE/ACM International conference on Computer-aided design, p.322-326, November 06-10, 2005, San Jose, CA Charles J. Alpert , Anirudh Devgan , Stephen T. Quay, Buffer insertion for noise and delay optimization, Proceedings of the 35th annual conference on Design automation, p.362-367, June 15-19, 1998, San Francisco, California, United States I-Min Liu , Adnan Aziz , D. F. Wong, Meeting delay constraints in DSM by minimal repeater insertion, Proceedings of the conference on Design, automation and test in Europe, p.436-440, March 27-30, 2000, Paris, France Hur , Ashok Jagannathan , John Lillis, Timing driven maze routing, Proceedings of the 1999 international symposium on Physical design, p.208-213, April 12-14, 1999, Monterey, California, United States Jason Cong , Tianming Kong , David Zhigang Pan, Buffer block planning for interconnect-driven floorplanning, Proceedings of the 1999 IEEE/ACM international conference on Computer-aided design, p.358-363, November 07-11, 1999, San Jose, California, United States Probir Sarkar , Vivek Sundararaman , Cheng-Kok Koh, Routability-driven repeater block planning for interconnect-centric floorplanning, Proceedings of the 2000 international symposium on Physical design, p.186-191, May 2000, San Diego, California, United States Jason Cong , Tianming Kong , Zhigang (David) Pan, Buffer block planning for interconnect planning and prediction, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.9 n.6, p.929-937, 12/1/2001 Feodor F. Dragan , Andrew B. Kahng , Ion Mndoiu , Sudhakar Muddu , Alexander Zelikovsky, Provably good global buffering using an available buffer block plan, Proceedings of the 2000 IEEE/ACM international conference on Computer-aided design, November 05-09, 2000, San Jose, California Ali Selamat , Sigeru Omatu, Web page feature selection and classification using neural networks, Information SciencesInformatics and Computer Science: An International Journal, v.158 n.1, p.69-88, January 2004 Dian Zhou , Rui-Ming Li, Design and verification of high-speed VLSI physical design, Journal of Computer Science and Technology, v.20 n.2, p.147-165, March 2005

Total Wire Length;MST;floorplanning;DBB-tree;SPT;elmore delay;buffer insertion;delay bounds

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Path-based next trace prediction.

The trace cache has been proposed as a mechanism for providing increased fetch bandwidth by allowing the processor to fetch across multiple branches in a single cycle. But to date predicting multiple branches per cycle has meant paying a penalty in prediction accuracy. We propose a next trace predictor that treats the traces as basic units and explicitly predicts sequences of traces. The predictor collects histories of trace sequences (paths) and makes predictions based on these histories. The basic predictor is enhanced to a hybrid configuration that reduces performance losses due to cold starts and aliasing in the prediction table. The Return History Stack is introduced to increase predictor performance by saving path history information across procedure call/returns. Overall, the predictor yields about a 26% reduction in misprediction rates when compared with the most aggressive previously proposed, multiple-branch-prediction methods.

Introduction Current superscalar processors fetch and issue four to six instructions per cycle - about the same number as in an average basic block for integer programs. It is obvious that as designers reach for higher levels of instruction level parallelism, it will become necessary to fetch more than one basic block per cycle. In recent years, there have been several proposals put forward for doing so [3,4,12]. One of the more promising is the trace cache [9,10], where dynamic sequences of instructions, containing embedded predicted branches, are assembled as a sequential "trace" and are saved in a special cache to be fetched as a unit. Trace cache operation can best be understood via an example. Figure 1 shows a program's control flow graph (CFG), where each node is a basic block, and the arcs represent potential transfers of control. In the figure, arcs corresponding to branches are labeled to indicate taken (T) and not taken (N) paths. The sequence ABD represents one possible trace which holds the instructions from the basic blocks A, B, and D. This would be the sequence of instructions beginning with basic block A where the next two branches are not taken and taken, respectively. These basic blocks are not contiguous in the original program, but would be stored as a contiguous block in the trace cache. A number of traces can extracted from the CFG - four possible traces are: 1: ABD 2: ACD 3: EFG 4: EG Of course, many other traces could also be chosen for the same CFG, and, in fact, a trace does not necessarily have to begin or end at a basic block boundary, which further increases the possibilities. Also, note that in a trace cache, the same instructions may appear in more than one trace. For example, the blocks A, D, E, and G each appear twice in the above list of traces. However, the mechanism that builds traces should use some heuristic to reduce the amount of redundancy in the trace cache; beginning and ending on basic block boundaries is a good heuristic for doing this. A F G Figure Associated with the trace cache is a trace fetch unit, which fetches a trace from the cache each cycle. To do this in a timely fashion, it is necessary to predict what the next trace will be. A straightforward method, and the one used in [9,10], is to predict simultaneously the multiple branches within a trace. Then, armed with the last PC of the preceding trace and the multiple predictions, the fetch unit can access the next trace. In our example, if trace 1 - ABD - is the most recently fetched trace, and a multiple branch predictor predicts that the next three branch outcomes will be T,T,N, then the next trace will implicitly be ACD. In this paper, we take a different approach to next trace prediction - we treat the traces as basic units and explicitly predict sequences of traces. For example, referring to the above list of traces, if the most recent trace is trace 1, then a next trace predictor might explicitly output "trace 2." The individual branch predictions T,T,N, are implicit. We propose and study next trace predictors that collect histories of trace sequences and make predictions based on these histories. This is similar to conditional branch prediction where predictions are made using histories of branch outcomes. However, each trace typically has more than two successors, and often has many more. Consequently, the next trace predictor keeps track of sequences of trace identifiers, each identifier containing multiple bits. We propose a basic predictor and then add enhancements to reduce performance losses due to cold starts, procedure call/returns, and interference due to aliasing in the prediction table. The proposed predictor yields substantial performance improvement over the previously proposed, multiple-branch-prediction methods. For the six benchmarks that we studied the average misprediction rate is 26% lower for the proposed predictor than for the most aggressive previously proposed multiple-branch predictor. 2. Previous work A number of methods for fetching multiple basic blocks per cycle have been proposed. Yeh et al. [12] proposed a Branch Address Cache that predicted multiple branch target addresses every cycle. Conte et al. [3] proposed an interleaved branch target buffer to predict multiple branch targets and detect short forward branches that stay within the same cache line. Both these methods use conventional instruction caches, and both fetch multiple lines based on multiple branch predictions. Then, after fetching, blocks of instructions from different lines have to be selected, aligned and combined - this can lead to considerable delay following instruction fetch. It is this complex logic and delay in the primary pipeline that the trace cache is intended to remove. Trace caches [9,10] combine blocks of instructions prior to storing them in the cache. Then, they can be read as a block and fed up the pipeline without having to pass through complex steering logic. Branch prediction in some form is a fundamental part of next trace prediction (either implicitly or explicitly). Hardware branch predictors predict the outcome of branches based on previous branch behavior. At the heart of most branch predictors is a Pattern History Table (PHT), typically containing two-bit saturating counters [11]. The simplest way to associate a counter with a branch instruction is to use some bits from the PC address of the branch, typically the least significant bits, to index into the PHT [11]. If the counter's value is two or three, the branch is predicted to be taken, otherwise the branch is predicted to be not taken. Correlated predictors can increase the accuracy of branch prediction because the outcome of a branch tends to be correlated with the outcome of previous branches [8,13]. The correlated predictor uses a Branch History Register (BHR). The BHR is a shift register that is usually updated by shifting in the outcome of branch instructions - a one for taken and a zero for not taken. In a global correlated predictor there is a single BHR that is updated by all branches. The BHR is combined with some bits (possibly zero) from a branch's PC address, either by concatenating or using an exclusive-or function, to form an index into the PHT. With a correlated predictor a PHT entry is associated not only with a branch instruction, but with a branch instruction in the context of a specific BHR value. When the BHR alone is used to index into the PHT, the predictor is a GAg predictor [13]. When an exclusive-or function is used to combine an equal number of bits from the BHR and the branch PC address, the predictor is a GSHARE predictor [6]. GSHARE has been shown to offer consistently good prediction accuracy. The mapping of instructions to PHT entries is essentially implemented by a simple hashing function that does not detect or avoid collisions. Aliasing occurs when two unrelated branch instructions hash to the same PHT entry. Aliasing is especially a problem with correlated predictors because a single branch may use many PHT entries depending on the value of the BHR, thus increasing contention. In order to support simultaneous fetching of multiple basic blocks, multiple branches must be predicted in a single cycle. A number of modifications to the correlated predictor discussed above have been proposed to support predicting multiple branches at once. Franklin and Dutta proposed subgraph oriented branch prediction mechanisms that uses local history to form a prediction that encodes multiple branches. Yeh, et al. [13] proposed modifications to a GAg predictor to multiport the predictor and produce multiple branch predictions per cycle. Rotenberg et al. [10] also used the modified GAg for their trace cache study. Recently, Patel et al. [9] proposed a multiple branch predictor tailored to work with a trace cache. The predictor attempts to achieve the advantages of a GSHARE predictor while providing multiple predictions. The predictor uses a BHR and the address of the first instruction of a trace, exclusive-ored together, to index into the PHT. The entries of the PHT have been modified to contain multiple two-bit saturating counters to allow simultaneous prediction of multiple branches. The predictor offers superior accuracy compared with the multiported GAg predictor, but does not quite achieve the overall accuracy of a single branch GSHARE predictor. Nair proposed "path-based" prediction, a form of correlated branch prediction that has a single branch history register and prediction history table. The innovation is that the information stored in the branch history register is not the outcome of previous branches, but their truncated PC addresses. To make a prediction, a few bits from each address in the history register as well as a few bits from the current PC address are concatenated to form an index into the PHT. Hence, a branch is predicted using knowledge of the sequence, or path, of instructions that led up to it. This gives the predictor more specific information about prior control flow than the taken/not taken history of branch outcomes. Jacobson et al. [5] refined the path-based scheme and applied it to next task prediction for multiscalar processors. It is an adaptation of the multiscalar predictor that forms the core of the path-based next trace predictor presented here. 3. Path-based next trace predictors We consider predictors designed specifically to work with trace caches. They predict traces explicitly, and in doing so implicitly predict the control instructions within the trace. Next trace predictors replace the conventional branch predictor, branch target buffer (BTB) and return address stack (RAS). They have low latency, and are capable of making a trace prediction every cycle. We show they also offer better accuracy than conventional correlated branch predictors. 3.1. Naming of traces In theory, a trace can be identified by all the PCs in the trace, but this would obviously be expensive. A cheaper and more practical method is to use the PC value for the first instruction in the trace combined with the outcomes of conditional branches embedded in the trace. This means that indirect jumps can not be internal to a trace. We use traces with a maximum length of 16 instructions. For accessing the trace cache we use the following method. We assume a 36 bit identifier, to identify the starting PC and six bits to encode up to six conditional branches. The limit of six branches is somewhat arbitrary and is chosen because we observed that length 16 traces almost never have more than six branches. It is important to note that this limit on branches is not required to simplify simultaneous multiple branch prediction, as is the case with trace predictors using explicit branch prediction. 3.2. Correlated predictor The core of the next trace predictor uses correlation based on the history of the previous traces. The identifiers of the previous few traces represent a path history that is used to form an index into a prediction table; see Figure 2. Each entry in the table consists of the identifier of the predicted trace (PC branch outcomes), and a two-bit saturating counter. When a prediction is correct the counter is incremented by one. When a prediction is incorrect and the counter is zero, the predicted trace will be replaced with the actual trace. Otherwise, the counter is decremented by two and the predicted trace entry is unchanged. We found that the increment-by-1, decrement-by-2 counter gives slightly better performance than either a one bit or a conventional two-bit counter. HISTORY REGISTER Predicted Trace ID Trace ID cnt Hashed ID Hashed ID Hashed ID Hashed ID Generation Hashing Function Figure Correlated predictor Path history is maintained as a shift register that contains hashed trace identifiers (Figure 2). The hashing function uses the outcome of the first two conditional branches in the trace identifier as the least significant two bits, the two least significant bits of the starting PC as the next two bits, the upper bits are formed by taking the outcomes of additional conditional branch outcomes and exclusive-oring them with the next least significant bits of the starting PC. Beyond the last conditional branch a value of zero is used for any remaining branch outcome bits. The history register is updated speculatively with each new prediction. In the case of an incorrect prediction the history is backed up to the state before the bad prediction. The prediction table is updated only after the last instruction of a trace is retired - it is not speculatively updated. O bits D back ID Width bits Figure 3 Index generation mechanism Ideally the index generation mechanism would simply concatenate the hashed identifiers from the history register to form the index. Unfortunately this is sometimes not practical because the prediction table is relatively small so the index must be restricted to a limited number of bits. The index generation mechanism is based on the method developed to do inter-task prediction for multiscalar processors [5]. The index generation mechanism uses a few bits from each of the hashed trace identifiers to form an index. The low order bits of the hashed trace identifiers are used. More bits are used from more recent traces. The collection of selected bits from all the traces may be longer than the allowable index, in which case the collection of bits is folded over onto itself using an exclusive-or function to form the index. In [5], the "DOLC" naming convention was developed for specifying the specific parameters of the index generation mechanism. The first variable 'D'epth is the number of traces besides the last trace that are used for forming the index. The other three variables are: number of bits from 'O'lder traces, number of bits from the `L'ast trace and the number of bits from the 'C'urrent. In the example shown in Figure 3 the collection of bits from the trace identifiers is twice as long as the index so it is folded in half and the two halves are combined with an exclusive- or. In other cases the bits may be folded into three parts, or may not need to be folded at all. 3.3. Hybrid predictor If the index into the prediction table reads an entry that is unrelated to the current path history the prediction will almost certainly be incorrect. This can occur when the particular path has never occurred before, or because the table entry has been overwritten by unrelated path history due to aliasing. We have observed that both are significant, but for realistically sized tables aliasing is usually more important. In branch prediction, even a randomly selected table entry typically has about a 50% chance of being correct, but in the case of next trace prediction the chances of being correct with a random table entry is very low. To address this issue we operate a second, smaller predictor in parallel with the first (Figure 4). The secondary predictor requires a shorter learning time and suffers less aliasing pressure. The secondary predictor uses only the hashed identifier of the last trace to index its table. The prediction table entry is similar to the one for the correlated predictor except a 4 bit saturating counter is used that decrements by 8 on a misprediction. The reason for the larger counter will be discussed at the end of this section. CORRELATING TABLE HISTORY REGISTER Prediction Trace ID cnt Hashed ID Hashed ID Hashed ID Hashed ID Hashing Function Figure 4 Hybrid predictor To decide which predictor to use for any given prediction, a tag is added to the table entry in the correlated predictor. The tag is set with the low 10 bits of the hashed identifier of the immediately preceding trace at the time the entry is updated. A ten bit tag is sofficient to eliminate practically all unintended aliasing When a prediction is being made, the tag is checked against the hashed identifier of the preceding trace, if they match the correlated predictor is used; otherwise the secondary predictor is used. This method increases the likelihood that the correlated predictor corresponds to the correct context when it is used. This method also allows the secondary table to make a prediction when the context is very limited, i.e. under startup conditions. The hybrid predictor naturally reduces aliasing pressure somewhat, and by modifying it slightly, aliasing pressure can be further reduced. If the 4-bit counter of the secondary predictor is saturated, its prediction is used, and more importantly, when it is correct the correlated predictor is not updated. This means if a trace is always followed by the same successor the secondary predictor captures this behavior and the correlated predictor is not polluted. This reduces the number of updates to the correlated predictor and therefore the chances of aliasing. The relatively large counter, 4-bits, is used to avoid giving up the opportunity to use the correlated predictor unless there is high probability that a trace has a single successor. 3.4. Return history stack (RHS) The accuracy of the predictor is further increased by a new mechanism, the return history stack (RHS). A field is added to each trace indicating the number of calls it contains. If the trace ends in a return, the number of calls is decremented by one. After the path history is updated, if there are any calls in the new trace, a copy of the most recent history is made for each call and these copies are pushed onto a special hardware stack. When there is a trace that ends in a return and contains no calls, the top of the stack is popped and is substituted for part of the history. One or two of the most recent entries from the current history within the subroutine are preserved, and the entries from the stack replace the remaining older entries of the history. When there are five or fewer entries in the history, only the most recent hashed identifier is kept. When there are more than five entries the two most recent hashed identifiers are kept. HISTORY REGISTER hashed ID hashed ID hashed ID hashed ID POP Figure 5 Return history stack implementation With the RHS, after a subroutine is called and has returned, the history contains information about what happened before the call, as well as knowledge of the last one or two traces of the subroutine. We found that the RHS can significantly increase overall predictor accuracy. The reason for the increased accuracy is that control flow in a program after a subroutine is often tightly correlated to behavior before the call. Without the RHS the information before the call is often overflowed by the control flow within a subroutine. We are trying to achieve a careful balance of history information before the call versus history information within the call. For different benchmarks the optimal point varies. We found that configurations using one or two entries from the subroutine provide consistently good behavior. The predictor does not use a return address stack (RAS), because it requires information on an instruction level granularity, which the trace predictor is trying to avoid. The RHS can partly compensate for the absence of the RAS by helping in the initial prediction after a return. If a subroutine is significantly long it will force any pre- call information out of the history register, hence determining the calling routine, and therefor where to return, would be much harder without the RHS. 4. Simulation methodology 4.1. Simulator To study predictor performance, trace driven simulation with the Simplescalar tool set is used [1]. Simplescalar uses an instruction set largely based on MIPS, with the major deviation being that delayed branches have been replaced with conventional branches. We use the Gnu C compiler that targets Simplescalar. The functional simulator of the Simplescalar instruction set is used to produce a dynamic stream of instructions that is fed to the prediction simulator. For most of this work we considered the predictor in isolation, using immediate updates. A prediction of the next trace is made and the predictor is updated with the actual outcome before the next prediction is made. We also did simulations with an execution engine. This allows updates to be performed taking execution latency into account. We modeled an 8-way out-of-order issue superscalar processor with a 64 instruction window. The processor had a 128KB trace cache, a 64KB instruction cache, and a 4-ported 64KB data cache. The processor has 8 symmetric functional units and supports speculative memory operations. 4.2. Trace selection For our study, we used traces that are a maximum of instructions in length and can contain up to six branches. The limit on the number of branches is imposed only by the naming convention of traces. Any control instruction that has an indirect target can not be embedded into a trace, and must be at the end of a trace. This means that some traces will be shorter than the maximum length. As mentioned earlier, instructions with indirect targets are not embedded to allow traces to be uniquely identified by their starting address and the outcomes of any conditional branches. We used very simple trace selection heuristics. More sophisticated trace selection heuristics are possible and would significantly impact the behavior of the trace predictor. A study of the relation of trace selection and trace predictability is beyond the scope of this paper. 4.3. Benchmarks We present results from six SpecInt95 benchmarks: compress, gcc, go, jpeg, m88ksim and xlisp. All results are based on runs of at least 100 million instructions. Table summary Benchmark Input number of instr. avg. trace length traces compress 400000 queens 7 first 100 million 12.4 1393 5. Performance 5.1. Sequential branch predictor For reference we first determined the trace prediction accuracy that could be achieved by taking proven control flow prediction components and predicting each control instruction sequentially. In sequential prediction each branch is explicitly predicted and at the time of the prediction the outcomes of all previous branches are known. This is useful for comparisons although it is not realizable because it would require multiple accesses to predict a single trace and requires knowledge of the branch addresses within the trace. The best multiple branch predictors to date have attempted to approximate the behavior of this conceptual sequential predictor. We used a 16-bit GSHARE branch predictor, a perfect branch target buffer for branches with PC-relative and absolute address targets, a 64K entry correlated branch target buffer for branches with indirect targets [2], and a perfect return address predictor. All of these predictors had ideal (immediate) updates. When simulating this mechanism, if one or more predictions within a trace was incorrect we counted it as one trace misprediction. This configuration represents a very aggressive, ideal predictor. The prediction accuracy of this idealized sequential prediction is given in Table 2. The mean of the trace misprediction rate is 12.1%. We show later that our proposed predictor can achieve levels of prediction accuracy significantly better than those achievable by this idealized sequential predictor. In the results section we refer to the trace prediction accuracy of the idealized sequential predictor as "sequential." The misprediction rate for traces tends to be lower than that obtained by simply multiplying the branch misprediction rate by the number of branches because branch mispredictions tend to be clustered. When a trace is mispredicted, multiple branches within the same trace are often mispredicted. Xlisp is the exception, with hard to predict branches tending to be in different traces. With the aggressive target prediction mechanisms none of the benchmarks showed substantial target misprediction. Table Prediction accuracy for sequential predictors Benchmark 16-bit Gshare branch misprediction Number of Branches per Trace Mispredic tion of traces compress 9.2 2.1 17.9 gcc 8.0 2.1 14.0 go 16.6 1.8 24.5 jpeg 6.9 1.0 6.7 xlisp 3.2 1.9 6.5 5.2. Performance with unbounded tables To determine the potential of path-based next trace prediction we first studied performance assuming unbounded tables. In this study, each unique sequence of trace identifiers maps to its own table entry. I.e. there is no aliasing. We consider varying depths of trace history, where depth is the number of traces, besides the most recent trace, that are combined to index the prediction table. For a depth of zero only the identifier of the most recent trace is used. We study history depths of zero through seven. Figure 6 presents the results for unbounded tables, the mean of the misprediction rate is 8.0% for the RHS predictor at the maximum depth. For comparisons, the "sequential" predictor is based on a 16-bit Gshare predictor that predicts all conditional branches sequentially. For all the benchmarks the proposed path-based predictor does better than the idealized sequential predictor. On average, the misprediction rate is 34% lower for the proposed predictor. In the cases of gcc and go the predictor has less than half the misprediction rate of the idealized sequential predictor. e Correlated Hybrid RHS Sequential Depth of History Misprediction Rate Correlated Hybrid RHS Sequential GO14182226 Depth of History Misprediction Rate Correlated Hybrid RHS Sequential Depth of History Misprediction Rate Correlated Hybrid RHS Sequential Depth of History Misprediction Rate Correlated Hybrid RHS Sequential Depth of History Misprediction Rate Correlated Hybrid RHS Sequential Figure 6 Next trace prediction with unbounded tables For all benchmarks, the hybrid predictor has a higher prediction accuracy than using the correlated predictor alone. The benchmarks with more static traces see a larger advantage from the hybrid predictor because they contain more unique sequences of traces. Because the table size is unbounded the hybrid predictor is not important for aliasing, but is important for making predictions when the correlated predictor entry is cold. For four out of the six benchmarks adding the return history stack (RHS) increases prediction accuracy. Furthermore, the four improved benchmarks see a more significant increase due to the RHS than the two benchmarks hurt by the RHS see a decrease. For benchmark compress the predictor does better without the RHS. For compress, the information about the subroutine being thrown away by the RHS is more important than the information before the subroutine that is being saved. Xlisp extensively uses recursion, and to minimize overhead it uses unusual control flow to backup quickly to the point before the recursion without iteratively performing returns. This behavior confuses the return history stack because there are a number of calls with no corresponding returns. However, it is hard to determine how much of the performance loss of RHS with xlisp is caused by this problem and how much is caused by loss of information about the control flow within subroutines. 5.3. Performance with bounded tables We now consider finite sized predictors. The table for the correlated predictor is the most significant component with respect to size. We study correlated predictors with tables of 2 14 , 2 15 and 2 16 entries. For each size we consider a number of configurations with different history depths. The configurations for the index generation function were chosen based on trial-and-error. Although better configurations are no doubt possible we do not believe differences would be significant. 2^14 entries entries entries Infinite Sequential Depth of History Misprediction Rate 2^14 entries entries entries Infinite Sequential Depth of History Misprediction Rate 2^14 entries entries entries Infinite Sequential Depth of History Misprediction Rate 2^14 entries entries entries Infinite Sequential Depth of History Misprediction Rate 2^14 entries entries entries Infinite Sequential Depth of History Misprediction Rate 2^14 entries entries entries Infinite Sequential Figure 7 Next trace prediction We use a RHS that has a maximum depth of 128. This depth is more than sufficient to handle all the benchmarks except for the recursive section of xlisp, where the predictor is of little use, anyway. Performance results are in Figure 7. Three of the benchmarks stress the finite-sized predictors: gcc, go and jpeg. In these predictors the deviation from the unbounded tables is very pronounced, as is the deviation between the different table sizes. As expected, the deviation becomes more pronounced with longer histories because there are more unique sequences of trace identifiers being used and, therefore, more aliasing. Go has the largest number of unique sequences of trace identifiers, and apparently suffers from aliasing pressure the most. At first, as history depth is increased the miss rate goes down. As the history depth continues to increase, the number of sequences competing for the finite size table increases aliasing. The detrimental effects of aliasing eventually starts to counter the gain of going to deeper histories and at some point dominates and causes a negative effect for increased history depth. The smaller the table size, the sooner the effects of aliasing start to become a problem. It is important to focus on the behavior of this benchmark and the other two larger benchmarks - gcc and jpeg, because in general the other benchmarks probably have relatively small working sets compared to most realistic programs. We see that for realistic tables, the predictor can achieve very high prediction accuracies. In most cases, the predictor achieves miss rates significantly below the idealized sequential predictor. The only benchmark where the predictor can not do better than sequential prediction is for a small, 2 14 entry, table for jpeg. But even in this case it can achieve performance very close to the sequential, and probably closer than a realistic implementation of Gshare modified for multiple branches per cycle throughput. For our predictor the means of the mispredict rates are 10.0%, 9.5% and 8.9% for the maximum depth configuration with 2 14 , 2 15 and 2 tables respectively. These are all significantly below the 12.1% misprediction rate of the sequential predictor, 26% lower for the 2 predictor. Table 3 Index generation configurations used Depth D-O-L-C for Thus far simulation results have used immediate updates. In a real processor the history register would be updated with each predicted trace, and the history would be corrected when the predictor backs up due to a misprediction. The table entry would not be updated until the last instruction of a trace has retired. Table 4 Impact of real updates Benchmark Misprediction with ideal updates Misprediction with real update compress 5.8 5.8 go 9.3 9.3 jpeg 3.5 3.6 2.4 2.1 xlisp 4.7 4.8 To make sure this does not make a significant impact on prediction accuracy, we ran a set of simulations where an execution engine was simulated. The configuration of the execution engine is discussed in section 4.1. The predictor being modeled has 2 16 entries and a 7-3-6-8 DOLC configuration. Table 4 shows the impact of delayed updates, and it is apparent that delayed updates are not significant to the performance of the predictor. In one case, m88ksim, the delayed updates actually increased prediction accuracy. The delayed updates has the effect of increasing the amount of hysteresis in the prediction table which in some cases can increase performance. 5.5. A cost-reduced predictor The cost of the proposed predictor is primarily a function of the size of the correlated predictor's table. The size of the correlated predictor's table is the number of entries multiplied by the size of an entry. The size of an entry is 48 bits: 36 bits to encode a trace identifier, two bits for the counter plus 10 bits for the tag. A much less expensive predictor can be constructed, however, by observing that before the trace cache can be accessed, the trace identifier read from the prediction table must be hashed to form a trace cache index. For practical sized trace caches this index will be in the range of 10 bits. Rather than storing the full trace identifier, the hashed cache index can be stored in the table, instead. This hashed index can be the same as the hashed identifier that is fed into the history register (Figure 2). That is, the Hashing Function can be moved to the input side of the prediction table to hash the trace identifier before it is placed into the table. This modification should not affect prediction accuracy in any significant way and reduces the size of the trace identifier field from 36 bits to bits. The full trace identifier is still stored in the trace cache as part of its entry and is read out as part of the trace cache access. The full trace identifier is used during execution to validate that the control flow implied by the trace is correct. 6. Predicting an alternate trace Along with predicting the next trace, an alternate trace can be predicted at the same time. This alternate trace can simplify and reduce the latency for recovering when it is determined that a prediction is incorrect. In some implementations this may allow the processor to find and fetch an alternate trace instead of resorting to building a trace from scratch. Alternate trace prediction is implemented by adding another field to the correlated predictor. The new field contains the identifier of the alternate prediction. When the prediction of the correlated predictor is incorrect the alternate prediction field is updated. If the saturating counter is zero the identifier in the prediction field is moved to the alternate field, the prediction field is then updated with the actual outcome. If the saturating counter is non-zero the identifier of the actual outcome is written into the alternate field. Figure 8 shows the performance of the alternate trace predictor for two representative benchmarks. The graphs show the misprediction rate of the primary 2 16 entry table predictor as well as the rate at which both the primary and alternate are mispredicted. A large percent of the mispredictions by the predictor are caught by the alternate prediction. For compress, 2/3 of the mispredictions are caught by the alternate, for gcc it is slightly less than half. It is notable that for alternate prediction the aliasing effect quickly dominates the benefit of more history because it does not require as much history to make a prediction of the two most likely traces, so the benefit of more history is significantly smaller. There are two reasons alternate trace prediction works well. First, there are cases where some branch is not heavily biased; there may be two traces with similar likelihood. Second, when there are two sequences of traces aliased to the same prediction entry, as one sequence displaces the other, it moves the other's likely prediction to the alternate slot. When a prediction is made for the displaced sequence of traces, and the secondary predictor is wrong, the alternate is likely to be correct. Depth of History Misprediction Rate Primary Alternate Depth of History Misprediction Rate Primary Alternate Figure Alternate trace prediction accuracy 7. Summary We have proposed a next trace predictor that treats the traces as basic units and explicitly predicts sequences of traces. The predictor collects histories of trace sequences and makes predictions based on these histories. In addition to the basic predictor we proposed enhancements to reduce performance losses due to cold starts, procedure call/returns, and the interference in the prediction table. The predictor yields consistent and substantial improvement over previously proposed, multiple-branch-prediction methods. On average the predictor had a 26% lower mispredict rate than the most aggressive previously proposed multiple-branch predictor. Acknowledgments This work was supported in part by NSF Grant MIP- 9505853 and the U.S. Army Intelligence Center and Fort Huachuca under Contract DAPT63-95-C-0127 and ARPA order no. D346. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsement, either expressed or implied, of the U.S. Army Intelligence Center and For Huachuca, or the U.S. Government. --R "Evaluating Future Microprocessors: The SimpleScalar Tool Set," "Target Prediction for Indirect Jumps," "Optimization of Instruction Fetch Mechanisms for High Issue Rates," "Control Flow Prediction with Tree-Like Subgraphs for Superscalar Processors," "Control Flow Speculation in Multiscalar Processors," "Combining Branch Predictors," "Dynamic Path-Based Branch Correlation," "Improving the Accuracy of Dynamic Branch Prediction Using Branch Correlation," "Critical Issues Regarding the Trace Cache Fetch Mechanism." "Trace Cache: a Low Latency Approach to High Bandwidth Instruction Fetching," "A Study of Branch Prediction Strategies," "Increasing the Instruction Fetch Rate via Multiple Branch Prediction and a Branch Address Cache," "Two-Level Adaptive Branch Prediction," --TR Two-level adaptive training branch prediction Improving the accuracy of dynamic branch prediction using branch correlation Increasing the instruction fetch rate via multiple branch prediction and a branch address cache Optimization of instruction fetch mechanisms for high issue rates Dynamic path-based branch correlation Control flow prediction with tree-like subgraphs for superscalar processors Trace cache Target prediction for indirect jumps A study of branch prediction strategies Control Flow Speculation in Multiscalar Processors --CTR Trace processors, Proceedings of the 30th annual ACM/IEEE international symposium on Microarchitecture, p.138-148, December 01-03, 1997, Research Triangle Park, North Carolina, United States independence in trace processors, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.4-15, November 16-18, 1999, Haifa, Israel Juan C. Moure , Domingo Bentez , Dolores I. Rexachs , Emilio Luque, Wide and efficient trace prediction using the local trace predictor, Proceedings of the 20th annual international conference on Supercomputing, June 28-July 01, 2006, Cairns, Queensland, Australia Michael Behar , Avi Mendelson , Avinoam Kolodny, Trace cache sampling filter, ACM Transactions on Computer Systems (TOCS), v.25 n.1, p.3-es, February 2007 Quinn Jacobson , James E. Smith, Trace preconstruction, ACM SIGARCH Computer Architecture News, v.28 n.2, p.37-46, May 2000 Ryan Rakvic , Bryan Black , John Paul Shen, Completion time multiple branch prediction for enhancing trace cache performance, ACM SIGARCH Computer Architecture News, v.28 n.2, p.47-58, May 2000 Bryan Black , Bohuslav Rychlik , John Paul Shen, The block-based trace cache, ACM SIGARCH Computer Architecture News, v.27 n.2, p.196-207, May 1999 Oliverio J. Santana , Alex Ramirez , Josep L. 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Weikle , Kevin Skadron, Evaluating trace cache energy efficiency, ACM Transactions on Architecture and Code Optimization (TACO), v.3 n.4, p.450-476, December 2006 Roni Rosner , Yoav Almog , Micha Moffie , Naftali Schwartz , Avi Mendelson, Power Awareness through Selective Dynamically Optimized Traces, ACM SIGARCH Computer Architecture News, v.32 n.2, p.162, March 2004 James R. Larus, Whole program paths, ACM SIGPLAN Notices, v.34 n.5, p.259-269, May 1999 Zhang , Rajiv Gupta, Whole execution traces and their applications, ACM Transactions on Architecture and Code Optimization (TACO), v.2 n.3, p.301-334, September 2005 Lucian Codrescu , D. Scott Wills , James Meindl, Architecture of the Atlas Chip-Multiprocessor: Dynamically Parallelizing Irregular Applications, IEEE Transactions on Computers, v.50 n.1, p.67-82, January 2001 Alex Ramirez , Oliverio J. Santana , Josep L. 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Gholamipour, On the performance of trace locality of reference, Performance Evaluation, v.60 n.1-4, p.51-72, May 2005 Shashidhar Mysore , Banit Agrawal , Timothy Sherwood , Nisheeth Shrivastava , Subhash Suri, Profiling over Adaptive Ranges, Proceedings of the International Symposium on Code Generation and Optimization, p.147-158, March 26-29, 2006 Matt T. Yourst , Kanad Ghose, Incremental Commit Groups for Non-Atomic Trace Processing, Proceedings of the 38th annual IEEE/ACM International Symposium on Microarchitecture, p.67-80, November 12-16, 2005, Barcelona, Spain Hao , Po-Yung Chang , Marius Evers , Yale N. Patt, Increasing the Instruction Fetch Rate via Block-Structured Instruction Set Architectures, International Journal of Parallel Programming, v.26 n.4, p.449-478, August 1998

trace cache;Return History Stack;Next Trace Prediction;Multiple Branch Prediction;Path-Based Prediction

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Run-time spatial locality detection and optimization.

As the disparity between processor and main memory performance grows, the number of execution cycles spent waiting for memory accesses to complete also increases. As a result, latency hiding techniques are critical for improved application performance on future processors. We present a microarchitecture scheme which detects and adapts to varying spatial locality, dynamically adjusting the amount of data fetched on a cache miss. The Spatial Locality Detection Table, introduced in this paper, facilitates the detection of spatial locality across adjacent cached blocks. Results from detailed simulations of several integer programs show significant speedups. The improvements are due to the reduction of conflict and capacity misses by utilizing small blocks and small fetch sizes when spatial locality is absent, and the prefetching effect of large fetch sizes when spatial locality exists.

Introduction This paper introduces an approach to solving the growing memory latency problem [2] by intelligently exploiting spatial locality. Spatial locality refers to the tendency for neighboring memory locations to be referenced close together in time. Traditionally there have been two main approaches used to exploit spatial locality. The first approach is to use larger cache blocks, which have a natural prefetching effect. However, large cache blocks can result in wasted bus bandwidth and poor cache utilization, due to fragmentation and underutilized cache blocks. Both negative effects occur when data with little spatial locality is cached. The second common approach is to prefech multiple blocks into the cache. However, prefetching is only beneficial when the prefetched data is accessed in cache, otherwise the prefetched data may displace more useful data from the cache, in addition to wasting bus bandwidth. Similar issues exist with allocate caches, which, in effect, prefetch the data in the cache block containing the written address. Particu- This technical report is a longer version of [1]. larly when using large block sizes and write allocation, the amount of prefetching is fixed. However, the spatial locality, and hence the optimal prefetch amount, varies across and often within programs. As the available chip area increases, it is meaningful to spend more resources to allow intelligent control over latency-hiding techniques, adapting to the variations in spatial locality. For numeric programs there are several known compiler techniques for optimizing data cache performance. In contrast, integer (non-numeric) programs often have irregular access patterns that the compiler cannot detect and optimize. For example, the temporal and spatial locality of linked list elements and hash table data are often difficult to determine at compile time. This paper focuses on cache performance optimization for integer programs. While we focus our attention on data caches, the techniques presented here are applicable to instruction caches. In order to increase data cache effectiveness for integer programs we are investigating methods of adaptive cache hierarchy management, where we intelligently control caching decisions based on the usage characteristics of accessed data. In this paper we examine the problem of detecting spatial locality in accessed data, and automatically control the fetch of multiple smaller cache blocks into all data caches and buffers. Not only are we able to reduce the conflict and capacity misses with smaller cache lines and fetch sizes when spatial locality is absent, but we also reduce cold start misses and prefetch useful data with larger fetch sizes when spatial locality is present. We introduce a new hardware mechanism called the Spatial Locality Detection Table (SLDT). Each SLDT entry tracks the accesses to multiple adjacent cache blocks, facilitating detection of spatial locality across those blocks while they are cached. The resulting information is later recorded in the Memory Address Table [3] for long-term tracking of larger regions called macroblocks. We show that these extensions to the cache microarchitecture significantly improve the performance of integer applications, achieving up to 17% and 26% improvements for 100 and 200-cycle memory latencies, respectively. This scheme is fully compatible with existing Instruction Set Architectures (ISA). The remainder of this paper is organized as follows: Section related work; Section 3 discusses general spatial locality issues, and a code example from a common application is used to illustrate the role of spatial locality and cache line sizes in determining application cache perfor- mance, as well as to motivate our spatial locality optimization techniques; Section 4 discusses hardware techniques; Section 5 presents simulation results; Section 6 performs a cost analysis of the added hardware; and Section 7 concludes with future directions. Related Work Several studies have examined the performance effects of cache block sizes [4][5]. One of the studies allowed multiple consecutive blocks to be fetched with one request [4], and found that for data caches the optimal statically-determined fetch size was generally twice the block size. In this work we also examine fetch sizes larger than the block size, however, we allow the fetch size to vary based on the detected spatial locality. Another method allows the number of blocks fetched on a miss to vary across program execution, but not across different data [6]. Hardware [7][8][9][10][11] and software [12][13][14] prefetching methods for uniprocessor machines have been proposed. However, many of these methods focus on prefetching regular array accesses within well-structured loops, which are access patterns primarily found in numeric codes. Other methods geared towards integer codes [15][16] focus on compiler-inserted prefetching of pointer targets, and could be used in conjunction with our techniques. The dual data cache [17] attempts to intelligently exploit both spatial and temporal locality, however the temporal and spatial data must be placed in separate structures, and therefore the relative amounts of each type of data must be determined a priori. Also, the spatial locality detection method was tuned to numeric codes with constant stride vectors. In integer codes, the spatial locality patterns may not be as regular. The split temporal/spatial cache [18] is similar in structure to the dual data cache, however, the run-time locality detection mechanism is quite different than that of both the dual data cache and this paper. 3 Spatial Locality Caches seek to exploit the principle of locality. By storing a referenced item, caches exploit temporal locality - the tendency for that item to be rereferenced soon. Additionally, by storing multiple items adjacent to the referenced item, they exploit spatial locality - the tendency for neighboring items to be referenced soon. While exploitation of temporal locality can result in cache hits for future accesses to a particular item, exploitation of spatial locality can result in cache hits for future accesses to multiple nearby items, thus avoiding the long memory latency for short-term accesses to these items as well. Traditionally, exploitation of spatial locality is achieved through either larger block sizes or prefetching of additional blocks. We define the following terms as they will be used throughout this paper: element A data item of the maximum size allowed by the ISA, which in our system is 8 bytes. spatial reuse A reference to a cached element other than the element which caused the referenced element to be fetched into the cache. The spatial locality in an application's data set can predict the effectiveness of spatial locality optimizations. Unfortu- nately, no quantitative measure of spatial locality exists, and we are forced to adopt indirect measures. One indirect measure of the amount of spatial locality is via its inverse rela- tioship to the distance between references in both space and time. With this in view, we measured the spatial reuses in a 64K-byte fully-associative cache with 32-byte lines. This gives us an approximate time bound (the time taken for a block to be displaced), and a space bound (within 32-byte block boundaries). We chose this block size because past studies have found that 16 or 32-byte block sizes maximize data cache performance [4]. These measurement techniques differ from those in [19], which explicitely measure the reuse distance (in time). Our goal is to measure both the reused and unused portions of the cache blocks, for different cache organizations. Figure 1(a) shows the spatial locality estimates for the fully-associative cache. The number of dynamic cache blocks is broken down by the number of 8-byte elements that were accessed during each block's cache lifetime. Blocks where only one element is accessed have no spatial locality within the measured context. This graph does not show the relative locations of the accessed elements within each 32-byte cache block. Figure 1(a) shows that between 13-83% of the cached blocks have no spatial reuse. Figure 1(b) shows how this distribution changes for a 16K-byte direct-mapped cache. In this case between 30-93% of the blocks have no spatial reuse. For a 32-byte cache block, over half the time the extra data fetched into the cache simply wastes bus bandwidth and cache space. Similar observations have been made for numeric codes [19]. Therefore, it would be beneficial to tune the amount of data fetched and cached on a miss to the spatial locality available in the data. This optimization is investigated in our work. We discuss several issues involved with varying fetch sizes, including cost efficient and accurate spatial locality detection, fetch size choice, and cache support for varying fetch sizes. 3.1 Code Example In this section we use a code example from SPEC92 gcc to illustrate the difficulties involved with static analysis and annotation of spatial locality information, motivating our dynamic approach. One of the main data structures used in gcc is an RTL ex- pression, or rtx, whose definition is shown in Figure 2. Each rtx structure contains a two-byte code field, a one-byte mode field, seven one-bit flags, and an array of operand fields. The operand array is defined to contain only one four-byte ele- ment, however, each rtx is dynamically allocated to contain as many array elements as there are operands, depending on the rtx code, or RTL expression type. Therefore, each rtx instance contains eight or more bytes. In the frequently executed rtx renumbered equal tine, which is used during jump optimization, two rtx 026.compress 072.sc 099.go 147.vortex Pcode lmdes2_customizer 085.cc1 130.li 134.perl 124.m88ksim Benchmark total blocks blocks with accesses to four elements blocks with accesses to three elements blocks with accesses to two elements blocks with accesses to one elements (a) 64K-byte fully-associative 0% 20% 40% 80% 100% 026.compress 072.sc 099.go 147.vortex Pcode lmdes2_customizer 085.cc1 130.li 134.perl 124.m88ksim Benchmark total blocks blocks with accesses to four elements blocks with accesses to three elements blocks with accesses to two elements blocks with accesses to one elements (b) 16K-byte direct-mapped Figure 1: Breakdown of blocks cached in L1 data cache by how many 8-byte elements were accessed while each block was cached. The results for two cache configurations are shown, each with 32-byte blocks. struct rtxdef /* The kind of expression this is. */ enum /* The kind of value the expression has. */ enum machinemode mode : 8; /* Various bit flags */ unsigned int unsigned int call : unsigned int unchanging : unsigned int volatil : unsigned int instruct : unsigned int used : unsigned integrated : /* The first element of the operands of this rtx. The number of operands and their types are controlled by the 'code' field, according to rtl.def. */ rtunion fld[1]; Common union for an element of an rtx. */ int rtint; char struct struct enum machinemode rttype; Figure 2: Gcc rtx Definition structures are compared to determine if they are equivalent Figure 3 shows a slightly abbreviated version of the renumbered equal routine. After checking if the code and mode fields of the two rtx structures are identical, the routine then compares the operands, to determine if they are also identical. Four branch targets in Figure 3 are annotated with their execution weights, derived from execution profiles using the SPEC reference input. Roughly 1% of the time only the code fields of the two rtx structures are compared before exiting. In this case, only the first two bytes in each rtx structure is accessed. About 46% of the time x and y are CONST INT rtx, and only the first operand is accessed. Therefore, only the first eight bytes of each rtx structure is accessed, and there is spatial locality within those eight bytes. For many other types of RTL expressions, the routine will use the for loop to iterate through the operands, from last to first, comparing them until a mismatch is found. In this case there will be spatial locality, but at a slightly larger distance (in space) than in the previous case. Most instruction types contain more than one operand. The most common operand type in this loop is an RTL expression, which results in a recursive call to rtx renumbered equal p. This routine illustrates that the amount of spatial locality can vary for particular load references, depending on the function arguments. Therefore, if the original access into each rtx structure in this routine is a miss, the optimal amount of data to fetch into the cache will vary correspond- ingly. For example, if the access GETCODE(y) on line 10 of Figure 3, which performs the access y-?code, misses in the L1 cache, the spatial locality in that data depends on whether the program will later fall into a case body of the switch statement on line 11 or into the body of the for loop on line 24, and on the rtx type of x which determines the initial value of i in the for loop. However, at the time of the cache miss on line 10 this information is not available, as it is highly data-dependent. As such, neither static analysis (if even possible) nor profiling will result in definitive or accurate spatial locality information for the load instructions. Dynamic analysis of the spatial locality in the data offers greater promise. For this routine, dynamic analysis of each instance accessed in the routine would obtain the most accurate spatial locality detection. Also, dynamic schemes do not require profiling, which many users are unwilling to perform, or ISA changes. case LABEL_REF: return (next_real_insn (x->fld[0].rtx) == next_real_insn (y->fld[0].rtx)); case SYMBOL_REF: 19 return x->fld[0].rtstr == y->fld[0].rtstr; 22 /* Compare the elements. If any pair of corresponding elements fail to match, return 0 for the whole thing. */ { register int j; 26 switch (fmt[i]) { case 'i': 28 if (x->fld[i].rtint != y->fld[i].rtint) return 0; 29 break; case 's': return 0; 36 break; 37 case 'E': 38 . /* Accesses *({x,y}->fld[i].rtvec) */ . return 43 } { 3 register int 4 register RTX_CODE 5 register char *fmt; 9 { . /* Rarely entered */ . } case 'e': switch (code) { case PC: case CC0: case ADDR_VEC: case ADDR_DIFF_VEC: return 0; 34 if (! rtx_renumbered_equal_p (x->fld[i].rtx, y->fld[i].rtx)) 14 case CONST_INT: return x->fld[0].rtint == y->fld[0].rtint; Exits here 448 times Exits here times Case matches times Exits here 30014 times Figure 3: Gcc rtx renumbered equal routine, executed 63173 times. 3.2 Applications Aside from varying the data cache load fetch sizes, our spatial locality optimizations could be used to control instruction cache fetch sizes, write allocate versus no-allocate poli- cies, and bypass fetch sizes when bypassing is employed. The latter case is discussed briefly in [3], and is greatly expanded in this paper. In this paper we examine the application of these techniques to control the fetch sizes into the L1 and L2 data caches. We also study these optimizations in conjunction with cache bypassing, a complementary optimization that also aims to improve cache performance. 4.1 Overview of Prior Work In this section we briefly overview the concept of a mac- roblock, as well as the Memory Address Table (MAT), introduced in an earlier paper [3] and utilized in this work. We showed that cache bypassing decisions could be effectively made at run-time, based on the previous usage of the memory address being accessed. Other bypassing schemes include [20][21][17][22]. In particular, our scheme dynamically kept track of the accessing frequencies of memory regions called macroblocks. The macroblocks are statically- defined blocks of memory with uniform size, larger than the cache block size. The macroblock size should be large enough so that the total number of accessed macroblocks is not excessively large, but small enough so that the access patterns of the cache blocks contained within each macroblock are relatively uniform. It was determined that 1K-byte macroblocks provide a good cost-performance tradeoff. In order to keep track of the macroblocks at run time we use an MAT, which ideally contains an entry for each macroblock, and is accessed with a macroblock address. To support dynamic bypassing decisions, each entry in the table contains a saturating counter, where the counter value represents the frequency of accesses to the corresponding mac- roblock. For details on the MAT bypassing scheme see [3]. Also introduced in that paper was an optimization geared towards improving the efficiency of L1 bypasses, by tracking the spatial locality of bypassed data using the MAT, and using that information to determine how much data to fetch on an L1 bypass. In this paper we introduce a more robust spatial locality detection and optimization scheme using the SLDT, which enables much more efficient detection of spatial locality. Our new scheme also supports fetching varying amounts of data into both levels of the data cache, both with and without bypassing. In practice this spatial locality optimization should be performed in combination with by- passing, in order to achieve the best possible performance, as well as to amortize the cost of the MAT hardware. The cost of the combined hardware is addressed in Section 6, set 31 set 00 set 01 set set tag data 8 bytes 0x000000 0x000000 . tag data 8 bytes 0x000000 0x000000 Figure 4: Layout of 8-byte subblocks from the 32-byte block starting at address 0x00000000 in a 512-byte 2-way set-associative cache with 8-byte lines. The shaded blocks correspond to the locations of the four 8-byte subblocks. following the presentation of experimental results. 4.2 Support for Varying Fetch Sizes The varying fetch size optimization could be supported using subblocks. In that case the block size is the largest fetch size and the subblock size is gcd(fetch where n is the number of fetch sizes supported. Currently, we only support two power-of-two fetch sizes for each level of cache, so the subblock size is simply the smaller fetch size. However, the cache lines will be underutilized when only the smaller size is fetched. Instead, we use a cache with small lines, equal to the smaller fetch size, and optionally fill in multiple, consecutive blocks when the larger fetch size is chosen. This approach is similar to that used in some prefetching strategies [23]. As a result, the cache can be fully utilized, even when the smaller sizes are fetched. It also eliminates conflict misses resulting from accesses to different subblocks. However, this approach makes detection of spatial reuses much more difficult, as will be described in Section 4.3. Also, smaller block sizes increase the tag array cost, which is addressed in Section 6. In our scheme, the max fetch size data is always aligned to boundaries. As a result, our techniques will fetch data on either side of the accessed element, depending on the location of the element within the max fetch size block. In our experience, spatial locality in the data cache can be in either direction (spatially) from the referenced element 4.3 Spatial Locality Detection Table To facilitate spatial locality tracking, a spatial counter, or sctr, is included in each MAT entry. The role of the sctr is to track the medium to long-term spatial locality of the corresponding macroblock, and to make fetch size decisions, as will be explained in Section 4.4. This counter will be incremented whenever a spatial miss is detected, which occurs when portions of the same larger fetch size block of data reside in the cache, but not the element currently being ac- cessed. Therefore, a hit might have occurred if the larger fetch size was fetched, rather than the smaller fetch size. In our implementation, where multiple cache blocks are filled when the larger fetch size is chosen, a spatial miss is not trivial to detect. If the cache is not fully-associative, the tags for different blocks residing in the same larger fetch size block will lie in consecutive sets, as shown in Figure 4, where the data in one 32-byte block is highlighted. Searching for other cache blocks in the same larger fetch size block of data will require access to the tags in these consecutive sets, and thus either additional cycles to access, or additional hardware support. One possibility is a restructured tag array design allowing efficient access to multiple consecutive sets of tags. Alternatively, a separate structure can be used to detect this information, which is the approach investigated in this work. This structure is called the Spatial Locality Detection Table (SLDT), and is designed for efficient detection of spatial reuses with low hardware overhead. The role of the SLDT is to detect spatial locality of data while it is in the cache, for recording in the MAT when the data is displaced. The SLDT is basically a tag array for blocks of the larger fetch size, allowing single-cycle access to the necessary information Figure 5 shows an overview of how the SLDT interacts with the MAT and L1 data cache, where the double-arrow line shows the correspondence of four L1 data cache entries with a single SLDT entry. In order to track all cache blocks, the SLDT would need N entries, where N is the number of blocks in the cache. This represents the worst case of having fetched only smaller (line) size blocks into the cache, all from different larger size blocks. However, in order to reduce the hardware overhead of the SLDT, we use a much smaller number of entries, which will allow us to capture only the shorter-term spatial reuses. The same SLDT could be used to track the spatial locality aspects of all structures at the same level in the memory hierarchy, such as the data cache, the instruction cache, and, when we perform bypassing, the bypass buffer. The SLDT tags correspond to maximum fetch size blocks. The sz field is one bit indicating if either the larger size block was fetched into the cache, or if only smaller blocks were fetched. The vc (valid count) field is log(max fetch size=min fetch size) bits in length, and indicates how many of the smaller blocks in the larger size block are currently valid in the data cache. The actual number of valid smaller blocks is vc+1. An SLDT entry will only be valid for a larger size block when some of its constituent blocks are currently valid in the data cache. A bit mask could be used to implement the vc, rather than the counter design, to reduce the operational complexity. However, for large maximum to minimum fetch size ratios, a bit mask will result in larger entries. Finally, the sr (spatial reuse) bit will be set if spatial reuse is detected, as will be discussed later. When a larger size block of data is fetched into the cache, an SLDT entry is allocated (possibly causing the replacement of an existing entry) and the values of sz and vc are set to 1 and max fetch size=min fetch size \Gamma 1, respectively. If a smaller size block is fetched and no SLDT entry currently exists for the corresponding larger size block, then an entry is allocated and sz and vc are both initialized to 0. If an entry already exists, vc is incremented to indicate that there is now an additional valid constituent block in the data cache. For both fetch sizes the sr bit is initialized to 0. When a cache block is replaced from the data cache, the corresponding SLDT entry is accessed and its vc value is decremented if it is greater than 0. If vc is already 0, then this was the only valid block, so the SLDT entry is invalidated. When s { spatial reuse? hit? MAT sctr fetch update sctr with hit and spatial reuse results L1 Data Cache addr tag sz vc sr Figure 5: SLDT and MAT Hardware an SLDT entry is invalidated its sr bit is checked to see if there was any spatial reuse while the data was cached. If not, the corresponding entry in the MAT is accessed and its sctr is decremented, effectively depositing the information in the MAT for longer-term tracking. Because the SLDT is managed as a cache, entries can be replaced, in which case the same actions are taken. An fi (fetch initiator) bit is added to each data cache tag to help detect spatial hits. The fi bit is set to 1 during the cache refill for the cache block containing the referenced element (i.e. the cache block causing the fetch), otherwise it is reset to 0. Therefore, a hit to any block with a 0 fi bit is a spatial hit, as this data was fetched into the cache by a miss to some other element. Table 1 summarizes the actions taken by the SLDT for memory accesses. The sr bit, which was initialized to zero, is set for all types of both spatial misses and spatial hits. Two types of spatial misses are detected. The first type of spatial miss occurs when other portions of the same larger fetch size block were fetched independently, indicated by a valid SLDT entry with a sz of 0. Therefore, there might have been a cache hit if the larger size block was fetched, so the corresponding entry in the MAT is accessed and its sctr is incremented. The second type can occur when the larger size block was fetched, but one of its constituent blocks was displaced from the cache, as indicated by a cache miss and a valid SLDT entry with a sz of 1. It is not trivial to detect if this miss is to the element which caused the original fetch, or to some other element in the larger fetch size block. The sr bit is conservatively set, but the sctr in the corresponding MAT entry is not incremented. A spatial hit can occur in two situations. If the larger size block was fetched, then the fi bit will only be set for one of the loaded cache blocks. A hit to any of the loaded cache blocks without the fi bit set is a spatial hit, as described earlier. We do not increment the sctr on spatial hits, because our fetch size was correct. We only update the sctr when the fetch size should be changed in the future. When multiple smaller blocks were fetched, a hit to one of these is also characterized as a spatial hit. This case is detected by checking if vc is larger than 0 when sz is 0. However, we do not increment the sctr in this case either because a spatial miss would have been detected earlier when a second element in the larger fetch size block was first accessed (and missed). Cache SLDT Access Access fi sz vc Action miss hit - 0 Cache entry vc ?0 replaced vc == 0 invalidate SLDT entry SLDT entry replaced sr == 0 or invalidated sr == 1 no action Table 1: SLDT Actions. A dash indicates that there is no corresponding value, and a blank indicates that the value does not matter. 4.4 Fetch Size Decisions On a memory access, a lookup in the MAT of the corresponding macroblock entry is performed in parallel with the data cache access. If an entry is found, the sctr value is compared to some threshold value. The larger size is fetched if the sctr is larger than the threshold, otherwise the smaller size is fetched. If no entry is found, a new entry is allocated and the sctr value is initialized to the threshold value, and the larger fetch size is chosen. In this paper the threshold is 50% of the maximum sctr value. 5 Experimental Evaluation 5.1 Experimental Environment We simulate ten benchmarks, including 026.compress, 072.sc and 085.cc1 from the SPEC92 benchmark suite using the reference inputs, and 099.go, 147.vortex, 130.li, 134.perl, and 124.m88ksim from the SPEC95 benchmark suite using the training inputs. The last two benchmarks consist of modules from the IMPACT compiler [24] that we felt were representative of many real-world integer applica- tions. Pcode, the front end of IMPACT, is run performing dependence analysis with the internal representation of the combine.c file from GNU CC as input. lmdes2 customizer, a machine description optimizer, is run optimizing the SuperSPARC machine description. These optimizations operate over linked list and complex data structures, and utilize hash tables for efficient access to the information. In order to provide a realistic evaluation of our technique for future high-performance, high-issue rate systems, we first optimized the code using the IMPACT compiler [24]. Classical optimizations were applied, then optimizations were performed which increase instruction level parallelism. The code was scheduled, register allocated and optimized for an eight-issue, scoreboarded, superscalar processor with register renaming. The ISA is an extension of the HP PA-RISC instruction set to support compile-time speculation. We perform cycle-by-cycle emulation-driven simulation on a Hewlett-Packard PA-RISC 7100 workstation, modelling the processor and the memory hierarchy (including all related busses). The instruction latencies used are those of a Hewlett-Packard PA-RISC 7100, as given in Table 2. The base machine configuration is described in Table 3. Since simulating the entire applications at this level of detail would be impractical, uniform sampling is used to reduce simulation time [25], however emulation is still performed Function Latency Function Latency memory load 2 FP multiply 2 memory store 1 FP divide (single prec.) 8 branch Table 2: Instruction latencies for simulation experiments. L1 Icache 32K-byte split-block, direct mapped, 64-byte block L1 Dcache 16K-byte non-blocking (50 max), direct mapped, 32-byte block, multiported, writeback, no write alloc L1-L2 Bus 8-byte bandwidth, split-transaction, 4-cycle latency, returns critical word first L2 Dcache same as L1 Dcache except: 256K-byte, 64-byte block System Bus same as L1-L2 Bus except: 100-cycle latency Issue 8-issue uniform, except 4 memory ops/cycle max Registers 64 integer, 64 double precision floating-point Table 3: Base Configuration. between samples. The simulated samples are 200,000 instructions in length and are spaced evenly every 20,000,000 instructions, yielding a 1% sampling ratio. For smaller ap- plications, the time between samples is reduced to maintain at least 50 samples (10,000,000 instructions). To evaluate the accuracy of this technique, we simulated several configurations both with and without sampling, and found that the improvements reported in this paper are very close to those obtained by simulating the entire application. 5.2 Macroblock Spatial Locality Variations Before presenting the performance improvements achieved by our optimizations, we first examine the accuracy of the macroblock granularity for tracking spatial locality. It is important to have accurate spatial locality information in the MAT for our scheme to be successful. This means that all data elements in a macroblock should have similar amounts of spatial locality at each phase of program execution. After dividing main memory into macroblocks, as described in Section 4.1, the macroblocks can be further subdivided into smaller sections, each the size of a 32-byte cache block. We will simply call these smaller sections blocks. In order to determine the dynamic cache block spatial locality behavior, we examined the accesses to each of these blocks, gathering information twice per simulation sample, or every 100,000 instructions. At the end of each 100,000-instruction phase, we determined the fraction of times that each block in memory had at least one spatial reuse each time it was cached during that phase. We call this the spatial reuse fraction for that block. Figure 6 shows a graphical representation of the resulting information for three programs. Each row in the graph represents a 1K-byte macroblock accessed in a particular phase. For every phase in which a particular macroblock was accessed, there will be a corresponding row. Each row contains one data point for every 32-byte block accessed during the corresponding phase that lies in that macroblock. For the purposes of clarity, the rows were sorted by the average of the block spatial reuse fractions per macroblock. The averages increase from the bottom to the top of the graphs. The cache blocks in each macroblock were also sorted so that their spatial reuse fractions increase from left to right. Some rows are not full, meaning that not all of their blocks were accessed during the corresponding phase. Finally, the cache blocks with spatial reuse fractions falling within the same range were plotted with the same marker. Figure 6(a) shows the spatial locality distribution for 026.compress. Most of the blocks, corresponding to the lighter gray points, have spatial reuse fractions between 0 and 0.25, meaning that there was spatial reuse to those blocks less than 25% of the time they were cached. Very few of the blocks, corresponding to the black points, had spatial reuse more than 75% of the time they were cached. This represents a fairly optimal scenario, because most of the macroblocks contain blocks which have approximately the same amount of reuse. Figure 6(b) shows the distribution for 134.perl. Around 34% of the macroblocks (IDs 0 to 6500) contain only blocks with little spatial reuse, their spatial reuse fractions all less than 0.25. About 29% of the macroblocks (IDs 13500 to 18900) contain only blocks with large fractions of spatial reuse, their spatial reuse fractions all over 0.75. About 37% of the macroblocks contain cache blocks with differing amounts of spatial reuse. The medium gray points in some of these rows correspond to blocks with spatial reuse fractions between 0.25 and 0.75. However, this information does not reveal the time intervals over which the spatial reuse in these blocks varies. It is possible that in certain small phases of program execution the spatial locality behavior is uniform, but that it changes drastically from one small phase of execution to another. This type of behavior is possible due to dynamically-allocated data, where a particular section of memory may be allocated as one type of data in one part of the program, then freed and reallocated as another type later. Finally, Figure 6(c) shows the distribution for 085.gcc, which has similar characteristics to 134.perl, but has more macroblocks with non-uniform spatial reuse fractions. 5.3 Performance Improvements In this section we examine the performance improvement, or the execution cycles eliminated, over the base 8-issue configuration described in Section 5.1. To support varying fetch sizes, we use an SLDT and an MAT at each level of the cache hierarchy. The L1 and L2 SLDTs are direct-mapped with entries. A large number of simulations showed that direct-mapped SLDTs perform as well as a fully-associative design, and that 32 entries perform almost as well as any larger power-of-two number of entries up to 1024 entries, which was the maximum size examined. The L1 and L2 MATs utilize 1K-byte macroblocks, and we examine both one and four-bit sctrs, We first present results for infinite- entry MATs, then study the effects of limiting the number of MAT entries. 5.3.1 Static versus Varying Fetch Sizes The left bar for each benchmark in Figure 7(a) shows the performance improvement achieved by using 8-byte L1 data cache blocks with a static 8-byte fetch size, over the base 32- byte block and fetch sizes. These bars show that the better choice of block size is highly application-dependent. The right bars show the improvement achieved by our spatial locality optimization at the L1 level only, using an 8-byte data cache block size, and fetching either 8 or 32-bytes on an L1 data cache miss, depending on the value of the (a) 026.compress (b) 134.perl (b) 085.gcc Figure reuse fractions (srf) for cache-block-sized-data in the accessed macroblocks for three applications. corresponding sctr. The results show that our scheme is able to obtain either almost all of the performance, or is able to outperform, the best static fetch size scheme. In most cases the 1 and 4-bit sctrs perform similarly, but in one case the 4-bit sctr achieves almost 2% greater performance improvement. The four leftmost bars for each benchmark in Figure 7(b) show the performance improvement using different L2 data cache block and (static) fetch sizes, and our L1 spatial locality optimization with a 4-bit sctr. The base configuration is again the configuration described in Section 5.1, which has 64-byte L2 data cache block and fetch sizes. These bars show that, again, the better static block/fetch size is highly application-dependent. For example, 134.perl achieves much better performance with a 256-byte fetch size, while 026.compress achieves its best performance with a 32- byte fetch size, obtaining over 14% performance degradation with 256-byte fetches. The rightmost two bars in Figure 7(b) show the performance improvement achieved with our L2 spatial locality optimization, which uses a 32-byte L2 data cache block size and fetches either 32 or 256 bytes on an L2 data cache miss, depending on the value of the corresponding L2 MAT sctr. Again, our spatial locality optimizations are able to obtain almost the same or better performance than the best static fetch size scheme for all benchmarks. Figure 8 shows the breakdown of processor stall cycles attributed to different types of data cache misses, as a percentage of the total base configuration execution cycles. The left and right bars for each benchmark are the stall cycle breakdown for the base configuration and our spatial locality optimization, respectively. The spatial locality optimizations were performed at both cache levels, using the same configuration as in Figure 7(b) with a 4-bit sctr. For the benchmarks that have large amounts of spatial locality, as indicated from the results of Figure 7, we obtain large reductions in L2 cold start stall cycles by fetching 256 bytes on L2 cache misses. The benchmarks with little spatial locality in the L1 data cache, such as 026.compress and Pcode, obtained reductions in L1 capacity miss stall cycles from fetching fewer small cache blocks on L1 misses. In some cases the L1 cold start stall cycles increase, indicating that the L1 optimizations are less aggressive in terms of fetching more data, however these increases are generally more than compensated by reductions in other types of L1 stall cycles. The conflict miss stall cycles increase for lmdes2 customizer, because it tends to fetch fewer blocks on an L1 miss, exposing some conflicts that were interpreted as capacity misses in the base configuration. Revisiting the example of Section 3.1, we found that the access y-?code on line 10 of Figure 3 missed 11,223 times, fetching bytes for 47% of the misses, and 8 bytes for the remaining 53%. We also found that on average, 0.99 spatial hits and only 0.02 spatial misses to the resulting data occurred per miss, illustrating that our techniques are successfully choosing the appropriate amount of data to fetch on a miss. 5.3.2 Set-associative Data Caches Increasing the set-associativity of the data caches can reduce the number of conflict misses, which may in turn reduce the advantage offered by our optimizations. However, the 8.00% 10.00% 026.compress 072.sc 099.go 147.vortex Pcode lmdes2_customizer 085.cc1 130.li 134.perl 124.m88ksim Benchmark Improvement over Base L1 static 8-byte block/fetch size varying fetch (1-bit sctr) varying fetch (4-bit sctr) (a) L1 Trends -15.00% -10.00% -5.00% 5.00% 10.00% 15.00% 20.00% 026.compress 072.sc 099.go 147.vortex Pcode lmdes2_customizer 085.cc1 130.li 134.perl 124.m88ksim Benchmark Improvement over Base L2 static 32-byte block/fetch size L2 static 64-byte block/fetch size L2 static 128-byte block/fetch size L2 static 256-byte block/fetch size varying fetch (1-bit sctr) varying fetch (4-bit sctr) (b) L2 Trends (with L1 varying fetches) Figure 7: Performance for various statically-determined block/fetch sizes and for our spatial locality optimizations using both 1 and 4-bit sctrs. 10.00% 20.00% 30.00% 40.00% 50.00% 70.00% 80.00% 90.00% 026.compress (base) (opti) (base) (opti) (base) (opti) 147.vortex (base) (opti) Pcode (base) (opti) lmdes2_customizer (base) (opti) (base) (opti) (base) (opti) 134.perl (base) (opti) (base) (opti) Benchmark %total base execution cycles cold start stall cycles L2 capacity miss stall cycles conflict miss stall cycles cold start stall cycles L1 capacity miss stall cycles conflict miss stall cycles Figure 8: Stall cycle breakdown for base and the spatial locality optimizations. reductions in capacity and cold start stall cycles that our optimizations achieve should remain. To investigate these effects, the data cache configuration discussed in Section 5.1 was modified to have a 2-way set-associative L1 data cache and a 4-way set-associative L2 data cache. Figure 9 shows the new performance improvements for our optimizations. The left bars show the result of applying our optimizations to the L1 data cache only, and the right bars show the result of applying our techniques to both the L1 and L2 data caches, using four-bit sctrs. The improvements have reduced significantly for some benchmarks over those shown in Figure 7. However, large improvements are still achieved for some benchmarks, particularly when applying the optimizations at the L2 data cache level, due to the reductions we achieve in L2 cold start stall cycles for data with spatial locality. -2.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 20.00% 026.compress 072.sc 099.go 147.vortex Pcode lmdes2_custom izer Benchmark Improvement over Base varying fetch (4-bit sctr) varying fetch (4-bit sctrs) Figure 9: Performance for the spatial locality optimizations with 2-way and 4-way set-associative L1 and L2 data caches, respectively. 5.3.3 Growing Memory Latency Effects As discussed in Section 1, memory latencies are increasing, and this trend is expected to continue. Figure 10 shows the improvements achieved by our optimizations when applied to direct-mapped caches for both 100 and 200-cycle la- tencies, each relative to a base configuration with the same memory latency. Most of the benchmarks see much larger improvements from our optimizations, with the exception of 026.compress. Because 026.compress has very little spatial locality to exploit, the longer latency cannot be hidden as effectively. Although the raw number of cycles we eliminate grows, as a percentage of the associated base execution cycle count it becomes smaller. 5.3.4 Comparison of Integrated Techniques to Doubled Data Caches As the memory latencies increase, intelligent cache management techniques will become increasingly important. We examined the performance improvement achieved by integrat- 026.compress 072.sc 099.go 147.vortex Pcode lmdes2_custom izer Benchmark Improvement over Base 100-cycle latency 200-cycle latency Figure 10: Performance for the spatial locality optimizations with growing memory latencies. 5.00% 10.00% 15.00% 20.00% 30.00% 026.compress 072.sc 099.go 147.vortex Pcode lmdes2_custom izer Benchmark Improvement over Base doubled (32K/512K) L1/L2 varying fetch (4-bit sctr) & bypass (infinite MAT) varying fetch (4-bit sctr) & bypass (1K-entry MAT) varying fetch (4-bit sctr) & bypass (512-entry MAT) Figure 11: Comparison of doubled caches to integrated spatial locality and bypassing optimizations. Infinite, 1024- entry, and 512-entry direct-mapped MATs are examined. ing our spatial locality optimizations with intelligent bypass- ing, using 8-bit access counters in each MAT entry [3]. The 4-way set-associative buffers used to hold the bypassed data at the L1 and L2 caches contain 128 8-byte entries and 512 32-byte entries, respectively. Then, the SLDT and MAT at each cache level are used to detect spatial locality and control the fetch sizes for both the data cache and the bypass buffer at that level. Figure 11 shows the improvements achieved by combining these techniques at both cache levels for a 100-cycle memory latency. We show results for three direct-mapped MAT sizes: infinite, 1K-entry, and 512-entry. Also shown are the performance improvements achieved by doubling both the L1 and L2 data caches. Doubling the caches is a brute-force technique used to improve cache performance. Figure 11 shows that performing our integrated optimizations at both cache levels can outperform simply doubling both levels of cache. The only case where the doubled caches perform significantly better than our optimizations is for 026.compress. This improvement mostly comes from doubling the L2 data cache, which results because its hash tables can fit into a Data Block Tag Tag Cache Cost Size Size Cost Level (bytes) (bytes) Sets (bits) (bytes) Table 4: Hardware Cost of Doubled Data Caches. 512K-byte cache. Pcode is the only benchmark for which the performance degrades significantly when reducing the MAT size, however, 1K-entry MATs can still outperform the doubled caches. Comparing Figure 11 to the bypassing improvements in [3] shows that often significant improvements can be achieved by intelligently controlling the fetch sizes into the data caches and bypass buffers. 6 Design Considerations In this section we examine the hardware cost of the spatial locality optimization scheme described in Section 4, and compare this to the cost of doubling the data caches at each level. As discussed in Section 4.1, the cost of the MAT hardware is amortized when performing both spatial locality and bypassing optimizations. For this reason, we compute the hardware cost of the hardware support for both of these opti- mizations, just as their combined performance was compared to the performance of doubling the caches in Section 5.3.4. The additional hardware cost incurred by our spatial locality optimization scheme is small compared to doubling the cache sizes at each level, particularly for the L2 cache. For the 16K-byte direct-mapped L1 cache used to generate the results of Section 5.3, bits of tag are used per entry (assuming 32-bit addresses). Doubling this cache will result in 17-bit tags. Because the line size is 32 bytes, the total additional cost of the increased tag array will be 17\Lambda2 which is 1K bytes 1 In addition, an extra 16K of data is needed. Similar computations will show that the cost of doubling the 256K-byte L2 cache is an extra 6144 bytes of tag and 256K bytes of data. The total tag and data costs of the doubled L1 and L2 caches is shown in Table 4. For a direct-mapped MAT with 8-bit access counters and 4-bit spatial counters, Table 12 gives the hardware cost of the data and tags for the MAT sizes discussed in Section 5.3.4. Since all addresses within a macroblock map to the same MAT counter, a number of lower address bits are discarded when accessing the MAT. The size of the resulting MAT address for 1K-byte macroblocks is shown in column 3 of Table 12(a). Table 12(b) shows the data and tag array costs for the direct-mapped data caches in our spatial locality optimization scheme. The data cost remains the same as the base configuration cost, but the tag array cost is increased due to the decreased line sizes and additional support for our scheme, which requires a 1-bit fetch initiator bit per tag entry The cost for the L1 buffer, which is a 4-way set-associative cache with 8-byte lines is shown in Table 12(c). As with the optimized data caches, the bypass buffers require a 1-bit fetch initiator bit, in addition to the address tag. The cost for the L2 bypass buffer is computed similarly in Table 12(c). 1 We are ignoring the valid bit and other state. MAT Data Cost Size of MAT Tag Size Tag Cost Entries (bytes) Address (bits) (bits) (bytes) (a) Hardware Cost of 512 and 1K entry MATs. Cost is same for both L1 and L2 cache levels. Data Fetch Block Tag Tag Cache Cost Size Size Size Cost Level (bytes) (bytes) (bytes) Sets (bits) (bytes) (b) Hardware Cost of Optimized Data Caches. Block Data Tag Tag Cache Fetch Size Cost Size Cost Level Entries Size (bytes) (bytes) (bytes) (bits) (bytes) (c) Hardware Cost of Bypass Buffers. Cache SLDT Tag Size Tag Cost Level Entries (bits) (bytes) (d) Hardware Cost of SLDTs. Figure 12: Hardware Cost Breakdown of Spatial Locality Optimizations. The final component of the spatial locality optimization scheme is the 32-entry SLDT, which can be organized as a direct-mapped tag array, with the vc, 1-bit sz and 1-bit sr fields included in each tag entry. The L1 SLDT requires a 2- bit vc because there are 4 8-byte lines per 32-byte maximum fetch, and the L2 SLDT requires a 3-bit vc due to the 8 32- byte lines per 256-byte maximum fetch. A bit mask could be used to implement the vc, rather than the counter design, to reduce the operational complexity. However, for large maximum to minimum fetch size ratios, such as the 8-to-1 ratio for the L2 cache, a bit mask will result in larger entries. Table 12(d) shows the total tag array costs of the L1 and L2 SLDTs. Finally, combining the costs of the MAT, the optimized data cache, the bypass buffer, and the SLDT results in a total L1 cost of 24376 bytes with a 512-entry MAT, and 25848 bytes with a 1K-entry MAT. Therefore, the savings over doubling the L1 data cache is over 10K and 8K bytes for the 512 and 1K-entry MATs, respectively. Similar calculations show that our L2 optimizations save over 247K bytes and 245K bytes for the 512 and 1K-entry MATs, re- spectively, over doubling the L2 data cache. This translates into 26% and 44% less tags and data than doubling the data caches at the L1 and L2 levels, respectively, for the larger 1K-entry MAT. Comparing the performance of the spatial locality and bypassing optimizations to the performance obtained by doubling the data caches at both levels, as shown in Figure 11, illustrates that for much smaller hardware costs our optimizations usually outperform simply doubling the caches. To reduce the hardware cost, we could potentially integrate the L1 MAT with the TLB and page tables. For a macroblock size larger than or equal to the page size, each TLB entry will need to hold only one 8-bit counter value. For a macroblock size less than the page size, each TLB entry needs to hold several counters, one for each of the macroblocks within the corresponding page. In this case a small amount of additional hardware is necessary to select between the counter values. However, further study is needed to determine the full effects of TLB integration. 7 Conclusion In this paper, we examined the spatial locality characteristics of integer applications. We showed that the spatial locality varied not only between programs, but also varied vastly between data accessed by the same application. As a result of varying spatial locality within and across applica- tions, spatial locality optimizations must be able to detect and adapt to the varying amount of spatial locality both within and across applications in order to be effective. We presented a scheme which meets these objectives by detecting the amount of spatial locality in different portions of memory, and making dynamic decisions on the appropriate number of blocks to fetch on a memory access. A Spatial Locality Detection Table (SLDT), introduced in this paper, facilitates spatial locality detection for data while it is cached. This information is later recorded in a Memory Address Table (MAT) for long-term tracking, and is then used to tune the fetch sizes for each missing access. Detailed simulations of several applications showed that significant speedups can be achieved by our techniques. The improvements are due to the reduction of conflict and capacity misses by utilizing small blocks and small fetch sizes when spatial locality is absent, and utilizing the prefetching effect of large fetch sizes when spatial locality exists. In ad- dition, we showed that the speedups achieved by this scheme increase as the memory latency increases. As memory latencies increase, the importance of cache performance improvements at each level of the memory hierarchy will continue to grow. Also, as the available chip area grows, it makes sense to spend more resources to allow intelligent control over the cache management, in order to adapt the caching decisions to the dynamic accessing behav- ior. We believe that our schemes can be extended into a more general framework for intelligent runtime management of the cache hierarchy. Acknowledgements The authors would like to thank Mark Hill, Santosh Abraham and Wen-Hann Wang, as well as all the members of the IMPACT research group, for their comments and suggestions which helped improve the quality of this research. This research has been supported by the National Science Foundation (NSF) under grant CCR-9629948, Intel Corpo- ration, Advanced Micro Devices Hewlett-Packard, SUN Mi- crosystems, NCR, and the National Aeronautics and Space Administration (NASA) under Contract NASA NAG 1-613 in cooperation with the Illinois Computer laboratory for Aerospace Systems and Software (ICLASS). --R "Run-time spatial locality detection and optimization," "Predicting and precluding problems with memory latency," "Run-time adaptive cache hierarchy management via reference analysis," "The performance impact of block sizes and fetch strategies," "Line (block) size choice for cpu cache memo- ries," "Fixed and adaptive sequential prefetching in shared memory multipro- cessors," "Cache memories," "Improving direct-mapped cache performance by the addition of a small fully-associative cache and prefetch buffers," "An effective on-chip preloading scheme to reduce data access penalty," "Quantifying the performance potential of a data prefetch mechanism for pointer-intensive and numeric programs," "Stride directed prefetching in scalar processors," Software Methods for Improvement of Cache Performance on Supercomputer Applications. "Design and evaluation of a compiler algorithm for prefetching," "Data access microarchitectures for superscalar processors with compiler-assisted data prefetching," "Compiler-based prefetching for recursive data structures," "SPAID: Software prefetching in pointer- and call- intensive environments," "A data cache with multiple caching strategies tuned to different types of local- ity," "The split temporal/spatial cache: Initial performance analysis," "A quantitative analysis of loop nest locality," "Efficient simulation of caches under optimal replacement with applications to miss characterization," "A modified approach to data cache management," "Reducing conflicts in direct-mapped caches with a temporality-based design," "Data prefetching in multi-processor vector cache memories," "IMPACT: An architectural framework for multiple-instruction-issue processors," "How to simulate 100 billion references cheaply," --TR Line (block) size choice for CPU cache memories Data prefetching in multiprocessor vector cache memories IMPACT Data access microarchitectures for superscalar processors with compiler-assisted data prefetching An effective on-chip preloading scheme to reduce data access penalty Design and evaluation of a compiler algorithm for prefetching directed prefetching in scalar processors A data cache with multiple caching strategies tuned to different types of locality A modified approach to data cache management Compiler-based prefetching for recursive data structures Run-time adaptive cache hierarchy management via reference analysis Cache Memories Predicting and Precluding Problems with Memory Latency Software methods for improvement of cache performance on supercomputer applications --CTR Guest Editors' Introduction-Cache Memory and Related Problems: Enhancing and Exploiting the Locality, IEEE Transactions on Computers, v.48 n.2, p.97-99, February 1999 Afrin Naz , Mehran Rezaei , Krishna Kavi , Philip Sweany, Improving data cache performance with integrated use of split caches, victim cache and stream buffers, ACM SIGARCH Computer Architecture News, v.33 n.3, June 2005 Jike Cui , Mansur. H. Samadzadeh, A new hybrid approach to exploit localities: LRFU with adaptive prefetching, ACM SIGMETRICS Performance Evaluation Review, v.31 n.3, p.37-43, December Sanjeev Kumar , Christopher Wilkerson, Exploiting spatial locality in data caches using spatial footprints, ACM SIGARCH Computer Architecture News, v.26 n.3, p.357-368, June 1998 Jie Tao , Wolfgang Karl, Detailed cache simulation for detecting bottleneck, miss reason and optimization potentialities, Proceedings of the 1st international conference on Performance evaluation methodolgies and tools, October 11-13, 2006, Pisa, Italy Srikanth T. Srinivasan , Roy Dz-ching Ju , Alvin R. Lebeck , Chris Wilkerson, Locality vs. criticality, ACM SIGARCH Computer Architecture News, v.29 n.2, p.132-143, May 2001 Gokhan Memik , Mahmut Kandemir , Alok Choudhary , Ismail Kadayif, An Integrated Approach for Improving Cache Behavior, Proceedings of the conference on Design, Automation and Test in Europe, p.10796, March 03-07, McCorkle, Programmable bus/memory controllers in modern computer architecture, Proceedings of the 43rd annual southeast regional conference, March 18-20, 2005, Kennesaw, Georgia Neungsoo Park , Bo Hong , Viktor K. Prasanna, Tiling, Block Data Layout, and Memory Hierarchy Performance, IEEE Transactions on Parallel and Distributed Systems, v.14 n.7, p.640-654, July Jaeheon Jeong , Per Stenstrm , Michel Dubois, Simple penalty-sensitive replacement policies for caches, Proceedings of the 3rd conference on Computing frontiers, May 03-05, 2006, Ischia, Italy Hur , Calvin Lin, Memory Prefetching Using Adaptive Stream Detection, Proceedings of the 39th Annual IEEE/ACM International Symposium on Microarchitecture, p.397-408, December 09-13, 2006 Prateek Pujara , Aneesh Aggarwal, Increasing cache capacity through word filtering, Proceedings of the 21st annual international conference on Supercomputing, June 17-21, 2007, Seattle, Washington Hantak Kwak , Ben Lee , Ali R. Hurson , Suk-Han Yoon , Woo-Jong Hahn, Effects of Multithreading on Cache Performance, IEEE Transactions on Computers, v.48 n.2, p.176-184, February 1999 Ben Juurlink , Pepijn de Langen, Dynamic techniques to reduce memory traffic in embedded systems, Proceedings of the 1st conference on Computing frontiers, April 14-16, 2004, Ischia, Italy Tony Givargis, Improved indexing for cache miss reduction in embedded systems, Proceedings of the 40th conference on Design automation, June 02-06, 2003, Anaheim, CA, USA Mirko Loghi , Paolo Azzoni , Massimo Poncino, Tag Overflow Buffering: An Energy-Efficient Cache Architecture, Proceedings of the conference on Design, Automation and Test in Europe, p.520-525, March 07-11, 2005 Timothy Sherwood , Brad Calder , Joel Emer, Reducing cache misses using hardware and software page placement, Proceedings of the 13th international conference on Supercomputing, p.155-164, June 20-25, 1999, Rhodes, Greece Soontae Kim , N. Vijaykrishnan , Mahmut Kandemir , Anand Sivasubramaniam , Mary Jane Irwin, Partitioned instruction cache architecture for energy efficiency, ACM Transactions on Embedded Computing Systems (TECS), v.2 n.2, p.163-185, May Razvan Cheveresan , Matt Ramsay , Chris Feucht , Ilya Sharapov, Characteristics of workloads used in high performance and technical computing, Proceedings of the 21st annual international conference on Supercomputing, June 17-21, 2007, Seattle, Washington Jonathan Weinberg , Michael O. McCracken , Erich Strohmaier , Allan Snavely, Quantifying Locality In The Memory Access Patterns of HPC Applications, Proceedings of the 2005 ACM/IEEE conference on Supercomputing, p.50, November 12-18, 2005

data cache;prefetching;block size;cache management;spatial locality

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The design and performance of a conflict-avoiding cache.

High performance architectures depend heavily on efficient multi-level memory hierarchies to minimize the cost of accessing data. This dependence will increase with the expected increases in relative distance to main memory. There have been a number of published proposals for cache conflict-avoidance schemes. We investigate the design and performance of conflict-avoiding cache architectures based on polynomial modulus functions, which earlier research has shown to be highly effective at reducing conflict miss ratios. We examine a number of practical implementation issues and present experimental evidence to support the claim that pseudo-randomly indexed caches are both effective in performance terms and practical from an implementation viewpoint.

Introduction On current projections the next 10 years could see CPU clock frequencies increase by a factor of twenty whereas DRAM row-address-strobe delays are projected to decrease by only a factor of two. This potential ten-fold increase in the distance to main memory has serious implications for the design of future cache-based memory hierarchies as well as for the architecture of memory devices themselves. There are many options for an architect to consider in the battle against memory latency. These can be partitioned into two broad categories - latency reduction and latency hiding. Latency * Departament d'Arquitectura de Computadors Universitat Polit-cnica de Catalunya c/ Jordi Girona 1-3, 08034 Barcelona (Spain) Email:{antonio, joseg}@ac.upc.es Department of Computer Science University of Edinburgh JCMB, Kings Buildings, Edinburgh (UK) reduction techniques rely on caches to exploit locality with the objective of reducing the latency of each individual memory reference. Latency hiding techniques exploit parallelism to overlap memory latency with other operations and thus "hide" it from a program's critical path. This paper addresses the issue of latency reduction and the degree to which future cache architectures can isolate their processor from increasing memory latency. We discuss the theory, and evaluate the practice, of using a particular class of conflict-avoidance indexing functions. We demonstrate how such a cache could be constructed and provide practical solutions to some previously un-reported problems, as well as some known problems, associated with unconventional indexing schemes. In section 2 we present an overview of the causes of conflict misses and summarise previous techniques that have been proposed to minimize their effect on performance. We propose a method of cache indexing which has demonstrably lower miss ratios than alternative schemes, and summarise the known characteristics of this method. In section 3 we discuss a number of implementation issues, such as the effect of using this novel indexing scheme on the processor cycle time. We present an experimental evaluation of the proposed indexing scheme in section 4. Our results show how the IPC of an out-of-order superscalar processor can be improved through the use of our proposed indexing scheme. Finally, in section 5, we draw conclusions from this study. 2 The problem of cache conflicts Whenever a block of main memory is brought into cache a decision must be made on which block, or set of blocks, in the cache will be candidates for storing that data. This is referred to as the placement policy. Conventional caches typically extract a field of bits from the address and use this to select one block from a set of . Whilst simple, and easy to implement, this indexing function is not robust. The principal weakness of this function is its susceptibility to repetitive conflict misses. For example, if is the cache capacity and is the block size, then addresses and map to the same cache set if . Condition 1 Repetitive collisions If and collide on the same cache block, then addresses and also collide in cache, for any integer , except when . Where and . There are two common cases when this happens: . when accessing a stream of addresses if collides with , then there may be up to conflict misses in this stream. . when accessing elements of two distinct arrays and , if collides with then collides with . -way set-associativity can help to alleviate such conflicts. However, if a working set contains conflicts on some cache block, set associativity can only eliminate at most of those conflicts. The following section proposes a remedy to the problem of cache conflicts by defining an improved method of block placement. 2.1 Conflict-resistant cache placement functions The objective of a conflict-resistant placement function is to avoid the repetitive conflicts defined by Condition 1. This is analogous to finding a suitable hash function for a hash table. Perhaps the most well-known alternative to conventional cache indexing is the skewed associative cache [21]. This involves two or more indexing functions derived by XORing two -bit fields from an address to produce an -bit cache index. In the field of interleaved memories it is well known that bank { } A i A i k conflicts can be reduced by using bank selection functions other than the simple modulo-power- of-two. Lawrie and Vora proposed a scheme using prime-modulus functions [16], Harper and Jump [11], and Sohi [24] proposed skewing functions. The use of XOR functions was proposed by Frailong et al. [5], and pseudo-random functions were proposed by Raghavan & Hayes [17] and Rau et al. [18], [19]. These schemes each yield a more or less uniform distribution of requests to banks, with varying degrees of theoretical predictability and implementation cost. In principle each of these schemes could be used to construct a conflict-resistant cache by using them as the indexing function. However, when considering conflict resistance in cache architectures two factors are critical. Firstly, the chosen placement function must have a logically simple implementation, and secondly we would like to be able to guarantee good behavior on all regular address patterns - even those that are pathological under a conventional placement function. In both respects the irreducible polynomial modulus (I-Poly) permutation function proposed by Rau [19] is an ideal candidate. The I-Poly scheme effectively defines families of pseudo-random hash functions which are implemented using exclusive-OR gates. They also have some useful behavioral characteristics which we discuss later. In [10] the miss ratio of the I-Poly indexing scheme is evaluated extensively in the context of cache indexing, and is compared with a number of different cache organizations including; direct-mapped, set-associative, victim, hash-rehash, column-associative and skewed-associative. The results of that study suggest that the I-Poly function is particularly robust. For example, on Spec95 an 8Kb two-way associative cache has an average miss ratio of 13.84%. An I-poly cache of identical capacity and associativity reduces that miss ratio to 7.14%, which compares well against a fully-associative cache which has a miss ratio of 6.80%. 2.1.1 Polynomial-modulus cache placement To define the most general form of conflict resistant cache indexing scheme let the placement of a block of data from an -bit address , in each of ways of a -way associative cache with sets, be determined by the set of indices . In I-Poly indexing each is given by the function , for . In this scheme is a member of a set of , possibly distinct, integer values , in the range . If we choose to use distinct values for each the cache will be skewed, though skewing is not an obligatory feature of this scheme. Each function is defined as follows. Consider the integers and in terms of their binary representations. For example, , and similarly for . We consider to be a polynomial defined over the field GF(2), and similarly . For best performance will be an irreducible polynomial, though it need not be so. The value of , also defined over GF(2), is given by of order less than such that Effectively is the polynomial modulus function ignoring the higher order terms of . Each bit of the index can be computed using an XOR tree, if is constant, or an AND-XOR tree if one requires a configurable index function. For best performance should be as close as possible to , though it may be as small as for this scheme to be distinct from conventional block placement. 2.1.2 Polynomial placement characteristics The class of polynomial hash functions described above have been studied previously in the context of stride-insensitive interleaved memories (see [18] and [19]). These functions have certain { { P A a n 1 - a n 2 - . a 0 A x - a n 2 - . a 0 - . a 1 x 1 a 0 x 0 R x provable characteristics which are of significant value in the context of cache indices. For example, all strides of the form produce address sequences that are free from conflicts - i.e. they do not Condition 1 set out in section 1. This is a fundamental result for polynomial indexing; if the addresses of a -strided sequence are partitioned into -long sub-sequences, where is the number of cache blocks, we can guarantee that there are no cache conflicts within each sub- sequence. Any conflicts between sub-sequences are due to capacity problems and only be solved by larger caches or tiling of the iteration space. The stride-insensitivity of the I-Poly index function can be seen in figure 1 which shows the behavior of four cache configurations, identical except in their indexing functions. All have capacity, two-way associativity. They were each driven from an address trace representing repeated accesses to a vector of 64 8-byte elements in which the elements were separated by stride . With no conflicts such a sequence would use at most half of the 128 sets in the cache. The experiment was repeated for all strides in the range to determine how many strides exhibited bad behavior for each indexing function. The experiment compares three different indexing schemes; conventional modulo power-of-2 (labelled a2), the function proposed in [21] for the skewed-associative cache (a2-Hx-Sk) and two I-Poly functions. The I-Poly scheme was simulated both with and without skewed index functions (a2-Hp and a2-Hp-Sk respectively). It is apparent that the I-Poly scheme with skewing displays a remarkable ability to tolerate all strides equally and without any pathological behavior. For all schemes the majority of strides yield low miss ratios. However, both the conventional and the skewed XOR functions display pathological behavior (miss ratio > 50%) on more than 6% of all strides. 3 Implementation Issues The logic of the polynomial modulus operation in GF(2) defines a class of hash functions which compute the cache placement of an address by combining subsets of the address bits using XOR gates. This means that, for example, bit 0 of the cache index may be computed as the exclusive-OR of bits 0, 11, 14, and 19 of the original address. The choice of polynomial determines which bits are included in each set. The implementation of such a function for a cache with an 8-bit index would require just eight XOR gates with fan-in of 3 or 4. Whilst this appears remarkably simple, there is more to consider than just the placement function. Firstly, the function itself uses address bits beyond the normal limit imposed by typical minimum page size restriction. Secondly, the use of pseudo-random placement in a multi-level memory hierarchy has implications for the maintenance of Inclusion. Here we briefly examine these two issues and show how the virtual-real two-level cache hierarchy proposed by Wang et al. [25] provides a clean solution to both problems. 3.1 Overcoming page size restrictions Typical operating systems permit pages to be as small as 8 or 16 Kbytes. In a conventional cache this places a limit on the first-level cache size if address translation is to proceed in parallel with tag lookup. Similarly, any novel cache indexing scheme which uses address bits beyond the Figure 1. Frequency distribution of miss ratios for conventional and pseudo-random indexing schemes. Columns represent I-Poly indexing and lines represent conventional and skewed-associative indexing. minimum page size boundary cannot use a virtually-indexed physically-tagged cache. There are four alternative options to consider: 1. Perform address translation prior to tag lookup (i.e. physical indices) 2. Enable I-Poly indexing only when data pages are known to be large enough 3. Use a virtually-indexed virtually-tagged level-1 cache 4. Index conventionally, but use a polynomial rehash on a level-1 miss. Option 1 is attractive if an existing processor pipeline performs address translation at least one stage prior to tag lookup. This might be the case in a processor which is able to hide memory latency through dynamic execution or multi-threading, for example. However, in many systems, performing address translation prior to tag lookup will either extend the critical path through a critical pipeline stage or introduce an extra cycle of untolerated latency via an additional pipeline stage. Option 2 could be attractive in high performance systems where large data sets and large physical memories are the norm. In such circumstances processes may typically have data pages of 256Kbytes or more. The O/S should be able to track the page sizes of segments currently in use by a process (and its kernel) and enable polynomial cache indexing at the first-level cache if all segments' page sizes are above a certain threshold. This will provide more unmapped bits to the hash function when possible, but revert to conventional indexing when this is not possible. For example, if the threshold is 256Kbytes and the cache is 8Kbytes two-way associative, then we may implement a polynomial function hashing 13 unmapped physical address bits to 7 cache index bits. This will be sufficient to produce good conflict-free behavior. Provided the level- 1 cache is flushed when the indexing function is changed, there is no reason why the indexing function needs to remain constant. The third option is not currently popular, primarily because of potential difficulties with aliases in the virtual address space as well as the difficulty of shooting down a level-1 virtual cache line when a physically-addressed invalidation operation is received from another processor. The two-level virtual-real cache hierarchy proposed by Wang et al. in [25] provides a way of implementing a virtually-tagged L1 cache, thus exposing more address bits to the indexing function without incurring address translation delays. The fourth option would be appropriate for a physically-tagged direct-mapped cache. It is similar in principle to the hash-rehash [1] and the column-associative caches [2]. The idea is to make an initial probe with a conventional integer-modulus indexing function, using only unmapped address bits. If this probe does not hit we probe again, but at a different index. By the time the second probe begins, the full physical address is available and can be used in a polynomial hashing function to compute the index of the second probe. Addresses which can be co-resident under a conventional index function will not collide on the first probe. Conversely, sets of addresses which do collide under a conventional indexing function collide under the second probe with negligible probability , due to the pseudo-random distribution of the polynomial hashing function. This provides a kind of pseudo-full associativity in what is effectively a direct-mapped cache. The hit time of such a cache on the first probe would be as good as any direct-mapped physically-indexed cache. However, the average hit time is lengthened slightly due the occasional need for a second probe. We have investigated this style of cache and devised a scheme for swapping cache lines between their "conventional" modulo- indexed location and their "alternative" polynomially-indexed location. This leads to a typical probability of around 90% that a hit is detected at the first probe. However, the slight increase in average hit time due to occasional double probes means that a column-associative cache is only attractive when miss penalties are comparatively large. Space restrictions prevent further coverage of this option. 3.2 Requirements for Inclusion Coherent cache architectures normally require that the property of Inclusion is maintained between all levels of the memory hierarchy. Thus, if represents the set of data present in cache at level , the property of Inclusion demands that for in an -level memory hierarchy. Whenever this property is maintained a snooping bus protocol need only compare addresses of global write operations with the tags of the lowest level of private cache. A line at index in the L2 cache is replaced when a line at index in the L1 cache is replaced with data at address if is not already present in L2. If line contains valid data we must be sure that after replacement its data is not still present in L1. In a conventionally-indexed cache this is not an issue because it is relatively easy to guarantee that the data at L2 index is always located at L1 index , thus ensuring that L1 replacement will automatically preserve Inclusion. In a pseudo-randomly indexed cache there is in general no way to make this guarantee. Instead, the cache replacement protocols must explicitly enforce Inclusion by invalidating data at L1 when required. This is guaranteed by the two-level virtual-real cache, but leads to the creation of holes at the upper level of the cache, in turn leading to the possibility of additional cache misses. 3.3 Performance implication of holes In a two-level virtual-real cache hierarchy there are three causes of holes at L1; these are: 1. Replacements at L2 2. Removal of virtual aliases at L1 3. Invalidations due to external coherency actions It is probable that the frequency of item 2 occurring will be low; for this kind of hole to cause a performance problem a process must issue interleaved accesses to two segments at distinct virtual addresses which map to the same physical address. We preserve a consistent copy of the data at these virtual addresses by ensuring that at most one such alias may be present in L1 at any instant. This does not prevent the physical copy from residing undisturbed at L2; it simply increases the traffic between L1 and L2 when accesses to virtual aliases are interleaved. Invalidations from external coherency actions occur regardless of the cache architecture so A A i 2 we do not consider them further in this analysis. The events that are of primary importance are invalidations at L1 due to the maintenance of Inclusion between L1 and L2. It is important to quantify their frequency and the effect they have on hit ratio at L1. Recall that the index function at L2 is based on a physical address whereas the index function at L1 uses a virtual address. Also, the number of bits included in the index function and the function itself will be different in both cases. As these functions are pseudo-random there will be no correlation between the indices at L1 and L2 for each particular datum. Consequently, when a line is replaced at L2 the data being replaced will also exist in L1 with probability (1) where and are the number of bits in the indices at L1 and L2 respectively. If the data being replaced at L2 does exist in L1, it is possible that the L1 index is coincidentally equal to the index of the data being brought into L1 (as the L2 replacement is actually caused by an L1 replacement). If this occurs a hole will not be created after all. Thus the probability that the elimination of a line at L1 to preserve inclusion will result in a hole is given by (2) The net probability that a miss at L2 will cause a hole to appear at L1 is , given by the product of and , thus: When the size ratio between L1 and L2 is large the value of is small. For example, an 8KB L1 cache and a 256KB L2 cache with lines yield . Slightly more than 3% of L2 will result in the creation of a hole. The expected increase in compulsory miss ratio at L1 can be modelled by the product of and the L2 miss ratio. When compared with simulated miss ratios we find that this approximation is accurate for L2:L1 cache size ratios of 16 or above. For instance simulations of the whole Spec95 suite with an 8Kb two-way skewed I-Poly L1 cache backed by a 1 Mb conventionally-indexed two-way set-associative L2 cache showed that the effect of holes on L1 miss ratio is negligible. The percentage of L2 misses that created a hole averaged less than 0.1% and was never greater than 1.2% for any program. The two-level virtual-real cache described in [25] implements a protocol between the L1 and L2 cache which effectively provides a mechanism for ensuring that inclusion is maintained, that coherence can be maintained without reverse address translation, and in our case that holes can be created at level-1 when required by the inclusion property. 3.4 Effect of polynomial mapping on critical path A cache memory access in a conventional organization normally computes its effective address by adding two registers or a register plus a displacement. I-poly indexing implies additional circuitry to compute the index from the effective address. This circuitry consists of several XOR gates that operate in parallel and therefore the total delay is just the delay of one gate. Each XOR gate has a number of inputs that depend on the particular polynomial being used. For the experiments reported in this paper the number of inputs is never higher than 5. Therefore, the delay due to the gates will be low compared with the delay of a complete pipeline stage. Depending on the particular design, it may happen that this additional delay can be hidden. For instance, if the memory access does not begin until the complete effective address has been computed, the XOR delay can be hidden since the address is computed from right to left and the gates use only the least-significant bits of the address (19 in the experiments reported in this paper). Notice that this is true even for carry look-ahead adders (CLA). A CLA with look-ahead blocks of size b bits computes first the b least-significant bits, which are available after a delay of approximately one look-ahead block. After a three-block delay the b 2 least-significant bits are available. In general, the b i least-significant bits have a delay of approximately 2i-1 blocks. For instance, for 64-bit addresses and a binary CLA, the 19 bits required by the I-poly functions used in the experiments of this paper have a delay of about 9 blocks whereas the whole address computation requires 11 block-delays. Once the 19 least-significant bits have been computed, it is reasonable to assume that the XOR gate delay is shorter than the time required to compute the remaining bits. However, since the cache access time usually determines the pipeline cycle, the fact that the least-significant bits are available early is sometimes exploited by designers in order to shorten the latency of memory instructions by overlapping part of the cache access (which requires only the least-significant bits) with the computation of the most significant address bits. This approach results in a pipeline with a structure similar to that shown in figure 2. Notice that this organization requires a pipelined memory (in the example we have assumed a two-stage pipelined memory). In this case, the polynomial mapping may cause some additional delay to the critical path. We will show later that even if the additional delay induces a one cycle penalty in the cache access time, the polynomial mapping provides a significant overall performance improvement. An additional delay in a load instruction may have a negative impact on the performance of the processor because the issue of dependent instructions may be delayed accordingly. On the other hand, this delay has a negligible effect, if any, on store instructions since these instructions are issued to memory when they are committed in order to have precise exceptions, and therefore the XOR functions can usually be performed while the instruction is waiting in the store buffer. Besides, only load instructions may depend on stores but these dependencies are resolved in current microprocessors (e.g. PA8000 [12]) by forwarding. This technique compares the effective address of load and store instructions in order to check a possible match but the cache index, which involves the use of the XOR gates, is not required by this operation. Memory address prediction can be also used to avoid the penalty introduced by the XOR delay when it lengthens the critical path. The effective address of memory references has been shown to be highly predictable. For instance, in [9] it has been shown that the address of about 75% of the dynamically executed memory instructions of the Spec95 suite can be predicted with a simple scheme based on a table that keeps track of the last address seen by a given instruction and its last stride. We propose to use a similar scheme to predict early in the pipeline the line that is likely to be accessed by a given load instruction. In particular, the scheme works as follows. The processor incorporates a table indexed by the instruction address. Each entry stores the last address and the predicted stride for some recently executed load instruction. In the fetch stage, Figure 2: A pipeline that overlaps part of the address computation with the memory access. least-significant bits most-significant bits Memory access Memory access ALU stage stage critical path this table is accessed with the program counter. In the decode stage, the predicted address is computed and the XOR functions are performed to compute the predicted cache line. Notice that this can be done in just one cycle since the XOR can be performed in parallel with the computation of the most-significant bits as discussed above, and the time to perform an integer addition is not higher than one cycle in the vast majority of processors. When the instruction is subsequently issued to the memory unit it uses the predicted line number to access the cache in parallel with the actual address and line computation. If the predicted line turns out to be incorrect, the cache access is repeated again with the actual address. Otherwise, the data provided by the speculative access can be loaded into the destination register. The scheme to predict the effective address early in the pipeline has been previously used for other purposes. In [7], a Load Target Buffer is presented, which predicts effective address adding a stride to the previous address. In [3] and [4] a Fast Address Calculation is performed by computing load addresses early in the pipeline without using history information. In those proposals the memory access is overlapped with the non-speculative effective address calculation in order to reduce the cache access time, though none of them execute speculatively the subsequent instructions that depend on the predicted load. Other authors have proposed the use of a memory address prediction scheme in order to execute memory instructions speculatively, as well as the instructions dependent on them. In the case of a miss-speculation, a recovery mechanism similar to that used by branch prediction schemes is utilized to squash the miss-speculated instructions. The most noteworthy papers on this topic are [8], [9] and [20]. 4 Experimental Evaluation In order to verify the impact of polynomial mapping on a realistic microprocessor architecture we have developed a parametric simulator of an out-of-order execution processor. A four-way superscalar processor has been simulated. Table 1 shows the different functional units and their latency considered for this experiment. The size of the reorder buffer is entries. There are two separate physical register files (FP and Integer), each one having 64 physical registers. The processor has a lockup-free data cache [14] that allows 8 outstanding misses to different cache lines. The cache size is either 8Kb or 16 Kb and is 2-way set-associative with 32-byte line size. The cache is write-through and no-write-allocate. The hit time of the cache is two cycles and the miss penalty is 20 cycles. An infinite L2 cache is assumed and a 64-bit data bus between L1 and L2 is considered (i.e., a line transaction occupies the bus during four cycles). There are two memory ports and dependencies thorough memory are speculated using a mechanism similar to the ARB of the Multiscalar [6] and PA8000 [12]. A branch history table with 2K entries and 2-bit saturating counters is used for branch prediction. The memory address prediction scheme has been implemented by means of a direct-mapped table with 1K entries and without tags in order to reduce cost at the expense of more interference in the table. Each entry contains the last effective address of the last load instruction that used this entry and the last observed stride. In addition, each entry contains a 2-bit saturating counter that assigns confidence to the prediction. Only when the most-significant bit of the counter is set is the prediction considered to be correct. The address field is updated for each new reference regardless of the prediction, whereas the stride field is only updated when the counter goes below Functional Unit Latency Repeat rate multiply Effective Address 1 1 Table 1: Functional units and instruction latency. Table 2 shows the IPC (instructions committed per cycle) and the miss ratio for different configurations. The baseline configuration is an 8 Kb cache with I-poly indexing and no address prediction (4th column). The average IPC of this configuration is 1.30 and the average miss ratio (6th column) is 16.53 1 . When I-poly indexing is used the average miss ratio goes down to 9.68 (8th column). If the XOR gates are not in the critical path this implies an increase in the IPC up to 1.35 (7th column). On the other hand, if the XOR gates are in the critical path and we assume a one cycle penalty in the cache access time, the resulting IPC is 1.32 (9th column). However, the use of the memory address prediction scheme when the XOR gates are in the critical path (10th column) provides about the same performance as a cache with the XOR gates not in the critical path (7th column). Thus, the main conclusion of this study is that the memory address prediction scheme can offset the penalty introduced by the additional delay of the XOR gates when they are in the critical path. Finally, table 2 also shows the performance of a 16 Kb 2-way set-associative cache (2nd and 3rd columns). Notice that the addition of I-poly indexing to an 8Kb cache yields over 60% of the IPC increase that can be obtained by doubling the cache size. The main benefit of polynomial mapping is to reduce the conflict misses. However, in the benchmark suite there are many benchmarks that exhibit a relatively low conflict miss ratio. In fact the Spec95 conflict miss ratio of a 2-way associative cache is less than 4% for all programs except tomcatv, swim and wave5. If we focus on those three benchmarks with the highest conflict miss ratios we can observe the ability of polynomial mapping to reduce the miss ratio and significantly boost the performance of these problem cases. This is shown in table 3. In this case we can see that the polynomial mapping provides a significant improvement in performance even if the XOR gates are in the critical path and the memory address prediction scheme is not used (27% increase in IPC). When memory address prediction is used the IPC is 33% higher than that of a conventional cache of the same capacity and 16% higher than that of a conventional cache with twice the capacity. Notice that the polynomial mapping scheme with 1. For each benchmark we considered 100M of instructions after skipping the first 2000M. prediction is even better than the organization with the XOR gates not in the critical path but without prediction. This is due to the fact that the memory address prediction scheme reduces by one cycle the effective cache hit time when the predictions are correct, since the address computation is overlapped with the cache access (the computed address is used to verify that the prediction was correct). However, the main benefits observed in table 3 come from the reduction in conflict misses. To isolate the different effects we have also simulated an organization with the Conventional indexing I-poly indexing Xor no CP Xor in CP IPC miss no pred. with pred. IPC miss no pred with pred IPC miss IPC IPC go 1.00 5.45 0.87 0.88 10.87 0.87 10.60 0.83 0.84 compress 1.13 12.96 1.12 1.13 13.63 1.11 14.17 1.07 1.10 li 1.40 4.72 1.30 1.32 8.01 1.33 7.10 1.26 1.31 ijpeg 1.31 0.94 1.28 1.28 3.72 1.29 2.17 1.28 1.30 perl 1.45 4.52 1.26 1.27 9.47 1.24 10.26 1.19 1.21 vortex 1.39 4.97 1.27 1.28 8.37 1.30 7.87 1.25 1.27 su2cor 1.28 13.74 1.24 1.26 14.69 1.24 14.66 1.21 1.25 hydro2d 1.14 15.40 1.13 1.15 17.23 1.13 17.22 1.11 1.15 applu 1.63 5.54 1.61 1.63 6.16 1.57 6.84 1.55 1.59 mgrid 1.51 4.91 1.50 1.53 5.05 1.50 5.31 1.46 1.52 turb3d 1.85 4.67 1.80 1.82 6.05 1.81 5.38 1.78 1.82 apsi 1.13 10.03 1.08 1.09 15.19 1.08 13.36 1.07 1.09 fpppp 2.14 1.09 2.00 2.00 2.66 1.98 2.47 1.93 1.94 wave5 1.37 27.72 1.26 1.28 42.76 1.51 14.67 1.48 1.54 Average 1.38 10.47 1.30 1.31 16.53 1.35 9.68 1.32 1.35 Table 2:IPC and load miss ratio for different cache configurations. memory address prediction scheme and conventional indexing for an 8Kb cache (column 5). If we compare this IPC with that in column 4 of table 3, we see that the benefits of the memory address prediction scheme due to the reduction of the hit time are almost negligible. This confirms that the improvement observed in the I-poly indexing scheme with address prediction derives from the reduction in conflict misses. Conclusions In this paper we have described a pseudo-random indexing scheme which is robust enough to eliminate repetitive cache conflicts. We have discussed the main implementation issues that arise from the use of novel indexing schemes. For example, I-poly indexing uses more address bits than a conventional cache to compute the cache index. Also, the use of different indexing functions at L1 and L2 results in the occasional creation of a hole at L1. Both of these problems can be solved using a two-level virtual-real cache hierarchy. Finally, we have proposed a memory address prediction scheme to avoid the penalty due to the potential delay in the critial path introduced by the I-poly indexing function. Detailed simulations of an o-o-o superscalar processor have demonstrated that programs with significant numbers of conflict misses in a conventional 8Kb 2-way set-associative cache Conventional indexing Polynomial mapping Xor no CP Xor in CP IPC miss no pred. with pred. IPC miss no pred. with pred. IPC miss IPC IPC wave5 1.37 27.72 1.26 1.28 42.76 1.51 14.67 1.48 1.54 Average 1.28 30.80 1.12 1.13 54.61 1.46 14.40 1.42 1.49 Table 3: IPC and load miss ratio for different cache configurations for selected bad programs. perceive IPC improvements of 33% (with address prediction) or 27% (without address prediction). This is up to 16% higher than the IPC improvements obtained simply by doubling the cache capacity. Furthermore, programs which do not experience significant conflict misses see a less than 1% reduction in IPC when I-poly indexing is used in conjunction with address prediction. An interesting by-product of I-poly indexing is an increase in the predictability of cache behaviour. In our experiments we see that I-poly indexing reduces the standard deviation of miss ratios across Spec95 from 18.49 to 5.16. If conflict misses are eliminated, the miss ratio depends solely on compulsory and capacity misses, which in general are easier to predict and control. Systems which incorporate an I-poly cache could be useful in the real-time domain, or in cache-based scientific computing where iteration-space tiling often introduces extra cache conflicts. 6 --R "Cache Performance of Operating Systems and Multiprogramming" "Column-Associative Caches: A Technique for Reducing the Miss Rate of Direct-Mapped Caches" "Streamling Data Cache Access with Fast Address Calculation" "Zero-Cycle Loads: Microarchitecture Support for Reducing Load Latency" "XOR-Schemes: A Flexible Data Organization in Parallel Memories" "ARB: A Hardware Mechanims for Dynamic Reordering of Memory References" "Hardware Support fot Hiding Cache Latency" "Memory Address Prediction for Data Speculation" "Speculative Execution via Address Prediction and Data Prefetching" "Eliminating Cache Conflict Misses Through XOR-based Placement Functions" "Vector Access Performance in Parallel Memories Using a Skewed Storage Scheme" "Advanced Performance Features of the 64-bit PA-8000" "Improving Direct-Mapped Cache Performance by the Addition of a Small Fully-Associative Cache and Prefetch Buffers" "Lockup-free instruction fetch/prefetch cache organization" "The Cache Performance and Optimization of Blocked Algorithms" "The Prime Memory System for Array Access" "On Randomly Interleaved Memories" "The Cydra 5 Stride-Insensitive Memory System" "Pseudo-Randomly Interleaved Memories" "The Performance Potential of Data Dependence Speculation & Collapsing" "A Case for Two-way Skewed-associative Caches" "Skewed-associative Caches" "Cache Memories" "Logical Data Skewing Schemes for Interleaved Memories in Vector Processors" "Organization and Performance of a Two-Level Virtual-Real Cache Hierarchy" --TR Vector access performance in parallel memories using skewed storage scheme Cache performance of operating system and multiprogramming workloads Organization and performance of a two-level virtual-real cache hierarchy The cache performance and optimizations of blocked algorithms On randomly interleaved memories Pseudo-randomly interleaved memory A case for two-way skewed-associative caches Column-associative caches Streamlining data cache access with fast address calculation Zero-cycle loads The performance potential of data dependence speculation MYAMPERSANDamp; collapsing Eliminating cache conflict misses through XOR-based placement functions Speculative execution via address prediction and data prefetching Cache Memories Skewed-associative Caches Memory Address Prediction for Data Speculation Advanced performance features of the 64-bit PA-8000 Lockup-free instruction fetch/prefetch cache organization --CTR Hans Vandierendonck , Koen De Bosschere, Highly accurate and efficient evaluation of randomising set index functions, Journal of Systems Architecture: the EUROMICRO Journal, v.48 n.13-15, p.429-452, May Steve Carr , Soner nder, A case for a working-set-based memory hierarchy, Proceedings of the 2nd conference on Computing frontiers, May 04-06, 2005, Ischia, Italy Mazen Kharbutli , Yan Solihin , Jaejin Lee, Eliminating Conflict Misses Using Prime Number-Based Cache Indexing, IEEE Transactions on Computers, v.54 n.5, p.573-586, May 2005 Nigel Topham , Antonio Gonzlez, Randomized Cache Placement for Eliminating Conflicts, IEEE Transactions on Computers, v.48 n.2, p.185-192, February 1999 Rui Min , Yiming Hu, Improving Performance of Large Physically Indexed Caches by Decoupling Memory Addresses from Cache Addresses, IEEE Transactions on Computers, v.50 n.11, p.1191-1201, November 2001 Jaume Abella , Antonio Gonzlez, Heterogeneous way-size cache, Proceedings of the 20th annual international conference on Supercomputing, June 28-July 01, 2006, Cairns, Queensland, Australia

multi-level memory hierarchies;cache architecture design;polynomial modulus functions;conflict-avoiding cache performance;high performance architectures;cache storage;data access cost minimization;main memory;conflict miss ratios

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A framework for balancing control flow and predication.

Predicated execution is a promising architectural feature for exploiting instruction-level parallelism in the presence of control flow. Compiling for predicated execution involves converting program control flow into conditional, or predicated, instructions. This process is known as if-conversion. In order to effectively apply if-conversion, one must address two major issues: what should be if-converted and when the if-conversion should be applied. A compiler's use of predication as a representation is most effective when large amounts of code are if-converted and if-conversion is performed early in the compilation procedure. On the other hand the final code generated for a processor with predicated execution requires a delicate balance between control flow and predication to achieve efficient execution. The appropriate balance is tightly coupled with scheduling decisions and detailed processor characteristics. This paper presents an effective compilation framework that allows the compiler to maximize the benefits of predication as a compiler representation while delaying the final balancing of control flow and predication to schedule time.

Introduction The performance of modern processors is becoming highly dependent on the ability to execute multiple instructions per cy- cle. In order to realize their performance potential, these processors demand that increasing levels of instruction-level parallelism (ILP) be exposed in programs. One of the major challenges to increasing the available ILP is overcoming the limitations imposed by branch instructions. ILP is limited by branches for several reasons. First, branches impose control dependences which often sequentialize the execution of surrounding instructions. Second, the uncertainty of branch outcomes forces compiler and hardware schedulers to make conservative decisions. Branch prediction along with speculative execution is generally employed to overcome these limitations [1][2]. However, branch misprediction takes away a significant portion of the potential performance gain. Third, traditional techniques only facilitate exploiting ILP along a single trajectory of control. The ability to concurrently execute instructions from multiple trajectories offers the potential to increase ILP by large amounts. Finally, branches often interfere with or complicate aggressive compiler transformations, such as optimization and scheduling. Predication is a model in which instruction execution conditions are not solely determined by branches. This characteristic allows predication to form the basis for many techniques which deal with branches effectively in both the compilation and execution of codes. It provides benefits in a compiler as a representation and in ILP processors as an architectural feature. The predicated representation is a compiler N-address program representation in which each instruction is guarded by a boolean source operand whose value determines whether the instruction is executed or nullified. This guarding boolean source operand is referred to as the predicate. The values of predicate registers can be manipulated by a predefined set of predicate defining instructions. The use of predicates to guard instruction execution can reduce or even completely eliminate the need for branch control dependences. When all instructions that are control dependent on a branch are predicated using the same condition as the branch, that branch can legally be removed. The process of replacing branches with appropriate predicate computations and guards is known as if-conversion [3][4]. The predicated representation provides an efficient and useful model for compiler optimization and scheduling. Through the removal of branches, code can be transformed to contain few, if any, control dependences. Complex control flow transformations can instead be performed in the predication domain as traditional straight-line code optimizations. In the same way, the predicated representation allows scheduling among branches to be performed in a domain without control dependences. The removal of these control dependences increases scheduling scope and affords new freedom to the scheduler [5]. Predicated execution is an architectural model which supports direct execution of the predicated representation [6][7][8]. With respect to a conventional instruction set architecture, the new features are an additional boolean source operand guarding each instruction and a set of compare instructions used to compute predicates. Predicated execution benefits directly from the advantages of compilation using the predicated representa- tion. In addition, the removal of branches yields performance benefits in the executed code, the most notable of which is the removal of branch misprediction penalties. In particular, the removal of frequently mispredicted branches yields large performance gains [9][10][11]. Predicated execution also provides an efficient mechanism for a compiler to overlap the execution of multiple control paths on the hardware. In this manner, processor performance may be increased by exploiting ILP across multiple program paths. Another, more subtle, benefit of predicated execution is that it allows height reduction along a single program path [12]. Supporting predicated execution introduces two compilation issues: what should be if-converted and when in the compilation procedure if-conversion should be applied. The first question to address is what should be if-converted or, more specifically, what branches should be removed via if-conversion. Traditionally, full if-conversion has led to positive results for compiling numerical applications [13]. However, for non-numeric applications, selective if-conversion is essential to achieve performance gains [14]. If-conversion works by removing branches and combining multiple paths of control into a single path of conditional instructions. However, when two paths are overlapped, the resultant path can exhibit increased constraints over those of the original paths. One important constraint is resources. Paths which are combined together must share processor resources. The compiler has the responsibility of managing the available resources when making if-conversion decisions so that an appropriate stopping point may be identified. Further if-conversion will only result in an increase in execution time for all the paths involved. As will be discussed in the next section, the problem of deciding what to if-convert is complicated by many factors, only one of which is resource consumption. The second question that must be addressed is when to apply if-conversion in the compilation procedure. At the broadest level, if-conversion may be applied early in the backend compilation procedure or delayed to occur in conjunction with scheduling. Applying if-conversion early enables the full use of the predicate representation by the compiler to facilitate ILP optimizations and scheduling. In addition, complex control flow transformations may be recast in the data dependence domain to make them practical and profitable. Examples of such transformations include branch reordering, control height reduction [12], and branch combining [15]. On the other hand, delaying if-conversion to as late as possible makes answering the first question much more practi- cal. Since many of the if-conversion decisions are tightly coupled to the scheduler and its knowledge of the processor characteris- tics, applying if-conversion at schedule time is the most natural choice. Also, applying if-conversion during scheduling alleviates the need to make the entire compiler backend cognizant of a predicated representation. An effective compiler strategy for predicated execution must address the "what" and "when" questions of if-conversion. The purpose of this paper is to present a flexible framework for if-conversion in ILP compilers. The framework enables the compiler to extract the full benefits of the predicate representation by applying aggressive if-conversion early in the compilation pro- cedure. A novel mechanism called partial reverse if-conversion then operates at schedule time to facilitate balancing the amount of control flow and predication present in the generated code, based on the characteristics of the target processor. The remainder of this paper is organized as follows. Section 2 details the compilation issues and challenges associated with compiling for predicated execution. Section 3 introduces our proposed compilation framework that facilitates taking full advantage of the predicate representation as well as achieving an efficient balance between branching and predication in the final code. The essential component in this framework, partial reverse if-conversion, is described in detail in Section 4. The effectiveness of this framework in the context of our prototype compiler for ILP processors is presented in Section 5. Finally, the paper concludes in Section 6. Compilation Challenges Effective use of predicated execution provides a difficult challenge for ILP compilers. Predication offers the potential for large performance gains when it is efficiently utilized. However, an imbalance of predication and control flow in the generated code can lead to dramatic performance losses. The baseline compilation support for predicated execution assumed in this paper is the hyperblock framework. Hyperblocks and the issues associated with forming quality hyperblocks are first summarized in this section. The remainder of this section focuses on an approach of forming hyperblocks early in the compilation procedure using heuristics. This technique is useful because it exposes the predicate representation throughout the backend optimization and scheduling pro- cess. However, this approach has several inherent weaknesses. Solving these weaknesses is the motivation for the framework presented in this paper. 2.1 Background The hyperblock is a structure created to facilitate optimization and scheduling for predicated architectures [14]. A hyperblock is a set of predicated basic blocks in which control may only enter from the top, but may exit from one or more locations. Hyperblocks are formed by applying tail duplication and if-conversion over a set of carefully selected paths. Inclusion of a path into a hyperblock is done by considering its profitability. The profitability is determined by four pieces of information: resource utilization, dependence height, hazard presence, and execution frequency. One can gain insights into effective hyperblock formation heuristics by understanding how each characteristic can lead to performance loss. The most common cause of poor hyperblock formation is excessive resource consumption. The resources required by overlapping the execution of multiple paths are the union of the resources required for each individual path. Consider if-converting a simple if-then-else statement. The resultant resource consumption for the hyperblock will be the combination of the resources required to separately execute the "then" and "else" paths. If each path alone consumes almost all of the processor resources, the resultant hyperblock would require substantially more resources than the processor has available. As a result, hyperblock formation results in a significant slowdown for both paths. Of course, these calculations do not account for the benefits gained by if-conversion. The important point is that resource over-subscription has the potential to negate all benefits of hyperblock formation or even degrade performance. Poor hyperblocks may also be formed by not carefully considering the dependence height of if-conversion. A hyperblock which contains multiple paths will not complete until all of its constituent paths have completed. Therefore, the overall height of the hyperblock is the maximum of all the original paths' dependence heights. Consider the if-conversion of a simple if-then- else statement with the "then" path having a height of two and the "else" path having a height of four. The height of the resultant hyperblock is the maximum of both paths, four. As a result, the "then" path is potentially slowed down by two times. The compiler must weigh this negative against the potential positive effects of if-conversion to determine whether this hyperblock is profitable to form. Another way poor hyperblocks may be formed is through the inclusion of a path with a hazard. A hazard is any instruction or set of instructions which hinders efficient optimization or scheduling of control paths other than its own. Two of the most common hazards are subroutine calls with unknown side effects and store instructions which have little or no alias information. Hazards degrade performance because they force the compiler to make conservative decisions in order to ensure correctness. For this reason, inclusion of a control path with a hazard into a hyperblock generally reduces the compiler's effectiveness for the entire hyperblock, not just for the path containing the hazard. Execution frequency is used as a measure of a path's importance and also provides insight into branch behavior. This information is used to weigh the trade-offs made in combining execution paths. For example, it may be wise to penalize an infrequently executing path by combining it with a longer frequently executing path and removing the branch joining the two. 2.2 Pitfalls of Hyperblock Selection The original approach used in the IMPACT compiler to support predicated execution is to form hyperblocks using heuristics based on the four metrics described in the previous section. Hyperblocks are formed early in the backend compilation procedure to expose the predicate representation throughout all the back-end compilation phases. Heuristic hyperblock formation has been shown to perform well for relatively regular machine models. In these machines, balancing resource consumption, balancing dependence height, and eliminating hazards are done effectively by the carefully crafted heuristics. However, experience shows that several serious problems exist that are difficult to solve with this approach. Three such problems presented here are optimizations that change code characteristics, unpredictable resource interfer- ence, and partial path inclusion. Optimization. The first problem occurs when code may be transformed after hyperblock formation. In general, forming hyperblocks early facilitates optimization techniques that take advantage of the predicate representation. However, the hyperblock formation decisions can change dramatically with compiler trans- formations. This can convert a seemingly bad formation decision into a good one. Likewise, it can convert a seemingly good formation decision into a bad one. Figure 1a shows a simple hammock to be considered for if- conversion. 1 The taken path has a dependence height of three cycles and consumes three instruction slots after if-conversion has removed instruction 5. The fall-through path consists of a depen- 1 For all code examples presented in this section, a simple machine model is used. The schedules are for a three issue processor with unit latencies. Any resource limitations for the processor that are assumed are specified with each example. These assumptions do not reflect the machine model or latencies used in the experimental evaluation (Section 5).2(8) (1) (a) (b) (1) branch Cond (2) jump Figure 1: Hyperblock formation of seemingly incompatible paths with positive results due to code transformations. The T and F annotations in (a) indicate the taken and fall-through path for the conditional branch. r2 is not referenced outside the T block. dence height of six cycles and a resource consumption of six instruction slots. A simple estimate indicates that combining these paths would result in a penalty for the taken path of three cycles due to the fall-through path's large dependence height. Figure 1b shows this code segment after hyperblock formation and further optimizations. The first optimization performed was renaming to eliminate the false dependences 7 ! 8 and 8 ! 10. This reduced the dependence height of the hyperblock to three cycles. If a heuristic could foresee that dependence height would no longer be an issue, it may still choose not to form this hyperblock due to resource considerations. An estimate of ten instructions after if-conversion could be made by inspecting Figure 1a. Un- fortunately, ten instructions needs at least four cycles to complete on a three issue machine, which would still penalize the taken path by one cycle, indicating that the combination of these paths may not be beneficial. After an instruction merging optimization in which instructions 2 and 6 were combined and 4 and 11 were combined, the instruction count becomes eight. The final schedule consists of only three cycles. Figure 1 shows that even in simple cases a heuristic which forms hyperblocks before some optimizations must anticipate the effectiveness of those optimizations in order to form profitable hyperblocks. In this example, some optimizations could potentially be done before hyperblock formation, such as renaming. However, others, like instruction merging, could not have been. In addition, some optimizations may have been applied differently when performed before if-conversion, because the different code characteristics will result in different trade-offs. Resource Interference. A second problem with heuristic hyperblock formation is that false conclusions regarding the resource compatibility of the candidate paths may often be reached. As a result, paths which seem to be compatible for if-conversion turn out to be incompatible. The problem arises because resource usage estimation techniques, such as the simple ones used in this section or even other more complex techniques, generally assume (1) branch Cond (2) jump (a)2 (b) Figure 2: Hyperblock formation of seemingly compatible paths that results in performance loss due to resource incompatibility. that resource usage is evenly distributed across the block. In prac- tice, however, few paths exhibit uniform resource utilization. Interactions between dependence height and resource consumption cause violations of the uniform utilization assumption. In gen- eral, most paths can be subdivided into sections that are either relatively parallel or relatively sequential. The parallel sections demand a large number of resources, while the sequential sections require few resources. When two paths are combined, resource interference may occur when the parallel sections of the paths overlap. For those sections, the demand for resources is likely to be larger than the available resources, resulting in a performance loss. To illustrate this problem, consider the example in Figure 2. The processor assumed for this example is again three issue, but at most one memory instruction may be issued each cycle. The original code segment, Figure 2a, consists of two paths with dependence heights of three cycles. The resource consumption of each path is also identical, four instructions. These paths are concluded to be good candidates for if-conversion. Figure 2b shows the hyperblock and its resulting schedule. Since there are no obvious resource shortages, one would expect the resultant schedule for the hyperblock to be identical in length to the schedules of each individual path, or four cycles. However, the hyperblock schedule length turns out to be six cycles. This increase is due to resource interference between the paths. Each path is parallel at the start and sequential at the end. In addition, the parallel sections of both paths have a high demand for the memory resource. With only one memory resource available, the paths are sequentialized in parallel sections. Note that if the requirements for the memory resource were uniformly distributed across both paths, this problem would not exist as the individual schedule lengths are four cycles and there are a total of four memory instructions. However, due to the characteristics of these paths, resource interference results in a performance loss for both paths selected for the hyperblock. (1) (2) F branch r1 > r10 jump (c) (a)2 (1) jump (b) Figure 3: An efficient hyperblock formed through the inclusion of a partial path. Partial Paths. The final problem with current heuristic hyperblock formation is that paths may not be subdivided when they are considered for inclusion in a hyperblock. In many cases, including part of a path may be more beneficial than including or excluding that entire path. Such an if-conversion is referred to as partial if-conversion. Partial if-conversion is generally effective when the resource consumption or dependence height of an entire candidate path is too large to permit profitable if-conversion, but there is a performance gain by overlapping a part of the candidate path with the other paths selected for inclusion in the hyperblock. To illustrate the effectiveness of partial if-conversion, consider the example in Figure 3. The three issue processor assumed for this example does not have any resource limitations other than the issue width. Figure 3a shows two paths which are not compatible due to mismatched dependence height. However, by including all of the taken path and four instructions from the fall-through path, an efficient hyperblock is created. This hyperblock is shown in Figure 3b. Notice that branch instruction 2 has been split into two instructions: the condition computation, labeled 2 0 , and a branch based on that computation, labeled 2 00 . The schedule did not benefit from the complete removal of branch instruction 2, as the branch instruction 2 00 has the same characteristics as the orig- inal. However, the schedule did benefit from the partial overlap of both paths. The destination of branch instruction 2 00 contains the code to complete the fall-through path is shown in Figure 3c. In theory, hyperblock formation heuristics may be extended to support partial paths. Since each path could be divided at any instruction in the path, the heuristics would have to consider many more possible selection alternatives. However, the feasibility of extending the selection heuristics to operate at the finer granularity of instructions, rather than whole paths, is questionable due the complex nature of the problem. 3 Proposed Compilation Framework Compilation for predicated execution can be challenging as described in Section 2. To create efficient code, a delicate balance between control flow and predication must be created. The desired balance is highly dependent on final code characteristics and the resource characteristics of the target processor. An effective compilation framework for predicated execution must provide a structure for making intelligent tradeoffs between control flow and predication so the desired balance can be achieved. Given the difficulties presented in Section 2.2 with forming hyperblocks early in the backend compilation process, a seemingly natural strategy is to perform if-conversion in conjunction with instruction scheduling. This can be achieved by integrating if-conversion within the scheduling process itself. A scheduler not only accurately models the detailed resource constraints of the processor but also understands the performance characteristics of the code. Therefore, the scheduler is ideally suited to make intelligent if-conversion decisions. In addition, all compiler optimizations are usually complete when scheduling is reached, thus the problem of the code changing after if-conversion does not exist However, a very serious problem associated with performing if-conversion during scheduling time is the restriction on the com- piler's use of the predicate representation to perform control flow transformations and predicate specific optimizations. With the schedule-time framework, the introduction of the predicate representation is delayed until schedule time. As a result, all transformations targeted to the predicate representation must either be foregone or delayed. If these transformations are delayed, much more complexity is added to a scheduler which must already consider many issues including control speculation, data speculation, and register pressure to achieve desirable code performance. Ad- ditionally, delaying only some optimizations until schedule time creates a phase ordering which can cause severe difficulties for the compiler. Generally, most transforms have profound effects on one another and must be repeatedly applied in turn to achieve desirable results. For example, a transformation, such as control height reduction [12], may subsequently expose a critical data dependence edge that should be broken by expression reformu- lation. However, until the control dependence height is reduced, there is no profitability to breaking the data dependence edge, so the compiler will not apply the transform. This is especially true since expression reformulation has a cost in terms of added in- structions. The net result of the schedule-time framework is a restriction in the use of the predicate representation which limits the effectiveness of back-end optimizations. Given that if-conversion at schedule time limits the use of the predicate representation for optimization and given that if-conversion at an early stage is limited in its ability to estimate the final code characteristics, it is logical to look to an alternative compilation framework. This paper proposes such a frame- work. This framework overcomes limitations of other schemes by utilizing two phases of predicated code manipulation to support predicated execution. Aggressive if-conversion is applied in an early compilation phase to create the predicate representation and to allow flexible application of predicate optimizations throughout the backend compilation procedure. Then at sched- Classical Optimizations Classical Optimizations Optimizations ILP Optimizations Register Allocataion Postpass Scheduling and Partial Reverse If-Conversion Integrated Prepass Scheduling Aggressive Hyperblock Formation Figure 4: Phase ordering diagram for the compilation framework. ule time, the compiler adjusts the final amount of predication to efficiently match the target architecture. The compilation frame- work, shown in Figure 4, consists of two phases of predicate manipulation surrounding classical, predicate specific, and ILP op- timizations. The first predicate manipulation phase, hyperblock formation, has been addressed thoroughly in [14]. The second predicate manipulation phase, adjustment of hyperblocks during scheduling, is proposed in this work and has been termed partial reverse if-conversion. The first phase of the compilation framework is to aggressively perform hyperblock formation. The hyperblock former does not need to exactly compute what paths, or parts of paths, will fit in the available resources and be completely compatible with each other. Instead, it forms hyperblocks which are larger than the target architecture can handle. The large hyperblocks increase the scope for optimization and scheduling, further enhancing their benefits. In many cases, the hyperblock former will include almost all the paths. This is generally an aggressive decision because the resource height or dependence height of the resulting hyperblock is likely to be much greater than the corresponding heights of any of its component paths. However, the if-converter relies on later compilation phases to ensure that this hyperblock is efficient. One criteria that is still enforced in the first phase of hyperblock formation is avoiding paths with haz- ards. As was discussed in Section 2, hazards reduce the com- piler's effectiveness for the entire hyperblock, thus they should be avoided to facilitate more aggressive optimization. The second phase of the compilation framework is to adjust the amount of predicated code in each hyperblock as the code is scheduled via partial reverse if-conversion. Partial reverse if-conversion is conceptually the application of reverse if-conversion to a particular predicate in a hyperblock for a chosen set of instructions [16]. Reverse if-conversion was originally proposed as the inverse process to if-conversion. Branching code that contains no predicates is generated from a block of predicated code. This allows code to be compiled using a predicate representation, but executed on a processor without support for predicated execution. The scheduler with partial reverse if-conversion operates by identifying the paths composing a hyperblock. Paths which are profitable to overlap remain unchanged. Conversely, a path that interacts poorly with the other paths is removed from the hyper- block. In particular, the partial reverse if-converter decides to eject certain paths, or parts of paths, to enhance the schedule. To do this, the reverse if-converter will insert a branch that is taken whenever the removed paths would have been executed. This has the effect of dividing the lower portion of the hyperblock into two parts, corresponding to the taken and fall-through paths of the inserted branch. The decision to reverse if-convert a particular path consists of three steps. First, the partial reverse if-converter determines the savings in execution time by inserting control flow and applying the full resources of the machine to two hyperblocks instead of only one. Second, it computes the loss created by any penalty associated with the insertion of the branch. Finally, if the gain of the reverse if-conversion exceeds the cost, it is ap- plied. Partial reverse if-conversion may be repeatedly applied to the same hyperblock until the resulting code is desirable. The strategy used for this compilation framework can be viewed analogously to the use of virtual registers in many compil- ers. With virtual registers, program variables are promoted from memory to reside in an infinite space of virtual registers early in the compilation procedure. The virtual register domain provides a more effective internal representation than do memory operations for compiler transformations. As a result, the compiler is able to perform more effective optimization and scheduling on the virtual register code. Then, at schedule time, virtual registers are assigned to a limited set of physical registers and memory operations are reintroduced as spill code when the number of physical registers was over-subscribed. The framework presented in this paper does for branches what virtual registers do for program variables. Branches are removed to provide a more effective internal representation for compiler transformations. At schedule time, branches are inserted according to the capabilities of the target processor. The branches reinserted have different conditions, targets, and predictability than the branches originally removed. The result is that the branches in the code are there for the benefit of performance for a particular processor, rather than as a consequence of code structure decisions made by the programmer. The key to making this predication and control flow balancing framework effective is the partial reverse if-converter. The mechanics of performing partial reverse if-conversion, as well as a proposed policy used to guide partial reverse if-conversion, are presented in the next section. 4 Partial Reverse If-Conversion The partial reverse if-conversion process consists of three components: analysis, transformation, and decision. Each of these steps is discussed in turn. 4.1 Analysis Before any manipulation or analysis of execution paths can be performed, these paths must be identified in the predicated code. Execution paths in predicated code are referred to as predicate paths. Immediately after hyperblock formation, the structure of the predicate paths is identical to the control flow graph of the (2) (1)000000111111111111 000000000000111111000000000000111111000000000000000000000011111111111000000000000000000000011111111111 (a) (2) (b) Figure 5: Predicate flow graph with partial dead code elimination given that r3 and r4 are not live out of this region. code before hyperblock formation. The structure of the predicate paths can be represented in a form called the predicate flow graph (PFG). The predicate flow graph is simply a control flow graph (CFG) in which predicate execution paths are also repre- sented. After optimizations, the structure of the PFG can change dramatically. For reasons of efficiency and complexity, the compiler used in this work does not maintain the PFG across opti- mizations, instead it is generated from the resulting predicated N-address code. The synthesis of a PFG from predicated N-address code is analogous to creating a CFG from N-address code. A simple example is presented to provide some insight into how this is done. Figure 5 shows a predicated code segment and its predicate flow graph. The predicate flow graph shown in Figure 5b is created in the following manner. The first instruction in Figure 5a is a predicate definition. At this definition, p1 can assume TRUE or FALSE.A path is created for each of these possibilities. The complement of p1, p2, shares these paths because it does not independently create new conditional outcomes. The predicate defining instruction 2 also creates another path. In this case, the predicates p3 and p4 can only be TRUE if p1 is TRUE because their defining instructions is predicated on p1, so only one more path is cre- ated. The creation of paths is determined by the interrelations of predicates, which are provided by mechanisms addressed in other work [17][18]. For the rest of the instructions, the paths that contain these instructions are determined by the predicate guarding their execution. For example, instruction 3 is based on predicate p1 and is therefore only placed in paths where p1 is TRUE. Instruction 4 is not predicated and therefore exists in all paths. The type of predicate defines used in all figures in this paper are un- conditional, meaning they always write a value [8]. Since they some value regardless of their predicate, their predicate can be ignored, and the instruction's destinations must be placed in all paths. Paths in a PFG can be merged when a predicate is no longer used and does not affect any other predicate later in the code. However, this merging of paths may not be sufficient to solve all (1) jump (b) (a) (2) Figure flow graph (a) and a partial reverse if-conversion of predicate p1 located after instructions 1 and 2 (b). potential path explosion problems in the PFG. This is because the number of paths in a PFG is exponentially proportional to the number of independent predicates whose live ranges overlap. Fortunately, this does not happen in practice until code schedul- ing. After code scheduling, a complete PFG will have a large number of paths and may be costly. A description of how the partial reverse if-converter overcomes this problem is located in Section 4.2. A more general solution to the path explosion problem for other aspects of predicate code analysis is currently being constructed by the authors. With a PFG, the compiler has the information necessary to know which instructions exist in which paths. In Figure 5, if the path in which p1 and p3 are TRUE is to be extracted, the instructions which would be placed into this path would be 3, 4 and 7. The instructions that remain in the other two paths seem to be 3, 4, 5, and 6. However, inspection of the dataflow characteristics of these remaining paths reveals that the results of instructions 3 and 4 are not used, given that r3 and r4 are not live out of this region. This fact makes these instructions dead code in the context of these paths. Performing traditional dead code removal on the PFG, instead of the CFG, determines which parts of these operations are dead. Since this application of dead code removal only indicates that these instructions are dead under certain predicate conditions, this process is termed predicate partial dead code removal and is related to other types of partial dead code removal [19]. The result of partial dead code removal indicates that instructions 3 and 4 would generate correct code and would not execute unnecessarily if they were predicated on p3. At this point, all the paths have been identified and unnecessary code has been removed by partial dead code removal. The analysis and possible ejection of these paths now becomes possible 4.2 Transformation Once predicate analysis and partial dead code elimination have been completed, performing reverse if-conversion at any point and for any predicate requires a small amount of additional processing. This processing determines whether each instruction belongs in the original hyperblock, the new block formed by re- (a) Figure 7: Simple code size reduction on multiple partial reverse if-conversions applied to an unrolled loop. Each square represents an unroll of the original loop. verse if-conversion, or both. Figure 6 is used to aid this discussion The partial reverse if-converted code can be subdivided into three code segments. These are: the code before the reverse if- converting branch, the code ejected from the hyperblock by reverse if-conversion, and the code which remains in the hyperblock below the reverse if-converting branch. Instructions before the location of the partial reverse if-converting branch are left untouched in the hyperblock. Figure 6b shows the partial reverse if-conversion created for p1 after instructions 1 and 2. This means that instructions 1 and 2 are left in their originally scheduled location and the reverse if-converting branch, predicated on p1, is scheduled immediately following them. The location of instructions after the branch is determined by the PFG. To use the PFG without experiencing a path explosion problem, the PFG's generated during scheduling are done only with respect to the predicate which is being reverse if-converted. This keeps the number of paths under control since a the single predicate PFG can contain no more than two paths. Figure 6a shows the PFG created for the predicate to be reverse if-converted, p1. Note that the partial dead code has already been removed as described in the previous section. Instructions which exist solely in the p1 is FALSE path, such as 5 and 6, remain in the original block. Instructions which exist solely in the p1 is TRUE path, such as 3, 4, and 7, are moved from the original block to the newly formed region. An instruction which exists in both paths must be placed in both regions. Notice that the hyperblock conditionally jumps to the code removed from the hyperblock but there is no branch from this code back into the original hyperblock. While this is possible, it was not implemented in this work. Branching back into the hyperblock would violate the hyperblock semantics since it would no longer be a single entry region. Violating hyperblock semantics may not be problematic since the benefits of the hyperblock have already been realized by the optimizer and prepass scheduler. However, the postpass hyperblock scheduler may experience reduced scheduling freedom since all re-entries into the hyperblock effectively divide the original hyperblock into two smaller hyperblocks The advantage of branching back into the original hyperblock is a large reduction in code size through elimination of unnecessarily duplicated instructions. However, as will be shown in the experimental section, code size was generally not a problem. One code size optimization which was performed merges targets of partial reverse if-conversion branches if the target blocks are identical. This resulted in a large code size reduction in codes where loop unrolling was performed. If a loop in an unrolled hyperblock needed to be reverse if-converted, it is likely that all iterations needed to be reverse if-converted. This creates many identical copies of the loop body subsequent to the loop being reverse if-converted. Figure 7a shows the original result of repeated reverse if-conversions on an unrolled loop. Figure 7b shows the result obtained by combining identical targets. While this simple method works well in reducing code growth, it does not eliminate all unnecessary code growth. To remove all unnecessary code growth, a method which jumps back into the hyperblock at an opportune location needs to be created. 4.3 Policy After creating the predicate flow graph and removing partial dead code, the identity and characteristics of all paths in a hyperblock are known. With this information, the compiler can make decisions on which transformations to perform. The decision process for partial reverse if-conversion consists of two parts: deciding which predicates to reverse if-convert and deciding where to reverse if-convert the selected predicates. To determine the optimal reverse if-conversion for a given architecture, the compiler could exhaustively try every possible reverse if-conversion, compute the optimal cycle count for each possibility, and choose the one with the best performance. Unfortunately, there are an enormous number of possible reverse if-conversions for any given hy- perblock. Consider a hyperblock with p predicates and n instruc- tions. This hyperblock has 2 p combinations of predicates chosen for reverse if-conversion. Each of these reverse if-conversions can then locate its branch in up to n locations in the worst case. Given that each of these possibilities must be scheduled to measure its cycle count, this can be prohibitively expensive. Obvi- ously, a heuristic is needed. While many heuristics may perform effective reverse if-conversions, only one is studied in this paper. This heuristic may not be the best solution in all applications, but for the machine models studied in this work it achieves a desirable balance between final code performance, implementation complexity, and compile time. The process of choosing a heuristic to perform partial reverse if-conversion is affected greatly by the type of scheduler used. Since partial reverse if-conversion is integrated into the prepass scheduler, the type of information provided by the scheduler and the structure of the code at various points in the scheduling process must be matched with the decision of what and where to if-convert. An operation-based scheduler may yield one type of heuristic while a list scheduler may yield another. The policy determining how to reverse if-convert presented here was designed to work within the context of an existing list scheduler. The algorithm with this policy integrated into the list scheduler is shown in Figure 8. The first decision addressed by the proposed heuristic is where to place a predicate selected for reverse if-conversion. If a location can be shown to be generally more effective than the rest, then the number of locations to be considered for each reverse if-conversion can be reduced from n to 1, an obvious improve- ment. Such a location exists under the assumption that the reverse if-converting branch consumes no resources and the code is scheduled by a perfect scheduler. It can be shown that there Number of operations; // Each trip through this loop is a new cycle 6 WHILE num unsched != 0 DO // Handle reverse if-converting branches first 7 FOREACH ric op IN ric queue DO 8 IF Schedule Op(ric op, cycle) THEN 9 Compute location for each unscheduled op; sched ric taken = Compute dynamic cycles in ric taken path; sched ric cycles in ric hyperblock; mipred ric = Estimate ric mispreds * miss penalty; 13 ric cycles = sched ric hb sched ric taken ; 14 ric cycles = ric cycles mispred ric ; (sched no ric ? ric cycles) THEN sched ric hb ; 17 Place all ops in their no ric schedule location; 19 Unschedule OP(ric op); Remove ric op from ric queue; // Then handle regular operations 22 IF Schedule Op(regular op, cycle) THEN Remove regular op from ready priority queue; 26 Add reverse if-converting branch to ric queue; Figure 8: An algorithm incorporating partial reverse if-conversion into a list scheduler is no better placement than the first cycle in which the value of the predicate to be reverse if-converted is available after its predicate defining instruction. 2 Since the insertion of the branch has the same misprediction or taken penalty regardless of its location, these effects do not favor one location over another. However, the location of the reverse if-converting branch does determine how early the paths sharing the same resources are separated and given the full machine bandwidth. The perfect scheduler will always do as well or better when the full bandwidth of the machine is divided among fewer instructions. Given this, the earlier the paths can be separated, the fewer the number of instructions competing for the same machine resources. Therefore, a best schedule will occur when the reverse if-converting branch is placed as early as possible. Despite this fact, placing the the reverse if-converting branch as early as possible is a heuristic. This is because the two assumptions made, a perfect scheduler and no cost for the reverse if-converting branch, are not valid in general. It seems reason- able, however, that this heuristic would do very well despite these imperfections. Another consideration is code size, since instructions existing on multiple paths must be duplicated when these paths are seperated. The code size can be reduced if the reverse if-converting branch is delayed. Depending on the characteristics There exist machines where the placement of a branch a number of cycles after the computation of its condition removes all of its mispredictions [20]. In these machines, there are two locations which should be considered, immediately after the predicate defining instruction and in the cycle in which the branch mispredictions are eliminated. of the code, this delay may have no cost or a small cost which may be less than the gain obtained by the reduction in code size. Despite these considerations, the placement of the partial reverse if-converting branch as early as possible is a reasonable choice. The second decision addressed by the heuristic is what to reverse if-convert. Without a heuristic, the number of reverse if- conversions which would need to be considered with the heuristic described above is 2 p . The only way to optimally determine which combination of reverse if-conversions yields the best results is to try them all. A reverse if-conversion of one predicate can affect the effectiveness of other reverse if-conversions. This interaction among predicates is caused by changes in code char- acteristecs after a reverse if-conversion has removed instructions from the hyperblock. In the context of a list scheduler, a logical heuristic is to consider each potential reverse if-conversion in a top-down fashion, in the order in which the predicate defines are scheduled. This heuristic is used in the algorithm shown in Figure 8. This has the desirable effect of making the reverse if-conversion process fit seemlessly into a list scheduler. It is also desirable because each reverse if-conversion is considered in the context of the decisions made earlier in the scheduling process. In order to make a decision on each reverse if-conversion, a method to evaluate it must be employed. For each prospective reverse if-conversion, three schedules must be considered: the code schedule without the reverse if-conversion, the code schedule of the hyperblock with the reverse if-converting branch inserted and paths excluded, and the code schedule of the paths excluded by reverse if-conversion. Together they yield a total of 3p schedules for a given hyperblock. Each of these three schedules needs to be compared to determine if a reverse if-conversion is prof- itable. This comparison can be written as: sched cyclesno ric ? sched cycles ric hb sched cycles ric taken miss penalty) where sched cyclesno ric is the number of dynamic cycles in the schedule without reverse if-conversion ap- plied, sched cycles ric hb is the number of dynamic cycles in the schedule of the transformed hyperblock, sched cycles ric taken is the number of dynamic cycles in the target of the reverse if-conversion, and mispredric is the number of mispredictions introduced by the reverse if-conversion branch. The mispredric can be obtained through profiling or static estimates. miss penalty is the branch misprediction penalty. This comparision is computed by lines 9 through 15 in Figure 8. While the cost savings due to the heuristic is quite significant, 3p schedules for more complicated machine models can still be quite costly. To reduce this cost, it is possible to reuse information gathered during one schedule in a later schedule. The first source of reuse is derived from the top-down property of the list scheduler itself. At the point each reverse if-conversion is considered, all previous instructions have been scheduled in their final location by lines 8 or 22 in Figure 8. Performing the scheduling on the reverse if-conversion and the original scenario only needs to start at this point. The number of schedules is still 3p, but the number of instructions in each schedule has been greatly reduced by the removal of instructions already scheduled. The second source of reuse takes advantage of the fact that, for the case in which the reverse if-conversion is not done, the schedule has already been computed. At the time the previous predicate was considered for reverse if-conversion, the schedule was computed for each outcome. Since the resulting code schedule in cycles is already known, no computation is necessary for the current predicate's sched cyclesno ric . This source of reuse takes the total schedules computed down to 2p with each schedule only considering the unscheduled instructions at each point due to the list scheduling effect. This reuse is implemented in Figure 8 by lines 5 and 16. Another way to reduce the total number of instructions scheduled is to take advantage of the fact that the code purged from the block is only different in the "then" and "else" blocks but not in the control equivalent split or join blocks. Once the scheduler has completely scheduled the "then" and "else" parts, no further scheduling is necessary since the remaining schedules are likely to be very similar. The only differences may be dangling latencies or other small differences in the available resources at the boundary. To be more accurate, the schedules can continue until they become identical, which is likely to occur at some point, though is not guaranteed to occur in all cases. An additional use for the detection of this point is code size reduction. This point is a logical location to branch from the ejected block back into the original hyperblock. With all of the above schedule reuse and reduction techniques, it can be shown that the number of times an instruction is scheduled is usually 1 d is that instruction's depth in its hammock. In the predication domain, this depth is the number of predicates defined in the chain used to compute that instruction's guarding predicate. If the cost of scheduling is still high, estimates may be used in- stead. There are many types of scheduling estimates which have been proposed and can be found in the literature. While many may do well for machines with regular structures, others do not. It is possible to create a hybrid scheduler/estimator which may balance good estimates with compile time cost. As mentioned previously, the schedule height of the two paths in the hammock must be obtained. Instead of purely scheduling both paths, which may be costly, or just estimating both paths, which may be inac- curate, a part schedule and part estimate may obtain more accurate results with lower cost. In the context of a list scheduler, one solution is the following. The scheduler could schedule an initial set of operations and estimate the schedule on those remain- ing. Accurate results will be obtained by the scheduled portion, in addition, the estimate may be able to benefit from information obtained from the schedule, as the characteristics of the scheduled code may be likely to match the characteristics of the code to be estimated. In the experiments presented in the next section, actual schedules are used in the decision to reverse if-convert because the additional compile time was acceptable. 5 Experimental Results This section presents an experimental evaluation of the partial reverse if-conversion framework. 5.1 Methodology The partial reverse if-conversion techniques described in this paper have been implemented in the second generation instruction scheduler of the IMPACT compiler. The compiler utilizes a machine description file to generate code for a parameterized superscalar processor. To measure the effectiveness of the partial reverse if-conversion technique, a machine model similar to many current processors was chosen. The machine modeled is a 4-issue superscalar processor with in-order execution that contains two integer ALU's, two memory ports, one floating point ALU, and one branch unit. The instruction latencies assumed match those of the HP PA-7100 microprocessor. The instruction set contains a set of non-trapping versions of all potentially excepting instruc- tions, with the exception of branch and store instructions, to support aggressive speculative execution. The instruction set also contains support for predication similar to that provided in the PlayDoh architecture [8]. The execution time for each benchmark is derived from the static code schedule weighted by dynamic execution frequencies obtained from profiling. Static branch prediction based on profiling is also utilized. Previous experience with this method of run time estimation has demonstrated that it accurately estimates simulations of an equivalent machine with perfect caches. The benchmarks used in this experiment consist of 14 non-numeric programs: the six SPEC CINT92 benchmarks, 008.espresso, 022.li, 023.eqntott, 026.compress, 072.sc, and 085.cc1; two SPEC CINT95 benchmarks, 132.ijpeg and 134.perl; and six UNIX utilities cccp, cmp, eqn, grep, wc, and yacc. 5.2 Results Figures compare the performance of the traditional hyperblock compilation framework and the new compilation framework with partial reverse if-conversion. The hyperblocks formed in these graphs represent those currently formed by the IMPACT compiler's hyperblock formation heuristic for the target machine. These same hyperblocks were also used as input to the partial reverse if-converter. The results obtained are therefore conservative since more aggressive hyperblocks would create the potential for better results. The bars represent the speedup achieved by these methods relative to superblock compilation. This is computed as follows: superblock cycles=technique cycles. Superblock compilation performance is chosen as the base because it represents the best possible performance currently obtainable by the IMPACT compiler without predication [21]. Figure 9 shows the performance of the hyperblock and partial reverse if-conversion compilation frameworks assuming perfect branch prediction. Since branch mispredictions are not factored in, benchmarks exhibiting performance improvement in this graph show that predication has performed well as a compilation model. In particular, the compiler has successfully overlapped the execution of multiple paths of control to increase ILP. Hyperblock compilation achieves some speedup for half of the benchmarks, most notably for 023.eqntott, cmp, 072.sc, grep, and wc. For these programs, the hyperblock techniques successfully overcome the problem superblock techniques were having in fully utilizing processor resources. On the other hand, hyperblock compilation results in a performance loss for half of the benchmarks. This dichotomy is a common problem experienced with hyperblocks and indicates that hyperblocks can do well, but often performance is victim to poor hyperblock selection. In all cases, partial reverse if-conversion improved upon or -40% -20% 0% 20% 40% 80% 100% 008.espresso 022.li 023.eqntott 026.compress 072.sc 085.cc1 132.ijpeg 134.perl cccp cmp eqn grep wc yacc Benchmark Hyperblock Framework Partial RIC Framework Figure 9: Performance increase over superblock exhibited by the hyperblock and partial reverse if-conversion frameworks with no misprediction penalty. -40% -20% 0% 20% 40% 80% 100% 008.espresso 022.li 023.eqntott 026.compress 072.sc 085.cc1 132.ijpeg 134.perl cccp cmp eqn grep wc yacc Benchmark Hyperblock Framework Partial RIC Framework Figure 10: Performance increase over superblock exhibited by the hyperblock and partial reverse if-conversion frameworks with a four cycle misprediction penalty. matched the performance of the hyperblock code. For six of the benchmarks, partial reverse if-conversion was able to change a loss in performance by hyperblock compilation into a gain. This is most evident for 008.espresso where a 28% loss was converted into a 39% gain. For 072.sc, 134.perl, and cccp, partial reverse if-conversion was able to significantly magnify relatively small gains achieved by hyperblock compilation. These results indicate that the partial reverse if-converter was successful at undoing many of the poor hyperblock formation decisions while capitalizing on the effective ones. For the four benchmarks where hyperblock techniques were highly effective, 023.eqntott, cmp, grep, and wc, partial reverse if-conversion does not have a large opportunity to increase performance since the hyperblock formation heuristics worked well in deciding what to if-convert. It is useful to examine the performance of two of the benchmarks more closely. The worst performing benchmark is 085.cc1, for which both frameworks result in a performance loss with respect to superblock compilation. Partial reverse if-conversion was not completely successful in undoing the bad hyperblock formation decisions. This failure is due to the policy that requires the list scheduler to decide the location of the reverse if-converting branch by its placement of the predicate defining instruction. Un- fortunately, the list scheduler may delay this instruction as it may not be on the critical path and is often deemed to have a low scheduling priority. Delaying the reverse if-conversion point can have a negative effect on code performance. To some extent this problem occurs in all benchmarks, but is most evident in 085.cc1. One of the best performing benchmarks was 072.sc. For this program, hyperblock compilation increased performance by a fair margin, but the partial reverse if-conversion increased this gain substantially. Most of 072.sc's performance gain was achieved by transforming a single function update. This function with superblock compilation executes in 25.6 million cycles. How- ever, the schedule is rather sparse due to a large number of data and control dependences. Hyperblock compilation increases the available ILP by eliminating a large fraction of the branches and overlapping the execution of multiple paths of control. This brings the execution time down to 19.7 million cycles. While the hyperblock code is much better than the superblock code, it has excess resource consumption on some paths which penalizes other paths. The partial reverse if-converter was able to adjust the amount of if-conversion to match the available resources to efficiently utilize the processor. As a result, the execution time for the update function is reduced to 16.8 million cycles with partial reverse if-conversion, a 52% performance improvement over the superblock code. Figure shows the performance of the benchmarks in the same manner as Figure 9 except with a branch misprediction penalty of four cycles. In general, the relative performance of hyperblock code is increased the most when mispredicts are considered because it has the fewest mispredictions. The relative performance of the partial reverse if-conversion code is also increased because it has fewer mispredictions than the superblock code. But, partial reverse if-conversion inserts new branches to accomplish its transformation, so this code contains more mispredictions than the hyperblock code. For several of the benchmarks, the number of mispredictions was actually larger for hyperblock and partial reverse if-conversion than that of superblock. When applying control flow transformations in the predicated repre- sentation, such as branch combining, the compiler will actually create branches with much higher mispredict rates than those re- moved. Additionally, the branches created by partial reverse if-conversion may be more unbiased than the the combination of branches in the original superblock they represent. The static code size exhibited by using the hyperblock and partial reverse if-conversion compilation frameworks with respect to the superblock techniques is presented in Figure 11. From the fig- ure, the use of predicated execution by the compiler has varying effects on the code size. The reason for this behavior is a tradeoff between increased code size caused by if-conversion with the decreased code size due to less tail duplication. With superblocks, tail duplication is performed extensively to customize individual execution paths. Whereas with predication, multiple paths are overlapped via if-conversion, so less tail duplication is required. The figure also shows that the code produced with the partial reverse if-conversion framework is consistently larger than hyper- block. On average, the partial reverse if-conversion code is 14% larger than the hyperblock code, with the largest growth occurring for yacc. Common to all the benchmarks which exhibit a -40% -30% -20% -10% 0% 10% 20% 30% 40% 50% 008.espresso 022.li 023.eqntott 026.compress 072.sc 085.cc1 132.ijpeg 134.perl cccp cmp eqn grep wc yacc Benchmark Code Growth Hyperblock Framework Partial RIC Framework Figure 11: Relative static code size exhibited by the hyperblock and partial reverse if-conversion frameworks compared with superblock Benchmark Reverse If-Conversions Opportunities 43 443 026.compress 11 56 132.ijpeg 134 1021 134.perl 42 401 cccp 77 1046 Table 1: Application frequency of partial reverse if-conversion. large code growth was a failure of the simple code size reduction mechanism presented earlier. Inspection of the resulting code indicates that many instructions are shared in the lower portion of the tail-duplications created by the partial reverse if-converter. For this reason, one can expect these benchmarks to respond well to a more sophisticated code size reduction scheme. Finally, the frequency of partial reverse if-conversions that were performed to generate the performance data is presented in Table 1. The "Reverse If-Conversions" column specifies the actual number of reverse if-conversions that occurred across the entire benchmark. The "Opportunities" column specifies the number of reverse if-conversions that could potentially have oc- curred. The number of opportunities is equivalent to the number of unique predicate definitions in the application, since each predicate define can be reverse if-converted exactly once. All data in Table are static counts. The table shows that the number of reverse if-conversions that occur is a relatively small fraction of the opportunities. This behavior is desirable as the reverse if- converter should try to minimize the number of branches it inserts to achieve the desired removal of instructions from a hy- perblock. In addition, the reverse if-converter should only be invoked when a performance problem exists. In cases where the performance of the original hyperblock cannot be improved, no reverse if-conversions need to be performed. The table also shows the expected correlation between large numbers of reverse if-conversions and larger code size increases of partial reverse if-conversion over hyperblock (Figure 11). 6 Conclusion In this paper, we have presented an effective framework for compiling applications for architectures which support predicated execution. The framework consists of two major parts. First, aggressive if-conversion is applied early in the compilation process. This enables the compiler to take full advantage of the predicate representation to apply aggressive ILP optimizations and control flow transformations. The second component of the framework is applying partial reverse if-conversion at schedule time. This delays the final if-conversion decisions until the point during compilation when the relevant information about the code content and the processor resource utilization are known. A first generation partial reverse if-converter was implemented and the effectiveness of the framework was measured for this paper. The framework was able to capitalize on the benefits of predication without being subject to the sometimes negative side effects of over-aggressive hyperblock formation. Furthermore, additional opportunities for performance improvement were exploited by the framework, such as partial path if-conversion. These points were demonstrated by the hyperblock performance losses which were converted into performance gains, and by moderate gains which were further magnified. We expect continuing development of the partial reverse if-converter and the surrounding scheduling infrastructure to further enhance performance. In addition, the framework provides an important mechanism to undo the negative effects of overly aggressive transformations at schedule time. With such a backup mechanism, unique opportunities are introduced for the aggressive use and transformation of the predicate representation early in the compilation process. Acknowledgments The authors would like to thank John Gyllenhaal, Teresa Johnson, Brian Deitrich, Daniel Connors, John Sias, Kevin Crozier and all the members of the IMPACT compiler team for their support, comments, and suggestions. This research has been supported by the National Science Foundation (NSF) under grant CCR-9629948, Intel Corporation, Advanced Micro De- vices, Hewlett-Packard, SUN Microsystems, and NCR. Additional support was provided by an Intel Foundation Fellowship. --R "A study of branch prediction strategies," "Two-level adaptive training branch predic- tion," "Conversion of control dependence to data dependence," "On predicated execution," Modulo Scheduling with Isomorphic Control Trans- formations "Highly concurrent scalar processing," "The Cydra 5 departmental supercomputer," "HPL PlayDoh architecture specification: Version 1.0," "Guarded execution and branch prediction in dynamic ILP processors," "Characterizing the impact of predicated execution on branch prediction," "The effects of predicated execution on branch pre- diction," "Height reduction of control recurrences for ILP processors," "Overlapped loop support in the Cydra 5," "Effective compiler support for predicated execution using the hyperblock," "A comparison of full and partial predicated execution support for ILP processors," "Reverse if- conversion," "Analysis techniques for predicated code," "Global predicate analysis and its application to register allocation," "Partial dead code elimina- tion," "Ar- chitectural support for compiler-synthesized dynamic branch prediction strategies: Rationale and initial results," "The Superblock: An effective technique for VLIW and superscalar compilation," --TR Highly concurrent scalar processing The Cydra 5 Departmental Supercomputer Overlapped loop support in the Cydra 5 Two-level adaptive training branch prediction Effective compiler support for predicated execution using the hyperblock Reverse If-Conversion The superblock Partial dead code elimination Guarded execution and branch prediction in dynamic ILP processors Height reduction of control recurrences for ILP processors The effects of predicated execution on branch prediction Characterizing the impact of predicated execution on branch prediction Modulo scheduling with isomorphic control transformations A comparison of full and partial predicated execution support for ILP processors Analysis techniques for predicated code Global predicate analysis and its application to register allocation Conversion of control dependence to data dependence A study of branch prediction strategies Architectural Support for Compiler-Synthesized Dynamic Branch Prediction Strategies --CTR Hyesoon Kim , Jos A. Joao , Onur Mutlu , Yale N. Patt, Profile-assisted Compiler Support for Dynamic Predication in Diverge-Merge Processors, Proceedings of the International Symposium on Code Generation and Optimization, p.367-378, March 11-14, 2007 Walter Lee , Rajeev Barua , Matthew Frank , Devabhaktuni Srikrishna , Jonathan Babb , Vivek Sarkar , Saman Amarasinghe, Space-time scheduling of instruction-level parallelism on a raw machine, ACM SIGPLAN Notices, v.33 n.11, p.46-57, Nov. 1998 Eduardo Quiones , Joan-Manuel Parcerisa , Antonio Gonzalez, Selective predicate prediction for out-of-order processors, Proceedings of the 20th annual international conference on Supercomputing, June 28-July 01, 2006, Cairns, Queensland, Australia Patrick Akl , Andreas Moshovos, BranchTap: improving performance with very few checkpoints through adaptive speculation control, Proceedings of the 20th annual international conference on Supercomputing, June 28-July 01, 2006, Cairns, Queensland, Australia Hyesoon Kim , Jose A. Joao , Onur Mutlu , Yale N. Patt, Diverge-Merge Processor (DMP): Dynamic Predicated Execution of Complex Control-Flow Graphs Based on Frequently Executed Paths, Proceedings of the 39th Annual IEEE/ACM International Symposium on Microarchitecture, p.53-64, December 09-13, 2006 David I. August , John W. Sias , Jean-Michel Puiatti , Scott A. Mahlke , Daniel A. Connors , Kevin M. Crozier , Wen-mei W. Hwu, The program decision logic approach to predicated execution, ACM SIGARCH Computer Architecture News, v.27 n.2, p.208-219, May 1999 Aaron Smith , Ramadass Nagarajan , Karthikeyan Sankaralingam , Robert McDonald , Doug Burger , Stephen W. Keckler , Kathryn S. McKinley, Dataflow Predication, Proceedings of the 39th Annual IEEE/ACM International Symposium on Microarchitecture, p.89-102, December 09-13, 2006 John W. Sias , Wen-Mei W. Hwu , David I. August, Accurate and efficient predicate analysis with binary decision diagrams, Proceedings of the 33rd annual ACM/IEEE international symposium on Microarchitecture, p.112-123, December 2000, Monterey, California, United States Mihai Budiu , Girish Venkataramani , Tiberiu Chelcea , Seth Copen Goldstein, Spatial computation, ACM SIGARCH Computer Architecture News, v.32 n.5, December 2004 Yuan Chou , Jason Fung , John Paul Shen, Reducing branch misprediction penalties via dynamic control independence detection, Proceedings of the 13th international conference on Supercomputing, p.109-118, June 20-25, 1999, Rhodes, Greece Spyridon Triantafyllis , Manish Vachharajani , Neil Vachharajani , David I. August, Compiler optimization-space exploration, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, March 23-26, 2003, San Francisco, California David I. August , Wen-Mei W. Hwu , Scott A. Mahlke, The Partial Reverse If-Conversion Framework for Balancing Control Flow and Predication, International Journal of Parallel Programming, v.27 n.5, p.381-423, Oct. 1999 Lori Carter , Beth Simon , Brad Calder , Larry Carter , Jeanne Ferrante, Path Analysis and Renaming for Predicated Instruction Scheduling, International Journal of Parallel Programming, v.28 n.6, p.563-588, December 2000

conditional instructions;compiler;instruction-level parallelism;parallel architecture;schedule time;program control flow;scheduling decisions;optimising compilers;predicated instructions;if-conversion;predicated execution

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Tuning compiler optimizations for simultaneous multithreading.

Compiler optimizations are often driven by specific assumptions about the underlying architecture and implementation of the target machine. For example, when targeting shared-memory multiprocessors, parallel programs are compiled to minimize sharing, in order to decrease high-cost, inter-processor communication. This paper reexamines several compiler optimizations in the context of simultaneous multithreading (SMT), a processor architecture that issues instructions from multiple threads to the functional units each cycle. Unlike shared-memory multiprocessors, SMT provides and benefits from fine-grained sharing of processor and memory system resources; unlike current multiprocessors, SMT exposes and benefits from inter-thread instruction-level parallelism when hiding latencies. Therefore, optimizations that are appropriate for these conventional machines may be inappropriate for SMT. We revisit three optimizations in this light: loop-iteration scheduling, software speculative execution, and loop tiling. Our results show that all three optimizations should be applied differently in the context of SMT architectures: threads should be parallelized with a cyclic, rather than a blocked algorithm; non-loop programs should not be software speculated and compilers no longer need to be concerned about precisely sizing tiles to match cache sizes. By following these new guidelines compilers can generate code that improves the performance of programs executing on SMT machines.

Introduction Compiler optimizations are typically driven by specific assumptions about the underlying architecture and implementation of the target machine. For example, compilers schedule long-latency operations early to minimize critical paths, order instructions based on the processor's issue slot restrictions to maximize functional unit utilization, and allocate frequently used variables to registers to benefit from their fast access times. When new processing paradigms change these architectural assumptions, however, we must reevaluate machine-dependent compiler optimizations in order to maximize performance on the new machines. Simultaneous multithreading (SMT) [32][31][21] [13] is a multithreaded processor design that alters several architectural assumptions on which compilers have traditionally relied. On an SMT processor, instructions from multiple threads can issue to the functional units each cycle. To take advantage of the simultaneous thread-issue capability, most processor resources and all memory subsystem resources are dynamically shared among the threads. This single feature is responsible for performance gains of almost 2X over wide-issue superscalars and roughly 60% over single-chip, shared memory multiprocessors on both multi-programmed (SPEC92, SPECint95) and parallel (SPLASH-2, SPECfp95) workloads; SMT achieves this improvement while limiting the slowdown of a single executing thread to under 2% [13]. Simultaneous multithreading presents to the compiler a different model for hiding operation latencies and sharing code and data. Operation latencies are hidden by instructions from all executing threads, not just by those in the thread with the long-latency operation. In addition, multi-thread instruction issue increases instruction-level parallelism (ILP) to levels much higher than can be sustained with a single thread. Both factors suggest reconsidering uniprocessor optimizations that Copyright 1997 IEEE. Published in the Proceedings of Micro-30, December 1-3, 1997 in Research Triangle Park, North Carolina. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. Contact: Manager, Copyrights and Permissions / IEEE Service Center / 445 Hoes Lane / P.O. Box 1331 / Piscataway, NJ hide latencies and expose ILP at the expense of increased dynamic instruction counts: on an SMT the latency-hiding benefits may not be needed, and the extra instructions may consume resources that could be better utilized by instructions in concurrent threads. Because multiple threads reside within a single SMT processor, they can cheaply share common data and incur no penalty from false sharing. In fact, they benefit from cross-thread spatial locality. This calls into question compiler-driven parallelization techniques, originally developed for distributed-memory multiprocessors, that partition data to physically distributed threads to avoid communication and coherence costs. On an SMT, it may be beneficial to parallelize programs so that they process the same or contiguous data. This paper investigates the extent to which simultaneous multithreading affects the use of several compiler optimizations. In particular, we examine one parallel technique (loop-iteration scheduling for compiler-parallelized applications) and two optimizations that hide memory latencies and expose instruction-level parallelism (software speculative execution and loop tiling). Our results prescribe a different usage of all three optimizations when compiling for an SMT processor. We found that, while blocked loop scheduling may be useful for distributing data in distributed-memory multiprocessors, cyclic iteration scheduling is more appropriate for an SMT architecture, because it reduces the TLB footprint of parallel applications. Since SMT threads run on a single processor and share its memory hierarchy, data can be shared among threads to improve locality in memory pages. Software speculative execution may incur additional instruction overhead. On a conventional wide-issue superscalar, instruction throughput is usually low enough that these additional instructions simply consume resources that would otherwise go unused. However, on an SMT processor, where simultaneous, multi-thread instruction issue increases throughput to roughly 6.2 on an 8-wide processor, software speculative execution can degrade performance, particularly for non-loop-based applications. Simultaneous multithreading also impacts loop tiling techniques and tile size selection. SMT processors are far less sensitive to variations in tile size than conventional processors, which must find an appropriate balance between large tiles with low instruction overhead and small tiles with better cache reuse and higher hit rates. processors eliminate this performance sweet spot by hiding the extra misses of larger tiles with the additional thread-level parallelism provided by multithreading. Tiled loops on an SMT should be decomposed so that all threads compute on the same tile, rather than creating a separate tile for each thread, as is done on multiprocessors. Tiling in this way raises the performance of SMT processors with moderately-sized memory subsystems to that of more aggressive designs. The remainder of this paper is organized as follows. Section 2 provides a brief description of an SMT processor. Section 3 discusses in more detail two architectural assumptions that are affected by simultaneous multithreading and their ramifications on compiler-directed loop distribution, software speculative execution, and loop tiling. Section 4 presents our experimental methodology. Sections 5 through 7 examine each of the compiler optimizations, providing experimental results and analysis. Section 8 briefly discusses other compiler issues raised by SMT. Related work appears in Section 9, and we conclude in Section 10. 2 The microarchitecture of a simultaneous multithreading processor Our SMT design is an eight-wide, out-of-order processor with hardware contexts for eight threads. Every cycle the instruction fetch unit fetches four instructions from each of two threads. The fetch unit favors high throughput threads, fetching from the two threads that have the fewest instructions waiting to be executed. After fetching, instructions are decoded, their registers are renamed, and they are inserted into either the integer or floating point instruction queues. When their operands become available, instructions (from any thread) issue to the functional units for execution. Finally, instructions retire in per-thread program order. Little of the microarchitecture needs to be redesigned to enable or optimize simultaneous multithreading - most components are an integral part of any conventional, dynamically-scheduled superscalar. The major exceptions are the larger register file (32 architectural registers per thread, plus 100 renaming registers), two additional pipeline stages for accessing the registers (one each for reading and writing), the instruction fetch scheme mentioned above, and several per-thread mechanisms, such as program counters, return stacks, retirement and trap logic, and identifiers in the TLB and branch target buffer. Notably missing from this list is special per-thread hardware for scheduling instructions onto the functional units. Instruction scheduling is done as in a conventional, out-of-order superscalar: instructions are issued after their operands have been calculated or loaded from memory, without regard to thread; the renaming hardware eliminates inter-thread register name conflicts by mapping thread-specific architectural registers onto the processor's physical registers (see [31] for more details). All large hardware data structures (caches, TLBs, and branch prediction tables) are shared among all threads. The additional cross-thread conflicts in the caches and branch prediction hardware are absorbed by SMT's enhanced latency-hiding capabilities [21], while TLB interference can be addressed with a technique described in Section 5. Rethinking compiler optimizations As explained above, simultaneous multithreading relies on a novel feature for attaining greater processor performance: the coupling of multithreading and wide- instruction issue by scheduling instructions from different threads in the same cycle. The new design prompts us to revisit compiler optimizations that automatically parallelize loops for enhanced memory performance and/or increase ILP. In this section we discuss two factors affected by SMT's unique design, data sharing among threads and the availability of instruction issue slots, in light of three compiler optimizations they affect. Inter-thread data sharing Conventional parallelization techniques target multiprocessors, in which threads are physically distributed on different processors. To minimize cache coherence and inter-processor communication overhead, data and loop distribution techniques partition and distribute data to match the physical topology of the multiprocessor. Parallelizing compilers attempt to decompose applications to minimize synchronization and communication between loops. Typically, this is achieved by allocating a disjoint set of data for each processor, so that they can work independently [34][10][7]. In contrast, on an SMT, multiple threads execute on the same processor, affecting performance in two ways. First, both real and false inter-thread data sharing entail local memory accesses and incur no coherence overhead, because of SMT's shared L1 cache. Consequently, sharing, and even false sharing, is beneficial. Second, by sharing data among threads, the memory footprint of a parallel application can be reduced, resulting in better cache and TLB behavior. Both factors suggest a loop distribution policy that clusters, rather then separates, data for multiple threads. Latency-hiding capabilities and the availability of instruction issue slots On most workloads, wide-issue processors typically cannot sustain high instruction throughput, because of low instruction-level parallelism in their single, executing thread. Compiler optimizations, such as software speculative execution and loop tiling (or blocking), try to increase ILP (by hiding or reducing instruction latencies, respectively), but often with the side effect of increasing the dynamic instruction count. Despite the additional instructions, the optimizations are often profitable, because the instruction overhead can be accommodated in otherwise idle functional units. Because it can issue instructions from multiple threads, an SMT processor has fewer empty issue slots; in fact, sustained instruction throughput can be rather high, roughly 2 times greater than on a conventional superscalar [13]. Furthermore, SMT does a better job of hiding latencies than single-threaded processors, because it uses instructions from one thread to mask delays in another. In such an environment, the aforementioned optimizations may be less useful, or even detrimental, because the overhead instructions compete with useful instructions for hardware resources. SMT, with its simultaneous multithreading capabilities, naturally tolerates high latencies without the additional instruction overhead. Before examining the compiler optimizations, we describe the methodology used in the experiments. We chose applications from the SPEC 92 [12], SPEC 95 [30] and SPLASH-2 [35] benchmark suites (Table 2). All programs were compiled with the Multiflow trace scheduling compiler [22] to generate DEC Alpha object files. Multiflow was chosen, because it generates high-quality code, using aggressive static scheduling for wide- issue, loop unrolling, and other ILP-exposing optimizations. Implicitly-parallel applications (the SPEC suites) were first parallelized by the SUIF compiler [15]; SUIF's C output was then fed to Multiflow. A blocked loop distribution policy commonly used for multiprocessor execution has been implemented in SUIF; because we used applications compiled with the latest version of SUIF [5], but did not have access to its source, we implemented an alternative algorithm (described in Section 5) by hand. SUIF also finds tileable loops, determines appropriate multiprocessor-oriented tile sizes for particular data sets and caches, and then generates tiled code; we experimented with other tile sizes with manual coding. Speculative execution was enabled/disabled by modifying the Multiflow compiler's machine description file, which specifies which instructions can be moved speculatively by the trace scheduler. We experimented with both statically- generated and profile-driven traces; for the latter, profiling information was generated by instrumenting the applications and then executing them with a training input data set that differs from the set used during simulation. The object files generated by Multiflow were linked with our versions of the ANL [4] and SUIF runtime libraries to create executables. Our SMT simulator processes these unmodified Alpha executables and uses emulation-based, instruction-level simulation to model in detail the processor pipelines, hardware support for out- of-order execution, and the entire memory hierarchy, including TLB usage. The memory hierarchy in our processor consists of two levels of cache, with sizes, latencies, and bandwidth characteristics, as shown in Application Data set Instruc- tions simulated f applu 33x33x33 array, 2 iterations 272 M X X mgrid su2cor 16x16x16x16, vector len. 4K, 2 iterations 5.4 B X X tomcatv 513x513 array, 5 iterations A -fft 64K data points LU 512x512 matrix 431 M X water- nsquared 512 molecules, 3 timesteps 870 M X water- spatial 512 molecules, 3 timesteps 784 M X compress train input set 64 M X go train input set, 2stone9 700 M X li train input set 258 M X test input set, dhrystone 164 M X perl train input set, scrabble 56 M X mxm from matrix multiply of 256x128 and 128x64 arrays gmt from 500x500 Gaussian elimination 354 M X adi integration stencil computation for solving partial differential equations Table 1: Benchmarks. The last three columns identify the studies in which the applications are used. speculative execution, and L1 I-cache L1 D-cache L2 cache Cache size (bytes) 128K / 32K 128K / Line size (bytes) 64 64 64 Banks 8 8 1 Transfer time/bank 1 cycle 1 cycle 4 cycles Cache fill time (cycles) Latency to next level 10 Table 2: Memory hierarchy parameters. When there is a choice of values, the first (the more aggressive) represents a forecast for an SMT implementation roughly three years in the future and is used in all experiments. The second set is more typical of today's memory subsystems and is used to emulate larger data set sizes [29]; it is used in the tiling studies only. Table 2. We model the cache behavior, as well as bank and bus contention. Two TLB sizes were used for the loop distribution experiments (48 and 128 entries), to illustrate how the performance of loop distribution policies is sensitive to TLB size. The larger TLB represents a probable configuration for a (future) general-purpose SMT; the smaller is more appropriate for a less aggressive design, such as an SMT multimedia co- processor, where page sizes are typically in the range of 2-8MB. For both TLB sizes, misses require two full memory accesses, incurring a 160 cycle penalty. For branch prediction, we use a McFarling-style hybrid predictor with a 256-entry, 4-way set-associative branch target buffer, and an 8K entry selector that chooses between a global history predictor (13 history bits) and a local predictor (a 2K-entry local history table that indexes into a 4K-entry, 2-bit local prediction table) [24]. Because of the length of the simulations, we limited our detailed simulation results to the parallel computation portion of the applications (the norm for simulating parallel applications). For the initialization phases of the applications, we used a fast simulation mode that only simulates the caches, so that they were warm when the main computation phases were reached. We then turned on the detailed simulation model. 5 Loop distribution To reduce communication and coherence overhead in distributed-memory multiprocessors, parallelizing compilers employ a blocked loop parallelization policy to distribute iterations across processors. A blocked distribution assigns each thread (processor) continuous array data and iterations that manipulate them (Figure 1). Figure presents SMT speedups for applications parallelized using a blocked distribution with two TLB sizes. Good speedups are obtained for many applications (as the number of threads is increased), but in the smaller TLB the performance of several programs (hydro2d, swim, and tomcatv) degrades with 6 or 8 threads. The 8- thread case is particularly important, because most applications will be parallelized to exploit all 8 hardware contexts in an SMT. Analysis of the simulation bottleneck metrics indicated that the slowdown was the result of thrashing in the data TLB, as indicated by the TLB miss rates of Table 3. The TLB thrashing is a direct result of blocked partitioning, which increases the total working set of an application because threads work on disjoint data sets. In the most severe cases, each of the 8 threads requires many TLB entries, because loops stride through several large arrays at once. Since the primary data sets are usually larger than a typical 8KB page size, at least one TLB entry is required for each array. The swim benchmark from SPECfp95 illustrates an extreme example. In one loop, 9 large arrays are accessed on each iteration of the loop. When the loop is parallelized using a blocked distribution, the data TLB footprint is 9 arrays * 8 excluding the entries required for other data. With any size less than 72, significant thrashing will occur and the parallelization is not profitable. The lesson here is that the TLB is a shared resource that needs to be managed efficiently in an SMT. At least three approaches can be considered: (1) using fewer than 8 threads when parallelizing, (2) increasing the data TLB size, or (3) parallelizing loops differently. The first alternative unnecessarily limits the use of the thread hardware contexts, and neither exploits SMT nor the parallel applications to their fullest potential. The second choice incurs a cost in access time and hardware, although with increasing chip densities, future processors may be able to accommodate. 1 Even with larger TLBs, 1. We found that 64 entries did not solve the problem. However, a 128- entry data TLB avoids TLB thrashing, and as Figure 2b indicates, achieves speedups, at least for the SPECfp95 data sets. Application Number of threads applu 0.7% 0.9% 1.0% 0.9% 1.0% hydro2d 0.1% 0.1% 0.1% 0.7% 6.3% mgrid 0.0% 0.0% 0.0% 0.0% 0.1% su2cor 0.1% 5.2% 7.7% 6.2% 5.5% tomcatv 0.1% 0.1% 0.1% 2.0% 10.7% Table 3: TLB miss rates. Miss rates are shown for a blocked distribution and a 48-entry data TLB. The bold entries correspond to decreased performance (see Figure 2) when the number of threads was increased. however, it is desirable to reduce the TLB footprint on an SMT. A true SMT workload would be multiprogrammed: for example, multiple parallel applications could execute together, comprising more threads than hardware contexts. The thread scheduler could schedule all 8 threads for the first parallel application, then context switch to run the second, and later switch back to the first. In this type of environment it would be performance-wise to minimize the data TLB footprint required by each application. (As an example, the TLB footprint of a multiprogrammed workload consisting of swim and hydro2d would be greater than 128 entries.) The third and most desirable solution relies on the compiler to reduce the data TLB footprint. Rather than distributing loop iterations in a blocked organization, it could use a cyclic distribution to cluster the accesses of multiple threads onto fewer pages. (With cyclic partitioning, swim would consume 9 rather than 72 TLB entries). Cyclic partitioning also requires less instruction overhead in calculating array partition bounds, a non- negligible, although much less important factor. (Compare the blocked and cyclic loop distribution code and data in Figure 1.) Figure 3 illustrates the speedups attained by a cyclic distribution over blocked, and Table 4 contains the corresponding changes in data TLB miss rates. With the 48-entry TLB all applications did better with a cyclic distribution. In most cases the significant decrease in data TLB misses, coupled with the long 160 cycle TLB miss penalty, was the major factor. Cyclic increased TLB conflicts in tomcatv at 2 and 4 threads, but, because the number of misses was so low, overall program performance did not suffer. At 6 and 8 threads, tomcatv's a) original loop for blocked parallelization for c) cyclic parallelization for Figure 1: A blocked and cyclic loop distribution example. The code for an example loop nest is shown in a). When using a blocked distribution, the code is structured as in b). The cyclic version is shown in c). On the right, d) and e) illustrate which portions of the array are accessed by each thread for the two policies. (For clarity, we assume 4 threads). Assume that each row of the array is 2KB (512 double precision elements). With blocked distribution (d), each thread accesses a different 8KB page in memory. With cyclic (e), however, the loop is decomposed in a manner that allows all four threads to access a single 8KB page at the same time, thus reducing the TLB footprint. dimension dimension Thread 0 Thread 1 Thread 2 Thread 3 d) blocked a) 48-entry data TLB26 Figure 2: Speedups over one thread for blocked parallelization. Number of threads applu hydro2d mgrid su2cor swim tomcatv average1.03.0 b) 128-entry data TLB applu hydro2d mgrid su2cor swim tomcatv average1.03.0Speedup blocked data TLB miss rate jumped to 2% and 11%, causing a corresponding hike in speedup for cyclic. Absolute miss rates in the larger data TLB are low enough (usually under 0.2%, except for applu and su2cor, which reached 0.9%) that most changes produced little or no benefit for cyclic. In contrast, su2cor saw degradation, because cyclic scheduling increased loop unrolling instruction overhead. This performance degradation was not seen with the smaller TLB size, because cyclic's improved TLB hit rate offset the overhead. Mgrid saw a large performance improvement for both TLB sizes, because of a reduction in dynamic instruction count. As Figures 1b and 1c illustrate, cyclic parallelization requires fewer computations and no long-latency divide. In summary, these results suggest using a cyclic loop distribution for SMT, rather than the traditional blocked distribution. For parallel applications with large data footprints, cyclic distribution increased program speedups. (We saw speedups as high as 4.1, even with the smallish SPECfp95 reference data sets.) For applications with smaller data footprints, cyclic broke even. Only in one application, where there was an odd interaction with the loop unrolling factor, did cyclic worsen performance. In a multiprocessor of SMT processors, a cyclic distribution would still be appropriate within each node. Application 48-entry TLB 128-entry TLB Number of threads Number of threads applu 0% 50% 58% 53% 15% 0% 91% 98% 85% 69% hydro2d 0% 0% 14% 91% 99% 0% 0% 0% 0% 14% mgrid 0% 0% 0% 0% 50% 0% 0% 0% 0% 0% su2cor 14% 99% 99% 99% 97% 0% 0% 98% 91% 94% tomcatv 0% -60% -60% 96% 99% 0% -60% -60% -60% -60% Table 4: Improvement (decrease) in TLB miss rates of cyclic distribution over blocked. 3.5 4.1 applu hydro2d mgrid su2cor swim tomcatv mean1.0Speedup versus blocked b) 128-entry data TLB applu hydro2d mgrid su2cor swim tomcatv mean1.0Speedup versus blocked a) 48-entry data TLB thread 4 thread 6 thread 8 thread Figure 3: Speedup attained by cyclic over blocked parallelization. For each application, the execution time for blocked is normalized to 1.0 for all numbers of threads. Thus, each bar compares the speedup for cyclic over blocked with the same number of threads. A hybrid parallelization policy might be desirable, though, with a blocked distribution across processors to minimize inter-processor communication. 6 Software speculative execution Today's optimizing compilers rely on aggressive code scheduling to hide instruction latencies. In global scheduling techniques, such as trace scheduling [22] or hyperblock scheduling [23], instructions from a predicted branch path may be moved above a conditional branch, so that their execution becomes speculative. If at runtime, the other branch path is taken, then the speculative instructions are useless and potentially waste processor resources. On in-order superscalars or VLIW machines, software speculation is necessary, because the hardware provides no scheduling assistance. On an SMT processor (whose execution core is an out-of-order superscalar), not only are instructions dynamically scheduled and speculatively executed by the hardware, but multithreading is also used to hide latencies. (As the number of SMT threads is increased, instruction throughput also increases.) Therefore, the latency-hiding benefits of software speculative execution may be needed less, or even be unnecessary, and the additional instruction overhead introduced by incorrect speculations may degrade performance. Our experiments were designed to evaluate the appropriateness of software speculative execution for an SMT processor. The results highlight two factors that determine its effectiveness for SMT: static branch prediction accuracy and instruction throughput. Correctly-speculated instructions have no instruction overhead; incorrectly-speculated instructions, however, add to the dynamic instruction count. Therefore, speculative execution is more beneficial for applications that have high speculation accuracy, e.g., loop-based programs with either profile-driven or state-of-the-art static branch prediction. Table 5 compares the dynamic instruction counts between (profile-driven) 2 speculative and non-speculative versions of our applications. Small increases in the dynamic instruction count indicate that the compiler (with the assistance of profiling information) has been able to accurately predict which paths will be executed. 3 Consequently, speculation may incur no penalties. Higher increases in dynamic instruction count, on the other hand, mean wrong-path speculations, and a probable loss in SMT performance. While instruction overhead influences the effectiveness of speculation, it is not the only factor. The level of instruction throughput in programs without speculation is also important, because it determines how easily speculative overhead can be absorbed. With sufficient instruction issue bandwidth (low IPC), incorrect speculations may cause no harm; with higher 2. We used profile-driven speculation to provide a best-case comparison to SMT. Without profiles, more mispredictions would have occurred and more overhead instructions would have been generated. Consequently, software speculation would have worse performance than we report, making its absence appear even more beneficial for SMT. 3. All the SPECfp95 programs, radix from SPLASH-2, and compress from SPECint95, are loop-based; all have small increases in dynamic instruction count with speculation. increase SPECint95 increase SPLASH-2 increase applu 2.1% compress 2.9% fft 13.7% hydro2d 1.9% go 12.6% LU 12.5% mgrid 0.5% li 7.3% radix 0.0% su2cor 0.1% m88ksim 4.0% water-nsquared 3.0% tomcatv 1.2% Table 5: Percentage increase in dynamic instruction count due to profile-driven software speculative execution. Data are shown for 8 threads. (One thread numbers were identical or very close). Applications in bold have high speculative instruction overhead and high IPC without speculation; those in italics have only the former. no spec SPECint95 spec no spec SPLASH-2 spec no spec applu 4.9 4.7 compress 4.1 4.0 fft 6.0 6.4 hydro2d 5.6 5.4 go 2.4 2.3 LU 6.7 6.8 mgrid 7.2 7.1 li 4.5 4.6 radix 5.4 5.4 su2cor 6.1 6.0 m88ksim 4.2 4.1 water- nsquared 6.4 6.1 tomcatv 6.2 5.9 spatial Table Throughput (instructions per cycle) with and without profile-driven software speculation for 8 threads. Programs in bold have high IPC without speculation, plus high speculation overhead; those in italics have only the former. per-thread ILP or more threads, software speculation should be less profitable, because incorrectly-speculated instructions are more likely to compete with useful instructions for processor resources (in particular, fetch bandwidth and functional unit issue). Table 6 contains the instruction throughput for each of the applications. For some programs IPC is higher with software speculation, indicating some degree of absorption of the speculation overhead. In others, it is lower, because of additional hardware resource conflicts, most notably L1 cache misses. Speculative instruction overhead (related to static branch prediction accuracy) and instruction throughput together explain the speedups (or lack thereof) illustrated in Figure 4. When both factors were high (the non-loop- based fft, li, and LU), speedups without software speculation were greatest, ranging up to 22%. 4 If one factor was low or only moderate, speedups were minimal or nonexistent (the SPECfp95 applications, radix and water-nsquared had only high IPC; go, m88ksim and perl had only speculation overhead). 5 Without either factor, software speculation helped performance, and for the same reasons it benefits other architectures - it hid latencies and executed the speculative instructions in 4. For these applications (and a few others as well), as more threads are used, the advantage of turning off speculation generally becomes even larger. Additional threads provide more parallelism, and therefore, speculative instructions are more likely to compete with useful instructions for processor resources. applu hydro2d tomcatv mgrid su2cor swim0.5Speedup over speculation a) SPECfp95 thread 4 thread 6 thread 8 thread LU go fft li perl over speculation b) SPLASH2 and SPEC95 int nsquared spatial Figure 4: Speedups of applications executing without software speculation over with speculation (speculative execution cycles / no speculation execution cycles). Bars that are greater than 1.0 indicate that no speculation is better. otherwise idle functional units. The bottom line is that, while loop-based applications should be compiled with software speculative execution, non-loop applications should be compiled without it. Doing so either improves SMT program performance or maintains its current level - performance is never hurt. 6 7 Loop tiling In order to improve cache behavior, loops can be tiled to take advantage of data reuse. In this section, we examine two tiling issues: tile size selection and the distribution of tiles to threads. If the tile size is chosen appropriately, the reduction in average memory access time more than compensates for the tiling overhead instructions [20][11][6]. (The code in Figures 6b and 6c illustrates the source of this overhead). On an SMT, however, tiling may be less beneficial. First, SMT's enhanced latency-hiding capabilities may render tiling unnecessary. Second, the additional tiling instructions may increase execution time, given SMT's higher (multithreaded) throughput. (These are the same factors that influence whether to software speculate.) To address these issues, we examined tileable loop nests with different memory access characteristics, executing on an SMT processor. The benefits of tiling vary when the size of the cache is changed. Smaller caches require smaller tiles, which naturally introduce more instruction overhead. On the other hand, smaller tiles also produce lower average memory latencies - i.e., fewer conflict misses - so the latency reducing benefit of tiling is better. We therefore varied tile sizes to measure the performance impact of a range of tiling overhead. We also simulated two memory hierarchies to gauge the interaction between cache size, memory latency and tile size. The larger memory configuration represents a probable SMT memory subsystem for machines in production approximately 3 years in the future (see Section 4). The other configuration is smaller, modeling today's memory hierarchies, and is designed to provide a more appropriate ratio between data set and cache size, modeling loops with larger, i.e., more realistic, data sets than those in our benchmarks. For these experiments, each thread was given a separate tile (the tiling norm). Figure 5 presents the performance (total execution cycles, average memory access time, and dynamic instruction count for a range of tile sizes and the larger memory configuration) of an 8-thread SMT execution of each application and compares it to a single-thread run 5. Even though it has few floating point computations, water-spatial had a high IPC without speculation (6.5). Therefore the speculative instructions bottlenecked the integer units, and execution without speculation was more profitable. (approximating execution on a superscalar [13]). The results indicate that tiling is profitable on an SMT, just as it is on conventional processors. Mxm may seem to be an exception, since tiling brings no improvement, but it is an exception that shows there is no harm in applying the optimization. Programs executing on an SMT appear to be insensitive to tile size; at almost all tile sizes examined, SMT was able to hide memory latencies (as indicated by the flat AMAT curves), while still absorbing tiling overhead. Therefore SMT is less dependent on static algorithms to determine optimal tile sizes for particular caches and working sets. In contrast, conventional processors are more likely to have a tile size sweet spot. Even with out-of-order execution, modern processors, as well as alternative single-die processor architectures, lack sufficient latency-hiding abilities; consequently, they require more exact tile size calculations from the compiler. Tile size is also not a performance determinant with the less aggressive memory subsystem (results not shown), indicating that tiling on SMT is robust across 6. Keep in mind that had we speculated without run-time support (the pro- filing), the relative benefit of no speculation (versus speculation) would have been higher. For example, at 8 threads water-nsquared breaks even with profile-driven speculation; however, relying only on Multiflow's static branch prediction gives no speculation a slight edge, with a speedup of 1.1. Nevertheless, the general conclusions still hold: both good branch prediction and low multi-thread IPC are needed for software speculation to benefit applications executing on an SMT. Figure 5: Tiling results with the larger memory subsystem and separate tiles/thread. All the horizontal axes are tile size. A tile size of 0 means no tiling; sizes greater than 0 are one dimension of the tile, measured in array elements. The vertical axes are metrics for evaluating tiling: dynamic instruction count, total execution cycles and AMAT. mxm Dynamic instruction count in millions Total execution cycles in millions Average memory access time in cycles (AMAT) Total execution cycles in millions Average memory access time in cycles (AMAT) adi 8 threads gmt 8 threads memory hierarchies (or, alternatively, a range of data set sizes). Execution time is, of course, higher, because performance is more dependent on AMAT parameters, rather than tiling overhead. Only adi became slightly less tolerant of tile size changes. At the largest tile size measured (32x32), its AMAT increased sharply, because of inter-thread interference in the small cache. For this loop nest, either tiles should be sized so that all fit in the cache, or an alternative tiling technique (described below) should be used. The second loop tiling issue is the distribution of tiles to threads. When parallelizing loops for multiprocessors, a different tile is allocated to each processor (thread) to maximize reuse and reduce inter-processor communication. On an SMT, however, tiling in this manner could be detrimental. Private, per-thread tiles discourage inter-thread tile sharing and increase the total- thread tile footprint on the single-processor SMT (the same factors that make blocked loop iteration scheduling inappropriate for SMT). Rather than giving each thread its own tile (called blocked tiling), a single tile can be shared by all threads, and loop iterations can be distributed cyclically across the threads (cyclic tiling). (See Figure 6 for a code explanation of blocked and cyclic tiling, and Figure 7 for the effect on the per-thread data layout). Because the tile is shared, cyclic tiling can be optimized by increasing the tile size to reduce overhead 7c). With larger tiles, cyclic tiling can drop execution times of applications executing on small memory SMTs closer to that of SMTs with more aggressive memory hierarchies. (Or, put another way, the performance of programs with large data sets can a) original loop for blocked tiling for (jT=lb; jT <= ub; jT+=jTsize) for for (iT=1; iT < M; iT+=iTsize) for (j=jT; j <= min(N,jT+jTsize-1);j++) for (k=max(1, kT); for (i=iT; i <= min(M,iT+iTsize-1);i++) c) cyclic tiling for for for for (k=max(1, kT); for Figure Code for blocked and cyclic versions of a tiled loop nest. approach those with smaller.) For example, Figure 8c illustrates that with larger tile sizes (greater than 8 array elements per dimension) cyclic tiling reduced mxm's AMAT enough to decrease average execution time on the smaller cache hierarchy by 51% (compare to blocked in Figure 8b) and to within 35% of blocked tiling on a memory subsystem several times the size (Figure 8a). Only at the very smallest tile size did an increase in tiling overhead overwhelm SMT's ability to hide memory latency. Cyclic tiling is still appropriate for a multiprocessor of SMTs. A hierarchical [8] or hybrid tiling approach might be most effective. Cyclic tiling could be used to maximize locality in each processor, while blocked tiling could distribute tiles across processors to minimize inter-processor Thread 1 Thread 2 Thread 3 dimension c) optimized cyclic0011 a) blocked223344 dimension b) cyclic21i dimension dimension Figure 7: A comparison of blocked and cyclic tiling techniques for multiple threads. The blocked tiling is shown in a). Each tile is a 4x4 array of elements. The numbers represent the order in which tiles are accessed by each thread. For cyclic tiling, each tile is still a 4x4 array, but now the tile is shared by all threads. In this example, each thread gets one row of the tile, as shown in b). With cyclic tiling, each thread works on a smaller chunk of data at a time, so the tiling overhead is greater. In c), the tile size is increased to 8x8 to reduce the overhead. Within each tile, each thread is responsible for 16 of the elements, as in the original blocked example. Total execution time in millions of cycles Average memory access time in cycles (AMAT) Dynamic instruction count in millions b) c) a) Figure 8: Tiling performance of 8-thread mxm. Tile sizes are along the x-axis. Results are shown for a) blocked tiling and the larger memory subsystem, b) blocked tiling with the smaller memory subsystem, and c) cyclic tiling, also with the smaller memory subsystem. 8 Other compiler optimizations In addition to the optimizations studied in this paper, compiler-directed prefetching, predicated execution and software pipelining should also be re-evaluated in the context of an SMT processor. On a conventional processor, compiler-directed prefetching [26] can be useful for tolerating memory latencies, as long as prefetch overhead (due to prefetch instructions, additional memory bandwidth, and/or cache interference) is minimal. On an SMT, this overhead is more detrimental: it interferes not only with the thread doing the prefetching, but also competes with other threads. Predicated execution [23][16][28] is an architectural model in which instruction execution can be guarded by boolean predicates that determine whether an instruction should be executed or nullified. Compilers can then use if-conversion [2] to transform control dependences into data dependences, thereby exposing more ILP. Like software speculative execution, aggressive predication can incur additional instruction overhead by executing instructions that are either nullified or produce results that are never used. Software pipelining [9][27][18][1] improves instruction scheduling by overlapping the execution of multiple loop iterations. Rather than pipelining loops, SMT can execute them in parallel in separate hardware contexts. Doing so alleviates the increased register pressure normally associated with software pipelining. Multithreading could also be combined with software pipelining if necessary. Most of the optimizations discussed in this paper were originally designed to increase single-thread ILP. While intra-thread parallelism is still important on an SMT processor, simultaneous multithreading relies on multiple threads to provide useful parallelism, and throughput often becomes a more important performance metric. SMT raises the issue of compiling for throughput or for a single-thread. For example, from the perspective of a single running thread, these optimizations, as traditionally applied, may be desirable to reduce the thread's running time. But from a global perspective, greater throughput (and therefore more useful work) can be achieved by limiting the amount of speculative work. 9 Related work The three compiler optimizations discussed in this paper have been widely investigated in non-SMT architectures. Loop iteration scheduling for shared-memory multiprocessors has been evaluated by Wolf and Lam [34], Carr, McKinley, and Tseng [7], Anderson, Amarasinghe, and Lam [3], and Cierniak and Li [10], among others. These studies focus on scheduling to minimize communication and synchronization overhead; all restructured loops and data layout to improve access locality for each processor. In particular, Anderson et al., discuss the blocked and cyclic mapping schemes, and present a heuristic for choosing between them. Global scheduling optimizations, like trace scheduling [22], superblocks [25] and hyperblocks [23], allow code motion (including speculative motion) across basic blocks, thereby exposing more ILP for statically-scheduled VLIWs and wide-issue superscalars. In their study on ILP limits, Lam and Wilson [19] found that speculation provides greater speedups on loop-based numeric applications than on non-numeric codes, but their study did not include the effects of wrong-path instructions. Previous work in code transformation for improved locality has proposed various frameworks and algorithms for selecting and applying a range of loop transformations [14][33][6][17][34][7]. These studies illustrate the effectiveness of tiling and also propose other loop transformations for enabling better tiling. Lam, Rothberg, and Wolf [20], Coleman and McKinley [11], and Carr et al., [6] show that application performance is sensitive to the tile size, and present techniques for selecting tile sizes based on problem-size and cache parameters, rather than targeting a fixed-size or fixed-cache occupancy. Conclusions This paper has examined compiler optimizations in the context of a simultaneous multithreading architecture. An SMT architecture differs from previous parallel architectures in several significant ways. First, SMT threads share processor and memory system resources of a single processor on a fine-grained basis, even within a single cycle. Optimizations for an SMT should therefore seek to benefit from this fine-grained sharing, rather than avoiding it, as is done on conventional shared-memory multiprocessors. Second, SMT hides intra-thread latencies by using instructions from other active threads; optimizations that expose ILP may not be needed. Third, instruction throughput on an SMT is high; therefore optimizations that increase instruction count may degrade performance. An effective compilation strategy for simultaneous multithreading processors must recognize these unique characteristics. Our results show specific cases where an SMT processor can benefit from changing the compiler optimization strategy. In particular, we showed that (1) cyclic iteration scheduling (as opposed to blocked scheduling) is more appropriate for an SMT, because of its ability to reduce the TLB footprint; (2) software speculative execution can be bad for an SMT, because it decreases useful instruction throughput; (3) loop tiling algorithms can be less concerned with determining the exact tile size, because SMT performance is less sensitive to tile size; and (4) loop tiling to increase, rather than reduce, inter-thread tile sharing, is more appropriate for an SMT, because it increases the benefit of sharing memory system resources. Acknowledgments We would like to thank John O'Donnell of Equator Technologies, Inc. and Tryggve Fossum of Digital Equipment Corp. for the source to the Alpha AXP version of the Multiflow compiler; and Jennifer Anderson of the DEC Western Research Laboratory for providing us with SUIF-parallelized copies of the benchmarks. We also would like to thank Jeffrey Dean of DEC WRL and the referees, whose comments helped improve this paper. This research was supported by the Washington Technology Center, NSF grants MIP-9632977, CCR-9200832, and CCR-9632769, DARPA grant F30602-97-2-0226, ONR grants N00014- 92-J-1395 and N00014-94-1-1136, DEC WRL, and a fellowship from Intel Corporation. --R Optimal loop parallelization. Conversion of control dependence to data dependence. Data and computation transformations for multiprocessors. Portable Programs for Parallel Processors. Compiler blockability of numerical algo- rithms Compiler optimizations for improving data locality. Hierarchical tiling for improved superscalar performance. An approach to scientific array processing: The architectural design of the AP-120B/FPS-164 family Unifying data and control transformations for distributed shared-memory machines Tile size selection using cache organization and data layout. New CPU benchmark suites from SPEC. Simultaneous multithreading: A platform for next-generation processors Strategies for cache and local memory management by global program transformation. Maximizing multiprocessor performance with the SUIF compiler. Highly concurrent scalar processing. Maximizing loop parallelism and improving data locality via loop fusion and distribution. Software pipelining: An effective scheduling technique for VLIW machines. Limits of control flow on parallelism. The cache performance and optimizations of blocked algorithms. Converting thread-level parallelism to instruction-level parallelism via simultaneous multithreading The Multiflow trace scheduling compiler. Effective compiler support for predicated execution using the hyperblock. Combining branch predictors. The superblock: An effective technique for VLIW and superscalar compilation. Design and evaluation of a compiler algorithm for prefetching. Some scheduling techniques and an easily schedulable horizontal architecture for high performance scientific computing. The Cydra 5 departmental supercomputer. Scaling parallel programs for multiprocessors: Methodology and examples. Exploiting choice: Instruction fetch and issue on an implementable simultaneous multithreading processor. Simultaneous multi- threading: Maximizing on-chip parallelism A data locality optimizing algorithm. A loop transformation theory and an algorithm to maximize parallelism. The SPLASH-2 programs: Characterization and methodological considerations --TR Highly concurrent scalar processing Strategies for cache and local memory management by global program transformation Optimal loop parallelization Software pipelining: an effective scheduling technique for VLIW machines The Cydra 5 Departmental Supercomputer The cache performance and optimizations of blocked algorithms A data locality optimizing algorithm New CPU benchmark suites from SPEC Limits of control flow on parallelism Design and evaluation of a compiler algorithm for prefetching Effective compiler support for predicated execution using the hyperblock Compiler blockability of numerical algorithms The multiflow trace scheduling compiler The superblock Compiler optimizations for improving data locality Unifying data and control transformations for distributed shared-memory machines Tile size selection using cache organization and data layout The SPLASH-2 programs Simultaneous multithreading Exploiting choice Compiler-directed page coloring for multiprocessors Converting thread-level parallelism to instruction-level parallelism via simultaneous multithreading Conversion of control dependence to data dependence Portable Programs for Parallel Processors Scaling Parallel Programs for Multiprocessors Maximizing Multiprocessor Performance with the SUIF Compiler Simultaneous Multithreading A Loop Transformation Theory and an Algorithm to Maximize Parallelism Hierarchical tiling for improved superscalar performance Maximizing Loop Parallelism and Improving Data Locality via Loop Fusion and Distribution Some scheduling techniques and an easily schedulable horizontal architecture for high performance scientific computing --CTR Mark N. Yankelevsky , Constantine D. Polychronopoulos, -coral: a multigrain, multithreaded processor architecture, Proceedings of the 15th international conference on Supercomputing, p.358-367, June 2001, Sorrento, Italy Nicholas Mitchell , Larry Carter , Jeanne Ferrante , Dean Tullsen, ILP versus TLP on SMT, Proceedings of the 1999 ACM/IEEE conference on Supercomputing (CDROM), p.37-es, November 14-19, 1999, Portland, Oregon, United States Jack L. Lo , Luiz Andr Barroso , Susan J. Eggers , Kourosh Gharachorloo , Henry M. Levy , Sujay S. Parekh, An analysis of database workload performance on simultaneous multithreaded processors, ACM SIGARCH Computer Architecture News, v.26 n.3, p.39-50, June 1998 Alex Settle , Joshua Kihm , Andrew Janiszewski , Dan Connors, Architectural Support for Enhanced SMT Job Scheduling, Proceedings of the 13th International Conference on Parallel Architectures and Compilation Techniques, p.63-73, September 29-October 03, 2004 Evangelia Athanasaki , Nikos Anastopoulos , Kornilios Kourtis , Nectarios Koziris, Exploring the performance limits of simultaneous multithreading for memory intensive applications, The Journal of Supercomputing, v.44 n.1, p.64-97, April 2008 Gary M. Zoppetti , Gagan Agrawal , Lori Pollock , Jose Nelson Amaral , Xinan Tang , Guang Gao, Automatic compiler techniques for thread coarsening for multithreaded architectures, Proceedings of the 14th international conference on Supercomputing, p.306-315, May 08-11, 2000, Santa Fe, New Mexico, United States Steven Swanson , Luke K. McDowell , Michael M. Swift , Susan J. Eggers , Henry M. Levy, An evaluation of speculative instruction execution on simultaneous multithreaded processors, ACM Transactions on Computer Systems (TOCS), v.21 n.3, p.314-340, August Calin Cacaval , David A. Padua, Estimating cache misses and locality using stack distances, Proceedings of the 17th annual international conference on Supercomputing, June 23-26, 2003, San Francisco, CA, USA James Burns , Jean-Luc Gaudiot, SMT Layout Overhead and Scalability, IEEE Transactions on Parallel and Distributed Systems, v.13 n.2, p.142-155, February 2002 Joshua A. Redstone , Susan J. Eggers , Henry M. Levy, An analysis of operating system behavior on a simultaneous multithreaded architecture, ACM SIGPLAN Notices, v.35 n.11, p.245-256, Nov. 2000 Joshua A. Redstone , Susan J. Eggers , Henry M. Levy, An analysis of operating system behavior on a simultaneous multithreaded architecture, ACM SIGARCH Computer Architecture News, v.28 n.5, p.245-256, Dec. 2000 Luke K. McDowell , Susan J. Eggers , Steven D. Gribble, Improving server software support for simultaneous multithreaded processors, ACM SIGPLAN Notices, v.38 n.10, October

simultaneous multithreading;compiler optimizations;processor architecture;software speculative execution;performance;loop-iteration scheduling;parallel architecture;cache size;inter-processor communication;memory system resources;latency hiding;parallel programs;optimising compilers;shared-memory multiprocessors;loop tiling;fine-grained sharing;instructions;cyclic algorithm;inter-thread instruction-level parallelism

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Trace processors.

Traces are dynamic instruction sequences constructed and cached by hardware. A microarchitecture organized around traces is presented as a means for efficiently executing many instructions per cycle. Trace processors exploit both control flow and data flow hierarchy to overcome complexity and architectural limitations of conventional superscalar processors by (1) distributing execution resources based on trace boundaries and (2) applying control and data prediction at the trace level rather than individual branches or instructions. Three sets of experiments using the SPECInt95 benchmarks are presented. (i) A detailed evaluation of trace processor configurations: the results affirm that significant instruction-level parallelism can be exploited in integer programs (2 to 6 instructions per cycle). We also isolate the impact of distributed resources, and quantify the value of successively doubling the number of distributed elements. (ii) A trace processor with data prediction applied to inter-trace dependences: potential performance improvement with perfect prediction is around 45% for all benchmarks. With realistic prediction, gcc achieves an actual improvement of 10%. (iii) Evaluation of aggressive control flow: some benchmarks benefit from control independence by as much as 10%.

Introduction Improvements in processor performance come about in two ways - advances in semiconductor technology and advances in processor microarchitecture. To sustain the historic rate of increase in computing power, it is important for both kinds of advances to continue. It is almost certain that clock frequencies will continue to increase. The microarchitectural challenge is to issue many instructions per cycle and to do so efficiently. We argue that a conventional superscalar microarchitecture cannot meet this challenge due to its complexity - its inefficient approach to multiple instruction issue - and due to its architectural limitations on ILP - its inability to extract sufficient parallelism from sequential programs. In going from today's modest issue rates to 12- or 16- way issue, superscalar processors face complexity at all phases of instruction processing. Instruction fetch band-width is limited by frequent branches. Instruction dis- patch, register renaming in particular, requires increasingly complex dependence checking among all instructions being dispatched. It is not clear that wide instruction issue from a large pool of instruction buffers or full result bypassing among functional units is feasible with a very fast clock. Even if a wide superscalar processor could efficiently exploit ILP, it still has fundamental limitations in finding the parallelism. These architectural limitations are due to the handling of control, data, and memory dependences. The purpose of this paper is to advocate a next generation microarchitecture that addresses both complexity and architectural limitations. The development of this microarchitecture brings together concepts from a significant body of research targeting these issues and fills in some gaps to give a more complete and cohesive picture. Our primary contribution is evaluating the performance potential that this microarchitecture offers. 1.1 Trace processor microarchitecture The proposed microarchitecture (Figure 1) is organized around traces. In this context, a trace is a dynamic sequence of instructions captured and stored by hard- ware. The primary constraint on a trace is a hardware- determined maximum length, but there may be a number of other implementation-dependent constraints. Traces are built as the program executes, and are stored in a trace cache [1][2]. Using traces leads to interesting possibilities that are revealed by the following trace properties: . A trace can contain any number and type of control transfer instructions, that is, any number of implicit control predictions. This property suggests the unit of control prediction should be a trace, not individual control transfer instructions. A next-trace predictor [3] can make predictions at the trace level, effectively ignoring the embedded control flow in a trace. . A trace uses and produces register values that are either live-on-entry, entirely local, or live-on-exit [4][5]. These are referred to as live-ins, locals, and live-outs, respectively. This property suggests a hierarchical register file implementation: a local register file per trace for holding values produced and consumed solely within a trace, and a global register file for holding values that are live between traces. The distinction between local dependences within a trace and global dependences between traces also suggests implementing a distributed instruction window based on trace boundaries. The result is a processor composed of processing elements (PE), each having the organization of a small-scale superscalar processor. Each PE has (1) enough instruction buffer space to hold an entire trace, (2) multiple dedicated functional units, (3) a dedicated local register file for holding local values, and (4) a copy of the global register file. Figure 1. A trace processor. 1.1.1 Hierarchy: overcoming complexity An organization based on traces reduces complexity by taking advantage of hierarchy. There is a control flow hierarchy - the processor sequences through the program at the level of traces, and contained within traces is a finer granularity of control flow. There is also a value hierarchy - global and local values - that enables the processor to efficiently distribute execution resources. With hierarchy we overcome complexity at all phases of processing: . Instruction predicting traces, multiple branches are implicitly predicted - a simpler alternative to brute-force extensions of single-branch predictors. Together the trace cache and trace predictor offer a solution to instruction fetch complexity. . Instruction dispatch: Because a trace is given a local register file that is not affected by other traces, local registers can be pre-renamed in the trace cache [4][5]. Pre-renaming by definition eliminates the need for dependence checking among instructions being dis- patched, because locals represent all dependences entirely contained within a trace. Only live-ins and live-outs go through global renaming at trace dispatch, thereby reducing bandwidth pressure to register maps and the free-list. . Instruction issue: By distributing the instruction window among smaller trace-sized windows, the instruction issue logic is no longer centralized. Furthermore, each PE has fewer internal result buses, and thus a given instruction monitors fewer result tag buses. . Result bypassing: Full bypassing of local values among functional units within a PE is now feasible, despite a possibly longer latency for bypassing global values between PEs. . Register file: The size and bandwidth requirements of the global register file are reduced because it does not hold local values. Sufficient read port bandwidth is achieved by having copies in each PE. Write ports cannot be handled this way because live-outs must be broadcast to all copies of the global file; however, write bandwidth is reduced by eliminating local value traffic. . Instruction retirement: Retirement is the "dual" of dispatch in that physical registers are returned to the free- list. Free-list update bandwidth is reduced because only live-outs are mapped to physical registers. 1.1.2 Speculation: exposing ILP To alleviate the limitations imposed by control, data, and memory dependences, the processor employs aggressive speculation. Control flow prediction at the granularity of traces can yield as good or better overall branch prediction accuracy than many aggressive single-branch predictors [3]. Value prediction [6][7] is used to relax the data dependence constraints among instructions. Rather than predict source or destination values of all instructions, we limit value predictions to live-ins of traces. Limiting predictions to a critical subset of values imposes structure on value prediction; predicting live-ins is particularly appealing because it enables traces to execute independently. Memory speculation is performed in two ways. First, all load and store addresses are predicted at dispatch time. Second, we employ memory dependence speculation - loads issue as if there are no prior stores, and disambiguation occurs after the fact via a distributed mechanism. 1.1.3 Handling misspeculation: selective reissuing Because of the pervasiveness of speculation, handling of misspeculation must fundamentally change. Misspeculation is traditionally viewed as an uncommon event and is treated accordingly: a misprediction represents a barrier for subsequent computation. However, data misspeculation in particular should be viewed as a normal aspect of computation. Data misspeculation may be caused by a mispredicted source register value, a mispredicted address, or a memory Preprocess Trace Construct Trace Instruction Branch Predict Cache Global Registers Live-in Value Predict Trace Cache Reorder Buffer segment per trace Next Trace Predict Maps Rename Global Registers Predicted Issue Buffers Registers Units Processing Element 1 Processing Element 2 Processing Element 3 Processing Element 0 Speculative State Data Cache dependence violation. If an instruction detects a mispre- diction, it will reissue with new values for its operands. A new value is produced and propagated to dependent instructions, which will in turn reissue, and so on. Only instructions along the dependence chain reissue. The mechanism for selective reissuing is simple because it is in fact the existing issue mechanism. Selective reissuing due to control misprediction, while more involved, is also discussed and the performance improvement is evaluated for trace processors. 1.2 Prior work This paper draws from significant bodies of work that either efficiently exploit ILP via distribution and hierarchy, expose ILP via aggressive speculation, or do both. For the most part, this body of research focuses on hardware- intensive approaches to ILP. Work in the area of multiscalar processors [8][9] first recognized the complexity of implementing wide instruction issue in the context of centralized resources. The result is an interesting combination of compiler and hard- ware. The compiler divides a sequential program into tasks, each task containing arbitrary control flow. Tasks, like traces, imply a hierarchy for both control flow and val- ues. Execution resources are distributed among multiple processing elements and allocated at task granularity. At run-time tasks are predicted and scheduled onto the PEs, and both control and data dependences are enforced by the hardware (with aid from the compiler in the case of register dependences). Multiscalar processors have several characteristics in common with trace processors. Distributing the instruction window and register file solves instruction issue and register file complexity. Mechanisms for multiple flows of control not only avoid instruction fetch and dispatch com- plexity, but also exploit control independence. Because tasks are neither scheduled by the compiler nor guaranteed to be parallel, these processors demonstrate aggressive control speculation [10] and memory dependence speculation [8][11]. More recently, other microarchitectures have been proposed that address the complexity of superscalar pro- cessors. The trace window organization proposed in [4] is the basis for the microarchitecture presented here. Con- ceivably, other register file and memory organizations could be superimposed on this organization; e.g. the original multiscalar distributed register file [12], or the distributed speculative-versioning cache [13]. So far we have discussed microarchitectures that distribute the instruction window based on task or trace boundaries. Dependence-based clustering is an interesting alternative [14][15]. Similar to trace processors, the window and execution resources are distributed among multiple smaller clusters. However, instructions are dispatched to clusters based on dependences, not based on proximity in the dynamic instruction stream as is the case with traces. Instructions are steered to clusters so as to localize dependences within a cluster, and minimize dependences between clusters. Early work [16] proposed the fill-unit for constructing and reusing larger units of execution other than individual instructions, a concept very relevant to next generation processors. This and subsequent research [17][18] emphasize atomicity, which allows for unconstrained instruction preprocessing and code scheduling. Recent work in value prediction and instruction collapsing [6][7] address the limits of true data dependences on ILP. These works propose exposing more ILP by predicting addresses and register values, as well as collapsing instructions for execution in combined functional units. 1.3 Paper overview In Section 2 we describe the microarchitecture in detail, including the frontend, the value predictor, the processing element, and the mechanisms for handling mis- speculation. Section 3 describes the performance evaluation method. Primary performance results, including a comparison with superscalar, are presented in Section 4, followed by results with value prediction in Section 5 and a study of control flow in Section 6. 2. Microarchitecture of the trace processor 2.1 Instruction supply A trace is uniquely identified by the addresses of all its instructions. Of course this sequence of addresses can be encoded in a more compact form, for example, starting addresses of all basic blocks, or trace starting address plus branch directions. Regardless of how a trace is identified, trace ids and derivatives of these trace ids are used to sequence through the program. The shaded region in Figure 2 shows the fast-path of instruction fetch: the next-trace predictor [3], the trace cache, and sequencing logic to coordinate the datapath. The trace predictor outputs a primary trace id and one alternate trace id prediction in case the primary one turns out to be incorrect (one could use more alternates, but with diminishing returns). The sequencer applies some hash function on the bits of the predicted trace id to form an index into the trace cache. The trace cache supplies the trace id (equivalent of a cache tag) of the trace cached at that location, which is compared against the full predicted trace id to determine if there is a hit. In the best case, the predicted trace is both cached and correct. If the predicted trace misses in the cache, a trace is constructed by the slow-path sequencer (non-shaded path in Figure 2). The predicted trace id encodes the instructions to be fetched from the instruction cache, so the sequencer uses the trace id directly instead of the conventional branch predictor. The execution engine returns actual branch outcomes. If the predicted trace is partially or completely incorrect, an alternate trace id that is consistent with the known branch outcomes can be used to try a different trace (trace cache hit) or build the trace remainder (trace cache miss). If alternate ids prove insufficient, the slow-path sequencer forms the trace using the conventional branch predictor and actual branch outcomes. Figure 2. Frontend of the trace processor. 2.1.1 Trace selection An interesting aspect of trace construction is the algorithm used to delineate traces, or trace selection. The obvious trace selection decisions involve either stopping at or embedding various types of control instructions: call directs, call indirects, jump indirects, and returns. Other heuristics may stop at loop branches, ensure that traces end on basic block boundaries, embed leaf functions, embed unique call sites, or enhance control independence. Trace selection decisions affect instruction fetch band- width, PE utilization, load balance between PEs, trace cache hit rate, and trace prediction accuracy - all of which strongly influence overall performance. Often, targeting trace selection for one factor negatively impacts another factor. We have not studied this issue extensively. Unless otherwise stated, the trace selection we use is: (1) stop at a maximum of 16 instructions, or (2) stop at any call indi- rect, jump indirect, or return instruction. 2.1.2 Trace preprocessing Traces can be preprocessed prior to being stored in the trace cache. Our processor model requires pre-renaming information in the trace cache. Register operands are marked as local or global, and locals are pre-renamed to the local register file [4]. Although not done here, preprocessing might also include instruction scheduling [17], storing information along with the trace to set up the reorder buffer quickly at dispatch time, or collapsing dependent instructions across basic block boundaries [7]. 2.1.3 Trace cache performance In this section we present miss rates for different trace cache configurations. The miss rates are measured by running through the dynamic instruction stream, dividing it into traces based on the trace selection algorithm, and looking up the successive trace ids in the cache. We only include graphs for go and gcc. Compress fits entirely within a 16K direct mapped trace cache; jpeg and xlisp show under 4% miss rates for a 32K direct mapped cache. There are two sets of curves, for two different trace selection algorithms. Each set shows miss rates for 1-way through 8-way associativity, with total size in kilobytes (instruction storage only) along the x-axis. The top four curves are for the default trace selection (Section 2.1.1). The bottom four curves, labeled with 'S' in the key, add two more stopping constraints: stop at call directs and stop at loop branches. Default trace selection gives average trace lengths of 14.8 for go and 13.9 for gcc. The more constraining trace selection gives smaller average trace lengths - 11.8 for go and 10.9 for gcc - but the advantage is much lower miss rates for both benchmarks. For go in particular, the miss rate is 14% with constrained selection and a 128kB trace cache, down from 34%. Figure 3. Trace cache miss rates. 2.1.4 Trace predictor The core of the trace predictor is a correlated predictor that uses the history of previous traces. The previous few trace ids are hashed down to fewer bits and placed in a shift register, forming a path history. The path history is used to form an index into a prediction table with 2 entries. Each table entry consists of the predicted trace id, hit logic fast-path sequencer Trace Next sequencer slow-path cached trace id alternate primary Function Hash predicted trace id trace pred's branch pred control targets, branch Construct Trace Trace new trace, trace id Preprocess outcomes from execution instr. block (optional path) miss rate GCC DM 2-way 4-way 8-way DM (S) 2-way (S) 4-way (S) 8-way (S)1030507048 miss rate size (K-bytes) GO DM 2-way 4-way 8-way DM (S) 2-way (S) 4-way (S) 8-way (S) an alternate trace id, and a 2-bit saturating counter for guiding replacement. The accuracy of the correlated predictor is aided by having a return history stack. For each call within a trace the path history register is copied and pushed onto a hardware stack. When a trace ends in a return, a path history value is popped from the stack and used to replace all but the newest trace in the path history register. To reduce the impact of cold-starts and aliasing, the correlated predictor is augmented with a second smaller predictor that uses only the previous trace id, not the whole path history. Each table entry in the correlated predictor is tagged with the last trace to use the entry. If the tag matches then the correlated predictor is used, otherwise the simpler predictor is used. If the counter of the simpler predictor is saturated its prediction is automatically used, regardless of the tag. A more detailed treatment of the trace predictor can be found in [3]. 2.1.5 Trace characteristics Important trace characteristics are shown in Table 1. Average trace length affects instruction supply bandwidth and instruction buffer utilization - the larger the better. We want a significant fraction of values to be locals, to reduce global communication. Note that the ratio of locals to live-outs tends to be higher for longer traces, as observed in [4]. 2.2 Value predictor The value predictor is context-based and organized as a two-level table. Context-based predictors learn values that follow a particular sequence of previous values [19]. The first-level table is indexed by a unique prediction id, derived from the trace id. A given trace has multiple prediction ids, one per live-in or address in the trace. An entry in the first-level table contains a pattern that is a hashed version of the previous 4 data values of the item being predicted. The pattern from the first-level table is used to look up a 32-bit data prediction in the second-level table. Replacement is guided by a 3-bit saturating counter associated with each entry in the second-level table. The predictor also assigns a confidence level to predictions [20][6]. Instructions issue with predicted values only if the predictions have a high level of confidence. The confidence mechanism is a 2-bit saturating counter stored with each pattern in the first-level table. The table sizes used in this study are very large in order to explore the potential of such an entries in the first-level, 2 20 entries in the second-level. Accuracy of context-based value prediction is affected by timing of updates, which we accurately model. A detailed treatment of the value predictor can be found in [19]. 2.3 Distributed instruction window 2.3.1 Trace dispatch The dispatch stage performs decode, renaming, and value predictions. Live-in registers of the trace are renamed by looking up physical registers in the global register rename map. Independently, live-out registers receive new names from the free-list of physical registers, and the global register rename map is updated to reflect these new names. The dispatch stage looks up value predictions for all live-in registers and all load/store addresses in the trace. The dispatch stage also performs functions related to precise exceptions, similar to the mechanisms used in conventional processors. First, a segment of the reorder buffer (ROB) is reserved by the trace. Enough information is placed in the segment to allow backing up rename map state instruction by instruction. Second, a snapshot of the register rename map is saved at trace boundaries, to allow backing up state to the point of an exception quickly. The processor first backs up to the snapshot corresponding to the excepting trace, and then information in that trace's ROB segment is used to back up to the excepting instruc- tion. The ROB is also used to free physical registers. 2.3.2 Freeing and allocating PEs For precise interrupts, traces must be retired in-order, requiring the ROB to maintain state for all outstanding traces. The number of outstanding traces is therefore limited by the number of ROB segments (assuming there are enough physical registers to match). Because ROB state handles trace retirement, a PE can be freed as soon as its trace has completed execution. Unfortunately, knowing when a trace is "completed" is not simple, due to our misspeculation model (a mechanism is needed to determine when an instruction has issued for the last time). Consequently, a PE is freed when its trace is retired, because retirement guarantees instructions are done. This is a lower performance solution because it effectively arranges the PEs in a circular queue, just like segments of the ROB. PEs are therefore allocated and freed in a fifo fashion, even though they might in fact complete out-of-order. Table 1. Trace characteristics. statistic comp gcc go jpeg xlisp trace length (inst) 14.5 13.9 14.8 15.8 12.4 live-ins 5.2 4.3 5.0 6.8 4.1 live-outs 6.2 5.6 5.8 6.4 5.1 locals 5.6 3.8 5.9 7.1 2.6 loads 2.6 3.6 3.1 2.9 3.7 stores 0.9 1.9 1.0 1.2 2.2 cond. branches 2.1 2.1 1.8 1.0 1.9 control inst 2.9 2.8 2.2 1.3 2.9 trace misp. rate 17.1% 8.1% 15.7% 6.6% 6.9% 2.3.3 Processing element detail The datapath for a processing element is shown in Figure 4. There are enough instruction buffers to hold the largest trace. For loads and stores, the address generation part is treated as an instruction in these buffers. The memory access part of loads and stores, along with address pre- dictions, are placed into load/store buffers. Included with the load/store buffers is validation hardware for validating predicted addresses against the result of address computa- tions. A set of registers is provided to hold live-in predic- tions, along with hardware for validating the predictions against values received from other traces. Figure 4. Processing element detail. Instructions are ready to issue when all operands become available. Live-in values may already be available in the global register file. If not, live-ins may have been predicted and the values are buffered with the instruction. In any case, instructions continually monitor result buses for the arrival of new values for its operands; memory access operations continually monitor the arrival of new computed addresses. Associated with each functional unit is a queue for holding completed results, so that instruction issue is not blocked if results are held up waiting for a result bus. The result may be a local value only, a live-out value only, or both; in any case, local and global result buses are arbitrated separately. Global result buses correspond directly with write ports to the global register file, and are characterized by two numbers: the total number of buses and the number of buses for which each PE can arbitrate in a cycle. The memory buses correspond directly with cache ports, and are characterized similarly. 2.4 Misspeculation In Section 1.1.3 we introduced a model for handling misspeculation. Instructions reissue when they detect mispredictions; selectively reissuing dependent instructions follows naturally by the receipt of new values. This section describes the mechanisms for detecting various kinds of mispredictions. 2.4.1 Mispredicted live-ins Live-in predictions are validated when the computed values are seen on the global result buses. Instruction buffers and store buffers monitor comparator outputs corresponding to live-in predictions they used. If the predicted and computed values match, instructions that used the predicted live-in are not reissued. Otherwise they do reissue, in which case the validation latency appears as a misprediction penalty, because in the absence of speculation the instructions may have issued sooner [6]. 2.4.2 Memory dependence and address misspeculation The memory system (Figure 5) is composed of a data cache and a structure for buffering speculative store data, distributed load/store buffers in the PEs, and memory buses connecting them. When a trace is dispatched, all of its loads and stores are assigned sequence numbers. Sequence numbers indicate the program order of all memory operations in the window. The store buffer may be organized like a cache [21], or integrated as part of the data cache itself [13]. The important thing is that some mechanism must exist for buffering speculative memory state and maintaining multiple versions of memory locations [13]. Figure 5. Abstraction of the memory system. Handling stores: . When a store first issues to memory, it supplies its address, sequence number, and data on one of the memory buses. The store buffer creates a new version for that memory address and buffers the data. Multiple versions are ordered via store sequence numbers. . If a store must reissue because it has received a new computed address, it must first "undo" its state at the old address, and then perform the store to the new address. Both transactions are initiated by the store sending its old address, new address, sequence number, and data on one of the memory buses. load/store buf FU FU tags values tags values Global buffers issue issue File Reg File Reg local result buses agen results store data FU FU global result buses (D$ ports) buses addr/data live-in value pred's . PEs . global memory buses dataN data2 address multiple versions . If a store must reissue because it has received new data, it simply performs again to the same address. Handling loads: . A load sends its address and sequence number to the memory system. If multiple versions of the location exist, the memory system knows which version to return by comparing sequence numbers. The load is supplied both the data and the sequence number of the store which created the version. Thus, loads maintain two sequence numbers: its own and that of the data. . If a load must reissue because it has received a new computed address, it simply reissues to the memory system as before with the new address. . Loads snoop all store traffic (store address and sequence number). A load must reissue if (1) the store address matches the load address, (2) the store sequence number is less than that of the load, and (3) the store sequence number is greater than that of the load data. This is a true memory dependence violation. The load must also reissue if the store sequence number simply matches the sequence number of the load data. This takes care of the store changing its address (a false dependence had existed between the store and load) or sending out new data. 2.4.3 Concerning control misprediction In a conventional processor, a branch misprediction causes all subsequent instructions to be squashed. How- ever, only those instructions that are control-dependent on the misprediction need to be squashed [22]. At least three things must be done to exploit control independence in the trace processor. First, only those instructions fetched from the wrong path must be replaced. Second, although not all instructions are necessarily replaced, those that remain may still have to reissue because of changes in register dependences. Third, stores on the wrong path must "undo" their speculative state in the memory system. Trace re-predict sequences are used for selective control squashes. After detecting a control misprediction within a trace, traces in subsequent PEs are not automatically squashed. Instead, the frontend re-predicts and re- dispatches traces. The resident trace id is checked against the re-predicted trace id; if there is a partial (i.e. common prefix) or total match, only instructions beyond the match need to be replaced. For those not replaced, register dependences may have changed. So the global register names of each instruction in the resident trace are checked against those in the new trace; instructions that differ pick up the new names. Reissuing will follow from the existing issue mechanism. This approach treats instructions just like data values in that they are individually "validated". If a store is removed from the window and it has already performed, it must first issue to memory again, but only an undo transaction is performed as described in the previous section. Loads that were false-dependent on the store will snoop the store and thus reissue. Removing or adding loads/stores to the window does not cause sequence number problems if sequence numbering is based on {PE #, buffer #}. 3. Simulation environment Detailed simulation is used to evaluate the performance of trace processors. For comparison, superscalar processors are also simulated. The simulator was developed using the simplescalar simulation platform [23]. This platform uses a MIPS-like instruction set (no delayed branches) and comes with a gcc-based compiler to create binaries. Table 2. Fixed parameters and benchmarks. Our primary simulator uses a hybrid trace-driven and execution-driven approach. The control flow of the simulator is trace-driven. A functional simulator generates the true dynamic instruction stream, and this stream feeds the processor simulator. The processor does not explicitly fetch instructions down the wrong path due to control mis- speculation. The data flow of the simulator is completely latency 2 cycles (fetch trace predictor see Section 2.1.4 value predictor see Section 2.2 trace cache total traces = 2048 trace line size = 16 instructions branch pred. predictor = 64k 2-bit sat counters no tags, 1-bit hyst. instr. cache line instructions 2-way interleaved miss global phys regs unlimited functional units n symmetric, fully-pipelined FUs (for n-way issue) memory unlimited speculative store buffering D$ line size = 64 bytes unlimited outstanding misses exec. latencies address generation memory integer ALU operations latencies validation latency *Compress was modified to make only a single pass. benchmark input dataset instr count compress * 400000 e 2231 104 million gcc -O3 genrecog.i 117 million go 9 9 133 million ijpeg vigo.ppm 166 million queens 7 202 million execution-driven. This is essential for accurately portraying the data misspeculation model. For example, instructions reissue due to receiving new values, loads may pollute the data cache (or prefetch) with wrong addresses, extra bandwidth demand is observed on result buses, etc. As stated above, the default control sequencing model is that control mispredictions cause no new traces to be brought into the processor until resolved. A more aggressive control flow model is investigated in Section 6. To accurately measure selective control squashing, a fully execution-driven simulator was developed - it is considerably slower than the hybrid approach and so is only applied in Section 6. The simulator faithfully models the frontend, PE, and memory system depicted in Figures 2, 4, and 5, respec- tively. Model parameters that are invariant for simulations are shown in Table 2. The table also lists the five SPEC95 integer benchmarks used, along with input datasets and dynamic instruction counts for the full runs. 4. Primary performance results In this section, performance for both trace processors and conventional superscalar processors is presented, without data prediction. The only difference between the superscalar simulator and the trace processor simulator is that superscalar has a centralized execution engine. All other hardware such as the frontend and memory system are identical. Thus, superscalar has the benefit of the trace predictor, trace cache, reduced rename complexity, and selective reissuing due to memory dependence violations. The experiments (Table 3) focus on three parameters: window size, issue width, and global result bypass latency. Trace processors with 4, 8, and 16 PEs are simulated. Each PE can hold a trace of 16 instructions. Conventional superscalar processors with window sizes ranging from 16 to 256 instructions are simulated. Curves are labeled with the model name - T for trace and SS for superscalar - followed by the total window size. Points on the same curve represent varying issue widths; in the case of trace proces- sors, this is the aggregate issue width. Trace processor curves come in pairs - one assumes no extra latency (0) for bypassing values between processing elements, and the other assumes one extra cycle (1). Superscalar is not penalized - all results are bypassed as if they are locals. Fetch bandwidth, local and global result buses, and cache buses are chosen to be commensurate with the configura- tion's issue width and window size. Note that the window size refers to all in-flight instructions, including those that have completed but not yet retired. The retire width equals the issue width for superscalar; an entire trace can be retired in the trace processor. From the graphs in Figure 6, the first encouraging result is that all benchmarks show ILP that increases nicely with window size and issue bandwidth, for both processor models. Except for compress and go, which exhibit poor control prediction accuracy, absolute IPC is also encouraging. For example, large trace processors average 3.0 to 3.7 instructions per cycle for gcc. The extra cycle for transferring global values has a noticeable performance impact, on the order of 5% to 10%. Also notice crossover points in the trace processor curves. For example, "T-64 2-way per PE" performs better than "T-128 1-way per PE". At low issue widths, it is better to augment issue capability than add more PEs. Superscalar versus Trace Processors One way to compare the two processors is to fix total window size and total issue width. That is, if we have a centralized instruction window, what happens when we divide the window into equal partitions and dedicate an equal slice of the issue bandwidth to each partition? This question focuses on the effect of load balance. Because of load balance, IPC for the trace processor can only approach that of the superscalar processor. For example, consider two points from the gcc, jpeg, and xlisp graphs: "T-128 (0) 2-way per PE" and "SS-128 16-way". The IPC performance differs by 16% to 19% - the effect of load balance. (Also, in the trace processor, instruction buffers are underutilized due to small traces, and instruction buffers are freed in discrete chunks.) The above comparison is rather arbitrary because it suggests an equivalence based on total issue width. In reality, total issue width lacks meaning in the context of trace processors. What we really need is a comparison method based on equivalent complexity, i.e. equal clock cycle. One measure of complexity is issue complexity, which goes as the product of window size and issue width [15]. With this equivalence measure, comparing the two previous datapoints is invalid because the superscalar processor is much more complex (128x16 versus 16x2). Unfortunately, there is not one measure of processor complexity. So instead we take an approach that demonstrates the philosophy of next generation processors: 1. Take a small-scale superscalar processor and maximize its performance. 2. Use this highly-optimized processor and replicate it, taking advantage of a hierarchical organization. In other words, the goal is to increase IPC while keeping clock cycle optimal and constant. The last graph in Figure 6 interprets data for gcc with this philosophy. Suppose we start with a superscalar processor with a instruction window and 1, 2, or 4-way issue as a basic building block, and successively add more copies to form a trace processor. Assume that the only penalty for having more than one PE is the extra cycle to bypass values between PEs; this might account for global result bus Figure 6. Trace processor and superscalar processor IPC. Note that the bottom-right graph is derived from the adjacent graph, as indicated by the arrow; it interprets the same data in a different way. Table 3. Experiments. PE window size -or- fetch/dispatch b/w number of PEs 4 8 issue b/w per PE total issue b/w 4 8 local result buses global result buses 4 4 global buses that can be used by a cache buses that can be used by a IPC total issue width T-128 (1) IPC total issue width T-128 (1) IPC total issue width T-128 (1) IPC total issue width T-128 (1) IPC total issue width T-128 (1) IPC number of PEs GCC 1-way issue 2-way issue 4-way issue arbitration, global bypass latency, and extra wakeup logic for snooping global result tag buses. One might then roughly argue that complexity, i.e. cycle time, remains relatively constant with successively more PEs. For gcc, 4- way issue per PE, IPC progressively improves by 58% (1 to 4 PEs), 19% (4 to 8 PEs), and 12% (8 to 16 PEs). 5. Adding structured value prediction This section presents actual and potential performance results for a trace processor configuration using data prediction. We chose a configuration with 8 PEs, each having 4-way issue, and 1 extra cycle for bypassing values over the global result buses. The experiments explore real and perfect value pre- diction, confidence, and timing of value predictor updates. There are 7 bars for each benchmark in Figure 7. The first four bars are for real value prediction, and are labeled R/*/ *, the first R denoting real prediction. The second qualifier denotes the confidence model: R (real confidence) says we use predictions that are marked confident by the predictor, O (oracle confidence) says we only use a value if it is correctly predicted. The third qualifier denotes slow (S) or immediate (I) updates of the predictor. The last three bars in the graph are for perfect value/no address prediction (PV), perfect address/no value prediction (PA), and perfect value and address prediction (P). Figure 7. Performance with data prediction. From the rightmost bar (perfect prediction), the potential performance improvement for data prediction is signif- icant, around 45% for all of the benchmarks. Three of the benchmarks benefit twice as much from address prediction than from value prediction, as shown by the PA/PV bars. Despite data prediction's potential, only two of the benchmarks - gcc and xlisp - show noticeable actual improvement, about 10%. However, keep in mind that data value prediction is at a stage where significant engineering remains to be done. There is still much to be explored in the predictor design space. Although gcc and xlisp show good improvement, it is less than a quarter of the potential improvement. For gcc, the confidence mechanism is not at fault; oracle confidence only makes up for about 7% of the difference. Xlisp on the other hand shows that with oracle confidence, over half the potential improvement is achievable. Unfortu- nately, xlisp performs poorly in terms of letting incorrect predictions pass as confident. The first graph in Figure 8 shows the number of instruction squashes as a fraction of dynamic instruction count. The first two bars are without value prediction, the last two bars are with value prediction (denoted by V). The first two bars show the number of loads squashed by stores (dependence misspeculation) and the total number of squashes that result due to a cascade of reissued instruc- tions. Live-in and address misspeculation add to these totals in the last two bars. Xlisp's 30% reissue rate explains why it shows less performance improvement than gcc despite higher accuracy. The second graph shows the distribution of the number of times an instruction issues while it is in the window. Figure 8. Statistics for selective reissuing. 6. Aggressive control flow This section evaluates the performance of a trace processor capable of exploiting control independence. Only instructions that are control dependent on a branch misprediction are squashed, and instructions whose register dependences change are selectively reissued as described in Section 2.4.3. Accurate measurement of this control flow model requires fetching instructions down wrong paths, primarily to capture data dependences that may exist on such paths. For this reason we use a fully execution-driven simulator in this section. For a trace processor with 16 PEs, 4-way issue per PE, two of the benchmarks show a significant improvement in IPC: compress (13%) and jpeg (9%). These benchmarks frequently traverse small loops containing simple, reconvergent control flow. Also important are small loops with a few and fixed number of iterations, allowing the processor to capture traces beyond the loop. Value Prediction Performance Results 5.0% 10.0% 15.0% 20.0% 30.0% 40.0% 45.0% 50.0% compress gcc go jpeg xlisp benchmark improvement over base IPC R/R/S R/R/I R/O/S R/O/I PA 10.0% 20.0% 30.0% 40.0% 50.0% 70.0% 80.0% 90.0% 100.0% number of times issued fraction of dynamic instr jpeg compress go gcc xlisp 0% 5% 10% 15% 20% 30% comp gcc go jpeg xlisp instruction squash rate load squash total load squash (V) total (V) 7. Conclusion Trace processors exploit the characteristics of traces to efficiently issue many instructions per cycle. Trace data characteristics - local versus global values - suggest distributing execution resources at the trace level as a way to overcome complexity limitations. They also suggest an interesting application of value prediction, namely prediction of inter-trace dependences. Further, treating traces as the unit of control prediction results in an efficient, high accuracy control prediction model. An initial evaluation of trace processors without value prediction shows encouraging absolute IPC values - e.g. gcc between 3 and 4 - reaffirming that ILP can be exploited in large programs with complex control flow. We have isolated the performance impact of distributing execution resources based on trace boundaries, and demonstrated the overall performance value of replicating fast, small-scale ILP processors in a hierarchy. Trace processors with structured value prediction show promise. Although only two of the benchmarks show noticeable performance improvement, the potential improvement is substantial for all benchmarks, and we feel good engineering of value prediction and confidence mechanisms will increase the gains. With the pervasiveness of speculation in next generation processors, misspeculation handling becomes an important issue. Rather than treating mispredictions as an afterthought of speculation, we discussed how data misspeculation can be incorporated into the existing issue mechanism. We also discussed mechanisms for exploiting control independence, and showed that sequential programs may benefit. Acknowledgments This work was supported in part by NSF Grant MIP- 9505853 and by the U.S. Army Intelligence Center and Fort Huachuca under Contract DABT63-95-C-0127 and ARPA order no. D346. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either express or implied, of the U.S. Army Intelligence Center and Fort Huachuca, or the U.S. Government. This work is also supported by a Graduate Fellowship from IBM. --R Trace cache: A low latency approach to high bandwidth instruction fetching. Critical issues regarding the trace cache fetch mechanism. Improving superscalar instruction dispatch and issue by exploiting dynamic code sequenc- es Facilitating superscalar processing via a combined static/dynamic register renaming scheme. Value Locality and Speculative Execution. The performance potential of data dependence speculation and collapsing. The Multiscalar Architecture. Multiscalar pro- cessors Control flow speculation in multiscalar processors. Dynamic speculation and synchronization of data dependences. The anatomy of the register file in a multiscalar processor. Data memory alternatives for multiscalar processors. The 21264: A superscalar alpha processor with out- of-order execution Hardware support for large atomic units in dynamically scheduled machines. Exploiting instruction level parallelism in processors by caching scheduled groups. Increasing the instruction fetch rate via block-structured instruction set ar- chitectures The predictability of data values. Assigning confidence to conditional branch predictions. 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Dally, Tradeoff between data-, instruction-, and thread-level parallelism in stream processors, Proceedings of the 21st annual international conference on Supercomputing, June 17-21, 2007, Seattle, Washington Roni Rosner , Micha Moffie , Yiannakis Sazeides , Ronny Ronen, Selecting long atomic traces for high coverage, Proceedings of the 17th annual international conference on Supercomputing, June 23-26, 2003, San Francisco, CA, USA Balasubramonian , Sandhya Dwarkadas , David H. Albonesi, Dynamically managing the communication-parallelism trade-off in future clustered processors, ACM SIGARCH Computer Architecture News, v.31 n.2, May J. Gregory Steffan , Christopher Colohan , Antonia Zhai , Todd C. Mowry, The STAMPede approach to thread-level speculation, ACM Transactions on Computer Systems (TOCS), v.23 n.3, p.253-300, August 2005 A. Mahjur , A. H. Jahangir , A. H. 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trace cache;selective reissuing;context-based value prediction;next trace prediction;trace processors;multiscalar processors

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Out-of-order vector architectures.

Register renaming and out-of-order instruction issue are now commonly used in superscalar processors. These techniques can also be used to significant advantage in vector processors, as this paper shows. Performance is improved and available memory bandwidth is used more effectively. Using a trace driven simulation we compare a conventional vector implementation, based on the Convex C3400, with an out-of-order, register renaming, vector implementation. When the number of physical registers is above 12, out-of-order execution coupled with register renaming provides a speedup of 1.24--1.72 for realistic memory latencies. Out-of-order techniques also tolerate main memory latencies of 100 cycles with a performance degradation less than 6%. The mechanisms used for register renaming and out-of-order issue can be used to support precise interrupts -- generally a difficult problem in vector machines. When precise interrupts are implemented, there is typically less than a 10% degradation in performance. A new technique based on register renaming is targeted at dynamically eliminating spill code; this technique is shown to provide an extra speedup ranging between 1.10 and 1.20 while reducing total memory traffic by an average of 15--20%.

Introduction Vector architectures have been used for many years for high performance numerical applications - an area where they still excel. The first vector machines were supercomputers using memory-to-memory operation, but vector machines only became commercially successful with the addition of vector registers in the [12]. Following the Cray-1, a number of vector machines have been designed and sold, from supercomputers with very high vector bandwidths [8] to more modest mini-supercomputers. More recently, This work was supported by the Ministry of Education of Spain under contract 0429/95, by CIRIT grant BEAI96/II/124 and by the CEPBA. y This work was supported in part by NSF Grant MIP- 9505853. the value of vector architectures for desktop applications is being recognized. In particular, many DSP and multimedia applications - graphics, compression, encryption - are very well suited for vector implementation [1]. Also, research focusing on new processor-memory organizations, such as IRAM [10], would also benefit from vector technology. Studies in recent years [13, 5, 11], however, have shown performance achieved by vector architectures on real programs falls short of what should be achieved by considering available hardware resources. Functional unit hazards and conflicts in the vector register file can make vector processors stall for long periods of time and result in latency problems similar to those in scalar processors. Each time a vector processor stalls and the memory port becomes idle, memory band-width goes unused. Furthermore, latency tolerance properties of vectors are lost: the first load instruction at the idle memory port exposes the full memory latency. These results suggest a need to improve the memory performance in vector architectures. Unfortunately, typical hardware techniques used in scalar processors to improve memory usage and reduce memory latency have not always been useful in vector architectures. For example, data caches have been studied [9, 6]; however, the results are mixed, with performance gain or loss depending on working set sizes and the fraction of non-unit stride memory access. Data caches have not been put into widespread use in vector processors (except to cache scalar data). Dynamic instruction issue is the preferred solution in scalar processors to attack the memory latency problem by allowing memory reference instructions to proceed when other instructions are waiting for memory data. That is, memory reference instructions are allowed to slip ahead of execution instructions. Vector processors have not generally used dynamic instruction issue (except in one recent design, the NEC SX-4 [14]). The reasons are unclear. Perhaps it has been thought that the inherent latency hiding advantages of vectors are sufficient. Or, it is possibly because the first successful vector machine, the Cray- issued instructions in order, and additional innovations in vector instruction issue were simply not pursued Besides in-order vector instruction issue, traditional vector machines have had a relatively small number of vector registers (8 is typical). The limited number of vector registers was initially the result of hardware costs when vector register instruction sets were originally being developed; today the small number of registers is generally recognized as a shortcoming. Register renaming, useful for out-of-order issue, can come to the rescue here as well. With register renaming more physical registers are made available, and vector register conflicts are reduced. Another feature of traditional vector machines is that they have not supported virtual memory - at least not in the fully flexible manner of most modern scalar processors. The primary reason is the difficulty of implementing precise interrupts for page faults - a difficulty that arises from the very high level of concurrency in vector machines. Once again, features for implementingdynamic instruction issue for scalars can be easily adapted to vectors. Register renaming and reorder buffers allow relatively easy recovery of state information after a fault condition has occurred. In this paper, we show that using out-of-order issue and register renaming techniques in a vector pro- cessor, performance can be greatly improved. Dynamic instruction scheduling allows memory latencies to be overlapped more completely - and uses the valuable memory resource more efficiently in the process. Moreover, once renaming has been introduced into the architecture, it enables straightforward implementations of precise exceptions, which in turn provide an easy way of introducing virtual memory, without much extra hardware and without incurring a great performance penalty. We also present a new technique aimed at dynamically eliminating redundant loads. Using this technique, memory traffic can be significantly reduced and performance is further increased. Vector Architectures and Implementation This study is based on a traditional vector processor and numerical applications, primarily because of the maturity of compilers and the availability of benchmarks and simulation tools. We feel that the general conclusions will extend to other vector applications, however. The renaming, out-of-order vector architecture we propose is modeled after a Convex C3400. In this section we describe the base C3400 architecture and implementation (henceforth, the reference archi- tecture), and the dynamic out-of-order vector architecture (referred to as OOOVA). 2.1 The C3400 Reference Architecture The Convex C3400 consists of a scalar unit and an independent vector unit. The scalar unit executes all instructions that involve scalar registers isters), and issues a maximum of one instruction per cycle. The vector unit consists of two computation units (FU1 and FU2) and one memory accessing unit Fetch Decode&Rename @ unit Reorder Buffer released regs S-regs A-regs V-regs mask-regs Figure 1: The Out-of-order and renaming version of the reference vector architecture. (MEM). The FU2 unit is a general purpose arithmetic unit capable of executing all vector instructions. The FU1 unit is a restricted functional unit that executes all vector instructions except multiplication, division and square root. Both functional units are fully pipelined. The vector unit has 8 vector registers which hold up to 128 elements of 64 bits each. The eight vector registers are connected to the functional units through a restricted crossbar. Pairs of vector registers are grouped in a register bank and share two read ports and one write port that links them to the functional units. The compiler is responsible for scheduling vector instructions and allocating vector registers so that no port conflicts arise. The reference machine implements vector chaining from functional units to other functional units and to the store unit. It does not chain memory loads to functional units, however. 2.2 The Dynamic Out-of-Order Vector Architecture (OOOVA) The out-of-order and renaming version of the reference architecture, OOOVA, is shown in figure 1. It is derived from the reference architecture by applying a renaming technique very similar to that found in the R10000 [16]. Instructions flow in-order through the Fetch and Decode/Rename stages and then go to one of the four queues present in the architecture based on instruction type. At the rename stage, a mapping table translates each virtual register into a physical register. There are 4 independent mapping tables, one for each type of register: A, S, V and mask registers. Each mapping table has its own associated list of free registers. When instructions are accepted into the decode stage, a slot in the reorder buffer is also allocated. Instructions enter and exit the reorder buffer in strict program order. When an instruction defines a new logical register, a physical register is taken from the Issue RF ALU Wb A-regs Issue RF S-regs Issue RF V-regs Rename Fetch Issue RF MEM . @ Range Calculation Dependency Calculation Figure 2: The Out-of-order and renaming main instruction pipelines. list, the mapping table entry for the logical register is updated with the new physical register number and the old mapping is stored in the reorder buffer slot allocated to the instruction. When the instruction commits, the old physical register is returned to its free list. Note that the reorder buffer only holds a few bits to identify instructions and register names; it never holds register values. Main Pipelines There are four main pipelines in the OOOVA architecture (see fig. 2), one for each type of instruction. After decoding and renaming, instructions wait in the four queues shown in fig. 1. The A, S and V queues monitor the ready status of all instructions held in the queue slots and as soon as an instruction is ready, it is sent to the appropriate functional unit for execution. Processing of instructions in the M queue proceeds in two phases. First, instructions proceed in-order through a 3 stage pipeline comprising the Issue/Rf stage, the range stage and the dependence stage. After they have completed these three steps, memory instructions can proceed out of order based on dependence information computed and operand availability (for stores). At the Range stage, the range of all addresses potentially modified by a memory instruction is com- puted. This range is used in the following stage for run-time memory disambiguation. The range is defined as all bytes falling between the base address (called Range Start) and the address defined as (called Range End), where V L is the vector length register and V S is the vector stride register. Note that the multiplier can be simplified because V L \Gamma 1 is short (never more than 7 bits), and the product (V can be kept in a non-architected register and implicitly updated when either VL or VS is modified. In the Dependence stage, using the Range Start/Range End addresses, the memory instruction is compared against all previous instructions found in the queue. Once a memory instruction is free of any dependences, it can proceed to issue memory requests. Machine Parameters Table 1 presents the latencies of the various functional units present in the architecture. Memory latency is not shown in the table because it will be varied. The memory system is modeled as follows. There is a single address bus shared by all types of memory trans- Parameters Latency Scal Vect (int/fp) (int/fp) read x-bar - 2 vector startup - (*) add 1/2 1/2 mul 5/2 5/2 logic/shift 1/2 1/2 div 34/9 34/9 sqrt 34/9 34/9 Table 1: Functional unit latencies (in cycles) for the two architectures.((*) 0 in OOOVA, 1 in REF) actions (scalar/vector and load/store), and physically separate data busses for sending and receiving data to/from main memory. Vector load instructions (and gather instructions) pay an initial latency and then receive one datum from memory per cycle. Vector store instructions do not result in observed latency We use a value of 50 cycles as the default memory latency. Section 4.3 will present results on the effects of varying this value. The V register read/write ports have been modified from the original C34 scheme. In the OOOVA, each vector register has 1 dedicated read port and 1 dedicated port. The original banking scheme of the register file can not be kept because renaming shuffles all the compiler scheduled read/write ports and, therefore, would induce a lot of port conflicts. All instruction queues are set at 16 slots. The reorder buffer can hold 64 instructions. The machine has a 64 entry BTB, where each entry has a 2-bit saturating counter for predicting the outcome of branches. Also, an 8-deep return stack is used to predict call/return sequences. Both scalar register files and S) have 64 physical registers each. The mask register file has 8 physical registers. The fetch stage, the decode stage and all four queues only process a maximum of 1 instruction per cycle. Committing instructions proceeds at a faster rate, and up to 4 instructions may commit per cycle. Commit Strategy For V registers we start with an aggressive implementation where physical registers are released at the time the vector instruction begins execution. Consider the vector instruction: add v0,v1-?v3. At the rename stage, v3 will be re-mapped to, say, physical register 9 (ph9), and the old mapping of v3, which was, say, physical register 12 (ph12), will be stored in the re-order buffer slot associated with the add instruction. When the add instruction begins execution, we mark the associated reorder buffer slot as ready to be com- mitted. When the slot reaches the head of the buffer, ph12 is released. Due to the semantics of a vector register, when ph12 is released, it is guaranteed that all instructions needing ph12 have begun execution at least one cycle before. Thus, the first element of ph12 is already flowing through the register file read cross- bar. Even if ph12 is immediately reassigned to a new logical register and some other instruction starts writ- #insns #ops % avg. Program Suite S V V Vect VL hydro2d Spec 41.5 39.2 3973.8 99.0 101 arc2d Perf. 63.3 42.9 4086.5 98.5 95 flo52 Perf. 37.7 22.8 1242.0 97.1 54 su2cor Spec 152.6 26.8 3356.8 95.7 125 bdna Perf. 239.0 19.6 1589.9 86.9 81 trfd Perf. 352.2 49.5 1095.3 75.7 22 dyfesm Perf. 236.1 33.0 696.2 74.7 21 Table 2: Basic operation counts for the Perfect Club and Specfp92 programs (Columns 3-5 are in millions). ing into ph12, the instructions reading ph12 are at the very least one cycle ahead and will always read the correct values. This type of releasing does not allow for precise exceptions, though. Section 5 will change the release algorithm to allow for precise exceptions. To assess the performance benefits of out-of-order issue and renaming in vector architectures we have taken a trace driven approach. A subset of the Perfect Club and Specfp92 programs is used as the benchmark set. These programs are compiled on a Convex C3480 machine and the tool Dixie [3] is used to modify the executable for tracing. Once the executables have been processed by Dixie, the modified executables are run on the Convex machine. This runs produce the desired set of traces that accurately represent the execution of the programs. This trace is then fed to two simulators for the reference and OOOVA architectures. 3.1 The benchmark programs Because we are interested in the benefits of out-of- order issue for vector instructions, we selected benchmark programs that are highly vectorizable. From all programs in the Perfect and Specfp92 benchmarks we chose the 10 programs that achieve at least 70% vec- torization. Table 2 presents some statistics for the selected Perfect Club and Specfp92 programs. Column number 2 indicates to what suite each program belongs. Next two columns present the total number of instructions issued by the decode unit, broken down into scalar and vector instructions. Column five presents the number of operations performed by vector instructions. The sixth column is the percentage of vectorization of each program (i.e., column five divided by the sum of columns three and five). Finally, column seven presents the average vector length used by vector instructions (the ratio of columns five and four, respectively). hydro2d10003000 Execution cycles dyfesm5001500 Figure 3: Functional unit usage for the reference ar- chitecture. Each bar represents the total execution time of a program for a given latency. Values on the x-axis represent memory latencies in cycles. Performance Results 4.1 Bottlenecks in the Reference Archi- tecture First we present an analysis of the execution of the ten benchmark programs when run through the reference architecture simulator. Consider the three vector functional units of the reference architecture (FU2, FU1 and MEM). The machine state can be represented with a 3-tuple that captures the individual state of each of the three units at a given point in time. For example, the 3-tuple represents a state where all units are working, while represents a state where all vector units are idle. Figure 3 presents the execution time for two of the ten benchmark programs (see [4] for the other 8 pro- grams). Space limitations prevents us from providing them all, but these two, hydro2d and dyfesm, are rep- resentative. During an execution the programs are in eight possible states. We have plotted the time spent in each state for memory latencies of 1, 20, 70, and 100 cycles. From this figure we can see that the number of cycles where the programs proceed at peak floating point speed (states hFU2; FU1;MEM i and low. The number of cycles in these states changes relatively little as the memory latency increases, so the fraction of fully used cycles decreases. Memory latency has a high impact on total execution time for programs dyfesm (shown in Figure 3), and trfd and flo52 (not shown), which have relatively small vector lengths. The effect of memory latency can be seen by noting the increase in cycles spent in state h ; ; i. The sum of cycles corresponding to states where the MEM unit is idle is quite high in all programs. These four states correspond to cycles where the mem- swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm2060 Idle Memory port% 170 Figure 4: Percentage of cycles where the memory port was idle, for 4 different memory latencies. ory port could potentially be used to fetch data from memory for future vector computations. Figure 4 presents the percentage of these cycles over total execution time. At latency 70, the port idle time ranges between 30% and 65% of total execution time. All benchmark programs are memory bound when run on a single port vector machine with two functional units. Therefore, these unused memory cycles are not the result of a lack of load/store work to be done. 4.2 Performance of the OOOVA In this section we present the performance of the OOOVA and compare it with the reference archi- tecture. We consider both overall performance in speedup and memory port occupation. The effects of adding out-of-order execution and renaming to the reference architecture can be seen in figure 5. For each program we plot the speedup over the reference architecture when the number of physical vector registers is varied from 9 to 64 (memory latency is set at 50 cycles). In each graph, we show the speedup for two OOOVA implementations: "OOOVA- 16" has length 16 instruction queues, and "OOOVA- 128" has length 128 queues. We also show the maximum ideal speedup that can theoretically be achieved ("IDEAL", along the top of each graph). To compute the IDEAL speedup for a program we use the total number of cycles consumed by the most heavily used vector unit (FU1, FU2, or MEM). Thus, in IDEAL we essentially eliminate all data and memory dependences from the program, and consider performance limited only by the most saturated resource across the entire execution. As can be seen from figure 5, the OOOVA significantly increases performance over the reference ma- chine. With physical registers, the lowest speedup is 1.24 (for tomcatv). The highest speedups are for trfd and dyfesm (1.72 and 1.70 resp.); the remaining programs give speedups of 1.3-1.45. For numbers of physical registers greater than 16, additional speedups are generally small. The largest speedup from going to physical registers is for bdna where the additional improvement is 8.3%. The improvement in bdna is due to an extremely large main loop, which generates a sequence of basic blocks with more than 800 vector instructions. More physical registers allow it to better match the large available ILP in these basic blocks. On the other hand, if the number of physical vector registers is a major concern, we observe that 12 physical registers still give speedups of 1.63 and 1.70 for trfd and dyfesm and that the other programs are in the range of 1.23 to 1.38. These results suggest that a physical vector register with as few as 12 registers is sufficient in most cases. A file with 16 registers is enough to sustain high performance in every case. When we increase the depth of the instruction queues to 128, the performance improvement is quite small (curve "OOOVA-128"). Analysis of the programs shows that two factors combine to prevent further improvements when increasing the number of issue queue slots. First, the spill code present in large basic blocks induces a lot of memory conflicts in the memory queue. Second, the lack of scalar registers sometimes prevents the dynamic unrolling of enough iterations of a vector loop to make full usage of the memory port. Memory The out-of-order issue feature allows memory access instructions to slip ahead of computation instructions, resulting in a compaction of memory access opera- tions. The presence of fewer wasted memory cycles is shown in figure 6. This figure contains the number of cycles where the address port is idle divided by the total number of execution cycles. Bars for the reference machine, REF, and for the out-of-order machine, OOOVA are shown. The OOOVA machines has physical vector registers and a memory latency of 50 cycles. With OOOVA, the fraction of idle memory cycles is more than cut in half in most cases. For all but two of the benchmarks, the memory port is idle less than 20% of the time. Resource Usage We now consider resource usage for the OOOVA machine and compare it with the reference machine. This is illustrated in figure 7. The same notation as in figure 3 is used for representing the execution state. As in the previous subsections, the OOOVA machine has physical vector registers and memory latency is set at 50 cycles. Figure 7 shows that the major improvement is in state h ; ; i, which has almost disappeared. Also, the fully-utilized state, hFU2; FU1;MEM i, is relatively more frequent due to the benefits of out-of- order execution. As we have already seen, the availability of more than one memory instruction ready to be launched in the memory queues allows for much higher usage of the memory port. 4.3 Tolerance of Memory Latencies One way of looking at the advantage of out-of-order execution and register renaming is that it allows long 1.11.3 9 arc2d1.29 flo521.21.6 9 nasa71.21.6 9 trfd1.5Speedup 9 IDEAL OOOVA-128 Figure 5: Speedup of the OOOVA over the REF architecture for different numbers of vector physical registers. swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm103050 Idle Memory REF OOOVA Figure Percentage of idle cycles in the memory port for the Reference architecture and the OOOVA archi- tecture. Memory latency is 50 cycles and the vector register file holds physical vector registers. memory latencies to be hidden. In previous subsections we showed the benefits of the OOOVA with a fixed memory latency of 50 cycles. In this subsection we consider the ability of the OOOVA machine to tolerate main memory latencies. Figure 8 shows the total execution time for the ten programs when executed on the reference machine and on the OOOVA machine for memory latencies of 1, 50, and 100 cycles. All results are for 16 physical vector registers. As shown in the figure, the reference machine is very sensitive to memory latency. Even though it is a vector machine, memory latency influences execution time considerably. On the other hand, the OOOVA machine is much more tolerant of the in- hydro2d dyfesm51525 Execution cycles Figure 7: Breakdown of the execution cycles for the REF (left bar) and OOOVA (right bar) machines. The OOOVA machine has 16 physical vector registers. For both architectures, memory latency was set at 50 cycles crease in memory latency. For most benchmarks the performance is flat for the entire range of memory la- tencies, from 1 to 100 cycles. Another important point is that even at a memory latency of 1 cycle the OOOVA machine typically obtains speedups over the reference machine in the range of 1.15-1.25 (and goes as high as 1.5 in the case of dyfesm). This speedup indicates that the effects of looking ahead in the instruction stream are good even in the absence of long latency memory operations. At the other end of the scale, we see that long memory latencies can be easily tolerated using out- of-order techniques. This indicates that the individ- 5cycles x cycles x trfd15cycles x dyfesm10REF IDEAL Figure 8: Effects of varying main memory latency for three memory models and for the 16 physical vector registers machines. ual memory modules in the memory system can be slowed down (changing very expensive SRAM parts for much cheaper DRAM parts) without significantly degrading total throughput. This type of technology change could have a major impact on the total cost of the machine, which is typically dominated by the cost of the memory subsystem. 5 Implementing Precise Traps An important side effect of introducing register renaming into a vector architecture is that it enables a straightforward implementation of precise exceptions. In turn, the availability of precise exceptions allows the introduction of virtual memory. Virtual memory has been implemented in vector machines [15], but is not used in many current high performance parallel vector processors [7]. Or, it is used in a very restricted form, for example by locking pages containing vector data in memory while a vector program executes [7, 14]. The primary problem with implementing precise page faults in a high performance vector machine is the high number of overlapped "in-flight" operations - in some machines there may be several hundred. Vector register renaming provides a convenient means for saving the large amount of machine state required for rollback to a precise state following a page fault or other exception. If the contents of old logical vector registers are kept until an instruction overwriting the logical register is known to be free of exceptions, then the architected state can be restored if needed. In order to implement precise traps, we introduce two changes to the OOOVA design: first, an instruction is allowed to commit only after it has fully completed (as opposed to the "early" commit scheme we have been using). Second, stores are only allowed to execute and update memorywhen they are at the head of the reorder buffer; that is, when they are the oldest uncommitted instructions. Figure 9 presents a comparison of the speedups over the reference architecture achieved by the OOOVA with early commit (labeled "early"), and by the OOOVA with late commit and execution of stores only at the head of the reorder buffer (labeled "late"). Again, all simulations are performed with a memory latency of 50 cycles. We can make two important observations about the graphs in Figure 9. First, the performance degradation due to the introduction of the late commit model is small for eight out of the ten programs. Programs hydro2d, arc2d, su2cor, tomcatv and bdna all degrade less than 5% with physical registers; programs flo52 and nasa7 degrade by 7% and 10.3%, respectively. Nevertheless, performance of the other two programs, trfd and dyfesm, is hurt rather severely when going to the late commit model (a 41% and 47% degradation, respectively). This behavior is explained by load-store dependences. The main loop in trfd has a memory dependence between the last vector store of iteration i and the first vector load of iteration are to the same address). In the early commit model, the store is done as soon as its input data is ready (with chaining between the producer and the store). In the late commit model, the store must wait until 2 intervening instructions between the producer and the store have committed. This delays the dispatching of the following load from the first iteration and explains the high slowdown. A similar situation explains the degradation in dyfesm. Second, in the late commit model, 12 registers are 1.11.3 9 arc2d1.29 flo521.21.6 9 nasa71.21.6 9 trfd1.5Speedup 9 IDEAL late Figure 9: Speedups of the OOOVA over the reference architecture for different numbers of vector physical registers under the early and late commit schemes. clearly not enough. The performance difference between 12 and 16 registers is much larger than in the early commit model. Thus, from a cost/complexity point of view, the introduction of late commit has a clear impact on the implementation of the vector registers 6 Dynamic Load Elimination Register renaming with many physical registers solves instruction issue bottlenecks caused by a limited number of logical registers. However, there is another problem caused by limited logical registers: register spilling. The original compiled code still contains register spills caused by the limited number of architected registers, and to be functionally correct these spills must be executed. Furthermore, besides the obvious store-load spills, limited registers also cause repeated loads from the same memory location. Limited registers are common in vector architec- tures, and the spill problem is aggravated because storing and re-loading a single vector register involves the movement of many words of data to and from memory. To illustrate the importance of spill code for vector ar- chitectures, table 3 shows the number of memory spill operations (number of words moved) in the ten benchmark programs. In some of the benchmarks relatively few of the loads and stores are due to spills, but in several there is a large amount of spill traffic. For ex- ample, over 69% of the memory traffic in bdna is due to spills. In this section we propose and study a method that uses register renaming to eliminate much of the memory load traffic due to spills. The method we propose also has significant performance advantages because a Vector load ops Vector store ops Total Program load spill % store spill % % hydro2d 1297 21 1.6 431 21 5 2.4 arc2d 1244 122 9 479 87 15 11 nasa7 1048 21 2.0 632 20 3 2.4 su2cor 786 201 20 404 103 20 20 bdna 142 266 Table 3: Vector memory spill operations. Columns 2, 3, 5 and 6 are in millions of operations. load for spilled data is executed in nearly zero time. We do not eliminate spill stores, however, because of the need to maintain strict binary compatibility. That is, the memory image should reflect functionally correct state. Relaxing compatibility could lead to removing some spill stores, but we have not yet pursued this approach. 6.1 Renaming under Dynamic Load Elim- ination To eliminate redundant load instructions we propose the following technique. A tag is associated with each physical register (A, S and V). This tag indicates the memory locations currently being held by the register. For vector registers, the tag is a 6-tuple: define a consecutive region of bytes in memory and vl, vs, and sz are the vector length, vector stride and access granularity used when the tag was created; v is a validity bit. For scalar registers, the tag is a 4-tuple - vl and vs are not needed. Although the problem of spilling scalar registers is somewhat tangential to our study, they are important in the Convex architecture because of its limited number of registers. Each time a memory operation is performed, its range of addresses is computed (this is done in the second stage of the memory pipeline). If the operation is a load, the tag associated with the destination physical register is filled with the appropriate address informa- tion. If the operation is a store, then the physical register being stored to memory has its tag updated with the corresponding address information. Thus, each time a memory operation is performed, we "alias" the register contents with the memory addresses used for loading or storing the physical register: the tag indicates an area in memory that matches the register data. To keep tag contents consistent with memory, when a store instruction is executed its tag has to be compared against all tags already present in the register files. If any conflict is found, that is, if the memory range defined by the store tag overlaps any of the existing tags, these existing tags must be invalidated (to simplify the conflict checking hardware, this invalidation may be done conservatively). By using the register tags, some vector load operations can be eliminated in the following manner. When a vector load enters the third stage of the memory pipeline, its tag is checked against all tags found in the vector register file. If an exact match is found (an exact match requires all tag fields to be identical), the destination register of the vector load is renamed to the physical register it matches. At this point the load has effectively been completed - in the time it takes to do the rename. Furthermore, matching is not restricted to live registers, it can also occur with a physical register that is on the free list. As long as the validity bit is set, any register (in the free list or in use) is eligible for matching. If a load matches a register in the free list, the register is taken from the list and added to the register map table. For scalar registers, eliminating loads is simpler. When a match involving two scalar registers is de- tected, the register value is copied from one register to the other. The scalar rename table is not affected. Note, however, that scalar store addresses still need to be compared against vector register tags and vector stores need to be compared against scalar tags to ensure full consistency. A similar memory tagging technique for scalar registers is described in [2]. There, tagging is used to store memory variables in registers in the face of potential aliasing problems. That approach, though, is complicated because data is automatically copied from register to register when a tag match is found. There- fore, compiler techniques are required to adapt to this implied data movement. In our application, a tag operation either (a) alters only the rename table or (b) invalidates a tag without changing any register value. Issue RF ALU Wb A-regs Issue RF S-regs Rename Fetch Issue RF Calculation @ Range Dependency Calculation V-regs V-RENAME RXBAR RXBAR EX0 Figure 10: The modified instruction pipelines for the Dynamic Load Elimination OOOVA. 6.2 Pipeline modifications With the scheme just described, when a vector load is eliminated at the disambiguation stage of the memory pipeline, the vector register renaming table is up- dated. Renaming is considerably complicated if vector registers are renamed in two different pipeline stages (at the decode and disambiguation stages). Therefore, the pipeline structure is modified to rename all vector registers in one and only one stage. Figure shows the modified pipeline. At the decode stage, all scalar registers are renamed but all vector registers are left untouched. Then, all instructions using a vector register pass in-order through the 3 stages of the memory pipeline. When they arrive at the disambiguation stage, renaming of vector registers is done. This ensures that all vector instruction see the same renaming table and that modifications introduced by the load elimination scheme are available to all following vector instructions. Moreover, this ensures that store tags are compared against all previous tags in order. 6.3 Performance of dynamic load elimina- tion In this section we present the performance of the OOOVA machine enhanced with dynamic load elimi- nation. As a baseline we use the late commit OOOVA described above, without dynamic load elimination. We also study the OOOVA with load elimination for scalar data only (SLE) and OOOVA with load elimination for both scalars and vectors (SLE+VLE). Figures 11 and 12 present the speedup of SLE and SLE+VLE over the baseline OOOVA for different numbers of physical vector registers (16, 32, 64). For SLE+VLE with 16 vector registers (figure 12), speedups over the base OOOVA are from 1.04 to 1.16 for most programs and are as high as 1.78 and 2.13 for dyfesm and trfd. At registers registers, the available storage space for keeping vector data doubles and allows more tag matchings. The speedups increase significantly and their range for most programs is between 1.10 and 1.20. For dyfesm and trfd, the speedups remain very high, but do not appreciably improve when going from 16 to registers. Doubling the number of vector registers again, to 64, does not yield much additional speedup. For most swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.11.3 over Figure 11: Speedup of SLE over the OOOVA machine for 3 different physical vector register file sizes. swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.5Speedup over Figure 12: Speedup of SLE+VLE over the OOOVA machine for 3 different physical vector register file sizes. programs, the improvement is below 5%, and only tomcatv and trfd seem to be able to take advantage of the extra registers (tomcatv goes from 1.19 up to 1.40). The results show that most of the data movement to be eliminated is captured with vector registers. The remarkably different performance behavior of dyfesm and trfd requires explanation. This can be done by looking at SLE (figure 11). Under SLE, all other programs have very low speedups (less than and, yet, trfd and dyfesm achieve speedups of and 1.36, respectively (for the configuration with vector registers). Our analysis of these two programs shows that the ability to bypass scalar data allows these programs to "see" more iterations of a certain loop at once. In particular, the ability to bypass data between loads and stores allows them to unroll the two most critical loops, whereas without SLE, the unrolling was not possible. 6.4 Traffic Reduction A very important effect of dynamic load elimination is that it reduces the total amount of traffic seen by the memory system. This is a very important feature swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm0.8Traffic Reduction SLE Figure 13: Traffic reduction under dynamic load elimination with physical vector registers. in multiprocessing environments, where less load on the memory modules usually translates into an overall system performance improvement. We have computed the traffic reduction of each of the programs for the two dynamic load elimination configurations considered. We define the traffic reduction as the ratio between the total number of requests (load and stores) sent over the address bus by the base-line OOOVA divided by the total number of requests done by either the SLE or the SLE+VLE configurations Figure 13 present this ratio for physical vector registers. As an example, figure 13 shows us that the SLE configuration for dyfesm performs 11% fewer memory requests than the OOOVA configuration. As can be seen, for SLE+VLE, the typical traffic reduction is between 15 and 20%. Programs dyfesm and trfd, due to their special behavior already men- tioned, have much larger reductions, as much as 40%. Summary In this paper we have considered the usefulness of out-of-order execution and register renaming for vector architectures. We have seen through simulation that the traditional in-order vector execution model is not enough to fully use the bandwidth of a single memory port and to cover up for main memory latency (even considering that the programs were memory bound). We have shown that when out-of-order issue and register renaming are introduced, vector performance is increased. This performance advantage can be realized even when adding only a few extra physical registers to be used for renaming. Out-of- order execution is as useful in a vector processor as it is widely recognized to be in current superscalar microprocessors Using only 12 physical vector registers and an aggressive commit model, we have shown significant speedups over the reference machine. At a modest cost of 16 vector registers, the range of speedups was 1.24-1.72. Increasing the number of vector registers up to 64 does not lead to significant extra improve- ments, however. Moreover, we have shown that large memory latencies of up to 100 cycles can be easily tolerated. The dynamic reordering of vector instructions and the disambiguation mechanisms introduced allow the memory unit to send a continuous flow of requests to the memory system. This flow is overlapped with the arrival of data and covers up main memory latency. The introduction of register renaming gives a powerful tool for implementing precise exceptions. By changing the aggressive commit model into a conservative model where an instruction only commits when it (and all its predecessors) are known to be free of exceptions, we can recover all the architectural state at any point in time. This allows the easy introduction of virtual memory. Our simulations have shown that the implementation of precise exceptions costs around 10% in application performance, though some programs may be much more sensitive than others. One problem not solved by register renaming is register spilling. The addition of extra physical registers, per se, does not reduce the amount of spilled data. We have introduced a new technique, dynamic load elimination, that uses the renaming mechanism to reduce the amount of load spill traffic. By tagging all our registers with memory information we can detect when a certain load is redundant and its required data is already in some other physical register. Under such conditions, the load can be performed through a simple rename table change. Our simulations have shown that this technique can further improve performance typically by factors of 1.07-1.16 (and as high as 1.78). The dynamic load elimination technique can benefit from more physical registers, since it can cache more data inside the vector register file. Simulations with physical vector registers show that load elimination yields improvements typically in the range 1.10-1.20. Moreover, at registers, load elimination can reduce the total traffic to the memory system by factors ranging between 15-20% and, in some cases, up to 40%. Finally, we feel that our results should be of use to the growing community of processor architectures implementing some kind of multimedia extensions. As graphics coprocessors and DSP functions are incorporated into general purpose microprocessors, the advantages of vector instruction sets will become more evident. In order to sustain high throughput to and from special purpose devices such as frame buffers, long memory latencies will have to be tolerated. These types of applications generally require high bandwidths between the chip and the memory system not available in current microprocessors. For both bandwidth and latency problems, out-of-order vector implementations can help achieve improved performance --R The T0 Vector Microprocessor. A new kind of memory for referencing arrays and pointers. Dixie: a trace generation system for the C3480. Decoupled vector architectures. Quantitative analysis of vector code. The performance impact of vector processor caches. The parallel processing feature of the NEC SX-3 supercomputer system Cache performance in vector supercomput- ers A Case for Intelligent DRAM: IRAM. Relationship between average and real memory behavior. The CRAY-1 computer system Explaining the gap between theoretical peak performance and real performance for super-computer architectures HNSX Supercomputers Inc. Architecture of the VPP500 Parallel Supercomputer. The Mips R10000 Superscalar Microprocessor. --TR CRegs: a new kind of memory for referencing arrays and pointers Distributed storage control unit for the Hitachi S-3800 multivector supercomputer Cache performance in vector supercomputers Architecture of the VPP500 parallel supercomputer Relationship between average and real memory behavior Explaining the gap between theoretical peak performance and real performance for supercomputer architectures The CRAY-1 computer system The MIPS R10000 Superscalar Microprocessor Decoupled vector architectures Quantitative analysis of vector code --CTR Roger Espasa , Mateo Valero, Exploiting Instruction- and Data-Level Parallelism, IEEE Micro, v.17 n.5, p.20-27, September 1997 Luis Villa , Roger Espasa , Mateo Valero, A performance study of out-of-order vector architectures and short registers, Proceedings of the 12th international conference on Supercomputing, p.37-44, July 1998, Melbourne, Australia Mark Hampton , Krste Asanovi, Implementing virtual memory in a vector processor with software restart markers, Proceedings of the 20th annual international conference on Supercomputing, June 28-July 01, 2006, Cairns, Queensland, Australia Christos Kozyrakis , David Patterson, Overcoming the limitations of conventional vector processors, ACM SIGARCH Computer Architecture News, v.31 n.2, May Francisca Quintana , Jesus Corbal , Roger Espasa , Mateo Valero, Adding a vector unit to a superscalar processor, Proceedings of the 13th international conference on Supercomputing, p.1-10, June 20-25, 1999, Rhodes, Greece Francisca Quintana , Jesus Corbal , Roger Espasa , Mateo Valero, A cost effective architecture for vectorizable numerical and multimedia applications, Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures, p.103-112, July 2001, Crete Island, Greece Karthikeyan Sankaralingam , Stephen W. Keckler , William R. Mark , Doug Burger, Universal Mechanisms for Data-Parallel Architectures, Proceedings of the 36th annual IEEE/ACM International Symposium on Microarchitecture, p.303, December 03-05, Christopher Batten , Ronny Krashinsky , Steve Gerding , Krste Asanovic, Cache Refill/Access Decoupling for Vector Machines, Proceedings of the 37th annual IEEE/ACM International Symposium on Microarchitecture, p.331-342, December 04-08, 2004, Portland, Oregon Banit Agrawal , Timothy Sherwood, Virtually Pipelined Network Memory, Proceedings of the 39th Annual IEEE/ACM International Symposium on Microarchitecture, p.197-207, December 09-13, 2006 Roger Espasa , Mateo Valero, A Simulation Study of Decoupled Vector Architectures, The Journal of Supercomputing, v.14 n.2, p.124-152, Sept. 1999 Christoforos Kozyrakis , David Patterson, Vector vs. superscalar and VLIW architectures for embedded multimedia benchmarks, Proceedings of the 35th annual ACM/IEEE international symposium on Microarchitecture, November 18-22, 2002, Istanbul, Turkey Roger Espasa , Mateo Valero , James E. Smith, Vector architectures: past, present and future, Proceedings of the 12th international conference on Supercomputing, p.425-432, July 1998, Melbourne, Australia Jesus Corbal , Roger Espasa , Mateo Valero, MOM: a matrix SIMD instruction set architecture for multimedia applications, Proceedings of the 1999 ACM/IEEE conference on Supercomputing (CDROM), p.15-es, November 14-19, 1999, Portland, Oregon, United States

memory latency;memory traffic elimination;vector architecture;register renaming;microarchitecture;precise interrupts;out-of-order execution

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Improving code density using compression techniques.

We propose a method for compressing programs in embedded processors where instruction memory size dominates cost. A post-compilation analyzer examines a program and replaces common sequences of instructions with a single instruction codeword. A microprocessor executes the compressed instruction sequences by fetching code words from the instruction memory, expanding them back to the original sequence of instructions in the decode stage, and issuing them to the execution stages. We apply our technique to the PowerPC, ARM, and i386 instruction sets and achieve an average size reduction of 39%, 34%, and 26%, respectively, for SPEC CINT95 programs.

Introduction According to a recent prediction by In-Stat Inc., the merchant processor market is set to exceed $60 billion by 1999, and nearly half of that will be for embedded processors. However, by unit count, embedded processors will exceed the number of general purpose microprocessors by a factor of 20. Compared to general purpose microprocessors, processors for embedded applications have been much less studied. The figures above suggest that they deserve more attention. Embedded processors are more highly constrained by cost, power, and size than general purpose microprocessors. For control oriented embedded applications, the most common type, a significant portion of the final circuitry is used for instruction memory. Since the cost of an integrated circuit is strongly related to die size, and memory size is proportional to die size, developers want their program to fit in the smallest memory possible. An additional pressure on program memory is the relatively recent adoption of high-level languages for embedded systems because of the need to control development costs. As typical code sizes have grown, these costs have ballooned at rates comparable to those seen in the desktop world. Thus, the ability to compress instruction code is important, even at the cost of execution speed. High performance systems are also impacted by program size due to the delays incurred by instruction cache misses. A study at Digital [Perl96] showed that an SQL server on a DEC 21064 Alpha, is bandwidth limited by a factor of two on instruction cache misses alone. This problem will only increase as the gap between processor performance and memory performance grows. Reducing program size is one way to reduce instruction cache misses and achieve higher performance [Chen97b]. This paper focuses on compression for embedded applications, where execution speed can be traded for compression. We borrow concepts from the field of text compression and apply them to the compression of instruction sequences. We propose modifications at the microarchitecture level to support compressed programs. A post-compilation analyzer examines a program and replaces common sequences of instructions with a single instruction codeword. A microprocessor executes the compressed instruction sequences by fetching codewords from the instruction memory, expanding them back to the original sequence of instructions in the decode stage, and issuing them to the execution stages. We demonstrate our technique by applying it to the PowerPC instruction set. 1.1 Code generation Compilers generate code using a Syntax Directed Translation Scheme (SDTS) [Aho86]. Syntactic source code patterns are mapped onto templates of instructions which implement the appropriate semantics. Consider, a simple schema to translate a subset of integer arithmetic: { { These patterns show syntactic fragments on the right hand side of the two productions which are replaced (or reduced) by a simpler syntactic structure. Two expressions which are added (or multiplied) together result in a single, new expression. The register numbers holding the operand expressions ($1 and $3) are encoded into the add (multiplication) operation and emitted into the generated object code. The result register ($1) is passed up the parse tree for use in the parent operation. These two patterns are reused for all arithmetic operations throughout program compilation More complex actions (such as translation of control structures) generate more instructions, albeit still driven by the template structure of the SDTS. In general, the only difference in instruction sequences for given source code fragments at different points in the object module are the register numbers in arithmetic instructions and operand offsets for load and store instructions. As a consequence, object modules are generated with many common sub-sequences of instructions. There is a high degree of redundancy in the encoding of the instructions in a program. In the programs we examined, only a small number of instructions had bit pattern encodings that were not repeated elsewhere in the same program. Indeed, we found that a small number of instruction encodings are highly reused in most programs. To illustrate the redundancy of instruction encodings, we profiled the SPEC CINT95 benchmarks [SPEC95]. The benchmarks were compiled for PowerPC with GCC 2.7.2 using -O2 opti- mization. Figure 1 shows that compiled programs consist of many instructions that have identical encodings. On average, less than 20% of the instructions in the benchmarks have bit pattern encodings which are used exactly once in the program. In the go benchmark, for example, 1% of the most frequent instruction words account for 30% of the program size, and 10% of the most frequent instruction words account for 66% of the program size. It is clear that the redundancy of instruction encodings provides a great opportunity for reducing program size through compression techniques. 1.2 Overview of compression method Our compression method finds sequences of instruction bytes that are frequently repeated throughout a single program and replaces the entire sequence with a single codeword. All rewritten (or encoded) sequences of instructions are kept in a dictionary which, in turn, is used at program execution time to expand the singleton codewords in the instruction stream back into the original sequence of instructions. All codewords assigned by the compression algorithm are merely indices into the instruction dictionary. The final compressed program consists of codewords interspersed with uncompressed instructions Figure 2 illustrates the relationship between the uncompressed code, the compressed code, and the dictionary. A complete description of our compression method is presented in Section 3. compress gcc go ijpeg li Benchmarks 0% 10% 20% 30% 40% Program Instructions Distinct instruction encodings used only once in program Distinct instruction encodings used multiple times in program Figure 1: Distinct instruction encodings as a percentage of entire program Uncompressed Code clrlwi r11,r9,24 addi r0,r11,1 cmplwi cr1,r0,8 ble cr1,000401c8 cmplwi cr1,r11,7 bgt cr1,00041d34 stb r18,0(r28) clrlwi r11,r9,24 addi r0,r11,1 cmplwi cr1,r0,8 bgt cr1,00041c98 Compressed Code CODEWORD #1 ble cr1,000401c8 cmplwi cr1,r11,7 bgt cr1,00041d34 CODEWORD #2 CODEWORD #1 bgt cr1,00041c98 Dictionary clrlwi r11,r9,24 addi r0,r11,1 cmplwi cr1,r0,8 stb r18,0(r28) Figure 2: Example of compression Background and Related Work In this section we will discuss strategies for text compression, and methods currently employed by microprocessor manufacturers to reduce the impact of RISC instruction sets on program size. 2.1 Text compression Text compression methods fall into two general categories: statistical and dictionary. Statistical compression uses the frequency of singleton characters to choose the size of the codewords that will replace them. Frequent characters are encoded using shorter codewords so that the overall length of the compressed text is minimized. Huffman encoding of text is a well-known example. Dictionary compression selects entire phrases of common characters and replaces them with a single codeword. The codeword is used as an index into the dictionary entry which contains the original characters. Compression is achieved because the codewords use fewer bits than the characters they replace. There are several criteria used to select between using dictionary and statistical compression techniques. Two very important factors are the decode efficiency and the overall compression ratio. The decode efficiency is a measure of the work required to re-expand a compressed text. The compression ratio is defined by the formula: (Eq. Dictionary decompression uses a codeword as an index into the dictionary table, then inserts the dictionary entry into the decompressed text stream. If codewords are aligned with machine words, the dictionary lookup is a constant time operation. Statistical compression, on the other hand, uses codewords that have different bit sizes, so they do not align to machine word bound- aries. Since codewords are not aligned, the statistical decompression stage must first establish the range of bits comprising a codeword before text expansion can proceed. It can be shown that for every dictionary method there is an equivalent statistical method which achieves equal compression and can be improved upon to give better compression [Bell90]. Thus statistical methods can always achieve better compression than dictionary methods albeit at the expense of additional computation requirements for decompression. It should be noted, how- ever, that dictionary compression yields good results in systems with memory and time constraints because one entry expands to several characters. In general, dictionary compression provides for faster (and simpler) decoding, while statistical compression yields a better compression ratio. 2.2 Compression for RISC instruction sets Although a RISC instruction set is easy to decode, its fixed-length instruction formats are wasteful of program memory. Thumb [ARM95][MPR95] and MIPS16 [Kissell97] are two compression ratio compressed size original size recently proposed instruction set modifications which define reduced instruction word sizes in an effort to reduce the overall size of compiled programs. Thumb is a subset of the ARM architecture consisting of 36 ARM 32-bit wide instructions which have been re-encoded to require only 16 bits. The instructions included in Thumb either do not require a full 32-bits, are frequently used, or are important to the compiler for generating small object code. Programs compiled for Thumb achieve 30% smaller code in comparison to the standard ARM instruction set [ARM95]. MIPS16 defines a 16-bit fixed-length instruction set architecture (ISA) that is a subset of MIPS-III. The instructions used in MIPS16 were chosen by statistically analyzing a wide range of application programs for the instructions most frequently generated by compilers. Code written for 32-bit MIPS-III is typically reduced 40% in size when compiled for MIPS16 [Kissell97]. Both Thumb and MIPS16 act as preprocessors for their underlying architectures. In each case, a 16-bit instruction is fetched from the instruction memory, expanded into a 32-bit wide instruc- tion, and passed to the base processor core for execution. Both the Thumb and MIPS16 shrink their instruction widths at the expense of reducing the number of bits used to represent register designators and immediate value fields. This confines Thumb and MIPS16 programs to 8 registers of the base architecture and significantly reduces the range of immediate values. As subsets of their base architectures, Thumb and MIPS16 are neither capable of generating complete programs, nor operating the underlying machine. Thumb relies on 32-bit instructions memory management and exception handling while MIPS16 relies on 32-bit instructions for floating-point operations. Moreover, Thumb cannot exploit the conditional execution and zero- latency shifts and rotates of the underlying ARM architecture. Both Thumb and MIPS16 require special branch instructions to toggle between 32-bit and 16-bit modes. The fixed set of instructions which comprise Thumb and MIPS16 were chosen after an assessment of the instructions used by a range of applications. Neither architecture can access all regis- ters, instructions, or modes of the underlying 32-bit core architecture. In contrast, we derive our codewords and dictionary from the specific characteristics of the program under execution. Because of this, a compressed program can access all the resources available on the machine, yet can still exploit the compressibility of each individual program. 2.3 CCRP The Compressed Code RISC Processor (CCRP) described in [Wolfe92][Wolfe94] has an instruction cache that is modified to run compressed programs. At compile-time the cache line bytes are Huffman encoded. At run-time cache lines are fetched from main memory, uncom- pressed, and put in the instruction cache. Instructions fetched from the cache have the same addresses as in the uncompressed program. Therefore, the core of the processor does not need modification to support compression. However, cache misses are problematic because missed instructions in the cache do not reside at the same address in main memory. CCRP uses a Line Address Table (LAT) to map missed instruction cache addresses to main memory addresses where the compressed code is located. The LAT limits compressed programs to only execute on processors that have the same line size for which they were compiled. One short-coming of CCRP is that it compresses on the granularity of bytes rather than full instructions. This means that CCRP requires more overhead to encode an instruction than our scheme which encodes groups of instructions. Moreover, our scheme requires less effort to decode a program since a single codeword can encode an entire group of instructions. In addition, our compression method does not need a LAT mechanism since we patch all branches to use the new instruction addresses in the compressed program. 2.4 Liao et al. A purely software method of supporting compressed code is proposed in [Liao96]. The author finds mini-subroutines which are common sequences of instructions in the program. Each instance of a mini-subroutine is removed from the program and replaced with a call instruction. The mini-subroutine is placed once in the text of the program and ends with a return instruction. Mini-subroutines are not constrained to basic blocks and may contain branch instructions under restricted conditions. The prime advantage of this compression method is that it requires no hardware support. However, the subroutine call overhead will slow program execution. [Liao96] suggests a hardware modification to support code compression consisting primarily of a call-dictionary instruction. This instruction takes two arguments: location and length. Common instruction sequences in the program are saved in a dictionary, and the sequence is replaced in the program with the call-dictionary instruction. During execution, the processor jumps to the point in the dictionary indicated by location and executes length instructions before implicitly returning. [Liao96] limits the dictionary to use sequences of instructions within basic blocks only. [Liao96] does not explore the trade-off of the field widths for the location and length arguments in the call-dictionary instruction. Only codewords that are 1 or 2 instruction words in size are considered. This requires the dictionary to contain sequences with at least 2 or 3 instructions, respectively, since shorter sequences would be no bigger than the call-dictionary instruction and no compression would result. Since single instructions are the most frequently occurring patterns, it is important to use a scheme that can compress them. In this paper we vary the parameters of dictionary size (the number of entries in the dictionary) and the dictionary entry length (the number of instructions at each dictionary entry) thus allowing us to examine the efficacy of compressing instruction sequences of any length. 3 Compression Method 3.1 Algorithm Our compression method is based on the technique introduced in [Bird96][Chen97a]. A dictionary compression algorithm is applied after the compiler has generated the program. We take advantage of SDTS and find common sequences of instructions to place in the dictionary. Our algorithm is divided into 3 steps: 1. Build the dictionary 2. Replace instruction sequences with codewords 3. Encode the codewords 3.1.1 Dictionary content For an arbitrary text, choosing those entries of a dictionary that achieve maximum compression is NP-complete in the size of the text [Storer77]. As with most dictionary methods, we use a greedy algorithm to quickly determine the dictionary entries 1 . On every iteration of the algorithm, we examine each potential dictionary entry and find the one that results in the largest immediate savings. The algorithm continues to pick dictionary entries until some termination criteria has been reached; this is usually the exhaustion of the codeword space. The maximum number of dictionary entries is determined by the choice of the encoding scheme for the codewords. Obviously, codewords with more bits can index a larger range of dictionary entries. We limit the dictionary entries to sequences of instructions within a basic block. We allow branch instructions to branch to codewords, but they may not branch within encoded sequences. We also do not compress branches with offset fields. These restrictions simplify code generation. 3.1.2 Replacement of instructions by codewords Our greedy algorithm combines the step of building the dictionary with the step of replacing instruction sequences. As each dictionary entry is defined, all of its instances in the program are replaced with a token. This token is replaced with an efficient encoding in the encoding step. 3.1.3 Encoding Encoding refers to the representation of the codewords in the compressed program. As discussed in Section 2.1, variable-length codewords, (such as those used in the Huffman encoding in are expensive to decode. A fixed-length codeword, on the other hand, can be used directly as an index into the dictionary making decoding a simple table lookup operation. Our baseline compression method uses a fixed-length codeword to enable fast decoding. We also investigate a variable-length scheme. However, we restrict the variable-length codewords to be a multiple of some basic unit. For example, we present a compression scheme with codewords that are 4 bits, 8 bits, 12 bits, and 16 bits. All instructions (compressed and uncompressed) are aligned to the size of the smallest codeword. The shortest codewords encode the most frequent dictionary entries to maximize the savings. This achieves better compression than a fixed-length encoding, but complicates decoding. 3.2 Related Issues 3.2.1 Branch instructions One side effect of any compression scheme is that it alters the locations of instructions in the program. This presents a special problem for branch instructions, since branch targets change as a result of program compression. 1. Greedy algorithms are often near-optimal in practice. For this study, we do not compress relative branch instructions (i.e. those containing an offset field used to compute a branch target). This makes it easy for us to patch the offset fields of the branch instruction after compression. If we allowed compression of relative branches, we might need to rewrite codewords representing relative branches after a compression pass; but this would affect relative branch targets thus requiring a rewrite of codewords, etc. The result is a NP-complete problem [Szymanski78]. Indirect branches are compressed in our study. Since these branches take their target from a register, the branch instruction itself does not need to be patched after compression, so it cannot create the codeword rewriting problem outlined above. However, jump tables (containing program addresses) need to be patched to reflect any address changes due to compression. GCC puts jump table data in the .text section immediately following the branch instruction. We assume that this table could be relocated to the .data section and patched with the post-compression branch target addresses. 3.2.2 Branch targets in fixed-length instruction sets Fixed-length instruction sets typically restrict branches to use targets that are aligned to instruction word boundaries. Since our primary concern is code size, we trade-off the performance advantages of aligned fixed-length instructions in exchange for more compact code. We use codewords that are smaller than instruction word boundaries and align them to the size of the smallest codeword (4 bits in this study). Therefore, we need to specify a method to address branch targets that do not fall at instruction word boundaries. One solution is to pad the compressed program so that all branch targets are aligned as defined by the original ISA. The obvious disadvantage of this solution is that it will decrease the compression ratio. A more complex solution (the one we have adopted for our experiments) is to modify the control unit of the processor to treat the branch offsets as aligned to the size of the smallest codeword. For example, if the size of a codeword is 8 bits, then a 32-bit aligned instruction set would have its branch offset range reduced by a factor of 4. Table 1 shows that most branches in the benchmarks do not use the entire range of their offset fields. The post-compilation compressor modifies all branch offsets to use the alignment of the codewords. Branches requiring larger ranges are modified to load their targets through jump tables. Of course, this will result in a slight increase in the code size for these branch sequences. 3.3 Compressed program processor The general design for a compressed program processor is given in Figure 3. We assume that all levels of the memory hierarchy will contain compressed instructions to conserve memory. Since the compressed program may contain both compressed and uncompressed instructions, there are two paths from the program memory to the processor core. Uncompressed instructions proceed directly to the normal instruction decoder. Compressed instructions must first be translated using the dictionary before being decoded and executed in the usual manner. The dictionary could be loaded in a variety of ways. If the dictionary is small, one possibility is to place it in a permanent on-chip memory. Alternatively, if the dictionary is larger, it might be kept as a data segment of the compressed program and each dictionary entry could be loaded as needed. 4 Experiments In this section we integrate our compression technique into the PowerPC instruction set. We compiled the SPEC CINT95 benchmarks with GCC 2.7.2 using -O2 optimization. The optimizations include common sub-expression elimination. They do not include function in-lining and loop unrolling since these optimizations tend to increase code size. Linking was done statically so Table 1: Usage of bits in branch offset field Bench number of relative Branches Branch offsets not wide enough to provide 2-byte resolution to branch targets Branch offsets not wide enough to provide 1-byte resolution to branch targets Branch offsets not wide enough to provide 4-bit resolution to branch targets Number Percent Number Percent Number Percent compress li 4,806 0 0.00% perl 14,578 15 0.10% 74 0.51% 191 1.31% vortex CPU core Figure 3: Overview of compressed program processor uncompressed instruction stream Dictionary Compressed program memory (usually ROM) that the libraries are included in the results. All compressed program sizes include the overhead of the dictionary. Recall that we are interested in the dictionary size (number of codewords) and dictionary entry length (number of instructions at each dictionary entry). 4.1 Baseline compression method In our baseline compression method, we use codewords of 2-bytes. The first byte is an escape byte that has an illegal PowerPC opcode value. This allows us to distinguish between normal instructions and compressed instructions. The second byte selects one of 256 dictionary entries. Dictionary entries are limited to a length of 16 bytes (4 PowerPC instructions). PowerPC has 8 illegal 6-bit opcodes. By using all 8 illegal opcodes and all possible patterns of the remaining 2 bits in the byte, we can have up to 32 different escape bytes. Combining this with the second byte of the codeword, we can specify up to 8192 different codewords. Since compressed instructions use only illegal opcodes, any processor designed to execute programs compressed with the base-line method will be able to execute the original programs as well. Our first experiments vary the parameters of the baseline method. Figure 4 shows the effect of varying the dictionary entry length. Interestingly, when dictionary entries are allowed to contain 8 instructions, the overall compression begins to decline. This can be attributed to our greedy selection algorithm for generating the dictionary. Selecting large dictionary entries removes some opportunities for the formation of smaller entries. The large entries are chosen because they result in an immediate reduction in the program size. However, this does not guarantee that they are the best entries to use for achieving good compression. When a large sequence is replaced, it destroys the small sequences that partially overlapped with it. It may be that the savings of using the multiple smaller sequences would be greater than the savings of the single large sequence. However, our greedy algorithm does not detect this case and some potential savings is lost. In general, dictionary entry sizes above 4 instructions do not improve compression noticeably Figure 5 illustrates what happens when the number of codewords (entries in the dictionary) increases. The compression ratio for each program continues to improve until a maximum amount of codewords is reached, after which only unique, single use encodings remain uncompressed. Table 2 lists the maximum number of codewords for each program under the baseline compression method, representing an upper bound on the size of the dictionary. compress gcc go ijpeg li Benchmarks 0% 10% 20% 30% 40% 50% 70% 80% 90% 100% Compression Ratio 1357Maximum number of instructions in each dictionary entry Figure 4: Effect of dictionary entry size on compression ratio The benchmarks contain numerous instructions that occur only a few times. As the dictionary becomes large, there are more codewords available to replace the numerous instruction encodings that occur infrequently. The savings of compressing an individual instruction is tiny, but when it is multiplied over the length of the program, the compression is noticeable. To achieve good compression, it is more important to increase the number of codewords in the dictionary rather than increase the length of the dictionary entries. A few thousand codewords is enough for most SPEC CINT95 programs. 4.1.1 Usage of the dictionary Since the usage of the dictionary is similar across all the benchmarks, we show results using ijpeg as a representative benchmark. We extend the baseline compression method to use dictionary entries with up to 8 instructions. Figure 6 shows the composition of the dictionary by the number of instructions the dictionary entries contain. The number of dictionary entries with only a single instruction ranges between 48% and 80%. Not surprisingly, the larger the dictionary, the higher the proportion of short dictionary entries. Figure 7 shows which dictionary entries contribute the most to compression. Dictionary entries with 1 instruction achieve between 48% and 60% of the compression savings. The short entries contribute to a larger portion of the savings as the size of the dictionary increases. The compression method in [Liao96] cannot take advantage Table 2: Maximum number of codewords used in baseline compression (max. dictionary entry Bench Maximum Number of Codewords Used compress 647 go 3123 ijpeg 2107 li 1104 perl 2970 vortex 3545 compress gcc go ijpeg li Benchmarks 0% 10% 20% 30% 40% 50% 70% 80% 90% 100% Compression Number of codewords Figure 5: Effect of number of codewords on compression ratio of this since the codewords are the size of single instructions, so single instructions are not compressed 4.1.2 Compression using small dictionaries Some implementations of a compressed code processor may be constrained to use small dic- tionaries. We investigated compression with dictionaries ranging from 128 bytes to 512 bytes in size. We present one compression scheme to demonstrate that compression can be beneficial even for small dictionaries. Our compression scheme for small dictionaries uses 1-byte codewords and dictionary entries of up to 4 instructions in size. Figure 8 shows results for dictionaries with 8, 16, and entries. On average, a dictionary size of 512 bytes is sufficient to get a code reduction of 15%. 4.1.3 Variable-length codewords In the baseline method, we used 2-byte codewords. We can improve our compression ratio by using smaller encodings for the codewords. Figure 9 shows that when the baseline compression uses 8192 codewords, 40% of the compressed program bytes are codewords. Since the baseline compression uses 2-byte codewords, this means 20% of the final compressed program size is due to escape bytes. We investigated several compression schemes using variable-length codewords Size of dictionary (number of entries) 0% 20% 40% 80% 100% Percentage of dictionary Figure Composition of dictionary for ijpeg (max. dictionary Length of dictionary entry (number of instructions) Size of dictionary (number of entries) 10.0% 20.0% 30.0% 40.0% Program bytes removed2468 Length of dictionary entry due to compression Figure 7: Bytes saved in compression of ijpeg according to instruction length of dictionary entry (number of instructions) aligned to 4-bits (nibbles). Although there is a higher decode penalty for using variable-length codewords, we are able to achieve better compression. By restricting the codewords to integer multiples of 4-bits, we have given the decoding process regularity that the 1-bit aligned Huffman encoding in [Wolfe94] lacks. Our choice of encoding is based on SPEC CINT95 benchmarks. We present only the best encoding choice we have discovered. We use codewords that are 4-bits, 8-bits, 12-bits, and 16-bits in length. Other programs may benefit from different encodings. For example, if many codewords are not necessary for good compression, then more 4-bit and 8-bit code words could be used to further reduce the codeword overhead. A diagram of the nibble aligned encoding is shown in Figure 10. This scheme is predicated on the observation that when an unlimited number of codewords are used, the final compressed program size is dominated by codeword bytes. Therefore, we use the escape code to indicate (less uncompressed instructions rather than codewords. The first 4-bits of the codeword determine the length of the codeword. With this scheme, we can provide 128 8-bit codewords, and a few thousand 12-bit and 16-bit codewords. This offers the flexibility of having many short codewords (thus minimizing the impact of the frequently used instructions), while allowing for a large overall number of codewords. One nibble is reserved as an escape code for uncompressed instruc- compress gcc go ijpeg li Benchmarks 0% 10% 20% 30% 40% 50% 70% 80% 90% 100% Compression Number of codewords Number of codewords Figure 8: Compression Ratio for 1-byte codewords with up to 4 instructions/entry Figure 9: Composition of Compressed Program (8192 2-byte codewords, 4 instructions/entry) compress gcc go ijpeg li Benchmarks 10.0% 20.0% 30.0% 40.0% 50.0% 70.0% 80.0% 90.0% 100.0% Compressed Program Size Dictionary Codewords: escape bytes Codewords: index bytes Uncompressed Instructions tions. We reduce the codeword overhead by encoding the most frequent sequences of instructions with the shortest codewords. Using this encoding technique effectively redefines the entire instruction set encoding, so this method of compression can be used in existing instruction sets that have no available escape bytes. Unfortunately, this also means that the original programs will no longer execute unmodified on processors that execute compressed programs without mode switching. Our results for the 4-bit aligned compression are presented in Figure 11. We obtain a code reduction of between 30% and 50% depending on the benchmark. For comparison, we extracted the instruction bytes from the benchmarks and compressed them with Unix Compress. Compress uses an adaptive dictionary technique (based on Ziv-Lempel coding) which can modify the dictionary in response to changes in the characteristics of the text. In addition, it also uses Huffman encoding on its codewords, and thus should be able to achieve better compression than our method. Figure 11 shows that Compress does indeed do better, but our compression ratio is still within 5% for all benchmarks. Figure 10: Nibble Aligned Encoding Instruction 128 8-bit codewords 1536 12-bit codewords 4096 16-bit codewords 36-bit uncompressed instruction Figure 11: Comparison of nibble aligned compression with Unix Compress compress gcc go ijpeg li Benchmarks 10.0% 20.0% 30.0% 40.0% 50.0% 70.0% 80.0% 90.0% 100.0% Compression Compression with nibble aligned codewords Unix Compress 5 Conclusions and Future Work We have proposed a method of compressing programs for embedded microprocessors where program size is limited. Our approach combines elements of two previous proposals. First we use a dictionary compression method (as in [Liao96]) that allows codewords to expand to several instructions. Second, we allow the codewords to be smaller than a single instruction (as in [Wolfe94]). We find that the size of the dictionary is the single most important parameter in attaining a better compression ratio. The second most important factor is reducing the codeword size below the size of a single instruction. We find that much of our savings comes from compressing patterns of single instructions. Our most aggressive compression for SPEC CINT95 achieves a 30% to 50% code reduction. Our compression ratio is similar to that achieved by Thumb and MIPS16. While Thumb and MIPS16 designed a completely new instruction set, compiler, and instruction decoder, we achieved our results only by processing compiled object code and slightly modifying the instruction fetch mechanism. There are several ways that our compression method can be improved. First, the compiler could attempt to produce instructions with similar byte sequences so they could be more easily compressed. One way to accomplish this is by allocating registers so that common sequences of instructions use the same registers. Another way is to generate more generalized STDS code sequences. These would be less efficient, but would be semantically correct in a larger variety of circumstances. For example, in most optimizing compilers, the function prologue sequence might save only those registers which are modified within the body of the function. If the prologue sequence were standardized to always save all registers, then all instructions of the sequence could be compressed to a single codeword. This space saving optimization would decrease code size at the expense of execution time. Table 3 shows that the prologue and epilogue combined typically account for 12% of the program size, so this type of compression would provide significant size reduction. We also plan to explore the performance aspects of our compression and examine the trade-offs in partitioning the on-chip memory for the dictionary and program. Table 3: Prologue and epilogue code in benchmarks Bench Static prologue instructions (percentage of entire program) Static epilogue instructions (percentage of entire program) compress 5.3% 6.2% gcc 4.2% 4.9% go 6.2% 6.8% ijpeg 6.9% 9.4% li 8.1% 9.9% perl 3.7% 4.3% vortex 6.3% 7.1% 6 --R Compiler: Principles Advanced RISC Machines Ltd. 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Prasanna, Configuration compression for FPGA-based embedded systems, Proceedings of the 2001 ACM/SIGDA ninth international symposium on Field programmable gate arrays, p.173-182, February 2001, Monterey, California, United States Montserrat Ros , Peter Sutton, A hamming distance based VLIW/EPIC code compression technique, Proceedings of the 2004 international conference on Compilers, architecture, and synthesis for embedded systems, September 22-25, 2004, Washington DC, USA Kelvin Lin , Chung-Ping Chung , Jean Jyh-Jiun Shann, Compressing MIPS code by multiple operand dependencies, ACM Transactions on Embedded Computing Systems (TECS), v.2 n.4, p.482-508, November Montserrat Ros , Peter Sutton, Compiler optimization and ordering effects on VLIW code compression, Proceedings of the international conference on Compilers, architecture and synthesis for embedded systems, October 30-November 01, 2003, San Jose, California, USA Shao-Yang Wang , Rong-Guey Chang, Code size reduction by compressing repeated instruction sequences, The Journal of Supercomputing, v.40 n.3, p.319-331, June 2007 Mats Brorsson , Mikael Collin, Adaptive and flexible dictionary code compression for embedded applications, Proceedings of the 2006 international conference on Compilers, architecture and synthesis for embedded systems, October 22-25, 2006, Seoul, Korea Talal Bonny , Joerg Henkel, Using Lin-Kernighan algorithm for look-up table compression to improve code density, Proceedings of the 16th ACM Great Lakes symposium on VLSI, April 30-May 01, 2006, Philadelphia, PA, USA Arvind Krishnaswamy , Rajiv Gupta, Dynamic coalescing for 16-bit instructions, ACM Transactions on Embedded Computing Systems (TECS), v.4 n.1, p.3-37, February 2005 Kelvin Lin , Jean Jyh-Jiun Shann , Chung-Ping Chung, Code compression by register operand dependency, Journal of Systems and Software, v.72 n.3, p.295-304, August 2004 Keith D. 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Govindarajan, Area and Power Reduction of Embedded DSP Systems using Instruction Compression and Re-configurable Encoding, Journal of VLSI Signal Processing Systems, v.44 n.3, p.245-267, September 2006 Chandra Krintz , Brad Calder , Han Bok Lee , Benjamin G. Zorn, Overlapping execution with transfer using non-strict execution for mobile programs, ACM SIGOPS Operating Systems Review, v.32 n.5, p.159-169, Dec. 1998 Yuan Xie , Wayne Wolf , Haris Lekatsas, A code decompression architecture for VLIW processors, Proceedings of the 34th annual ACM/IEEE international symposium on Microarchitecture, December 01-05, 2001, Austin, Texas Jeremy Lau , Stefan Schoenmackers , Timothy Sherwood , Brad Calder, Reducing code size with echo instructions, Proceedings of the international conference on Compilers, architecture and synthesis for embedded systems, October 30-November 01, 2003, San Jose, California, USA Marc L. Corliss , E. Christopher Lewis , Amir Roth, DISE: a programmable macro engine for customizing applications, ACM SIGARCH Computer Architecture News, v.31 n.2, May Marc L. Corliss , E. Christopher Lewis , Amir Roth, The implementation and evaluation of dynamic code decompression using DISE, ACM Transactions on Embedded Computing Systems (TECS), v.4 n.1, p.38-72, February 2005 Luca Benini , Francesco Menichelli , Mauro Olivieri, A Class of Code Compression Schemes for Reducing Power Consumption in Embedded Microprocessor Systems, IEEE Transactions on Computers, v.53 n.4, p.467-482, April 2004 Y. Larin , Thomas M. Conte, Compiler-driven cached code compression schemes for embedded ILP processors, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.82-92, November 16-18, 1999, Haifa, Israel Stephen Roderick Hines , Gary Tyson , David Whalley, Addressing instruction fetch bottlenecks by using an instruction register file, ACM SIGPLAN Notices, v.42 n.7, July 2007 Oliver Rthing , Jens Knoop , Bernhard Steffen, Sparse code motion, Proceedings of the 27th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.170-183, January 19-21, 2000, Boston, MA, USA Christopher W. Fraser, An instruction for direct interpretation of LZ77-compressed programs, SoftwarePractice & Experience, v.36 n.4, p.397-411, April 2006 rpd Beszdes , Rudolf Ferenc , Tibor Gyimthy , Andr Dolenc , Konsta Karsisto, Survey of code-size reduction methods, ACM Computing Surveys (CSUR), v.35 n.3, p.223-267, September

embedded systems;compression;code density;Code Space Optimization

266820

Procedure based program compression.

Cost and power consumption are two of the most important design factors for many embedded systems, particularly consumer devices. Products such as personal digital assistants, pagers with integrated data services and smart phones have fixed performance requirements but unlimited appetites for reduced cost and increased battery life. Program compression is one technique that can be used to attack both of these problems. Compressed programs require less memory, thus reducing the cost of both direct materials and manufacturing. Furthermore, by relying on compressed memory, the total number of memory references is reduced. This reduction saves power by lowering the traffic on high-capacitance buses. This paper discusses a new approach to implementing transparent program compression that requires little or no hardware support. Procedures are compressed individually, and a directory structure is used to bind them together at run-time. Decompressed procedures are explicitly cached in ordinary RAM as complete units, thus resolving references within each procedure. This approach has been evaluated on a set of 25 embedded multimedia and communications applications, and results in an average memory reduction of 40% with a run-time performance overhead of 10%.

Introduction We will present a technique for saving power and reducing cost in embedded systems. We are concerned primarily with data-rich consumer devices used for computation and communications, e.g. the so-called information appliance. Currently this product category includes devices such as simple Personal Digital Assistants, pagers and cell phones. In the future, there is an emerging industry vision of ubiquitous multimedia devices and the Java appliance. These products have extremely tight constraints on component cost. It is not at all uncommon for memory to be one of the most expensive components in these products, thus providing the need to reduce the size of stored programs. A second important design goal is low power consumption. Each of these products is battery powered, and reduced power consumption can be directly translated into extended battery life. Battery life is often the most important factor for this product class: once a pager or cellular phone is functionally verified, battery life is one of the few effective techniques for product differentiation. For a large number of embedded systems, the power used to access memory on the processor bus is the dominant factor in the system power consumption. Unlike desktop computing, performance often is not a primary factor for these devices. While information appliances are not the classic forms of mission-critical computing, they tend to have important real-time aspects. For example, a certain amount of processor performance is necessary to decode a paging message; there is little benefit in providing more. Faced with these factors, we have decided to investigate the benefits storing programs in a compressed form. The compressed programs may reside in any type of memory, often depending on whether the system supports software field upgrades. The basic approach is to store the program image in a compressed form, and dynamically decompress it on demand. An effective compression scheme would reduce the amount of system memory required for various applications, thus saving cost, board space and some static power consumption. An additional benefit involves significantly reduced power consumption due to dynamic memory references. While we believe that an effective compression scheme will reduce power consumption, we cannot currently provide any direct evidence of the relationship. 1.1 Previous Approaches The relevant previous work can be divided into four groups: whole program transformation, cache-based dictionary schemes and highly encoded instruction set architectures. The most direct technique for compressing programs involves explicit compression and decompression of the complete program. A portion of RAM is dedicated for a decompressed buffer, and programs are expanded from their compressed form into this RAM prior to execution. This approach has been applied to file systems [1] for saving disk space. Virtual memory and explicit file caches are usually effective at reducing the impact of the decompression algorithms on latency. Unfortunately, this approach is not well suited to embedded systems. Some information appliances have only a single application, for example a data-rich pager with a vertical-market application such as the Motorola SportsTrax news device. Whole program decompression for such systems would result in RAM size exceeding ROM size, which increases both cost and power consumption. Wolfe, Chanin and Kozuch presented a scheme for block based decompression in response to dynamic demands [2, 3]. Their goal was to improve code density of general-purpose processor architectures. They considered a number of compression algorithms, and concluded that a Huffman code [4], which reduced code size to 74% of the original, was most effective. Programs are decompressed as they are brought into the instruction cache, and thus the compression is transparent to the executing application. One problem that Wolfe et al. identified involves translating memory addresses between the program space (i.e. decompressed) and the compressed program in the backing store. For example, if the program does a PC relative jump, which hits in the cache, the ordinary cache hardware will resolve the reference. However, in the case of a cache miss the refill hardware must determine where in the compressed space the target is stored. This problem requires a set of jump tables to patch references from the program space to the compressed space. A link time tool can be used to automatically generate the necessary jump tables. Liao, Devadas and Keutzer developed a dictionary approach to reduce code size for DSP processors [5]. A correlation is done across all basic blocks in a program after compilation but before linking. The purpose of this correlation is to identify common instruction sequences that exceed some minimal length. These sequences are written into a dictionary, and each of the original occurrences is replaced with a mini- subroutine call (sic), i.e. a procedure call with no arguments. The application code is a skeleton of procedure calls and bits of uncommon code sequences. This approach can be implemented with no hardware support, and results were presented which indicated that it might achieve code size reductions of approximately 12%. With a minor modification of the hardware an additional 4% code size reduction can also be achieved. While this approach will result in an extremely high rate of procedure calls, there is no discussion of the impact of these calls on performance. This issue could be particularly significant in the face of the tight code scheduling constraints of their target machine, the Texas Instruments TMS320C25 DSP. Ernst et al investigated the use of byte-coded machine language [6] for compression. This approach hearkens back to the original goal of tightly encoded ISA formats for CISC processors, and much of their work is focused on minimizing the impact on performance. 2. The Procedure Cache While Wolfe et al. have developed an effective technique for transparent code compression, their approach has two specific features which may prove disadvantageous. First, since the compression process is transparent to the supervisor code as well as the application, the entire decompression and translation process must be implemented in hardware. While dedicated decompression hardware has the benefit of low overhead, there is no option to use the approach on stock hardware. We are interested in schemes that can leverage existing hardware, with the option (but not necessity) of hardware accelerators. Secondly, the mapping problem between compressed space and program space complicates the hardware as well as the linking process. Pcache Application Directory ROM Compression utilities Compression Tables Processor Core ROM Hardware Accelerator Figure 1: Embedded system architecture for pcache system In place of dedicated hardware that transparently code blocks, we propose using demand driven decompression triggered by procedure invocations. Procedures are decompressed as atomic units, and are stored into a dedicated region of RAM that is explicitly managed by the runtime system (Figure 1). This approach efficiently solves the problem of address mapping for all references that are contained within one procedure (e.g. loops and conditional code), and experience shows that these are the most common forms of branching. The remaining inter-procedure and global references must be resolved through the use of a directory service. We call this software cache the Procedure Cache (pcache). The pcache should be able to store any procedure that is small enough to fit within it. As a result of this goal the pcache algorithms must manage variable size objects which may not be aligned on a boundary that is convenient for addressing, unlike conventional hardware caches which manage fixed size lines and blocks. The issue of maximum procedure size is problematic, and while a number of solutions present themselves we have not decided yet upon a recommended path. 2.1 Runtime Binding Procedure calls are bound together at runtime by consulting a directory service. A linking tool translates each call into a request through a unique identifier. The directory service looks up the location of the call target and activates the target with the proper linkage needed for the return operation. A table stored with each program is used to translate procedure identifiers into addresses in the compressed memory. As was the case with the scheme from Wolfe et al., this table must be generated when the program is linked. The process used for runtime binding can be broken down into the following stages: 1. Source invokes directory service with unique identifier of the target procedure 2. If the target is in the pcache go to step 9 3. Find the target address in compressed memory by consulting the directory service 4. If enough contiguous free space exists in the pcache for the target go to step 8 5. If enough fragmented free space exists in the pcache for the target go to step 7 6. Mark procedures for eviction until enough space is available 7. Coalesce fragmented space into contiguous block 8. Decompress target procedure into assigned pcache location 9. Patch the state to allow the target to return to the caller 10. Invoke the target procedure The traditional execution environment binds one procedure to another through a call instruction, which typically executes one memory reference and updates the program counter. Two call instructions are used in the process above, one at step 1 and one at step 10. Let l represent the time required to lookup the target procedure identifier in the directory service data structure in step 2. Let m represent the time required for management, which should be relatively stable at steady state, and let d t required to decompress procedure t . The pcache hit rate is represented by h , and is significant in the calculation of the expected case performance. The worst-case execution time involves compacting the free space and identifying procedures to replace, followed by the time required to decompress the target procedure. The worst case call time is . In the expected case, i.e. that of a cache hit, the call time is . In this case it is clearly important to increase the hit rate. However, in the limiting case where the hit rate is high, the directory scheme still imposes a cost of c l on every procedure invocation. A better approach is to cache the address of the target procedure at the call site, and then avoid the directory service overhead for subsequent calls to the target. A similar approach has been used in high performance object-oriented runtimes systems to speculatively bind method invocations to typed methods [7]. The runtime directory binds call sites to targets once and then patches the call site. Subsequent invocations jump straight to the target, and then test the runtime type information to determine if the processor ended up at the right location. If the test succeeds the execution continues and if it fails the directory service is consulted. This approach works if the cached target address is guaranteed to be the start of a valid code sequence, which is difficult to guarantee when code blocks move within the address space after they are loaded. Such an approach would not work for the pcache because cache replacement and compaction make alignment restrictions prohibitively expensive. Procedures need not be aligned on any standard boundary, and thus the risk exists that a jump would end up in the middle of a procedure or in free space. An alternate approach, which we are advocating, is to test the validity of the cached target address at the call site. This scheme involves more test operations in the pcache than does testing at the destination, since each procedure has a single entry point but may call multiple targets. In order to test at the call site, each procedure must have a prologue that contains the procedure identifier. Conceptually, the call site loads the word that precedes the cached target address, compares it to the target it wishes to invoke, and jumps to the cached address on a match. This sequence changes the best case invocation sequence from two jumps and the directory service lookup to one load, a test and a conditional jump. For a pcache with a large number of procedures, and thus an expensive directory service lookup, the cached target should result in a performance improvement over the pure directory scheme. There is the possibility that some procedure will have an identifier that corresponds to a legal code sequence, introducing the danger of a false positive test. This problem is avoided by introducing a tag byte that identifies the word as a procedure identifier. The specific tag byte to use depends on the processor architecture, as it must correspond to an illegal opcode and be placed at the proper word alignment. Unfortunately, this approach is not foolproof for processors with variable length instructions, such as the Intel x86 and the 68k. In these cases it is not possible to guarantee that the target identifier will not match some code sequence or embedded data, though the likelihood of this event can be reduced. Tagging procedure identifiers also makes it easy to implement a reference scheme in order to approximate LRU data for pcache management. The runtime system will periodically clear each of the tag bytes, thus forcing the cached targets to fail and invoking the directory service. For machines with 32-bit words the use of tag bytes does reduce the procedure identifier space to 24 bits, but we feel that this range will prove more than sufficient for the needs of embedded systems. 2.2 Procedure Returns Return instructions are a bit complicated because the traditional code sequence cannot explicitly name the destination of the return operation. The pcache runtime system solves this problem by storing three pieces of data: the source procedure identifier, the predicted address of the start of then return procedure (that is the address that the caller had at the time it invoked the active procedure), and the offset of the call site from the start of the source procedure. The regular return procedure is then replaced by a test of the predicted prologue for the source, and a jump to that address plus displacement in the event of success. A failure causes the directory service to lookup (and possibly reload) the destination of the return operation and then do a jump to the address plus displacement. 2.3 Replacement Algorithms The task of allocating space in the pcache for a new procedure involves a two step process. First, the pcache is searched for a free block that is large enough to satisfy the new demand. We have experimented with both best fit and first-fit for this stage, and have found that both approaches produce similar results. All of the results presented below will be with respect to first-fit, because of the simplified implementation. If there is enough free space available, but not in any single block, then the runtime system must invoke the pcache compactor. The pcache is compacted by moving all of the live procedures to the start of the pcache and all of the free fragments to the end of the pcache. In the event that the runtime system does not find must invoke a replacement algorithm to identify procedures that should be evicted from the pcache. We have experimented with two algorithms for replacement: least recently used (LRU) and Neighbor. With LRU the runtime system scans an LRU list and marks each procedure in sequence for eviction, until enough space has been freed. Once this is accomplished, the compactor is invoked to coalesce the free space into a single block large enough for the new procedure. While LRU is easy to implement and conceptually simple, it will tend to cause a significant amount of memory traffic in the pcache to coalesce the fragmented free space. For example, consider the case where together the two LRU procedures have enough space to satisfy a new request but happen to reside in the first and third quadrant of the pcache. The subsequent compacting stage will need to move approximately half of the pcache data in order to combine the space freed up by evicting these two procedures. It may be the case that the third LRU procedure could be combined with one of the first two and resides in an adjoining region of memory. In this case, the compacting procedure is trivial, since no intervening procedures need to be moved in the pcache. While the primary benefit of such an optimization is reducing the data movement in the pcache, a secondary benefit is avoiding subsequent tag misses on the moved procedures, and the resulting lookup events at the directory service. We have experimented with two schemes to reduce the amount of data movement within the pcache in response to pcache misses. The first scheme is called Neighbor, and involves looking for sets of adjacent procedures that are good candidates for eviction. Each set of procedures is evaluated on the basis of the sum of squares of the LRU values, where the least recently used procedure has an LRU value of 1 and all other increase sequentially. This approach is biased toward avoiding those procedures that were used recently, though it does not explicitly exclude them from consideration. Neighbor scans the pcache for adjacent blocks of memory that are large enough for the new request, and considers both free and occupied space. The algorithm then selects the set of blocks that have the lowest sum of squares for the LRU values. A more general form of Neighbor involves evaluating a function F(S) for each set of neighbors, where F() is some arbitrary function. The advantage of using a sum of squares is that it is easy to evaluate at runtime and it provides a strong bias against selecting the most recently used procedures. A modification of this approach is to use a limiting term to cap the value for each set of adjacent blocks, and exclude from consideration a set that exceeds this limit. The goal of this approach is to make it impossible to remove the most frequently used procedures, even in those cases where one of these procedures would otherwise be selected, e.g. when it neighbors a large procedure which is the least frequently used. 3. Experiments 3.1 Approach Trace driven simulation is used to evaluate the effectiveness of caching whole procedures. The pcache simulator must know when each procedure is activated, either through a directed call, the result of a return operation or an asynchronous transfer (e.g. exception or UNIX longjmp). The traces are collected with a special augmented version of Lsim from the IMPACT compilation system [8]. This tool allows us to dynamically generate a large set of activation events, with support for sophisticated trace sampling. 3.2 Applications There currently exists a significant void with regards to effective benchmarks for embedded systems. While a number of industrial and academic efforts have been proposed, to date there has been little progress towards a suite of representative programs and workloads. One part of the problem is that the field of embedded systems covers an extremely wide range of computing systems. It is difficult to imagine a benchmark suite that would reveal useful information to the designers of machines and cellular phones, because of the drastically different uses for these products. While there is some hope for emerging unification in the area of information appliances, because of the more cohesive focus to the devices, there currently are no options to choose from. This unfortunate state of affairs is best reflected by the continued use of the Dhrystones benchmark, and the derivative metric Dhrystones per milli-Watt. For the purposes of this paper we have adopted a number of programs from the MediaBench benchmark suite [9]. Six additional programs have been selected, five from the SPEC95 benchmark set along with the Backwater basic interpreter. Figure 2 and Figure 3 show the cumulative distribution functions (CDF) of procedure sizes for bwbasic and go, which represent the typical distribution and worst case (widest spread) respectively. Data is presented for both the static program image in memory, as well as the dynamic distribution seen during execution. This data suggests that a modest size pcache will often succeed in capturing the working set. In general the dynamic data exhibits a slightly slower growth than the static data and show sharper breaks; both phenomena are a result of the skewed distribution of call frequency among the procedures.0.10.30.50.70.9Static Dynamic Figure 2: Procedure size distribution for bwbasic 3.3 Pcache Miss Rates Table 1 shows the raw miss rates for LRU replacement, while Table 2 presents the same data for when Neighbor is used for replacement. Miss rates are calculated by counting each reference generated by the program, regardless of whether the procedure is actually cacheable (i.e. smaller than the simulated pcache size). The significance of low cache miss rates is much more difficult to evaluate than for a traditional hardware cache with fixed size objects. For example, there is a significant difference between missing on the average static procedure for go, which is under 1k, and missing on the most frequent procedure which is over 12k. Nevertheless, the simulator marks both as a single miss event.0.10.30.50.70.9Static Dynamic Figure 3: Procedure size distribution for go Several application, in particular the raw audio encoder and decoder, achieve extremely low miss rates. The general trends are for both LRU and Neighbor to have very similar hit rates, with two notable exceptions. Both djpeg and mpeg2enc show high miss rates with 1k pcaches with both LRU and Neighbor. The rates stay relatively high for 2k for Neighbor, while the drop for LRU. In both cases there is a specific procedure which is frequently called and which is captured by the LRU dynamics, though not for Neighbor. The procedure size CDF of go (Figure suggests that the dynamic reference stream has a much larger footprint than bwbasic (Figure 2), and thus it is not surprising that bwbasic shows reduced miss rates for comparable pcache sizes. On the other hand, a relatively small pcache size (e.g. 16k or 32k) can still be very effective. The miss rate data displays an interesting result: Neighbor generally achieves a lower miss rate than LRU. While it is certainly possible for pathological or pedagogical cases to exhibit behavior like this, in general LRU is expected to achieve the highest hit rates. This behavior is a consequence of the caching of variable size objects. Consider the case of a hardware cache with a fixed block size. If a certain amount of space must be made available and the entire cache is allocated, a specific number of blocks must be evicted from the cache. For such a cache LRU has proven to be effective at providing the best guess for which blocks should be evicted. With the software pcache used here the number of procedures evicted from the pcache depends upon the specific procedures selected. We explain this result by considering three different types of pcaches: small, medium and large. For a small pcache Neighbor will tend to approach LRU performance because of the smaller spread in LRU values between the least and the most recently used procedures. For a large pcache evictions will be rare so performance will approach the compulsory miss regardless of the replacement algorithm. For the medium case some number of references will be satisfied by the available free space, and for these the replacement algorithm is inconsequential. Assume that the CDF of procedure size is not correlated to LRU value 1 , and a new procedure activation causes the runtime to invoke the replacement algorithm. For half of these cases the size of the newly activated procedure will be equal to or less than the size of the LRU procedure, and thus both LRU and Neighbor will select the LRU procedure. For the remaining cases, the LRU algorithm will traverse the list of least frequently used procedures and mark each for eviction until the amount of space freed up is at least equal to the new request. Neighbor, however, looks for contiguous blocks that are good candidates according to a specific cost function. By relying on the SQU(LRU) cost function, Neighbor is biased against combining multiple procedures into the best set. Figure 4 illustrates this phenomenon. Assume that three "units" of space must be freed up. The LRU algorithm will choose the first three procedures on the LRU list (procedures 23, 17 and 87) for eviction and 1 While this may not hold in all cases, we believe that the assumption is valid in general. then invoke the compactor to coalesce the space. On the other hand, because Neighbor is using the square of the LRU counts, it will select procedure 12 for eviction 2 . We have found that, in general, for those cases where LRU and Neighbor select different sets of procedures for replacement Neighbor tends to select fewer procedures. Continuing the example in Figure 4, while it may be a good idea to evict either procedure 17, or 87 before procedure 12, it seems intuitive that it would not be best to evict all three rather than 12. Proc 12: LRU: 4: Cost: Proc Proc 23: LRU: 3: Cost: 9 Proc 35: LRU: 12: Cost: 144 Proc 87: LRU: 2: Cost: 4 Selected by LRU Selected by Neighbor Figure 4: Example: LRU chooses "optimal" set while Neighbor evicts fewer procedures The pcache miss rate for gcc is a direct result of the dynamic program size CDF. The most frequently used procedure in gcc is over 12k bytes and corresponds to 8% of the procedure activations, while the second most common procedure is 52 bytes and corresponds to 2% of the procedure activations. Although procedures that exceed the pcache size are excluded they still contribute to the count of procedure activations: thus the gcc simulations have low hit rates for pcaches below 64k bytes. An interesting phenomenon may occur when one of these large and common procedures is finally admitted to the pcache. The impact of introducing the large procedure into the pcache causes LRU to evict a huge number of procedures, while as was just discussed Neighbor tends to evict fewer. The resulting impact on pcache miss rate for LRU can be dramatic, as seen for gsmencode between 8k and 16k. 3.4 Compacting Events Table 3 and Table 4 show the rate of compaction events per procedure activation, rather than raw event counts, in order to make the presentation consistent across the test programs. This data illustrates the effectiveness of the Neighbor algorithm. While both Neighbor and LRU achieve roughly comparable results for pcache hit rates, Neighbor is much more effective at reducing the number of compacting events. For this data, the Neighbor algorithm generally produces a much 2 Note that the position of procedure 35 (with a high LRU) in the cache blocks Neighbor from merging procedures 17 and 35 along with the free space adjacent to 35. flatter response as a function of pcache size. This performance is a consequence of Neighbor compacting the pcache only when there is already enough free space and no procedure eviction is required. The amount of expected free space in a pcache is not monotonic with pcache size, but rather is a complicated function of the size of dynamic references. This fact is illustrated for pcache size of 8k for go, illustrating that the rate of compacting events can actually increase in response to an increase in pcache size. Again, gcc shows a sharp response soon after the admission of the 12k procedure. 126.gcc cjpeg djpeg gsmdecode mipmap pgpdecode rasta rawdaudio unepic Figure 5: Performance impact of pcache operations, relative to the base case 4. Performance The pcache structure will reduce performance due to three types of events: cache management including LRU management and directory service, memory movement due to compaction events within the pcache, and decompression of data transferred from memory. The impact of these factors was evaluated, and the resulting performance is shown in Figure 5. Because of the large volume of memory traffic into and within the pcache, the management component has comparatively little impact on performance. Each byte of memory moved within the pcache due to compaction was charged half of a clock cycle, assuming that two 32-bit memory operations are required and loop unrolling can be used to hide loop management overhead. The compression technique used is based on an algorithm that requires an average of 22 cycles on a SPARC to compress each byte of SPARC binaries and achieves a 60% compression ratio [10]. However, a number of algorithms could be selected to balance the demands of performance against compression. In particular, an application of Huffman coding (in hardware) will be briefly discussed later. The average slowdown for all of the applications is 166% with a 64k-byte pcache. However, when go and gcc are excluded from the pcache, the average slowdown is only 11%. These numbers climb to 600% and 36% for 32k-byte caches. Clearly, it is important to exclude ill-behaved applications from the pcache, but this problem is easy to manage with an embedded system where the software is generally more highly tuned to the execution environment. 5. Discussion While there is a benefit to considering schemes that require no additional hardware, the pcache architecture can still take advantage of hardware acceleration if available. A number of researchers have designed hardware for instruction decompression. A particularly good example is [11], which provides 480Mb/s Huffman decompression on a Mips instruction stream. This device requires 1 cm 2 in 0.8-micron process technology. When used with hardware decompression, the pcache is still effective at increasing compression rates, by increasing significantly increasing the block size. Furthermore, the dictionary size is reduced, since instead of having an entry for every possible jump target there is only an entry for each subroutine. It is impossible to say how the pcache will interact with traditional hardware caches without discussing the specific hardware configuration. If the pcache memory is nearby on high-speed SRAM the cache should ignore it, as transparent caching in hardware will provide no direct latency benefit while consuming valuable resources. On the other hand, if the pcache memory is relatively slow it should be cached by the hardware as well. Almost all sophisticated embedded processors include memory control hardware for functions such as chip-select, wait-state insertion and bus sizing. This hardware should be augmented to allow blocks of memory to become non-cacheable, which is a common feature in high-performance processors. 6. Conclusions We have presented a new approach to applying compression to stored program images. This technique can easily reduce the program storage by approximately 40%, which can correspond to a significant cost reduction for embedded products targeted to the consumer market. By compressing complete procedures, rather than smaller sub-blocks, we are able to avoid the cost of dedicated hardware. The resulting performance impact has been measured to be approximately 10% for a wide range of sophisticated embedded applications. Trace driven simulation has been used to evaluate the opportunity of using compression and the associated tradeoff points. The results suggest that a small software controlled cache, perhaps 16k bytes of standard SRAM, can be effective at caching the working set and reducing dynamic memory traffic. The additional effect of compressing traffic on the system bus further decreases main memory traffic, and thus helps to attack the problem of power consumption. --R "Combining the Concepts of Compression and Caching for a Two-Level Filesystem," "Executing Compressed Programs on an Embedded RISC Architecture," "Compression of Embedded System Programs," "A Method for the Construction of Minimum Redundancy Codes," "Code Density Optimization for Embedded DSP Processors Using Data Compression Techniques," "Code Compression," "Efficient Implementation of the Smalltalk-80 System," "IMPACT: An Architectural Framework for Multiple-Instruction- Issue Processors," A Tool for Evaluating Multimedia and Communications Systems," "An Extremely Fast Ziv-Lempel Data Compression Algorithm," "A High-Speed Asynchronous Decompression Circuit for Embedded Processors," --TR Combining the concepts of compression and caching for a two-level filesystem IMPACT Executing compressed programs on an embedded RISC architecture Code compression Compression of Embedded System Programs Code density optimization for embedded DSP processors using data compression techniques A High-Speed Asynchronous Decompression Circuit for Embedded Processors Efficient implementation of the smalltalk-80 system --CTR Israel Waldman , Shlomit S. Pinter, Profile-driven compression scheme for embedded systems, Proceedings of the 3rd conference on Computing frontiers, May 03-05, 2006, Ischia, Italy Saumya Debray , William S. Evans, Cold code decompression at runtime, Communications of the ACM, v.46 n.8, August Youtao Zhang , Jun Yang , Rajiv Gupta, Frequent value locality and value-centric data cache design, ACM SIGPLAN Notices, v.35 n.11, p.150-159, Nov. 2000 Youtao Zhang , Jun Yang , Rajiv Gupta, Frequent value locality and value-centric data cache design, ACM SIGOPS Operating Systems Review, v.34 n.5, p.150-159, Dec. 2000 Stacey Shogan , Bruce R. Childers, Compact Binaries with Code Compression in a Software Dynamic Translator, Proceedings of the conference on Design, automation and test in Europe, p.21052, February 16-20, 2004 G. Hallnor , Steven K. Reinhardt, A compressed memory hierarchy using an indirect index cache, Proceedings of the 3rd workshop on Memory performance issues: in conjunction with the 31st international symposium on computer architecture, p.9-15, June 20-20, 2004, Munich, Germany Keith D. Cooper , Nathaniel McIntosh, Enhanced code compression for embedded RISC processors, ACM SIGPLAN Notices, v.34 n.5, p.139-149, May 1999 Jun Yang , Youtao Zhang , Rajiv Gupta, Frequent value compression in data caches, Proceedings of the 33rd annual ACM/IEEE international symposium on Microarchitecture, p.258-265, December 2000, Monterey, California, United States Marc L. Corliss , E. Christopher Lewis , Amir Roth, A DISE implementation of dynamic code decompression, ACM SIGPLAN Notices, v.38 n.7, July Charles Lefurgy , Eva Piccininni , Trevor Mudge, Evaluation of a high performance code compression method, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.93-102, November 16-18, 1999, Haifa, Israel Bita Gorjiara , Daniel Gajski, FPGA-friendly code compression for horizontal microcoded custom IPs, Proceedings of the 2007 ACM/SIGDA 15th international symposium on Field programmable gate arrays, February 18-20, 2007, Monterey, California, USA O. Ozturk , H. Saputra , M. Kandemir , I. Kolcu, Access Pattern-Based Code Compression for Memory-Constrained Embedded Systems, Proceedings of the conference on Design, Automation and Test in Europe, p.882-887, March 07-11, 2005 Susan Cotterell , Frank Vahid, Synthesis of customized loop caches for core-based embedded systems, Proceedings of the 2002 IEEE/ACM international conference on Computer-aided design, p.655-662, November 10-14, 2002, San Jose, California Susan Cotterell , Frank Vahid, Tuning of loop cache architectures to programs in embedded system design, Proceedings of the 15th international symposium on System Synthesis, October 02-04, 2002, Kyoto, Japan Oliver Rthing , Jens Knoop , Bernhard Steffen, Sparse code motion, Proceedings of the 27th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.170-183, January 19-21, 2000, Boston, MA, USA Marc L. Corliss , E. Christopher Lewis , Amir Roth, The implementation and evaluation of dynamic code decompression using DISE, ACM Transactions on Embedded Computing Systems (TECS), v.4 n.1, p.38-72, February 2005 Guilin Chen , Mahmut Kandemir, Optimizing Address Code Generation for Array-Intensive DSP Applications, Proceedings of the international symposium on Code generation and optimization, p.141-152, March 20-23, 2005 Milenko Drini , Darko Kirovski , Hoi Vo, Code optimization for code compression, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, March 23-26, 2003, San Francisco, California Bjorn De Sutter , Bruno De Bus , Koen De Bosschere, Sifting out the mud: low level C++ code reuse, ACM SIGPLAN Notices, v.37 n.11, November 2002 Milenko Drini , Darko Kirovski , Hoi Vo, PPMexe: Program compression, ACM Transactions on Programming Languages and Systems (TOPLAS), v.29 n.1, p.3-es, January 2007 Bjorn De Sutter , Ludo Van Put , Dominique Chanet , Bruno De Bus , Koen De Bosschere, Link-time compaction and optimization of ARM executables, ACM Transactions on Embedded Computing Systems (TECS), v.6 n.1, February 2007 rpd Beszdes , Rudolf Ferenc , Tibor Gyimthy , Andr Dolenc , Konsta Karsisto, Survey of code-size reduction methods, ACM Computing Surveys (CSUR), v.35 n.3, p.223-267, September Bjorn De Sutter , Bruno De Bus , Koen De Bosschere, Link-time binary rewriting techniques for program compaction, ACM Transactions on Programming Languages and Systems (TOPLAS), v.27 n.5, p.882-945, September 2005

pagers;run-time performance overhead;procedural reference resolution;multimedia applications;procedure-based program compression;compressed memory;cached procedures;battery life;power consumption;high-capacitance bus traffic;consumer devices;embedded systems;RAM;memory references;performance requirements;source coding;transparent program compression;smart telephones;design factors;directory structure;memory reduction;integrated data services;communications applications;personal digital assistants

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Improving the accuracy and performance of memory communication through renaming.

As processors continue to exploit more instruction-level parallelism, a greater demand is placed on reducing the effects of memory access latency. In this paper, we introduce a novel modification of the processor pipeline called memory renaming. Memory renaming applies register access techniques to load instructions, reducing the effect of delays caused by the need to calculate effective addresses for the load and all preceding stores before the data can be fetched. Memory renaming allows the processor to speculatively fetch values when the producer of the data can be reliably determined without the need for an effective address. This work extends previous studies of data value and dependence speculation. When memory renaming is added to the processor pipeline, renaming can be applied to 30% to 50% of all memory references, translating to an overall improvement in execution time of up to 41%. Furthermore, this improvement is seen across all memory segments-including the heap segment, which has often been difficult to manage efficiently.

Introduction Two trends in the design of microprocessors combine to place an increased burden on the implementation of the memory system: more aggressive, wider instruction issue and higher clock speeds. As more instructions are pushed through the pipeline per cycle, there is a proportionate increase in the processing of memory operations - which account for approximately 1/3 of all instructions. At the same time, the gap between processor and DRAM clock speeds has dramatically increased the latency of the memory operations. Caches have been universally adopted to reduce the average memory access latency. Aggressive out-of-order pipeline execution along with non-blocking cache designs have been employed to alleviate some of the remaining latency by bypassing stalled instructions. Unfortunately, instruction reordering is complicated by the necessity of calculating effective addresses in memory operations. Whereas, dependencies between register-register instructions can be identified by examining the operand fields, memory dependencies cannot be determined until much later in the pipeline (when the effective address is calculated). As a result, mechanisms specific to loads and stores (e.g., the MOB in the Pentium Pro [1]) are required to resolve these memory dependencies later in the pipeline and enforce memory access semantics. To date, the only effective solution for dealing with ambiguous memory dependencies requires stalling loads until no earlier unknown store address exists. This approach is, however, overly conservative since many loads will stall awaiting the addresses of stores that they do not depend on, resulting in increased load instruction latency and reduced program performance. This paper proposes a technique called memory renaming that effectively predicts memory dependencies between store and load instructions, allowing the dynamic instruction scheduler to more accurately determine when loads should commence execution In addition to reordering independent memory ref- erences, further flexibility in developing an improved dynamic schedule can be achieved through our tech- nique. Memory renaming enables load instructions to retrieve data before their own effective address have been calculated. This is achieved by identifying the relationship between the load and the previous store instruction that generated the data. A new mechanism is then employed which uses an identifier associated with the store-load pair to address the value, bypassing the normal addressing mechanism. The term memory renaming comes from the similarity this approach has to the abstraction of operand specifiers performed in register renaming. [7]. In this paper, we will examine the characteristics of the memory reference stream and propose a novel architectural modification to the pipeline to enable speculative execution of load instructions early in the pipeline (before address calculation). By doing this, true dependencies can be eliminated, in particular those true dependencies supporting the complex address calculations used to access the program data. This will be shown to have a significant impact on overall performance (as much as 41% speedup for the experiments presented). The remainder of this paper is organized as follows: Section 2 examines previous approaches to speculating load instructions. Section 3 introduces the memory reordering approach to speculative load execution and evaluates the regularity of memory activity in order to identify the most successful strategy in executing loads speculatively. In section 4, we show one possible integration of memory renaming into an out-of-order pipeline implementation. Section 5 provides performance analysis for a cycle level simulation of the tech- niques. In section 6 we state conclusions and identify future research directions for this work. Background A number of studies have targeted the reduction of memory latency. Austin and Sohi [2] employed a sim- ple, fast address calculation early in the pipeline to effectively hide the memory latency. This was achieved by targeting the simple base+offset addressing modes used in references to global and stack data. Dahl and O'Keefe [5] incorporated address bits associated with each register to provide a hardware mechanism to disambiguate memory references dynamically. This allowed the compiler to be more aggressive in placing frequently referenced data in the register file (even when aliasing may be present), which can dramatically reduce the number of memory operations that must be executed. Lipasti and Shen [8] described a mechanism in which the value of a load instruction is predicted based on the previous values loaded by that instruction. In their work, they used a load value prediction unit to hold the predicted value along with a load classification table for deciding whether the value is likely to be correct based on past performance of the predictor. They observed that a large number of load instructions are bringing in the same value time after time. By speculatively using the data value that was last loaded by this instruction before all dependencies are resolved, they are able to remove those dependencies from the critical path (when speculation was accurate). Using this approach they were able to achieve a speedup in execution of between 3% (for a simple implementation) to 16% (with infinite resources and perfect prediction). Sazeides, Vassiliadis and Smith [10] used address speculation on load instructions to remove the dependency caused by the calculation of the effective address. This enables load instructions to proceed speculatively without their address operands when effective address computation for a particular load instruction remains constant (as in global variable references). Finally, Moshovos, Breach, Vijaykumar and Sohi [9] used a memory reorder buffer incorporating data dependence speculation. Data dependence speculation allows load instructions to bypass preceding stores before ambiguous dependencies are resolved; this greatly increases the flexibility of the dynamic instruction scheduling to find memory instruction ready to exe- cute. However, if the speculative bypass violates a true dependency between the load and store instructions in flight, the state of the machine must be restored to the point before the load instruction was mis-speculated and all instructions after the load must be aborted. To reduce the number of times a mis-speculation occurs, a prediction confidence circuit was included controlling when bypass was allowed. This confidence mechanism differs from that used in value prediction by locating dependencies between pairs of store and load instructions instead of basing the confidence on the history of the load instruction only. When reference prediction was added to the Multiscalar architecture, execution performance was improved by an average of 5-10%. Our approach to speculation extends both value prediction and dependence prediction to perform targeted speculation of load instructions early in the architectural pipeline. Renaming Memory Operations Memory renaming is an extension to the processor pipeline designed to initiate load instructions as early as possible. It combines dependence prediction with value prediction to achieve greater performance than possible with either technique alone. The basic idea behind memory renaming is to make a load instruction look more like a register reference and thereby process the load in a similar manner. This is difficult to achieve because memory reference instructions, unlike the simple operand specifiers used to access a register, require an effective address calculations before dependence analysis can be performed. To eliminate the need to generate an effective address from the critical path for accessing load data, we perform a load speculatively, using the program counter (PC) of the load instruction as an index to retrieve the speculative value. This is similar to the approach used in Lipasti and Shen's value prediction except that in memory renaming this is performed indirectly; when a load instruction is en- countered, the PC address associated with that load (LDPC) is used to index a dependence table (called the store-load cache) to determine the likely producer of the data (generally a store instruction). When it is recog- Value File Store/Load Cache Speculative Load Data Figure 1: Support for Memory Renaming. nized that the data referenced by a load instruction is likely produced by a single store instruction, the data from the last execution of that store can be retrieved from a value file. Accessing this value file entry can be performed (speculatively) without the need to know the effective address of the load or the store instruction, instead the value file is indexed by a unique identifier associated with the store-load pairing (this mechanism will be described in the next section). Store instructions also use the store-load cache to locate entries in the value file which are likely to be referenced by future loads. A store and load instruction pair which are determined to reference the same locations will map to the same value file index. Figure 1 shows an overview of the memory renaming mechanism. This approach has the advantage that when a pro- ducer/consumer relationship exists, the load can proceed very early in the pipeline. The effective address calculation will proceed as usual, as will the memory operation to retrieve the value from the Dcache; this is done to validate the speculative value brought in from the value file. If the data loaded from the Dcache matches that from the value file, then speculation was successful and processing continues unfettered. If the values differ, then the state of the processor must be corrected. In order for memory renaming to improve processor performance, it may be prudent to include a prediction confidence mechanism capable of identifying when speculation is warranted. As in value prediction and dependence prediction, we use a history based scheme to identify when speculation is likely to succeed. Unlike value renaming, we chose to identify stable store-load pairings instead of relying on static value references. To see the advantage of using a producer/consumer relationship as a confidence mechanism, analysis is shown for some of the SPEC95 benchmarks. All programs were compiled with GNU GCC (ver- sion 2.6.2), GNU GAS (version 2.5), and GNU GLD (version 2.5) with maximum optimization (-O3) and loop unrolling enabled (-funroll-loops). The Fortran codes were first converted to C using AT&T F2C version 1994.09.27. All experiments were performed on an extended version of the SimpleScalar [3] tool set. The tool set employs the SimpleScalar instruction set, which is a (virtual) MIPS-like architecture [6]. Table 1: Benchmark Application Descriptions Bench- Instr Loads Value Addr Prod mark (Mil.) (Mil.) Locality Loc. Loc. go 548 157 25 % 28 % 62 % gcc 264 97 compress 3.5 1.3 15 % 37 % 50 % li 956 454 24 % 23 % 55 % tomcatv 2687 772 43 % 48 % 66 % su2cor hydro2d 967 250 mgrid 4422 1625 42 % There are several approaches to improving the processing of memory operations by exploiting regularity in the reference stream. Regularity can be found in the stream of values loaded from memory, in the effective address calculations performed and in the dependence chains created between store and load instructions. Table 1 shows the regularity found in these differing characteristics of memory traffic. The first three column show the benchmark name, the total number of instructions executed and the total number of loads. The forth column shows the percentage of load executions of con- stant, or near constant values. The percentage shown is how often a load instruction fetches the same data value in two successive executions. (this is a measure of value locality). As shown in the table, a surprising number of load instruction executions bring in the same values as the last time, averaging 29% for SPEC integer benchmarks and 44% for SPECfp benchmarks. While it is surprising that so much regularity exists in value reuse, these percentages cover only about a third of all loads. Column 5 shows the percentage of load executions that reference the same effective address as last time. This shows about the same regularity in effective address reuse. The final column shows the percentage of time that the producer of the value remains unchanged over successive instances of a load instruction - this means that the same store instruction generated the data for the load. Here we see that this relationship is far more stable - even when the values transferred change, or when a different memory location is used for the transfer, the relationship between the sourcing store and the load remains stable. These statistics led us to the use of dependence pairings between store and load instructions to identify when speculation would be most profitable. 4 Experimental Pipeline Design To support memory renaming, the pipeline must be extended to identify store/load communication pairs, promote their communications to the register communication infrastructure, verify speculatively forwarded values, and recover the pipeline if the speculative store/load forward was unsuccessful. In the following text, we detail the enhancements made to the baseline out-of-order issue processor pipeline. An overview of the extensions to the processor pipeline and load/store queue entries is shown in Figure 2. 4.1 Promoting Memory Communication to Registers The memory dependence predictor is integrated into the front end of the processor pipeline. During de- code, the store/load cache is probed (for both stores and loads) for the index of the value file entry assigned to the dependence edge. If the access hits in the store/load cache, the value file index returned is propagated to the rename stage. Otherwise, an entry is allocated in the store/load cache for the instruction. In addition, a value file entry is allocated, and the index of the entry is stored in the store/load cache. It may seem odd to allocate an entry for a load in the value file, however, we have found in all our simulations that this is a beneficial optimization; it promotes constants and rarely stored variables into the value file, permitting these accesses to also benefit from faster, more accurate communication and synchronization. In addition, the decode stage holds confidence counters for renamed loads. These counters are incremented for loads when their sourcing stores are predicted correctly, and decremented (or reset) when they are predicted incorrectly. In the rename stage of the pipeline, loads use the value file index, passed in from the decode stage, to access an entry in the value file. The value file returns either the value last stored into the predicted dependence edge, or if the value is in the process of being computed (i.e. in flight), a load/store queue reservation station index is returned. If a reservation station index is returned, the load will stall until the sourcing store data is written to the store's reservation station. When a renamed load completes, it broadcasts its result to dependent instructions; the register and memory scheduler operate on the speculative load result as before without modification. All loads, speculative or otherwise, access the memory system. When a renamed load's value returns from the memory system, it is compared to the predicted value. If the values do not match, a load data misspeculation has occurred and pipeline recovery is initiated Unlike loads, store instructions do not access the value file until retirement. At that time, stores deposit their store data into the value file and the memory sys- tem. Later renamed loads that reference this value will be able to access it directly from the value file. No attempt is made to maintain coherence between the value file and main memory. If their contents diverge (due to, for example, external DMAs), the pipeline will continue to operate correctly. Any incoherence will be detected when the renamed load values are compared to the actual memory contents. The initial binding between stores and loads is created when a load that was not renamed references the data produces by a renamed store. We explored two approaches to detecting these new dependence edges. The simplest approach looks for renamed stores that forward to loads in the load/store queue forwarding network (i.e., communications between instructions in flight). When these edges are detected, the store/load cache entry for loads is updated accordingly. A slightly more capable approach is to attach value file indices to renamed store data, and propagate these indices into the memory hierarchy. This approach performs better because it can detect longer-lived dependence edges, however, the extra storage for value file indices makes the approach more expensive. 4.2 Recovering from Mis-speculations When a renamed load injects an incorrect value into the program computation, correct program execution requires that, minimally, all instructions that used the incorrect value and dependent instructions be re-executed. To this end, we explored two approaches to recovering the pipeline from data mis-speculations: squash and re-execution recovery. The two approaches exhibit varying cost and complexity - later we will see that lower cost mis-speculation recovery mechanisms enable higher levels of performance, since they permit the pipeline to promote more memory communication into the register infrastructure. Squash recovery, while expensive in performance penalty, is the simplest approach to implement. The approach works by throwing away all instructions after a mis-speculated load instruction. Since all dependent instructions will follow the load instruction, the restriction that all dependent instructions be re-executed will indeed be met. Unfortunately, this approach can Decode Rename Schedule Writeback Load/Store Queue Entry store/load cache value id conf. to rename to writeback value file LSQ entry LRU (opt) value id to schedule to LRU logic resv stations all loads and stores other insts speculative forwards resv stations load/store queue ld pred ldd st std load/store data (STD/LDD) faults load/store address value file index load/store queue ld pred ldd st std speculative STD forward result non-speculative LDD (from LSQ or mem) if (pred != non-spec), then recover load/store addrs Commit load/store queue ld pred ldd st std to memory system to value file Figure 2: Pipeline Support for Memory Renaming. Shown are the additions made to the baseline pipeline to support memory renaming. The solid edges in the writeback stage represent forwarding through the reservation stations, the dashed lines represent forwarding through the load/store queue. Also shown are the fields added (shown in gray) to the instruction re-order buffer entries. throw away many instructions independent of the mis- speculated load result, requiring many unnecessary re- executions. The advantage of this approach is that it requires very little support over what is implemented today. Mis-speculated loads may be treated the same as mis-speculated branches. Re-execution recovery, while more complex, has significantly lower cost than squash recovery. The approach leverages dependence information stored in the reservation stations of not-yet retired instruction to permit re-execution of only those instructions dependent on a speculative load value. The cost of this approach is added pipeline complexity. We implemented re-execution by injecting the correct result of mis-speculated loads onto the result bus - all dependent instructions receiving the correct load result will re-execute, and re-broadcast their results, forcing dependent instructions to re-execute, and so on. Since it's non-trivial for an instruction to know how many of its operands will be re-generated through re- execution, an instruction may possibly re-execute multiple times, once for every re-generated operand that arrives. In addition, dependencies through memory may require load instructions to re-execute. To accommodate these dependencies, the load/store queue also re-checks memory dependencies of any stores that re-execute, re-issuing any dependent load instructions. Additionally, loads may be forced to re-execute if they receive a new address via instruction re-execution. At retirement, any re-executed instruction will be the oldest instruction in the machine, thus it cannot receive any more re-generated values, and the instruction may be safely retired. In Section 5, we will demonstrate through simulation that re-execution is a much less expensive approach to implementing load mis-speculation recovery. 5 Experimental Evaluation We evaluated the merits of our memory renaming designs by extending a detailed timing simulator to support the proposed designs and by examining the performance of programs running on the extended sim- ulator. We varied the the confidence mechanism, misspeculation recovery mechanism, and key system parameters to see what affect these parameters had on performance. 5.1 Methodology Our baseline simulator is detailed in Table 2. It is from the SimpleScalar simulation suite (simulator sim-outorder) [3]. The simulator executes only user-level instructions, performing a detailed timing simulation of an 4-way superscalar microprocessor with two levels of instruction and data cache memory. The simulator implements an out-of-order issue execution model. Simulation is execution-driven, including execution down any speculative path until the detection of a fault, TLB miss, or mis-prediction. The model employs a 256 entry re-order buffer that implements re-named register storage and holds results of pending in- structions. Loads and stores are placed into a 128 entry load/store queue. In the baseline simulator, stores execute when all operands are ready; their values, if spec- ulative, are placed into the load/store queue. Loads may execute when all prior store addresses have been computed; their values come from a matching earlier store in the store queue (i.e., a store forward) or from the data cache. Speculative loads may initiate cache misses if the address hits in the TLB. If the load is subsequently squashed, the cache miss will still com- plete. However, speculative TLB misses are not per- mitted. That is, if a speculative cache access misses in the TLB, instruction dispatch is stalled until the instruction that detected the TLB miss is squashed or committed. Each cycle the re-order buffer issues up to 8 ready instructions, and commits up to 8 results in-order to the architected register file. When stores are committed, the store value is written into the data cache. The data cache modeled is a four-ported 32k two-way set-associative non-blocking cache. We found early on that instruction fetch bandwidth was a critical performance bottleneck. To mitigate this problem, we implemented a limited variant of the collapsing buffer described in [4]. Our implementation supports two predictions per cycle within the same instruction cache block, which provides significantly more instruction fetch bandwidth and better pipeline resource utilization. When selecting benchmarks, we looked for programs with varying memory system performance, i.e., programs with large and small data sets as well as high and low reference locality. We analyzed 10 programs from the SPEC'95 benchmark suite, 6 from the integer codes and 4 from the floating point suite. All memory renaming experiments were performed with a 1024 entry, 2-way set associative store/load cache and a 512 entry value file with LRU replacement. To detect initial dependence edge bindings, we propagate the value file indices of renamed store data into the top-level data cache. When loads (that were not renamed) access renamed store data, the value file index stored in the data cache is used to update the load's store/load cache entry. 1 5.2 Predictor Performance Figure 3 shows the performance of the memory dependence predictor. The graph shows the hit rate of 1 Due to space restrictions, we have omitted experiments which explore predictor performance sensitivity to structure sizes. The structure sizes selected eliminates most capacity problems in the predictor, allowing us to concentrate on how effectively we can leverage the predictions to improve program performance 10305070CC1 Comp Go Hydro-2D Mgrid Su2Cor Tomcatv Hit Rate Hit Rate Figure 3: Memory Dependence Predictor Performance. the memory dependence predictor for each benchmark, where the hit rate is computed as the number of loads whose sourcing store value was correctly identified after probing the value file. The predictor works quite well, predicting correctly as many as 76% of the pro- gram's memory dependencies - an average of 62% for all the programs. Unlike many of the value predictor mechanisms [8], dependence predictors work well, even better, on floating point programs. To better understand where the dependence predictor was finding its dependence locality, we broke down the correct predictions by the segment in which the reference data resided. Figure 4 shows the breakdown of correct predictions for data residing in the global, stack, and heap segments. A large fraction of the correct dependence predictions, as much as 70% for Mgrid and 41% overall on the average, came from stack references. This result is not surprising considering the frequency of stack segment references and their semi-static na- ture, i.e., loads and stores to the stack often reference the same variable many times. (Later we leverage this property to improve the performance of the confidence mechanisms.) Global accesses also account for many of the correct predictions, as much as 86% for Tomcatv and 43% overall on the average. Finally, a significant number of correct predictions come from the heap seg- ment, as much as 40% for Go and 15% overall on the average. To better understand what aspects of the program resulted in these correct predictions, we profiled top loads and then examined their sourcing stores, we found a number of common cases where heap accesses exhibited dependence locality These examples are typical of program constructs that challenge even the most sophisticated register allocators. As a result, only significant advances in compiler technology will eliminate these memory accesses. The same assertion holds for global ac- Fetch Interface fetches any 4 instructions in up to two cache block per cycle, separated by at most two branches Instruction Cache 32k 2-way set-associative, latency Branch Predictor 8 bit global history indexing a 4096 entry pattern history table (GAp [11]) with 2-bit saturating counters, 8 cycle mis-prediction penalty Out-of-Order Issue out-of-order issue of up to 8 operations per cycle, 256 entry re-order buffer, 128 entry Mechanism load/store queue, loads may execute when all prior store addresses are known Architected Registers floating point Functional Units 8-integer ALU, 4-load/store units, 4-FP adders, 1-integer MULT/DIV, 1-FP MULT/DIV Functional Unit Latency integer ALU-1/1, load/store-2/1, integer MULT-3/1, integer DIV-12/12, FP adder-2/1 Data Cache 32k 2-way set-associative, write-back, write-allocate, latency four-ported, non-blocking interface, supporting one outstanding miss per physical register 4-way set-associative, unified L2 cache, 64 byte blocks, Virtual Memory 4K byte pages, fixed TLB miss latency after earlier-issued instructions complete Table 2: Baseline Simulation Model. 10% 20% 30% 40% 50% 70% 80% 90% 100% CC1 Comp Go Hydro-2D Mgrid Su2Cor Tomcatv Breakdown by Segment Global Stack Heap Figure 4: Predictor Hits by Memory Segment. repeated accesses to aliased data, which cannot be allocated to registers ffl accesses to loop data with a loop dependence distance of one 3 ffl accesses to single-instance dynamic storage, e.g., a variable allocated at the beginning of the program, pointed to by a few, immutable global pointers As discussed in Section 3, a pipeline implementation can also benefit from a confidence mechanism. Figure 5 shows the results of experiments exploring the efficacy of attaching confidence counters to load instruc- cesses, all of which the compiler must assume are aliased. Stack accesses on the other hand, can be effectively register allocated, thereby eliminating the memory accesses, given that the processor has enough registers. 3 Note that since we always predict the sourcing store to be that last previous one, our predictors will not work with loop dependence distances greater than one, even if they are regular accesses. Support for these cases are currently under investigation tions. The graphs show the confidence and coverage for a number of predictors. Confidence is the success rate of high-confidence loads. Coverage is the fraction of correctly predicted loads, without confidence, covered by the high-confidence predictions of a particular predictor. Confidence and coverage are shown for 6 pre- dictors. The notation used for each is as follows: XYZ, where X is the count that must be reached before the predictor considers the load a high-confidence load. By default, the count is incremented by one when the predictor correctly predicts the sourcing store value, and reset to zero when the predictor fails. Y is the count increment used when the opcode of the load indicates an access off the stack pointer. Z is the count increment used when the opcode of the load indicates an access off the global pointer. Our analyses showed that stack and global accesses are well behaved, thus we can increase coverage, without sacrificing much confidence, by incrementing their confidence counters with a value greater than one. As shown in Figure 5, confidence is very high for the configurations examined, as much as 99.02% for Hydro-2D and at least 69.22% for all experiments that use confidence mechanisms. For most of the experiments we tried, increasing the increments for stack and global accesses to half the confidence counter performed best. While this configuration usually degrades confidence over the baseline case (an increment of one for all accesses), coverage is improved enough to improve program performance. Coverage varies significantly, a number of the programs, e.g., Compress and Hydro- 2D, have very high coverage, while others, such as CC1 and Perl do not gain higher coverage until a significant amount of confidence is sacrificed. Another interesting feature of our confidence measurements is the relative insensitivity of coverage to the counter threshold once the confidence thresholds levels rise above 2. This rein- 4Coverage of loads) CC1 Comp Go Perl Hydro-2D507090000 111 211 411 422 844 Confidence of loads) CC1 Comp Go Perl Hydro-2D Figure 5: Confidence and Coverage for Predictors with Confidence Counters. -2261014CC1 Comp Go Hydro-2D Mgrid Su2Cor Tomcatv SQ-422 SQ844 RE-422 RE-211 Figure Performance with Varied Predic- tor/Recovery Configuration. forces our earlier observation that the memory dependencies in a program are relatively static - once they occur a few times, they very often occur in the same fashion for much of the program execution. 5.3 Pipeline Performance Predictor hit rates are an insufficient tool for evaluating the usefulness of a memory dependence predictor. In order to fully evaluate it, we must integrate it into a modern processor pipeline, leverage the predictions it produces, and correctly handle the cases when the predictor fails. Figure 6 details the performance of the memory dependence predictor integrated into the base-line out-of-order issue performance simulator. For each experiment, the figure shows the speedup (in percent, measured in cycles to execute the entire program) with respect to the baseline simulator. Four experiments are shown for each benchmark in Figure 6. The first experiment, labeled SQ-422, shows the speedup found with a dependence predictor utilizing a 422 confidence configuration and squash recovery for load mis-speculations. Experiment SQ-844 is the same experiment except with a 844 confidence mecha- nism. The RE-422 configuration employs a 422 confidence configuration and utilizes the re-execution mechanism described in Section 3 to recover from load mis- speculations. Finally, the RE-211 configuration also employs the re-execution recovery mechanism, but utilizes a lower-confidence 211 confidence configuration. The configuration with squash recovery and the 422 confidence mechanism, i.e., SQ-422, shows small speedups for many of the programs, and falls short on others, such as CC1 which saw a slowdown of more than 5%. A little investigations of these slowdowns quickly revealed that the high-cost of squash recovery, i.e., throwing away all instructions after the mis-speculated load, often completely outweighs the benefits of memory renaming. (Many of the programs had more data mis-speculations than branch mis-predictions!) One remedy to the high-cost of mis-speculation is to permit renaming only for higher confidence loads. The experiment labeled SQ-844 renames higher-confidence loads. This configuration performs better because it suffers from less mis-speculation, however, some exper- iments, e.g., CC1, show very little speedup because they are still plagued with many high-cost load mis- speculations. A better remedy for high mis-speculation recovery costs is a lower cost mis-speculation recovery mech- anism. The experiment labeled RE-422 adds re-execution support to a pipeline with memory renaming support with a 422 confidence mechanism. This design has lower mis-speculation costs, allowing it to to show speedups for all the experiments run, as much as 14% for M88ksim and an average overall speedup of over 6%. To confirm our intuitions as to the lower cost of re-execution, we measured directly the cost of squash recovery and re-execution for all runs by counting the number of instructions thrown away due to load mis-speculations. We found that overall, re-execution consumes less than 1/3 of the execution bandwidth required by squash recovery - in other words, less than 1/3 of the instructions in flight after a load misspeculation are dependent on the mis-speculated load, on average. Additionally, re-execution benefits from not having to re-fetch, decode, and issue instructions after the mis-speculated load. Given the lower cost of cost of re-execution, we explored whether speedups would be improved if we also renamed lower-confidence loads. The experiment labeled RE-211 employs re-execution recovery with a lower-confidence 211 confidence configuration. This configuration found better performance for most of the experiments, further supporting the benefits of re-execution. We also explored the use of yet even lower-confidence (111) and no-confidence (000) config- urations, however, mis-speculation rates rise quickly for these configurations, and performance suffered accordingly for most experiments. Figure 7 takes our best-performing configuration, i.e., RE-211, and varies two key system parameters to see their effect on the efficacy of memory renaming. The first experiment, labeled FE/2, cuts the peak instruction delivery bandwidth of the fetch stage in half. This configuration can only deliver up to four instructions from one basic block per cycle. For many of the experiments, this cuts the average instruction delivery B/W by nearly half. As shown in the results, the effects of memory renaming are severely attenuated. With half of the instruction delivery bandwidth the machine becomes fetch bottlenecked for many of the experiments. Once fetch bottlenecked, improving execution performance with memory renaming does little to improve the performance of the program. This is especially true for the integer codes where fetch bandwidth is very limited due to many small basic blocks. The second experiment in Figure 7, labeled SF*3, increases the store forward latency three-fold to three cy- cles. The store forward latency is the minimumlatency, in cycles, between any two operations that communicate a value to each other through memory. In the base-line experiments of Figure 6, the minimum store forward latency is one cycle. As shown in the graph, performance improvements due to renaming rise sharply, to as much as 41% for M88ksim and more than 16% overall. This sharp rise is due to the increased latency for communicationthrough memory - this latency must CC1 Comp Go Hydro-2D Mgrid Su2Cor Tomcatv Figure 7: Program Performance with Varied System Configuration. be tolerated, which consumes precious parallelism. Re-named memory accesses, however, may communication through the register file in potentially zero cycles (via bypass), resulting in significantly lower communication latencies. Given the complexity of load/store queue dataflow analysis and the requirement that it be performed in one cycle for one-cycle store forwards (since addresses in the computation may arrive in the previous cycle), designers may soon resort to larger load/store queues with longer latency store forwards. This trend will make memory renaming more attractive. A fitting conclusion to our evaluation is to grade ourselves against the goal set forth in the beginning of this paper: build a renaming mechanism that maps all memory communication to the register communication and synchronization infrastructure. It is through this hoisting of the memory communication into the registers that permits more accurate and faster memory communication. To see how successful we were at this goal, we measured the breakdown of communication handled by the load/store queue and the data cache. Memory communications handled by the load/store queue are handled "in flight", thus this communication can benefit from renaming. How did we do? Figure 8 shows for each benchmark, the fraction of references serviced by the load/store queue in the base configu- ration, labeled Base, and the fraction of the references serviced by the load/store queue in the pipeline with renaming support, labeled RE-422. As shown in the figure, a significant amount of the communication is now being handled by the register communication in- frastructure. Clearly, much of the short-term communication is able to benefit from the renamer support. However, a number of the benchmarks, e.g., CC1, Xlisp and Tomcatv, still have a non-trivial amount of CC1 Comp Go Hydro-2D Mgrid Su2Cor Tomcatv of all loads Base RE-422 Figure 8: Percent of Memory Dependencies Serviced by Load/Store Queue. short-term communication that was not identified by the dependence predictor. For these programs, the execution benefits from the load/store queues ability to quickly compute load/store dependencies once addresses are available. One goal of this work should be to improve the performance of the dependence predictor until virtually all short-term communication is captured in the high-confidence predictions. Not only will this continue to improve the performance of memory communication, but once this goal has been attained, the performance of the load/store queue will become less important to overall program performance. As a re- sult, less resources will have to be devoted to load/store queue design and implementation. 6 Conclusions In this paper we have described a new mechanism designed to improve memory performance. This was accomplished by restructuring the processor pipeline to incorporate a speculative value and dependence predictor to enable load instructions to proceed much earlier in the pipeline. We introduce a prediction confidence mechanism based on store-load dependence history to control speculation and a value file containing load and store data which can be efficiently accessed without performing complex address calculations. Simulation results validate this approach to improving memory performance, showing an average application speedup of 16%. We intend to extend this study in a number of ways. The most obvious extension of this work is to identify new mechanisms to improve the confidence mechanism and increase the applicability of this scheme to more load instructions. To do this we are exploring integrating control flow information into the confidence mecha- nism. Another architectural modification is to improve the efficiency of squashing instructions effected by a mis-prediction. This is starting to become important in branch prediction, but becomes more important in value prediction because of the lower confidence in this mechanism. Also, the number of instructions that are directly effected by a misprediction in a load value is less than for a branch prediction allowing greater benefit from a improvement in identifying only those instructions that need to be squashed. Acknowledgments Finally, we would like to acknowledge the help of Haitham Akkary, who offered numerous suggestions which have greatly improved the quality of this work. We are also grateful to the Intel Corporation for its support of this research through the Intel Technology for Education 2000 grant. --R Intel boosts pentium pro to 200 mhz. Evaluating future microprocessors: The simplescalar tool set. Optimization of instruction fetch mechanisms for high issue rates. Reducing memory traffic with cregs. MIPS RISC Architecture. Value locality and load value prediction. Dynamic speculation and synchronization of data dependences. The performance potential of data dependence speculation and collapsing. --TR MIPS RISC architectures Two-level adaptive training branch prediction Reducing memory traffic with CRegs Optimization of instruction fetch mechanisms for high issue rates Zero-cycle loads Value locality and load value prediction The performance potential of data dependence speculation MYAMPERSANDamp; collapsing Dynamic speculation and synchronization of data dependences Look-Ahead Processors --CTR Adi Yoaz , Mattan Erez , Ronny Ronen , Stephan Jourdan, Speculation techniques for improving load related instruction scheduling, ACM SIGARCH Computer Architecture News, v.27 n.2, p.42-53, May 1999 Daniel Ortega , Eduard Ayguad , Mateo Valero, Dynamic memory instruction bypassing, Proceedings of the 17th annual international conference on Supercomputing, June 23-26, 2003, San Francisco, CA, USA Ben-Chung Cheng , Daniel A. Connors , Wen-mei W. Hwu, Compiler-directed early load-address generation, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.138-147, November 1998, Dallas, Texas, United States George Z. Chrysos , Joel S. Emer, Memory dependence prediction using store sets, ACM SIGARCH Computer Architecture News, v.26 n.3, p.142-153, June 1998 Andreas Moshovos , Gurindar S. Sohi, Speculative Memory Cloaking and Bypassing, International Journal of Parallel Programming, v.27 n.6, p.427-456, 1999 Daniel Ortega , Mateo Valero , Eduard Ayguad, Dynamic memory instruction bypassing, International Journal of Parallel Programming, v.32 n.3, p.199-224, June 2004 Gokhan Memik , Mahmut T. Kandemir , Arindam Mallik, Load elimination for low-power embedded processors, Proceedings of the 15th ACM Great Lakes symposium on VLSI, April 17-19, 2005, Chicago, Illinois, USA Jinsuo Zhang, The predictability of load address, ACM SIGARCH Computer Architecture News, v.29 n.4, September 2001 Matt T. Yourst , Kanad Ghose, Incremental Commit Groups for Non-Atomic Trace Processing, Proceedings of the 38th annual IEEE/ACM International Symposium on Microarchitecture, p.67-80, November 12-16, 2005, Barcelona, Spain Anastas Misev , Marjan Gusev, Visual simulator for ILP dynamic OOO processor, Proceedings of the 2004 workshop on Computer architecture education: held in conjunction with the 31st International Symposium on Computer Architecture, June 19, 2004, Munich, Germany Sanjay Jeram Patel , Marius Evers , Yale N. Patt, Improving trace cache effectiveness with branch promotion and trace packing, ACM SIGARCH Computer Architecture News, v.26 n.3, p.262-271, June 1998 Vlad Petric , Anne Bracy , Amir Roth, Three extensions to register integration, Proceedings of the 35th annual ACM/IEEE international symposium on Microarchitecture, November 18-22, 2002, Istanbul, Turkey Dean M. Tullsen , John S. Seng, Storageless value prediction using prior register values, ACM SIGARCH Computer Architecture News, v.27 n.2, p.270-279, May 1999 Enric Morancho , Jos Mara Llabera , ngel Oliv, A comparison of two policies for issuing instructions speculatively, Journal of Systems Architecture: the EUROMICRO Journal, v.53 n.4, p.170-183, April, 2007 Jian Huang , David J. Lilja, Extending Value Reuse to Basic Blocks with Compiler Support, IEEE Transactions on Computers, v.49 n.4, p.331-347, April 2000 Andreas Moshovos , Gurindar S. Sohi, Read-after-read memory dependence prediction, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.177-185, November 16-18, 1999, Haifa, Israel Glenn Reinman , Brad Calder , Dean Tullsen , Gary Tyson , Todd Austin, Classifying load and store instructions for memory renaming, Proceedings of the 13th international conference on Supercomputing, p.399-407, June 20-25, 1999, Rhodes, Greece Daniel Ortega , Mateo Valero , Eduard Ayguad, A novel renaming mechanism that boosts software prefetching, Proceedings of the 15th international conference on Supercomputing, p.501-510, June 2001, Sorrento, Italy Jos Gonzlez , Antonio Gonzlez, The potential of data value speculation to boost ILP, Proceedings of the 12th international conference on Supercomputing, p.21-28, July 1998, Melbourne, Australia Tingting Sha , Milo M. K. Martin , Amir Roth, Scalable Store-Load Forwarding via Store Queue Index Prediction, Proceedings of the 38th annual IEEE/ACM International Symposium on Microarchitecture, p.159-170, November 12-16, 2005, Barcelona, Spain Stephen Jourdan , Ronny Ronen , Michael Bekerman , Bishara Shomar , Adi Yoaz, A novel renaming scheme to exploit value temporal locality through physical register reuse and unification, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.216-225, November 1998, Dallas, Texas, United States Amir Roth , Andreas Moshovos , Gurindar S. Sohi, Dependence based prefetching for linked data structures, ACM SIGOPS Operating Systems Review, v.32 n.5, p.115-126, Dec. 1998 Gabriel Loh, A time-stamping algorithm for efficient performance estimation of superscalar processors, ACM SIGMETRICS Performance Evaluation Review, v.29 n.1, p.72-81, June 2001 S. Stone , Kevin M. Woley , Matthew I. Frank, Address-Indexed Memory Disambiguation and Store-to-Load Forwarding, Proceedings of the 38th annual IEEE/ACM International Symposium on Microarchitecture, p.171-182, November 12-16, 2005, Barcelona, Spain Peng , Jih-Kwon Peir , Qianrong Ma , Konrad Lai, Address-free memory access based on program syntax correlation of loads and stores, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.11 n.3, p.314-324, June Tingting Sha , Milo M. K. Martin , Amir Roth, NoSQ: Store-Load Communication without a Store Queue, Proceedings of the 39th Annual IEEE/ACM International Symposium on Microarchitecture, p.285-296, December 09-13, 2006 Glenn Reinman , Brad Calder, Predictive techniques for aggressive load speculation, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.127-137, November 1998, Dallas, Texas, United States Craig B. Zilles , Gurindar S. Sohi, Understanding the backward slices of performance degrading instructions, ACM SIGARCH Computer Architecture News, v.28 n.2, p.172-181, May 2000 Pedro Marcuello , Antonio Gonzlez , Jordi Tubella, Speculative multithreaded processors, Proceedings of the 12th international conference on Supercomputing, p.77-84, July 1998, Melbourne, Australia Sangyeun Cho , Pen-Chung Yew , Gyungho Lee, Access region locality for high-bandwidth processor memory system design, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.136-146, November 16-18, 1999, Haifa, Israel V. Krishna Nandivada , Jens Palsberg, Efficient spill code for SDRAM, Proceedings of the international conference on Compilers, architecture and synthesis for embedded systems, October 30-November 01, 2003, San Jose, California, USA Andreas Moshovos , Gurindar S. Sohi, Reducing Memory Latency via Read-after-Read Memory Dependence Prediction, IEEE Transactions on Computers, v.51 n.3, p.313-326, March 2002 Sangyeun Cho , Pen-Chung Yew , Gyungho Lee, Decoupling local variable accesses in a wide-issue superscalar processor, ACM SIGARCH Computer Architecture News, v.27 n.2, p.100-110, May 1999 J. Gonzlez , A. Gonzlez, Control-Flow Speculation through Value Prediction, IEEE Transactions on Computers, v.50 n.12, p.1362-1376, December 2001 Sangyeun Cho , Pen-Chung Yew , Gyungho Lee, A High-Bandwidth Memory Pipeline for Wide Issue Processors, IEEE Transactions on Computers, v.50 n.7, p.709-723, July 2001 Lieven Eeckhout , Koen De Bosschere, Quantifying behavioral differences between multimedia and general-purpose workloads, Journal of Systems Architecture: the EUROMICRO Journal, v.48 n.6-7, p.199-220, January

address calculation;heap segment;instruction-level parallelism;stores;storage allocation;performance;data fetching;execution time;data value speculation;data dependence speculation;instruction loading;memory renaming;memory references;processor pipeline;memory communication;memory segments;delays;memory access latency;register access techniques;accuracy

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The predictability of data values.

The predictability of data values is studied at a fundamental level. Two basic predictor models are defined: Computational predictors perform an operation on previous values to yield predicted next values. Examples we study are stride value prediction (which adds a delta to a previous value) and last value prediction (which performs the trivial identity operation on the previous value); Context Based} predictors match recent value history (context) with previous value history and predict values based entirely on previously observed patterns. To understand the potential of value prediction we perform simulations with unbounded prediction tables that are immediately updated using correct data values. Simulations of integer SPEC95 benchmarks show that data values can be highly predictable. Best performance is obtained with context based predictors; overall prediction accuracies are between 56% and 91%. The context based predictor typically has an accuracy about 20% better than the computational predictors (last value and stride). Comparison of context based prediction and stride prediction shows that the higher accuracy of context based prediction is due to relatively few static instructions giving large improvements; this suggests the usefulness of hybrid predictors. Among different instruction types, predictability varies significantly. In general, load and shift instructions are more difficult to predict correctly, whereas add instructions are more predictable.

Introduction There is a clear trend in high performance processors toward performing operations speculatively, based on predic- tions. If predictions are correct, the speculatively executed instructions usually translate into improved performance. Although program execution contains a variety of information that can be predicted, conditional branches have received the most attention. Predicting conditional branches provides a way of avoiding control dependences and offers a clear performance advantage. Even more prevalent than control dependences, however, are data dependences. Virtually every instruction depends on the result of some preceding instruction. As such, data dependences are often thought to present a fundamental performance barrier. However, data values may also be predicted, and operations can be performed speculatively based on these data predictions. An important difference between conditional branch prediction and data value prediction is that data are taken from a much larger range of values. This would appear to severely limit the chances of successful prediction. How- ever, it has been demonstrated recently [1] that data values exhibit "locality" where values computed by some instructions tend to repeat a large fraction of the time. We argue that establishing predictability limits for program values is important for determining the performance potential of processors that use value prediction. We believe that doing so first requires understanding the design space of value predictors models. Consequently, the goals of this paper are twofold. Firstly, we discuss some of the major issues affecting data value prediction and lay down a framework for studying data value prediction. Secondly, for important classes of predictors, we use benchmark programs to establish levels of value predictability. This study is somewhat idealized: for example, predictor costs are ignored in order to more clearly understand limits of data predictability. Furthermore, the ways in which data prediction can be used in a processor microarchitecture are not within the scope of this paper, so that we can concentrate in greater depth on the prediction process, itself. 1.1 Classification of Value Sequences The predictability of a sequence of values is a function of both the sequence itself and the predictor used to predict the sequence. Although it is beyond the scope of this paper to study the actual sources of predictability, it is useful for our discussion to provide an informal classification of data sequences. This classification is useful for understanding the behavior of predictors in later discussions. The following classification contains simple value sequences that can also be composed to form more complex sequences. They are best defined by giving examples: 28 -13 -99 107 23 456. Constant sequences are the simplest, and result from instructions that repeatedly produce the same result. Lipasti and Shen show that this occurs surprisingly often, and forms the basis for their work reported in [1]. A stride sequence has elements that differ by some constant delta. For the example above, the stride is one, which is probably the most common case in programs, but other strides are pos- sible, including negative strides. Constant sequences can be considered stride sequences with a zero delta. A stride sequence might appear when a data structure such as an array is being accessed in a regular fashion; loop induction variables also have a stride characteristic. The non-stride category is intended to include all other sequences that do not belong to the constant or stride cat- egory. This classification could be further divided, but we choose not to do so. Non-strides may occur when a sequence of numbers is being computed and the computation is more complex than simply adding a constant. Traversing a linked list would often produce address values that have a non-stride pattern. Also very important are sequences formed by composing stride and non-stride sequences with themselves. Repeating sequences would typically occur in nested loops where the inner loop produces either a stride or a non-stride sequence, and the outer loop causes this sequence to be repeated Repeated Repeated 7. Examination of the above sequences leads naturally to two types of prediction models that are the subject of discussion throughout the remainder of this paper: Computational predictors that make a prediction by computing some function of previous values. An example of a computational predictor is a stride predictor. This predictor adds a stride to the previous value. Context based predictors learn the value(s) that follow a particular context - a finite ordered sequence of values - and predict one of the values when the same context repeats. This enables the prediction of any repeated sequence, stride or non-stride. 1.2 Related Work In [1], it was reported that data values produced by instructions exhibit "locality" and as a result can be pre- dicted. The potential for value predictability was reported in terms of "history depth", that is, how many times a value produced by an instruction repeats when checked against the most recent n values. A pronounced difference is observed between the locality with history depth 1 and history depth 16. The mechanism proposed for prediction, how- ever, exploits the locality of history depth 1 and is based on predicting that the most recent value will also be the next. In [1], last value prediction was used to predict load values and in a subsequent work to predict all values produced by instructions and written to registers [2]. Address prediction has been used mainly for data prefetching to tolerate long memory latency [3, 4, 5], and has been proposed for speculative execution of load and store instructions [6, 7]. Stride prediction for values was proposed in [8] and its prediction and performance potential was compared against last value prediction. Value prediction can draw from a wealth of work on the prediction of control dependences [9, 10, 11]. The majority of improvements in the performance of control flow predictors has been obtained by using correlation. The correlation information that has been proposed includes local and global branch history [10], path address history [11, 12, 13], and path register contents [14]. An interesting theoretical observation is the resemblance of the predictors used for control dependence prediction to the prediction models for text compression [15]. This is an important observation because it re-enforces the approach used for control flow prediction and also suggests that compression-like methods can also be used for data value prediction. A number of interesting studies report on the importance of predicting and eliminating data dependences. Moshovos [16] proposes mechanisms that reduce misspeculation by predicting when dependences exist between store and load instructions. The potential of data dependence elimination using prediction and speculation in combination with collapsing was examined in [17]. Elimination of redundant computation is the theme of a number of software/hardware proposals [18, 19, 20]. These schemes are similar in that they store in a cache the input and output parameters of a function and when the same inputs are detected the output is used without performing the function. Virtually all proposed schemes perform predictions based on previous architected state and values. Notable exceptions to this are the schemes proposed in [6], where it is predicted that a fetched load instruction has no dependence and the instruction is executed "early" without dependence checking, and in [21], where it is predicted that the operation required to calculate an effective address using two operands is a logical or instead of a binary addition. In more theoretical work, Hammerstrom [22] used information theory to study the information content (en- tropy) of programs. His study of the information content of address and instruction streams revealed a high degree of redundancy. This high degree of redundancy immediately suggests predictability. 1.3 Paper Overview The paper is organized as follows: in Section 2, different data value predictors are described. Section 3 discusses the methodology used for data prediction simulations. The results obtained are presented and analyzed in Section 4. We conclude with suggestions for future research in Section 5. 2 Data Value Prediction Models A typical data value predictor takes microarchitecture state information as input, accesses a table, and produces a prediction. Subsequently, the table is updated with state information to help make future predictions. The state information could consist of register values, PC values, instruction fields, control bits in various pipeline stages, etc. The variety and combinations of state information are almost limitless. Therefore, in this study, we restrict ourselves to predictors that use only the program counter value of the instruction being predicted to access the prediction table(s). The tables are updated using data values produced by the instruction - possibly modified or combined with other information already in the table. These restrictions define a relatively fundamental class of data value predic- tors. Nevertheless, predictors using other state information deserve study and could provide a higher level of predictability than is reported here. For the remainder of this paper, we further classify data value predictors into two types - computational and context-based. We describe each in detail in the next two subsections. 2.1 Computational Predictors Computational predictors make predictions by performing some operation on previous values that an instruction has generated. We focus on two important members of this class. Last Value Predictors perform a trivial computational operation: the identity function. In its simplest form, if the most recent value produced by an instruction is v then the prediction for the next value will also be v. However, there are a number of variants that modify replacement policies based on hysteresis. An example of a hysteresis mechanism is a saturating counter that is associated with each table entry. The counter is incremented/decremented on prediction success/failure with the value held in the table replaced only when the count is below some threshold. Another hysteresis mechanism does not change its prediction to a new value until the new value occurs a specific number of times in succession. A subtle difference between the two forms of hysteresis is that the former changes to a new prediction following incorrect behavior (even though that behavior may be inconsistent), whereas the latter changes to a new prediction only after it has been consistently observed Stride Predictors in their simplest form predict the next value by adding the sum of the most recent value to the difference of the two most recent values produced by an instruction. That is if vn\Gamma1 and vn\Gamma2 are the two most recent values, then the predictor computes As with the last value predictors, there are important variations that use hysteresis. In [7] the stride is only changed if a saturating counter that is incre- mented/decremented on success/failure of the predictions is below a certain threshold. This reduces the number of mispredictions in repeated stride sequences from two per repeated sequence to one. Another policy, the two-delta method, was proposed in [6]. In the two-delta method, two strides are maintained. The one stride (s1) is always up-dated by the difference between the two most recent val- ues, whereas the other (s2) is the stride used for computing the predictions. When stride s1 occurs twice in a row then it is used to update the prediction stride s2. The two-delta strategy also reduces mispredictions to one per iteration for repeated stride sequences and, in addition, only changes the stride when the same stride occurs twice - instead of changing the stride following mispredictions. Other Computational Predictors using more complex organizations can be conceived. For example, one could use two different strides, an "inner" one and an "outer" one - typically corresponding to loop nests - to eliminate the mispredictions that occur at the beginning of repeating stride sequences. This thought process illustrates a significant limitation of computational prediction: the designer must anticipate the computation to be used. One could carry this to ridiculous extremes. For example, one could envision a Fibonacci series predictor, and given a program that happens to compute a Fibonacci series, the predictor would do very well. Going down this path would lead to large hybrid predictors that combine many special-case computational predictors with a "chooser" - as has been proposed for conditional branches in [23, 24]. While hybrid prediction for data values is in general a good idea, a potential pitfall is that it may yield an ever-escalating collection of computational predictors, each of which predicts a diminishing number of additional values not caught by the others. In this study, we focus on last value and stride methods as primary examples of computational predictors. We also consider hybrid predictors involving these predictors and the context based predictors to be discussed in the next section. 2.2 Context Based Predictors Context based predictors attempt to "learn" values that follow a particular context - a finite ordered sequence of previous values - and predict one of the values when the same context repeats. An important type of context based predictors is derived from finite context methods used in text compression [25]. Finite Context Method Predictors (fcm) rely on mechanisms that predict the next value based on a finite number of preceding values. An order k fcm predictor uses k preceding values. Fcms are constructed with counters that count the occurrences of a particular value immediately following a certain context (pattern). Thus for each context there must be, in general, as many counters as values that are found to follow the context. The predicted value is the one with the maximum count. Figure 1 shows fcm models of different orders and predictions for an example sequence. In an actual implementation where it may be infeasible to maintain exact value counts, smaller counters may be used. The use of small counters comes from the area of text compression. With small counters, when one counter reaches the maximum count, all counters for the same context are reset by half. Small counters provide an advantage if heavier weighting should be given to more recent history instead of the entire history. In general, n different fcm predictors of orders 0 to n- 1 can be used for predicting the next value of a sequence, with the highest order predictor that has a context match being used to make the prediction. The combination of more than one prediction model is known as blending [25]. There are a number of variations of blending algorithms, depending on the information that is updated. Full blending updates all contexts, and lazy exclusion selects the prediction with the longer context match and only updates the counts for the predictions with the longer match or higher. Other variations of fcm predictors can be devised by reducing the number of values that are maintained for a given context. For example, only one value per context might be maintained along with some update policy. Such policies can be based on hysteresis-type update policies as discussed above for last value and stride prediction. Correlation predictors used for control dependence prediction strongly resemble context based prediction. As far as we know, context based prediction has not been considered for value prediction, though the last value predictor can be viewed as a 0th order fcm with only one prediction maintained per context. 2.3 An Initial Analysis At this point, we briefly analyze and compare the proposed predictors using the simple pattern sequences shown in Section 1.1. This analysis highlights important issues as a ca b c a a a b a c b a c a c ca b c a b c0 0 a a a a a b a b c b c ac a a 0th order Model a b c Context Next Symbol FrequencySequence:a a a b c a a a b c a a a ? 1st order Model 2nd order Model 3rd order Model Prediction: a Prediction: a Prediction: a Figure 1: Finite Context Models well as advantages and disadvantages of the predictors to be studied. As such, they can provide a basis for analyzing quantitative results given in the following sections. We informally define two characteristics that are important for understanding prediction behavior. One is the Learning Time (LT) which is the number of values that have to be observed before the first correct prediction. The second is the Learning Degree (LD) which is the percentage of correct predictions following the first correct prediction We quantify these two characteristics for the classes of sequences given earlier in Section 1.1. For the repeating sequences, we associate a period (p), the number of values between repetitions, and frequency, the number of times a sequence is repeated. We assume repeating sequences where p is fixed. The frequency measure captures the finiteness of a repeating sequence. For context predictors, the order (o) of a predictor influences the learning time. Table summarizes how the different predictors perform for the basic value sequences. Note that the stride predictor uses hysteresis for updates, so it gets only one incorrect prediction per iteration through a sequence. A row of the table with a "-" indicates that the given predictor is not suitable for the given sequence, i.e., its performance is very low for that sequence. As illustrated in the table, last value prediction is only useful for constant sequences - this is obvious. Stride prediction does as well as last value prediction for constant sequences because a constant sequence is essentially zero stride. The fcm predictors also do very well on constant sequences, but an order predictor must see a length sequence before it gets matches in the table (unless some form of blending is used). For (non-repeating) stride sequences, only the stride Prediction Model Sequence Last Value Stride FCM Table 1: Behavior of various Prediction Models for Different Value Sequences predictor does well; it has a very short learning time and then achieves a 100% prediction rate. The fcm predictors cannot predict non-repeating sequences because they rely on repeating patterns. For repeating stride sequences, both stride and fcm predictors do well. The stride predictor has a shorter learning time, and once it learns, it only gets a misprediction each time the sequence begins to repeat. On the other hand, the fcm predictor requires a longer learning time - it must see the entire sequence before it starts to predict correctly but once the sequence starts to repeat, it gets 100% accuracy (Figure 2). This example points out an important tradeoff between computational and context based predic- tors. The computational predictor often learns faster - but the context predictor tends to learn "better" when repeating sequences occur. Finally, for repeating non-stride sequences, only the fcm predictor does well. And the flexibility this provides is clearly the strong point of fcm predictors. Returning to our Fibonacci series example - if there is a sequence containing a repeating portion of the Fibonacci series, then an fcm predictor will naturally begin predicting it correctly following the first pass through the sequence. Of course, in reality, value sequences can be complex combinations of the simple sequences in Section 1.1, and a given program can produce about as many different sequences as instructions are being predicted. Consequently, in the remainder of the paper, we use simulations to get a more realistic idea of predictor performance for programs. 3 Simulation Methodology We adopt an implementation-independent approach for studying predictability of data dependence values. The reason for this choice is to remove microarchitecture and other implementation idiosyncrasies in an effort to develop a basic understanding of predictability. Hence, these results can best be viewed as bounds on performance; it will take additional engineering research to develop realistic implementations Steady State Repeats Same Mistake Repeated Steady State Misspredictions period Learn Prediction CONTEXT BASED Figure 2: Computational vs Context Based Prediction We study the predictability of instructions that write results into general purpose registers (i.e. memory addresses, stores, jumps and branches are not considered). Prediction was done with no table aliasing; each static instruction was given its own table entry. Hence, table sizes are effectively unbounded. Finally, prediction tables are updated immediately after a prediction is made, unlike the situation in practice where it may take many cycles for the actual data value to be known and available for prediction table updates We simulate three types of predictors: last value prediction (l) with an always-update policy (no hysteresis), stride prediction using the 2-delta method (s2), and a finite context method (fcm) that maintains exact counts for each value that follows a particular context and uses the blending algorithm with lazy exclusion, described in Section 2. Fcm predictors are studied for orders 1, 2 and 3. To form a context for the fcm predictor we use full concatenation of history values so there is no aliasing when matching contexts. Trace driven simulation was conducted using the Simplescalar toolset [26] for the integer SPEC95 benchmarks shown in Table 2 1 . The benchmarks were compiled using the simplescalar compiler with -O3 optimization. Integer benchmarks were selected because they tend to have less data parallelism and may therefore benefit more from data predictions. For collecting prediction results, instruction types were grouped into categories as shown in Table 3. The ab- 1 For ijpeg the simulations used the reference flags with the following changes: compression.quality 45 and compression.smoothing factor 45. Benchmark Input Dynamic Instructions Flags Instr. (mil) Predicted (mil) compress 30000 e 8.2 5.8 (71%) gcc gcc.i 203 137 (68%) ijpeg specmun.ppm 129 108 (84%) m88k ctl.raw 493 345 (70%) xlisp 7 queens 202 125 (62%) Table 2: Benchmarks Characteristics Instruction Types Code Addition, Subtraction AddSub Loads Loads And, Or, Xor, Nor Logic Shifts Shift Compare and Set Set Multiply and Divide MultDiv Load immediate Lui Floating, Jump, Other Other Table 3: Instruction Categories breviations shown after each group will be used subsequently when results are presented. The percentage of predicted instructions in the different benchmarks ranged between 62%-84%. Recall that some instructions like stores, branches and jumps are not predicted. A breakdown of the static count and dynamic percentages of predicted instruction types is shown in Tables 4-5. The majority of predicted values are the results of addition and load instructions. We collected results for each instruction type. However, we do not discuss results for the other, multdiv and lui instruction types due to space limitations. In the benchmarks we studied, the multdiv instructions are not a significant contributor to dynamic instruction count, and the lui and "other" instructions rarely generate more than one unique value and are over 95% predictable by all predictors. We note that the effect of these three types of instructions is included in the calculations for the overall results. For averaging we used arithmetic mean, so each benchmark effectively contributes the same number of total predictions 4 Simulation Results 4.1 Predictability Figure 3 shows the overall predictability for the selected benchmarks, and Figures 4-7 show results for the important instruction types. From the figures we can draw a number Type com gcc go ijpe m88k perl xlis Loads 686 29138 9929 3645 2215 3855 1432 Logic 149 2600 215 278 674 460 157 MultDi 19 313 196 222 77 26 25 Other 108 5848 1403 517 482 778 455 Table 4: Predicted Instructions - Static Count Type com gcc go ijpe m88k perl xlis Loads 20.5 38.6 26.2 21.4 24.8 43.1 48.6 Logic 3.1 3.1 0.5 1.9 5.0 3.1 3.4 Shift 17.4 7.7 13.3 16.4 3.2 8.2 3.2 Set 7.4 5.4 4.9 4.2 15.2 5.6 3.2 Lui 3.3 3.7 11.4 0.2 6.9 2.4 0.8 Other 5.7 2.1 1.3 0.3 2.1 3.3 4.8 Table 5: Predicted Instructions - Dynamic(%) of conclusions. Overall, last value prediction is less accurate than stride prediction, and stride prediction is less accurate than fcm prediction. Last value prediction varies in accuracy from about 23% to 61% with an average of about 40%. This is in agreement with the results obtained in [2]. Stride prediction provides accuracy of between 38% and 80% with an average of about 56%. Fcm predictors of orders 1, 2, and 3 all perform better than stride prediction; and the higher the order, the higher the accuracy. The order 3 predictor is best and gives accuracies of between 56% and over 90% with an average of 78%. For the three fcm predictors studied, improvements diminish as the order is increased. In particular, we observe that for every additional value in the context the performance gain is halved. The effect on predictability with increasing order is examined in more detail in Section 4.4. Performance of the stride and last value predictors varies significantly across different instruction types for the same benchmark. The performance of the fcm predictors varies less significantly across different instruction types for the same benchmark. This reflects the flexibility of the fcm predictors - they perform well for any repeating sequence, not just strides. In general both stride and fcm prediction appear to have higher predictability for add/subtracts than loads. Logical instructions also appear to be very predictable especially by the fcm predictors. Shift instructions appear to be the most difficult to predict. Stride prediction does particularly well for add/subtract compress cc1 go ijpeg m88k perl xlisp %of Predictions Figure 3: Prediction Success for All Instructions instructions. But for non-add/subtract instructions the performance of the stride predictor is close to last value pre- diction. This indicates that when the operation of a computational predictor matches the operation of the instruction (e.g. addition), higher predictability can be expected. This suggests new computational predictors that better capture the functionality of non-add/subtract instructions could be useful. For example, for shifts a computational predictor might shift the last value according to the last shift distance to arrive at a prediction. This approach would tend to lead to hybrid predictors, however, with a separate component predictor for each instruction type. 4.2 Correlation of Correctly Predicted Sets In effect, the results in the previous section essentially compare the sizes of the sets of correctly predicted values. It is also interesting to consider relationships among the specific sets of correctly predicted values. Primarily, these relationships suggest ways that hybrid predictors might be constructed - although the actual construction of hybrid predictors is beyond the scope of this paper. The predicted set relationships are shown in Figure 8. Three predictors are used: last value, stride (delta-2), and fcm (order 3). All subsets of predictors are represented. Specifically: l is the fraction of predictions for which only the last value predictor is correct; s and f are similarly defined for the stride and fcm predictors respectively; ls is the fraction of predictions for which both the last value and the stride predictors are correct but the fcm predictor is not; lf and sf are similarly defined; lsf is the fraction of predictions for which all predictors are correct; and np is the fraction for which none of the predictors is correct. In the figure results are averaged over all benchmarks, but the qualitative conclusions are similar for each of the1030507090compress cc1 go ijpeg m88k perl xlisp %of Predictions Figure 4: Prediction Success for Add/Subtract Instructions1030507090compress cc1 go ijpeg m88k perl xlisp %of Predictions Figure 5: Prediction Success for Loads Instructions1030507090compress cc1 go ijpeg m88k perl xlisp %of Predictions (Logic) Figure Prediction Success for Logic Instructions1030507090compress cc1 go ijpeg m88k perl xlisp %of Predictions Figure 7: Prediction Success for Shift Instructions All AddSu Loads Logic Shift Set Predictions s ls lf sf lsf Figure 8: Contribution of different Predictors individual benchmarks. Overall, Figure 8 can be briefly summarized: small number, close to 18%, of values are not predicted correctly by any model. ffl A large portion, around 40%, of correct predictions is captured by all predictors. ffl A significant fraction, over 20%, of correct predictions is only captured by fcm. ffl Stride and last value prediction capture less than 5% of the correct predictions that fcm misses. The above confirms that data values are very pre- dictable. And it appears that context based prediction is necessary for achieving the highest levels of predictabil- ity. However, almost 60% of the correct predictions are also captured by the stride predictor. Assuming that context based prediction is the more expensive approach, this suggest that a hybrid scheme might be useful for enabling high prediction accuracies at lower cost. That is, one should try to use a stride predictor for most predictions, and use fcm prediction to get the remaining 20%. Another conclusion is that last value prediction adds very little to what the other predictors achieve. So, if either stride or fcm prediction is implemented, there is no point in adding last value prediction to a hybrid predictor. The important classes of load and add instructions yield results similar to the overall average. Finally, we note that for non-add/subtract instructions the contribution of stride prediction is smaller, this is likely due to the earlier observation that stride prediction does not match the func-20406080100 % of Static Instructions that FCM does better than Stride Normalized AddSub Loads Logic Figure 9: Cumulative Improvement of FCM over Stride tionality of other instruction types. This suggests a hybrid predictor based on instruction types. Proceeding along the path of a hybrid fcm-stride pre- dictor, one reasonable approach would be to choose among the two component predictors via the PC address of the instruction being predicted. This would appear to work well if the performance advantage of the fcm predictor is due to a relatively small number of static instructions. To determine if this is true, we first constructed a list of static instructions for which the fcm predictor gives better performance. For each of these static instructions, we determined the difference in prediction accuracy between fcm and stride. We then sorted the static instructions in descending order of improvement. Then, in Figure 9 we graph the cumulative fraction of the total improvement versus the accumulated percentage of static instructions. The graph shows that overall, about 20% of the static instructions account for about 97% of the total improvement of fcm over stride prediction. For most of individual instruction types, the result is similar, with shifts showing slightly worse performance. The results do suggest that improvements due to context based prediction are mainly due to a relatively small fraction of static instructions. Hence, a hybrid fcm-stride predictor with choosing seems to be a good approach. 4.3 Value Characteristics At this point, it is clear that context based predictors perform well, but may require large tables that store history values. We assume unbounded tables in our study, but when real implementations are considered, of course this will not be possible. To get a handle on this issue, we study the value characteristics of instructions. In particu- Predicted Instructions >65536163841024644 Figure 10: Values and Instruction Behavior lar we report on the number of unique values generated by predicted instructions. The overall numbers of different values could give a rough indication of the numbers of values that might have to be stored in a table. In the left half of Figure 10, we show the number different values produced by percentages of static instructions (an s prefix). In the right half, we determine the fractions of dynamic instructions (a d prefix) that correspond to each of the static categories. From the figure, we observe: ffl A large number, 50%, of static instructions generate only one value. ffl The majority of static instructions, 90%, generate fewer than 64 values. ffl The majority, 50%, of dynamic instructions correspond to static instructions that generate fewer than values. ffl Over 90% of the dynamic instructions are due to static instructions that generate at most 4096 unique values. ffl The number of values generated varies among instruction types. In general add/subtract and load instructions generate more values as compared with logic and shift operations. ffl The more frequently an instruction executes the more values it generates. The above suggest that a relatively small number of values would be required to predict correctly the majority of dynamic instructions using context based prediction - a positive result. From looking at individual benchmark results (not shown) there appears to be a positive correlation between programs that are more difficult to predict and the programs that produce more values. For example, the highly predictable m88ksim has many more instructions that produce few values as compared with the less predictable gcc and go. This would appear to be an intuitive result, but there may be cases where it does not hold; for example if values are generated in a fashion that is predictable with computational predictors or if a small number of values occur in many different sequences. 4.4 Sensitivity Experiments for Context Based Prediction In this section we discuss the results of experiments that illustrate the sensitivity of fcm predictors to input data and predictor order. For these experiments, we focus on the gcc benchmark and report average correct predictions among all instruction types. Sensitivity to input data: We studied the effects of different input files and flags on correct prediction. The fcm predictor used in these experiments was order 2. The prediction accuracy and the number of predicted instructions for the different input files is shown in Table 6. The fraction of correct predictions shows only small variations across the different input files. We note that these results are for unbounded tables, so aliasing affects caused by different data set sizes will not appear. This may not be the case with fixed table sizes. In Table 7 we show the predictability for gcc for the same input file, but with different compilation flags, again using an order 2 fcm predictor. The results again indicate that variations are very small. Sensitivity to the order: experiments were performed for increasing order for the same input file (gcc.i) and flags. The results for the different orders are shown in Figure 11. The experiment suggests that higher order means better performance but returns are diminishing with increasing order. The above also indicate that few previous values are required to predict well. Conclusions We considered representatives from two classes of prediction models: (i) computational and (ii) context based. Simulations demonstrate that values are potentially highly predictable. Our results indicate that context based prediction outperforms previously proposed computational prediction (stride and last value) and that if high prediction correctness is desired context methods probably need to be used either alone or in a hybrid scheme. The obtained results also indicate that the performance of computational prediction varies between instruction types indicating that File Predictions (mil) Correct (%) recog.i 192 78.6 stmt.i 372 77.8 Table of 126.gcc to Different Input Files Flags Predictions (mil) Correct (%) none ref flags 137 77.1 Table 7: Sensitivity of 126.gcc to Input Flags with input file gcc.i72768084 Order Prediction Accuracy Figure 11: Sensitivity of 126.gcc to the Order with input file gcc.i its performance can be further improved if the prediction function matches the functionality of the predicted instruc- tion. Analysis of the improvements of context prediction over computational prediction suggest that about 20% of the instructions that generate relatively few values are responsible for the majority of the improvement. With respect to the value characteristics of instructions, we observe that the majority of instructions do not generate many unique values. The number of values generated by instructions varies among instructions types. This result suggests that different instruction types need to be studied separately due to the distinct predictability and value behavior. We believe that value prediction has significant potential for performance improvement. However, a lot of innovative research is needed for value prediction to become an effective performance approach. 6 Acknowledgements This work was supported in part by NSF Grants MIP- 9505853 and MIP-9307830 and by the U.S. Army Intelligence Center and Fort Huachuca under Contract DABT63- 95-C-0127 and ARPA order no. D346. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or im- plied, of the U. S. Army Intelligence Center and Fort Huachuca, or the U.S. Government. The authors would like to thank Stamatis Vassiliadis for his helpful suggestions and constructive critique while this work was in progress. --R "Value locality and data speculation," "Exceeding the dataflow limit via value prediction," "Effective hardware-based data prefetching for high performance processors," "Examination of a memory access classification scheme for pointer intensive and numeric programs," "Prefetching using markov pre- dictors," "A load instruction unit for pipelined processors," "Speculative execution via address prediction and data prefetching," "Speculative execution based on value prediction," "A study of branch prediction strategies," "Alternative implementations of two-level adaptive branch prediction," "Target prediction for indirect jumps," "Improving the accuracy of static branch prediction using branch correlation," "Dynamic path-based branch correlation," "Compiler synthesized dynamic branch prediction," "Analysis of branch prediction via data compression," "Dynamic speculation and synchronization of data depen- dences," "The performance potential of data dependence speculation & collaps- ing," "An architectural alternative to optimizing compilers," "Caching function results: Faster arithmetic by avoiding unnecessary computation," "Dynamic instruction reuse," "Zero-cycle loads: Microarchitecture support for reducing load latency," "Information content of cpu memory referencing behavior," "Combining branch predictors," "Using hybrid branch predictors to improve branch prediciton in the presence of context switches," "Evaluating future microprocessors: The simplescalar tool set," --TR Text compression Alternative implementations of two-level adaptive branch prediction Improving the accuracy of static branch prediction using branch correlation Dynamic path-based branch correlation Zero-cycle loads Using hybrid branch predictors to improve branch prediction accuracy in the presence of context switches Analysis of branch prediction via data compression Value locality and load value prediction Examination of a memory access classification scheme for pointer-intensive and numeric programs Compiler synthesized dynamic branch prediction Exceeding the dataflow limit via value prediction The performance potential of data dependence speculation MYAMPERSANDamp; collapsing Speculative execution via address prediction and data prefetching Dynamic speculation and synchronization of data dependences Dynamic instruction reuse Prefetching using Markov predictors Target prediction for indirect jumps Effective Hardware-Based Data Prefetching for High-Performance Processors An architectural alternative to optimizing compilers A study of branch prediction strategies Information content of CPU memory referencing behavior --CTR G. 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Smith, Improving branch predictors by correlating on data values, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.28-37, November 16-18, 1999, Haifa, Israel Youfeng Wu , Dong-Yuan Chen , Jesse Fang, Better exploration of region-level value locality with integrated computation reuse and value prediction, ACM SIGARCH Computer Architecture News, v.29 n.2, p.98-108, May 2001 Freddy Gabbay , Avi Mendelson, The effect of instruction fetch bandwidth on value prediction, ACM SIGARCH Computer Architecture News, v.26 n.3, p.272-281, June 1998 Parthasarathy Ranganathan , Sarita Adve , Norman P. Jouppi, Reconfigurable caches and their application to media processing, ACM SIGARCH Computer Architecture News, v.28 n.2, p.214-224, May 2000 Huiyang Zhou , Thomas M. 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Sam , Martin Burtscher, Improving memory system performance with energy-efficient value speculation, ACM SIGARCH Computer Architecture News, v.33 n.4, November 2005 Madhu Mutyam , Vijaykrishnan Narayanan, Working with process variation aware caches, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France Tanaus Ramrez , Alex Pajuelo , Oliverio J. 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Conte, Detecting global stride locality in value streams, ACM SIGARCH Computer Architecture News, v.31 n.2, May Ilya Ganusov , Martin Burtscher, Future execution: A prefetching mechanism that uses multiple cores to speed up single threads, ACM Transactions on Architecture and Code Optimization (TACO), v.3 n.4, p.424-449, December 2006 Afrin Naz , Krishna Kavi , JungHwan Oh , Pierfrancesco Foglia, Reconfigurable split data caches: a novel scheme for embedded systems, Proceedings of the 2007 ACM symposium on Applied computing, March 11-15, 2007, Seoul, Korea Brad Calder , Glenn Reinman , Dean M. Tullsen, Selective value prediction, ACM SIGARCH Computer Architecture News, v.27 n.2, p.64-74, May 1999 Pedro Marcuello , Antonio Gonzlez, Clustered speculative multithreaded processors, Proceedings of the 13th international conference on Supercomputing, p.365-372, June 20-25, 1999, Rhodes, Greece Andreas Moshovos , Gurindar S. Sohi, Reducing Memory Latency via Read-after-Read Memory Dependence Prediction, IEEE Transactions on Computers, v.51 n.3, p.313-326, March 2002 Martin Burtscher , Amer Diwan , Matthias Hauswirth, Static load classification for improving the value predictability of data-cache misses, ACM SIGPLAN Notices, v.37 n.5, May 2002 Jian Huang , David J. Lilja, Balancing Reuse Opportunities and Performance Gains with Subblock Value Reuse, IEEE Transactions on Computers, v.52 n.8, p.1032-1050, August Smruti R. Sarangi , Wei Liu, Josep Torrellas , Yuanyuan Zhou, ReSlice: Selective Re-Execution of Long-Retired Misspeculated Instructions Using Forward Slicing, Proceedings of the 38th annual IEEE/ACM International Symposium on Microarchitecture, p.257-270, November 12-16, 2005, Barcelona, Spain Timothy Sherwood , Suleyman Sair , Brad Calder, Predictor-directed stream buffers, Proceedings of the 33rd annual ACM/IEEE international symposium on Microarchitecture, p.42-53, December 2000, Monterey, California, United States Daehyun Kim , Mainak Chaudhuri , Mark Heinrich, Leveraging cache coherence in active memory systems, Proceedings of the 16th international conference on Supercomputing, June 22-26, 2002, New York, New York, USA Zhang , Rajiv Gupta, Whole execution traces and their applications, ACM Transactions on Architecture and Code Optimization (TACO), v.2 n.3, p.301-334, September 2005 Sangyeun Cho , Pen-Chung Yew , Gyungho Lee, A High-Bandwidth Memory Pipeline for Wide Issue Processors, IEEE Transactions on Computers, v.50 n.7, p.709-723, July 2001 Sang-Jeong Lee , Pen-Chung Yew, On Augmenting Trace Cache for High-Bandwidth Value Prediction, IEEE Transactions on Computers, v.51 n.9, p.1074-1088, September 2002 Lucian Codrescu , D. Scott Wills , James Meindl, Architecture of the Atlas Chip-Multiprocessor: Dynamically Parallelizing Irregular Applications, IEEE Transactions on Computers, v.50 n.1, p.67-82, January 2001 Martin Burtscher, VPC3: a fast and effective trace-compression algorithm, ACM SIGMETRICS Performance Evaluation Review, v.32 n.1, June 2004 Glenn Reinman , Brad Calder , Todd Austin, Optimizations Enabled by a Decoupled Front-End Architecture, IEEE Transactions on Computers, v.50 n.4, p.338-355, April 2001 Yiannakis Sazeides , James E. Smith, Limits of Data Value Predictability, International Journal of Parallel Programming, v.27 n.4, p.229-256, Aug. 1999 Martin Burtscher , Nana B. Sam, Automatic Generation of High-Performance Trace Compressors, Proceedings of the international symposium on Code generation and optimization, p.229-240, March 20-23, 2005 Suleyman Sair , Timothy Sherwood , Brad Calder, A Decoupled Predictor-Directed Stream Prefetching Architecture, IEEE Transactions on Computers, v.52 n.3, p.260-276, March S. Subramanya Sastry , Rastislav Bodk , James E. Smith, Rapid profiling via stratified sampling, ACM SIGARCH Computer Architecture News, v.29 n.2, p.278-289, May 2001 Martin Burtscher , Ilya Ganusov , Sandra J. Jackson , Jian Ke , Paruj Ratanaworabhan , Nana B. Sam, The VPC Trace-Compression Algorithms, IEEE Transactions on Computers, v.54 n.11, p.1329-1344, November 2005

Context Based Prediction;Last Value Prediction;stride prediction;prediction;value prediction

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Value profiling.

variables as invariant or constant at compile-time allows the compiler to perform optimizations including constant folding, code specialization, and partial evaluation. Some variables, which cannot be labeled as constants, may exhibit semi-invariant behavior. A "semi-invariant" variable is one that cannot be identified as a constant at compile-time, but has a high degree of invariant behavior at run-time. If run-time information was available to identify these variables as semi-invariant, they could then benefit from invariant-based compiler optimizations. In this paper we examine the invariance found from profiling instruction values, and show that many instructions have semi-invariant values even across different inputs. We also investigate the ability to estimate the invariance for all instructions in a program from only profiling load instructions. In addition, we propose a new type of profiling called "Convergent Profiling". Estimating the invariance from loads and convergent profiling are used to reduce the profiling time needed to generate an accurate value profile. The value profile can then be used to automatically guide code generation for dynamic compilation, adaptive execution, code specialization, partial evaluation and other compiler optimizations.

Introduction Many compiler optimization techniques depend upon analysis to determine which variables have invariant be- havior. Variables which have invariant run-time behav- ior, but cannot be labeled as such at compile-time, do not fully benefit from these optimizations. This paper examines using profile feedback information to identify which variables have semi-invariant behavior. A semi-invariant variable is one that cannot be identified as a constant at compile-time, but has a high degree of invariant behavior at run-time. This occurs when a variable has one to N (where N is small) possible values which account for most of the variable's values at run-time. Value profiling is an approach that can identify these semi-invariant variables. The goal of value profiling is different from value pre- diction. Value prediction is used to predict the next result value (write of a register) for an instruction. This has been shown to provide predictable results by using previously cached values to predict the next value of the variable using a hardware buffer [5, 9, 10]. This approach was shown to work well for a hardware value predictor, since values produced by an instruction have a high degree of temporal locality. Our research into the semi-invariance of variables is different from these previous hardware predication stud- ies. For compiler optimizations, we are more concerned with the invariance of a variable, the top N values of the variable, or a popular range of values for the variable over the life-time of the program, although the temporal relationship between values can provide useful information. The value profiling techniques presented in this paper keep track of the top N values for an instruction and the number of occurrences for each of those values. This information can then be used to automatically guide compilation and optimization. In the next section, we examine motivation for this paper and related work. Section 3 describes a method for value profiling. Section 4 describes the methodology used to gather the results for this paper. Section 5 examines the semi-invariant behavior of all instruction types, parame- ters, and loads, and shows that there is a high degree of invariance for several types of instructions. In order to reduce the time to generate a value profile for optimization, x6 investigates the ability to estimate the invariance for all non-load instructions by value profiling only load instructions and propagating their invariance. Section 7 examines a new type of profiler called the Convergent Profiler and its use for value profiling. The goal of a convergent profiler is to reduce the amount of time it takes to gather detailed profile information. For value profiling, we found that the data being profiled, the invariance of instructions, often reaches a steady state, and at that point profiling can be turned off or sampled less often. This reduces the profiling time, while still creating an accurate value profile. We conclude by summarizing the paper in x8. Motivation and Related Work This paper was originally motivated by a result we found when examining the input values for long latency instruc- tions. A divide on a DEC Alpha 21064 processor can take cycles to execute, and a divide on the Intel Pentium processor can take up to 46 cycles. Therefore, it would be beneficial to special case divide instructions with optimizable numerators or denominators. In profiling hydro2d from the SPEC92 benchmark suite, we found that 64% of the executed divide instructions had either a 0 for its numerator or a 1 for its denominator. In conditioning these divide instructions on the numerator or denominator, with either 0 or 1 based on profiling information, we were able to reduce the execution time of hydro2d by 15% running on a DEC Alpha 21064 processor. In applying the same optimization to a handful of video games (e.g., Fury3 and Pitfall) on the Intel Pentium processor, we were able to reduce the number of cycles executed by an estimated 5% 1 for each of these programs. These results show that value profiling can be very effective for reducing the execution time of long latency instructions. The recent publications on Value Prediction in hardware provided further motivation for our research into value profiling [5, 9, 10]. The recent paper by Lipasti et al. [9] showed that on average 49% of the instructions wrote the same value as they did the last time, and 61% of the executed instructions produced the same value as one of the last 4 values produced by that instruction using a 16K value prediction table. These results show that there is a high degree of temporal locality in the values produced by instructions, but this does not necessarily equal the in- struction's degree of invariance, which is needed for certain compiler optimizations. 2.1 Uses for Value Profiling Value profiling can benefit several areas of research including dynamic compilation and adaptive execution, performing compiler optimizations to specialize a program for certain values, and providing hints for value prediction hardware This estimation is a static calculation using a detailed pipeline architecture model of the Pentium processor. The estimation takes into consideration data dependent and resource conflict stalls. 2.1.1 Dynamic Compilation, Adaptive Execution and Code Specialization Dynamic compilation and adaptive execution are emerging directions for compiler research which provide improved execution performance by delaying part of the compilation process to run-time. These techniques range from filling in compiler generated specialized templates at run-time to fully adaptive code generation. For these techniques to be effective the compiler must determine which sections of code to concentrate on for the adaptive execution. Existing techniques for dynamic compilation and adaptive execution require the user to identify run-time invariants using user guided annotations [1, 3, 4, 7, 8]. One of the goals of value profiling is to provide an automated approach for identifying semi-invariant variables and to use this to guide dynamic compilation and adaptive execution. Staging analysis has been proposed by Lee and Leone [8] and Knoblock and Ruf [7] as an effective means for determining which computations can be performed early by the compiler and which optimizations should be performed late or postponed by the compiler for dynamic code generation. Their approach requires programmers to provide hints to the staging analysis to determine what arguments have semi-invariant behavior. In addition, Autrey and Wolfe have started to investigate a form of staging analysis for automatic identification of semi-invariant variables [2]. Consel and Noel [3] use partial evaluation techniques to automatically generate templates for run-time code generation, although their approach still requires the user to annotate arguments of the top-level procedures, global variables and a few data structures as run-time con- stants. Auslander et.al. [1] proposed a dynamic compilation system that uses a unique form of binding time analysis to generate templates for code sequences that have been identified as semi-invariant. Their approach currently uses user defined annotations to indicate which variables are semi-invariant. The annotations needed to drive the above techniques require the identification of semi-invariant variables, and value profiling can be used to automate this process. To automate this process, these approaches can use their current techniques for generating run-time code to identify code regions that could potentially benefit from run-time code generation. Value profiling can then be used to determine which of these code regions have variables with semi- invariant behavior. Then only those code regions identified as profitable by value profiling would be candidates for dynamic compilation and adaptive execution. The above approaches used for dynamic compilation, to determine optimizable code regions, can also be applied to static optimization. These regions can benefit from code specialization if a variable or instruction has the same value across multiple inputs. If this is the case, the code could be duplicated creating a specialized version optimized to treat the variable as a constant. The execution of the specialized code would then be conditioned on that value. Value profiling can be used to determine if these potential variables or instructions have the same value across multiple inputs, in order to guide code specialization. 2.1.2 Hardware-based Optimizations In predicting the most recent value(s) seen, an instruction's future value has been shown to have good predictability using tag-less hardware buffers [9, 10]. Our results show that value profiling can be used to classify the invariance of instructions, so a form of value profiling could potentially be used to improve hardware value prediction. Instructions that are shown to be variant can be kept out of the value prediction buffer, reducing the number of conflicts and aliasing effects, resulting in a more accurate prediction using smaller tables. Instructions shown to have a high invariance with the value profiler could even be given a "sticky" replacement policy. The Memory Disambiguation Buffer [6] (MDB) is an architecture that allows a load and its dependent instructions to be hoisted out of a loop, by checking if store addresses inside the loop conflict with the load. If a store inside the loop is to the same address, the load and its dependent instructions are re-executed. A similar hardware mechanism can be used to take advantage of values, by checking not only the store address, but also its value. In this architecture, only when the value of the load hoisted out of the loop changes should the load and its dependent instructions be re-executed. Value profiling can be used to identify these semi-invariant load instructions. 3 Value Profiling In this section we will discuss a straight forward approach to value profiling. This study concentrates on profiling at the instruction level; finding the invariance of the written register values for instructions. The value profiling information at this level can be directly mapped back to the corresponding variables by the compiler for optimization. There are two types of information needed for value profiling to be used for compiler optimizations: (1) how invariant is an instruction's value over the life-time of the program, and (2) what were the top N result values for an instruction. Determining the invariance of an instruction's resulting value can be calculated in many different ways. The value prediction results presented by Lipasti et al. [9, 10] used a tag-less table to store a cache of the most recently used values to predict the next result value for an instruction. Keeping track of the number of correct predictions equates to the number of times an instruction's destination register void InstructionProfile::collect stats (Reg cur value) f total executed ++; if (cur value == last value) f num times profiled ++; else f LFU insert into tnv table(last value, num times profiled); last Figure 1: A simple value profiler keeping track of the N most frequent occurring values, along with the most recent value (MRV-1) metric. was assigned a value that was the most recent value or one of the most recent M values. We call this the Most Recent is the history depth of the most recent values kept. The MRV metric provides an indication of the temporal reuse of values for an instruction, but it does not equate exactly to the invariance of an instruction. By Invariance - M (Inv-M) we mean the percent of time an instruction spends executing its most frequent M values. For example, an instruction may write a register with values X and Y in the following repetitive pattern :::XY XYXYXY:::. This pattern would result in a MRV-1 (which stores only the most recent value) of 0%, but the instruction has an invariance Inv-1 of 50% and Inv-2 of 100%. Another example is when 1000 different values are the result of an instruction each 100 times in a row before switching to the next value. In this case the MRV-1 metric would determine that the variable used its most recent value 99% of the time, but the instruction has only a 0.1% invariance for Inv-1. The MRV differs from invariance because it does not have state associated with each value indicating the number of times the value has occurred. Therefore, the replacement policy it uses, least recently used, cannot tell what value is the most common. We found the MRV metric is at times a good prediction of invariance, but at other times it is not because of the examples described above. 3.1 A Value Profiler The value profiling information required for compiler optimization ranges from needing to know only the invariance of an instruction to also having to know the top N values or a popular range of values. Figure 1 shows a simple profiler to keep track of this information in pseudo-code. The value profiler keeps a Top N Value (TNV) table for the register being written by an instruction. Therefore, there is a TNV table for every register being profiled. The TNV table stores (value, number of occurrences) pairs for each entry with a least frequently used (LFU) replacement policy. When inserting a value into the table, if the entry already exists its occurrence count is incremented by the number of recent profiled occurrences. If the value is not found, the least frequently used entry is replaced. 3.2 Replacement Policy for Top N Value Table We chose not to use an LRU replacement policy, since replacing the least recently used value does not take into consideration number of occurrences for that value. Instead we use a LFU replacement policy for the TNV table. A straight forward LFU replacement policy for the TNV table can lead to situations where an invariant value cannot make its way into the TNV table. For example, if the TNV table already contains N entries, each profiled more than once, then using a least frequently used replacement policy for a sequence of :::XY XYXYXY::: (where X and Y are not in the table) will make X and Y battle with each other to get into the TNV table, but neither will succeed. The TNV table can be made more forgiving by either adding a "temp" TNV table to store the current values for a specified time period which is later merged into a final TNV table, or by just clearing out the bottom entries of the TNV table every so often. In this paper we use the approach of clearing out the bottom half of the TNV table after profiling the instruction for a specified clear-interval. After an instruction has been profiled more than the clear-interval, the bottom half of the table is cleared and the clear-interval counter is reset. We made the number of times sampled for the clear- interval dependent upon the frequency of the middle entry in the TNV table. This middle entry is the LFU entry in the top half of the table. The clear-interval needs to be larger than the frequency count of this entry, otherwise a new value could never work its way into the top half of the table. In our profiling, we set the clear-interval to be twice the frequency of the middle entry each time the table is cleared, with a minimum clear-interval of 2000 times. 4 Evaluation Methodology To perform our evaluation, we collected information for the SPEC95 programs. The programs were compiled on a DEC Alpha AXP-21164 processor using the DEC C and FORTRAN compilers. We compiled the SPEC benchmark suite under OSF/1 V4.0 operating system using full compiler optimization (-O4 -ifo). Table 1 shows the two data sets we used in gathering results for each program, and the number of instructions executed in millions. We used ATOM [11] to instrument the programs and gather the value profiles. The ATOM instrumentation tool has an interface that allows the elements of the program executable, such as instructions, basic blocks, and proce- dures, to be queried and manipulated. In particular, ATOM Data Set 1 Data Set 2 Program Name Exe M Name Exe M compress ref 93 short 9 gcc 1cp-decl 1041 1stmt 337 ijpeg specmun 34716 vigo 39483 li ref (w/o puzzle) 18089 puzzle 28243 perl primes 17262 scrabble 28243 vortex ref 90882 train 3189 applu ref 46189 train 265 apsi ref 29284 train 1461 fpppp ref 122187 train 234 hydro2d ref 42785 train 4447 mgrid ref 69167 train 9271 su2cor ref 33928 train 10744 tomcatv ref 27832 train 4729 turb3d ref 81333 train 8160 wave5 ref 29521 train 1943 Table 1: Data sets used in gathering results for each pro- gram, and the number of instructions executed in millions for each data set. allows an "instrumentation" program to navigate through the basic blocks of a program executable, and collect information about registers used, opcodes, branch conditions, and perform control-flow and data-flow analysis. 5 Invariance of Instructions This section examines the invariance and predictability of values for instruction types, procedure parameters, and loads. When reporting invariance results we ignored instructions that do not need to be executed for the correct execution of the program. This included a reasonable number of loads for a few programs. These loads can be ignored since they were inserted into the program for code alignment or prefetching for the DEC Alpha 21164 processor. For the results we used two sizes for the TNV table when profiling. For the breakdown of the invariance for the different instruction types (Table 2), we used a TNV table of size 50. For all the other results we used a TNV table of size 10 for each instruction (register). 5.1 Metrics We now describe some of the metrics we will be using throughout the paper. When an instruction is said to have an "Invariance-M" of X%, this is calculated by taking the number of times the top M values for the instruction occurred during profiling, as found in the final TNV table after profiling, and dividing this by the number of times the instruction was executed (profiled). In order to examine the invariance for an instruction we look at Inv-1 and Inv-5. For Inv-1, the frequency count of Program ILd FLd LdA St IMul FMul FDiv IArth FArth Cmp Shft CMov FOps compress 44(27) 0( li perl 70(24) 54( vortex hydro2d 76( su2cor 37( turb3d 54( Avg Table 2: Breakdown of invariance by instruction types. These categories include integer loads (ILd), floating point loads load address calculations (LdA), stores (St), integer multiplication (IMul), floating point multiplication floating point division (FDiv), all other integer arithmetic (IArth), all other floating point arithmetic (Cmp), shift (Shft), conditional moves (CMov), and all other floating point operations (FOps). The first number shown is the percent invariance of the top most value (Inv-1) for a class type, and the number in parenthesis is the dynamic execution frequency of that type. Results are not shown for instruction types that do not write a register (e.g., branches). the most frequently occurring value in the final TNV table is divided by the number of times the instruction was profiled. For Inv-5, the number of occurrences for the top 5 values in the final TNV table are added together and divided by the number of times the instruction was profiled. When examining the difference in invariance between the two profiles, for either the two data sets or between the normal and convergent profile, we examine the difference in invariance and the difference in the top values encountered for instructions executed in both profiles. Diff-1 and Diff-5 are used show the weighted difference in invariance between two profiles for the top most value in the TNV table and the top 5 values. The difference in invariance is calculated on an instruction by instruction basis and is included into a weighted average based on the first input, for only instructions that are executed in both profiles. The metric Same-1 shows the percent of instructions profiled in the first profile that had the same top value in the second profile. To calculate Same-1 for an instruction, the top value in the TNV table for the first profile is compared to the top value in the second profile. If they are the same, then the number of times that value occurred in the TNV table for the first profile is added to a sum counter. This counter is then divided by the total number of times these instructions were profiled based on the first input. Two other metrics, Find-1 and Find-5, are calculated in a similar manner. They show the percent of time the top 1 element or the top 5 elements in the first profile for an instruction appear in the top 5 values for that instruction in the second profile. When calculating the results for Same-1, Find-1, and Find-5 we only look at instructions whose invariance in the first profile are greater than 30%. The reason for only looking at instructions with an Inv-1 invariance larger than 30% is to ignore all the instructions with random invari- ance. For variant instructions there is a high likelihood that the top values in the two profiles are different, and we are not interested in these instructions. Therefore, we arbitrarily chose 30% since it is large enough to avoid variant instructions when looking at the top 5 values. For these results two numbers are shown, the first number is the percent match in values found between the two profiles, and the second number in parenthesis is the percent of profiled instructions the match corresponds to because of the 30% invariance filter. Therefore, the number in parenthesis is the percent of instructions profiled that had an invariance greater than 30%. When comparing the two different data sets, Overlap represents the percent of instructions, weighted by execu- tion, that were profiled in the first data set that were also profiled in the second data set. 5.2 Breakdown of Instruction Type Invariance Table 2 shows the percent invariance for each program broken down into 14 different and disjoint instruction categories using data set 1. The first number represents the average percent invariance of the top value (Inv-1) for a given instruction type. The number next to it in parenthesis is the percent of executed instructions that this class Data Set 1 Data Set 2 Comparing Params in Data Set 1 to Data Set 2 Procedure Calls Params Params Over- Invariance Top Values Program %Instr 30% 50% 70% 90% Inv1 Inv5 Inv1 Inv5 lap diff1 diff5 same1 find1 find5 compress gcc 1.23 54 48 34 17 31 43 31 43 li 2.45 perl 1.23 54 hydro2d mgrid su2cor average 0.53 73 67 61 44 54 69 54 70 Table 3: Invariance of parameter values and procedure calls. Instr is the percent of executed instructions that are procedure calls. The next four columns show the percent of procedure calls that had at least one parameter with an Inv-1 invariance greater than 30, 50, 70 and 90%. The rest of the metrics are in terms of parameters and are described in detail in x5.1. type accounts for when executing the program. For the store instructions, the invariance reported is the invariance of the value being stored. The results show that for the integer programs, that the integer loads (ILd), the calculation of the load addresses (LdA), and the integer arithmetic instructions have a high degree of invariance and are frequently executed. For the floating point instructions the invariance found for the types are very different from one program to the next. Some programs mgrid, swim, and tomcatv show very low invariance, while hydro2d has very invariant instructions. 5.3 Invariance of Parameters Specializing procedures based on procedure parameters is a potentially beneficial form of specialization, especially if the code is written in a modular fashion for general purpose use, but is used in a very specialized manner for a given run of an application. Table 3 shows the predictability of parameters. Instr shows the percent of instructions executed which were procedure calls for data set 1. The next four columns show the percent of procedure calls that had at least one parameter with an Inv-1 invariance greater than 30, 50, 70, and 90%. These first five columns show results in terms of proce- dures, and the remaining columns show results in terms of parameter invariance and values. The remaining metrics are described in detail in x5.1. The results show that the invariance of parameters is very predictable between the different input sets. The Table also shows that on average the top value for 44% of the parameters executed (passed to procedures) for data set 1 had the same value 84% of the time when that same parameter was passed in a procedure for the second data set. 5.4 Invariance of Loads The graphs in Figure 2 show the invariance for loads in terms of the percent of dynamically executed loads in each program. The left graph shows the percent invariance calculated for the top value (Inv-1) in the final 10 entry TNV table for each instruction, and the right graph shows the percent invariance for the top 5 values (Inv-5). The invariance shown is non-accumulative, and the x-axis is weighted by frequency of execution. Therefore, if we were interested in optimizing all instructions that had an Inv-1 invariance greater than 50% for li, this would account for around 40% of the executed loads. The Figure shows that some of the programs compress, vortex, m88ksim, and perl have 100% Inv-1 invariance for around 50% of their executed loads, and m88ksim and perl have a 100% Inv-5 invariance for almost 80% of their loads. It is interesting to note from these graphs the bi-modal nature of the load invariance for many of the programs. Most of the loads are either completely invariant or very variant. Table 4 shows the value invariance for loads. The invariance Inv-1 and Inv-5 shown in this Table for data set 1 is the average of the invariance shown in Figure 2. Mrv-1 is the percentage of time the most recent value was the next value encountered by the load. Diff M/I is the weighted difference in Mrv-1 and Inv-1 percentages on an instruction by instruction basis. The rest of the metrics are described in x5.1. The results show that the MRV-1 metric has a 10% difference in invariance on average, but the difference is Percent Executed Loads Percent Invariance for compress gcc go ijpeg li perl vortex applu apsi hydro2d mgrid su2cor turb3d Percent Executed Loads Percent Invariance for Inv-5 Figure 2: Invariance of loads. The graph on the left shows the percent invariance of the top value (Inv-1) in the TNV table, and graph on the right shows the percent invariance of the top 5 values (Inv-5) in the TNV table. The percent invariance is shown on the y-axis, and the x-axis is the percent of executed load instructions. The graph is formed by sorting all the instructions by their invariance, and then putting the instructions into 100 buckets filling the buckets up based on each load's execution frequency. Then the average invariance, weighted by execution frequency, of each bucket is graphed. Comparing Data Set 1 and Data Set 2 Data Set 1 Data Set 2 % Invariance Top Values Program Mrv1 Inv1 Inv5 diff M/I Mrv1 Inv1 Inv5 diff M/I Overlap diff1 diff5 same1 find1 find5 compress go ijpeg 26 28 47 19 26 li 37 perl vortex 28 hydro2d su2cor turb3d 36 38 48 8 40 42 52 8 average 38 Table 4: Invariance of load values using a TNV table of size 10. Mrv1 is the average percent of time the current value for a load was the last value for the load. Diff M/I is the difference between Mrv1 and Inv1 calculated instruction by instruction. The rest of the metrics are described in detail in x5.1. large for a few of the programs. The difference in invariance of instructions between data sets is very small. The results show that 27% of the loads executed in both data sets (using the 30% invariance filter) have the same top invariant value 90% of the time. Not only is the invariance between inputs similar, but a certain percentage (24%) of their values are the same. The clearing interval and table size parameters we used affect the top values found for the TNV table more than the invariance. When profiling the loads with a 10 entry TNV table, if clearing the bottom half of the table is turned off, the average results showed a 1% difference in invariance and the top value was different 8% of the time in each TNV table using the 30% filter. In examining different table sizes (with clearing on), a TNV table of size 4 had on average a 1% difference in invariance from a TNV table of size 10, and the top value found was different 2% of the time. When using a table size of 50 for the load profile, on average there was a 0% difference in invariance and the top value was different 4% of the time when compared to the entry TNV table when only examining loads that had an invariance above 30%. 6 Estimating Invariance Out of all the instructions, loads are really the "unknown quantity" when dealing with a program's execution. If the value and invariance for all loads are known, then it is reasonable to believe that the invariance and values for many of the other instructions can be estimated through invariance and value propagation. This would significantly reduce the profiling time needed to generate a value profile for all instructions. To investigate this, we used the load value profiles from the previous section, and propagated the load invariance through the program using data flow and control flow analysis deriving an invariance for the non-load instructions that write a register. We achieved reasonable results using a simple inter-procedural analysis algorithm. Our estimation algorithm first builds a procedure call graph, and each procedure contains a basic block control flow graph. To propagate the invariance, each basic block has an OUT RegMap associated with it, which contains the invariance of all the registers after processing the basic block. When a basic block is processed, the OUT RegMaps of all of its predecessors in the control flow graph are merged together and are used as the IN RegMap for that basic block. The RegMap is then updated processing each instruction in the basic block to derive the OUT RegMap for the basic block. To calculate the invariance for the instructions within a basic block we developed a set of simple heuristics. The default heuristic used for instructions with two input registers is to set the def register invariance to the invariance of first use register times the invariance of second use register. If one of the two input registers is undefined, the invariance of def register is left undefined in the RegMap. For instructions with only one input register (e.g., MOV), the invariance of the def register is assigned the invariance of the use. Other heuristics used to propagate the invariance included the loop depth, induction variables, stack pointer, and special instructions (e.g., CMOV), but for brevity we will not go into these. Table 5 shows the invariance using our estimation algorithm for non-load instructions that write a register. The second column in the table shows the percent of executed instructions to which these results apply. The third column Prof shows the overall invariance (Inv-1) for these instructions using the profile used to form Table 2. The fourth column is the overall estimated invariance for these instruc- tions, and the fifth column is the weighted difference in invariance Inv-1 between the real profile and the estimation on an instruction by instruction basis. The next 7 columns show the percent of executed instructions that have an average invariance above the threshold of 10, 30, 50, 60, 70, and 90%. Each column contains three numbers, the first number is the percent of instructions executed that had an invariance above the threshold. The second number is the percent of these invariant instructions that the estimation also classified above the invariant threshold. The last number in the column shows the percent of these instructions (normalized to the invariant instructions found above the threshold) the estimation thought were above the invariant threshold, but were not. Therefore, the last number in the column is the normalized percent of instructions that were over estimated. The results show that our estimated propagation has an 8% difference on average in invariance from the real profile. In terms of actually classifying variables above an invariant threshold our estimation finds 83% of the instructions that have an invariance of 60% or more, and the estimation over estimates the invariant instructions above this threshold by 7%. Our estimated invariance is typically lower than the real profile. There are several reasons for this. The first is the default heuristic which multiplies the invariance of the two uses together to arrive at the invariance for the def. At times this estimation is correct, although a lot of time it provides a conservative estimation of the invariance for the written register. Another reason is that at times the two uses for an instruction were variant but their resulting computation was invariant. This was particularly true for logical instructions (e.g, AND, OR, Shift) and some arithmetic instructions. 7 Convergent Value Profiling The amount of time a user will wait for a profile to be generated will vary depending on the gains achievable from using value profiling. The level of detail required from a % of % Inv-1 % Instructions Found Above Invariance Threshold Program Instrs Prof Est Diff-1 10% 30% 50% 60% 70% 80% 90% compress 50 go ijpeg 71 li mgrid su2cor turb3d 56 average 53 29 24 8 20 (75, 5) 17 (76, Table 5: Invariance found for instructions computed by propagating the invariance from the load value profile. Instrs shows the percent of instructions which are non-load register writing instructions to which the results in this table apply. Prof and Est are the the percent invariance found for the real profile and the estimated profile. Diff-1 is the percent difference between the profile and estimation. The last 7 columns show the percent of executed instructions that have an average invariance above the threshold of 10, 30, 50, 60, 70, 80 and 90%, and the percentage of these that the estimation profile found and the percent that were over estimated. value profiler determines the impact on the time to profile. The problem with a straight forward profiler, as shown in Figure 1, is it could run hundreds of times slower than the original application, especially if all of the instructions are profiled. One solution we propose in this paper is to use a somewhat intelligent profiler that realizes the data (invari- ance and top N values) being profiled is converging to a steady state and then profiling is turned off on an instruction by instruction basis. In examining the value invariance of instructions, we noticed that most instructions converge in the first few percent of their execution to a steady state. Once this steady state is reached, there is no point to further profiling the instruction. By keeping track of the percent change in invariance one can classify instructions as either "converged" or "changing". The convergent profiler stops profiling the instructions that are classified as converged based on a convergence criteria. This convergence criteria is tested after a given time period (convergence-interval) of profiling the instruction. To model this behavior, the profiling code is conditioned on a boolean to test if profiling is turned off or on for an instruction. If profiling is turned on, normal profiling occurs, and after a given convergence interval the convergence criteria is tested. The profiling condition is then set to false if the profile has converged for the instruction. If profiling is turned off, periodically the execution counter is checked to see if a given retry time period has elapsed. When profiling is turned off the retry time period is set to a number total executed backoff , where back-off can either be a constant or a random number. This is used to periodically turn profiling back on to see if the invariance is at all changing. In this paper we examine the performance of two heuristics for the convergence criteria for value profiling. The first heuristic concentrates on the instructions with an increasing invariance. For instructions whose invariance is changing we are more interested in instructions that are increasing their final invariance than those that are decreasing their final invariance for compiler optimization pur- poses. Therefore, we continue to profile the instruction's whose final invariance is increasing, but choose to stop profiling those instructions whose invariance is decreas- ing. When the percent invariance for the convergence test is greater than the percent invariance in the previous inter- val, then the invariance is increasing so profiling contin- ues. Otherwise, profiling is stopped. When calculating the invariance the total frequency of the top half of the TNV table is examined. For the results, we use a convergence- interval for testing the criteria of 2000 instruction executions The second heuristic examined for the convergence cri- teria, is to only continue profiling if the change in invariance for the current convergence interval is greater than an inv-increase bound or lower than an inv-decrease bound. If the percent invariance is changing above or below these bounds, profiling continues. Otherwise profiling stops because the invariance has converged to be within these bounds. Convergent Profile Comparing Full Load Profile to Convergent Convergence % Invariance Invariance Top Values Program Prof % Conv % Inc Backoff % Inv-1 % Inv-5 % diff-1 % diff-5 % same-1 % find-1 % find-5 compress li Table Convergent profiler, where profiling continues if invariance is increasing, otherwise it is turned off. Prof is percent of time the executable was profiled. Conv and Inc are the percent of time the convergent criteria decided that the invariance had converged or was still increasing. Backoff is the percent of time spent profiling after turning profiling back on. 7.1 Performance of the Convergent Profiler Table 6 shows the performance of the convergent profiler, which stops profiling the first instance the change in invariance decreases. The second column, percent of instructions profiled, shows the percentage of time profiling was turned on for the program's execution. The third column (Conv) shows the percent of time profiling converged when the convergence criteria was tested, and the next column (Inc) is the percent of time the convergence test decided that the invariance was increasing. The fifth column (Back- off) shows the percent of time spent profiling after turning profiling back on using the retry time period. The rest of the metrics are described in x5.1 and they compare the results of profiling the loads for the program's complete execution to the convergent profile results. The results show that on average convergent profiling spent 2% of its time profiling and profiling was turned off for the other 98% of the time. In most of the programs the time to converge was 1% or less. gcc was the only outlier, taking 24% of its execution to converge. The reason is gcc executes more than 60,000 static load instructions for our inputs and many of these loads do not execute for long. Therefore, most of these loads were fully profiled since their execution time fit within the time interval of sampling for convergence (2000 invocations). These results show that the convergent pro- filer's invariance differed by only 10% from the full profile, and we were able to find the top value of the full length profile in the top 5 values in the convergent profile 98% of the time. Table 7 shows the performance of the convergent pro- filer, when using the upper and lower change in invariance Convergence Back- Invariance Program Prof Conv Inc Dec off diff1 diff5 compress gcc li perl 0 43 38 19 19 3 0 average 4 22 28 50 57 3 0 Table 7: Convergent profiler, where profiling continues as long as the change in invariance is either above the inv- increase or below the inv-decrease bound. The new column Dec shows the percent of time the invariance was decreasing when testing for convergence. bounds for determining convergence. A new column (Dec) shows the percent of time the test for convergence decided to continue profiling because the invariance was decreas- ing. For these results we use an inv-increase threshold of 2% and an inv-decrease threshold of 4%. If the invariance is not increasing by more than 2%, or decreasing by more than 4% then profiling is turned off. The results show that this heuristic spends more time profiling, 4% on average, but has a lower difference in invariance (3%) in comparison to the first heuristic (10%). In terms of values this new heuristic only increased the matching of the top values by 1%. Therefore, the only advantage of using this second heuristic is to obtain a more accurate invariance. Table 7 shows a lot of the time is spent on profiling the decrease in invariance. The reason is that a variant instruction can start out looking invariant with just a couple of values at first. It then can take awhile for the overall invariance of the instruction to reach its final variant behavior. The results also show that more of the time profiling, 57%, is spent after profiling is turned back on than using our first convergence criteria, 24%. One problem is that after an instruction is profiled for a long time, it takes awhile for its overall invariance to change. If the invariance for an instruction converges after profiling for awhile and then it changes into a new steady state, it will take a lot of profiling to bring the overall invariance around to the new steady state. One possible solution is to monitor if this is happening, and if so dump the current profile information and start a new TNV table for the instruction. This would then converge faster to the new steady state. Examining this, sampling techniques, and other approaches to convergent profiling is part of future research. Summary In this paper we explored the invariant behavior of values for loads, parameters, and all register defining instructions. The invariant behavior was identified by a value profiler, which could then be used to automatically guide compiler optimizations and dynamic code generation. We showed that value profiling is an effective means for finding invariant and semi-invariant instructions. Our results show that the invariance found for instructions, when using value profiling, is very predictable even between different input sets. In addition we examined two techniques for reducing the profiling time to generate a value profile. The first technique used the load value profile to estimate the invariance for all non-load instructions with an 8% invariance difference from a real profile. The second approach we proposed for reducing profiling time, is the idea of creating a convergent profiler that identifies when profiling information reaches a steady state and has converged. The convergent profiler we used for loads, profiled for only 2% of the program's execution on average, and recorded an invariance within 10% of the full length profiler and found the top values 98% of the time. The idea of convergent profiling proposed in this paper can potentially be used for decreasing the profiling time needed for other types of detailed profilers. We view value profiling as an important part of future compiler research, especially in the areas of dynamic compilation and adaptive execution, where identifying invariant or semi-invariant instructions at compile time is es- sential. A complementary approach for trying to identify semi-invariant variables, is to use data-flow and staging analysis to try and prove that a variable's value will not change often or will hold only a few values over the life-time of the program. This type of analysis should be used in combination with value profiling to identify optimizable code regions. Acknowledgments We would like to thank Jim Larus, Todd Austin, Florin Baboescu, Barbara Kreaseck, Dean Tullsen, and the anonymous reviewers for providing useful comments. This work was funded in part by UC MICRO grant No. 97-018, DEC external research grant No. US-0040-97, and a generous equipment and software grant from Digital Equipment Corporation. --R Initial results for glacial variable analy- sis A general approach for run-time specialization and its application to C Speculative execution based on value prediction. Dynamic memory disambiguation using the memory conflict buffer. Data specialization. Optimizing ml with run-time code generation Exceeding the dataflow limit via value prediction. Value locality and load value prediction. ATOM: A system for building customized program analysis tools. --TR ATOM Dynamic memory disambiguation using the memory conflict buffer Optimizing ML with run-time code generation Fast, effective dynamic compilation Data specialization Value locality and load value prediction C: a language for high-level, efficient, and machine-independent dynamic code generation A general approach for run-time specialization and its application to C Exceeding the dataflow limit via value prediction Initial Results for Glacial Variable Analysis --CTR Dean M. Tullsen , John S. Seng, Storageless value prediction using prior register values, ACM SIGARCH Computer Architecture News, v.27 n.2, p.270-279, May 1999 Characterization of value locality in Java programs, Workload characterization of emerging computer applications, Kluwer Academic Publishers, Norwell, MA, 2001 Daniel A. Connors , Wen-mei W. 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profiling;invariance;compiler optimization

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Can program profiling support value prediction?.

This paper explores the possibility of using program profiling to enhance the efficiency of value prediction. Value prediction attempts to eliminate true-data dependencies by predicting the outcome values of instructions at run-time and executing true-data dependent instructions based on that prediction. So far, all published papers in this area have examined hardware-only value prediction mechanisms. In order to enhance the efficiency of value prediction, it is proposed to employ program profiling to collect information that describes the tendency of instructions in a program to be value-predictable. The compiler that acts as a mediator can pass this information to the value-prediction hardware mechanisms. Such information can be exploited by the hardware in order to reduce mispredictions, better utilize the prediction table resources, distinguish between different value predictability patterns and still benefit from the advantages of value prediction to increase instruction-level parallelism. We show that our new method outperforms the hardware-only mechanisms in most of the examined benchmarks.

Introduction Modern microprocessor architectures are increasingly designed to employ multiple execution units that are capable of executing several instructions (retrieved from a sequential instruction stream) in parallel. The efficiency of such architectures is highly dependent on the instruction-level parallelism (ILP) that they can extract from a program. The extractable ILP depends on both processor's hardware mechanisms as well as the program's characteristics ([6], [7]). Program's characteristics affect the ILP in the sense that instructions cannot always be eligible for parallel execution due to several constraints. These constraints have been classified into three classes: true-data dependencies, name dependencies (false dependencies) and control dependencies ([6], [7], [15]). Both control dependencies and name dependencies are not considered an upper bound on the extractable ILP since they can be handled or even eliminated in several cases by various hardware and software techniques ([1], [2], [3], [6], As opposed to name dependencies and control dependencies, only true-data dependencies were considered to be a fundamental limit on the extractable ILP since they reflect the serial nature of a program by dictating in which sequence data should be passed between instructions. This kind of extractable parallelism is represented by the dataflow graph of the program ([7]). Recent works ([9], [10], [4], [5]) have proposed a novel hardware-based paradigm that allows superscalar processors to exceed the limits of true-data dependencies. This paradigm, termed value prediction, attempted to collapse true-data dependencies by predicting at run-time the outcome values of instructions and executing the true-data dependent instructions based on that prediction. Within this concept, it has been shown that the limits of true-data dependencies can be exceeded without violating the sequential program correctness. This claim breaks two accepted fundamental principles: 1. the ILP of a sequential program is limited by its dataflow graph representation, and 2. in order to guarantee the correct execution of the program, true-data dependent instructions cannot be executed in parallel. It was also indicated that value prediction can cause the execution of instructions to become speculative. Unlike branch prediction that can also cause instructions to be executed speculatively since they are control dependent, value prediction may cause instructions to become speculative since it is not assured that they were fed with the correct input values. All recent works in the area of value prediction considered hardware-only mechanisms. In this paper we provide new opportunities enabling the compiler to support value prediction by using program profiling. Profiling techniques are being widely used in different compilation areas to enhance the optimization of programs. In general, the idea of profiling is to study the behavior of the program based on its previous runs. In each of the past runs, the program can be executed based on different sets of input parameters and input files (training inputs). During these runs, the required information (profile image) can be collected. Once this information is available it can be used by the compiler to optimize the program's code more efficiently. The efficiency of program profiling is mainly based on the assumption that the characteristics of the program remain the same under different runs as well. In this paper we address several new open questions regarding the potential of profiling and the compiler to support value prediction. Note that we do not attempt to replace all the value prediction hardware mechanisms in the compiler or the profiler. We aim at revising certain parts of the value prediction mechanisms to exploit information that is collected by the profiler. In the profile phase, we suggest collecting information about the instructions' tendency to be value-predictable (value predictability) and classify them accordingly (e.g., we can detect the highly predictable instructions and the unpredictable ones). Classifying instructions according to their value predictability patterns may allow us to avoid the unpredictable instructions from being candidates for value prediction. In general, this capability introduces several significant advantages. First, it allows us to better utilize the prediction table by enabling the allocation of highly predictable instructions only. In addition, in certain microprocessors, mispredicted values may cause some extra misprediction penalty due to their pipeline organization. Therefore the classification allows the processor to reduce the number of mispredictions and saves the extra penalty. Finally, the classification increases the effective prediction accuracy of the predictor. previous works have performed the classification by employing a special hardware mechanism that studies the tendency of instructions to be predictable at run-time ([9], [10], [4], [5]). Such a mechanism is capable of eliminating a significant part of the mispredictions. However, since the classification was performed at run-time, it could not allocate in advance the predictable instructions in the prediction table. As a result unpredictable instructions could have uselessly occupied entries in the prediction table and evacuated the predictable instructions. In this work we propose an alternative technique to perform the classification. We show that profiling can provide the compiler with accurate information about the tendency of instructions to be value-predictable. The role of the compiler in this case is to act as a mediator and to pass the profiling information to the value prediction hardware mechanisms through special opcode directives. We show that such a classification methodology outperforms the hardware-based classification in most of the examined benchmarks. In particular, the performance improvement is most observable when the pressure on the prediction table, in term of potential instructions to be allocated, is high. Moreover, we indicate that the new classification method introduces better utilization of the prediction table resources and avoidance of value mispredictions. The rest of this paper is organized as follows: Section summarizes previous works and results in the area of value prediction. Section 3 presents the motivation and the methodology of this work. Section 4 explores the potential of program profiling through various quantitative measurements. Section 5 examines the performance gain of the new technique. Section 6 concludes this paper. 2. Previous works and results This section summarizes some of the experimental results and the hardware mechanisms of the previous familiar works in the area of value prediction ([9], [10], [4], [5]). These results and their significance have been broadly studied by these works, however we have chosen to summarize them since they provide substantial motivation to our current work. Subsections 2.1 and 2.2 are dedicated to value prediction mechanisms: value predictors and classification mechanisms. Subsection 2.3, 2.4 and 2.5 describe the statistical characteristics of the phenomena related to value prediction. The relevance of these characteristics to this work is presented in Section 3. 2. 1.Value predictors Previous works have introduced two different hardware-based value predictors: the last-value predictor and the stride predictor. For simplicity, it was assumed that the predictors only predict destination values of register operands, even though these schemes could be generalized and applied to memory storage operands, special registers, the program counter and condition codes. Last-value predictor: ([9], [10]) predicts the destination value of an individual instruction based on the last previously seen value it has generated (or computed). The predictor is organized as a table (e.g., cache table - see figure 2.1), and every entry is uniquely associated with an individual instruction. Each entry contains two fields: tag and last-value. The tag field holds the address of the instruction or part of it (high-order bits in case of an associative cache table), and the last-value field holds the previously seen destination value of the corresponding instruction. In order to obtain the predicted destination value of a given instruction, the table is searched by the absolute address of the instruction. Stride predictor: ([4], [5]) predicts the destination value of an individual instruction based on its last previously seen value and a calculated stride. The predicted value is the sum of the last value and the stride. Each entry in this predictor holds an additional field, termed stride field that stores the previously seen stride of an individual instruction (figure 2.1). The stride field value is always determined upon the subtraction of two recent consecutive destination values. Tag Last value Predicted value hit/miss Instruction address Last-value predictor Tag Last value Stride hit/miss Instruction address Predicted value predictor Figure 2.1 - The "last value" and the "stride" predictors. 2. 2.Classification of value predictability Classification of value predictability aims at distinguishing between instructions which are likely to be correctly predicted and those which tend to be incorrectly predicted by the predictor. A possible method of classifying instructions is using a set of saturated counters ([9], [10]). An individual saturated counter is assigned to each entry in the prediction table. At each occurrence of a successful or unsuccessful prediction the corresponding counter is incremented or decremented respectively. According to the present state of the saturated counter, the processor can decide whether to take the suggested prediction or to avoid it. In Section 5 we compare the effectiveness of this hardware-based classification mechanism versus the proposed mechanism. 2. 3.Value prediction accuracy The benefit of using value prediction is significantly dependent on the accuracy that the value predictor can accomplish. The previous works in this field ([4], [5], [9] and [10]) provided substantial evidence to support the observation that outcome values in programs tend to be predictable (value predictability). The prediction accuracy measurements of the predictors that were described in Subsection 2.1 on Spec-95 benchmarks are summarized in table 2.1. Note that in the floating point benchmarks (Spec-fp95) the prediction accuracy was measured in each benchmark for two execution phases: initialization (when the program reads its input data) and computation (when the actual computation is made). Broad study and analysis of these measurements can be found in [4] and [5]. Prediction accuracy [%] Integer loads ALU instructions FP loads FP computation instructions Init. phase Comp. phase 28 Notations predictor L Last-value predictor Table 2.1 - Value prediction accuracy measurements. 2. 4.Distribution of value prediction accuracy Our previous studies ([4], [5]) revealed that the tendency of instruction to be value-predictable does not spread uniformly among the instructions in a program (we only refer to those instructions that assign outcome value to a destination register). Approximately 30% of the instructions are very likely to be correctly predicted with prediction accuracy greater than 90%. In addition, approximately 40% of the instructions are very unlikely to be correctly predicted (with a prediction accuracy less than 10%). This observation is illustrated by figure 2.2 # for both integer and floating point benchmarks. The importance of this observation and its implication are discussed in Subsection 3.1. 2. 5.Distribution of non-zero strides In our previous works ([4], [5]) we examined how efficiently the stride predictor takes advantage of the additional "stride" field (in its prediction table) beyond the last-value predictor that only maintains a single field (per entry) of the "last value". We considered the stride fields to be utilized efficiently only when the predictor accomplishes a correct value prediction and the stride field is not equal to zero (non-zero stride). In order to grade this efficiency we used a measure that we term stride efficiency ratio (measured in percentages). The stride efficiency ratio is the ratio of successful non-zero stride-based value predictions to overall successful predictions. # 1. The initialization phase of the floating-point benchmarks is denoted by #1 and the computation phase by #2. 2. gcc1 and gcc2 denotes the measurements when the benchmark was run with different input files (the same for perl1 and perl2). JR P#NVL JFF# JFF# FRPSUH OL LMSHJ SHUO# SHUO# YRUWH[ 7KH#GLVWULEXWLRQ#RI#SUHGLFWLRQ#DFFXUDF\# Y# Y# VX#FRU VX#FRU K\GUR# G# K\GUR# G# 7KH#GLVWULEXWLRQ#RI#SUHGLFWLRQ#DFFXUDF\# Figure 2.2 - The spread of instructions according to their value prediction accuracy. Our measurements indicated that in the integer benchmarks the stride efficiency ratio is approximately 16%, and in the floating point benchmarks it varies from 12% in the initialization phase to 43% in the computation phase. We also examined the stride efficiency ratio of each instruction in the program that was allocated to the prediction table. We observed that most of these instructions could be divided into two major subsets: a small subset of instructions which always exhibits a relatively high stride efficiency ratio and a large subset of instructions which always tend to reuse their last value (with a very low stride efficiency ratio). Figure 2.3 draws histograms of our experiments and illustrates how instructions in the program are scattered according to their stride efficiency ratio. 0% 20% 40% 80% 100% sim 126.gcc 129.comp ress 130.li 132.ijpeg 134.perl 147.vorte x efficiency ratio % of instructions Figure 2.3 - The spread of instructions according to their stride efficiency ratio. 3. The proposed methodology Profiling techniques are broadly being employed in various compilation areas to enhance the optimization of programs. The principle of this technique is to study the behavior of the program based on one set of train inputs and to provide the gathered information to the compiler. The effectiveness of this technique relies on the assumption that the behavioral characteristics of the program remain consistent with other program's runs as well. In the first subsection we present how the previous knowledge in the area of value prediction motivated us towards our new approach. In the second subsection we present our methodology and its main principles. 3. 1.Motivation The consequences of the previous results described in Section 2 are very significant, since they establish the basis and motivation for our current work with respect to the following aspects: 1. The measurements described in Subsection 2.3 indicated that a considerable portion of the values that are computed by programs tends to be predictable (either by stride or last-value predictors). It was shown in the previous works that exploiting this property allows the processor to exceed the dataflow graph limits and improve ILP. 2. Our measurements in Subsection 2.4 indicated that the tendency of instructions to be value predictable does not spread uniformly among the instructions in the program. In most programs exhibit two sets of instructions, highly value-predictable instructions and highly unpredictable ones. This observation established the basis for emlpoying classification mechanisms. 3. Previous experiments ([4], [5]) have also provided preliminary indication that different input files do not dramatically affect the prediction accuracy of several examined benchmarks. If this observation is found to be common enough, then it may have a tremendous significance when considering the involvement of program profiling. It may imply that the profiling information which is collected in previous runs of the program (running the application with training input files) can be correlated to the true situation where the program runs with its real input files (provided by the user). This property is extensively examined in this paper. 4. We have also indicated that the set of value-predictable instructions in the program is partitioned into two subsets: a small subset of instructions that exhibit stride value predictability (predictable only by the stride predictor) and a large subset of instructions which tend to reuse their last value (predictable by both predictors). Our previous works ([4], [5]) showed that although the first subset is relatively smaller than the second subset, it appears frequently enough to significantly affect the extractable ILP. On one hand, if we only use the last-value predictor then it cannot exploit the predictability of the first subset of instructions. On the other hand, if we only use the stride predictor, then in a significant number of entries in the prediction table, the extra stride field is useless because it is assigned to instructions that tend to reuse their most recently produced value (zero strides). This observation motivates us to employ a hybrid predictor that combines both the stride prediction table and the last-value prediction table. For instance we may consider a relatively small stride prediction table only for the instructions that exhibit stride patterns and a larger table for the instructions that tend to reproduce their last value. The combination of these schemes may allow us utilize the extra stride field more efficiently. 3. 2. A classification based on program profiling and compiler support The methodology that we are introducing in this work combines both program profiling and compiler support to perform the classification of instructions according to their tendency to be value predictable. All familiar previous works performed the classification by using a hardware mechanism that studies the tendency of instructions to be predictable at run-time ([4], [5], [9], [10]). Such a mechanism was capable of eliminating a significant part of the mispredictions. However, since the classification was performed dynamically, it could not allocate in advance the highly value predictable instructions in the prediction table. As a result unpredictable instructions could have uselessly occupied entries in the prediction table and evacuated useful instructions. The alternative classification technique, proposed in this paper, has two tasks: 1. identify the highly predictable instructions and 2. indicate whether an instruction is likely to repeat its last value or whether it exhibits stride patterns. Our methodology consists of three basic phases (figure 3.1). In the first phase the program is ordinarily compiled (the compiler can use all the available and known optimization methods) and the code is generated. In the second phase the profile image of the program is collected. The profile image describes the prediction accuracy of each instruction in the program (we only refer to instructions which write a computed value to a destination register). In order to collect this information, the program can be run on a simulation environment (e.g., the SHADE simulator - see [12]) where the simulator can emulate the operation of the value predictor and measure for each instruction its prediction accuracy. If the simulation emulates the operation of the stride predictor it can also measure the stride efficiency ratio of each instruction. Such profiling information could not only indicate which instructions tend to be value-predictable or not, but also which ones exhibit value predictability patterns in form of "strides" or "last-value". The output of the profile phase can be a file that is organized as a table. Each entry is associated with an individual instruction and consists of three fields: the instruction's address, its prediction accuracy and its stride efficiency ratio. Note that in the profile phase the program can be run either single or multiple times, where in each run the program is driven by different input parameters and files. Compiler Program (C or FORTRAN) Binary executable Simulator Train input parameters and files Profile image file Phase #1 Phase #2 Compiler New binary executable with opcode directives Phase #3 threshold value (user) Figure 3.1 - The three phases of the proposed classification methodology. In the final phase the compiler only inserts directives in the opcode of instructions. It does not perform instruction scheduling or any form of code movement with respect to the code that was generated in the first phase. The inserted directives act as hints about the value predictability of instructions that are supplied to the hardware. Note, that we consider such use of opcode directives as feasible, since recent processors, such as the PowerPC 601, made branch predictions based on opcode directives too ([11]). Our compiler employs two kinds of directives: the "stride" and the "last-value". The "stride" directive indicates that the instruction tends to exhibit stride patterns, and the "last-value" directive indicates that the instruction is likely to repeat its recently generated outcome value. By default, if none of these directives are inserted in the opcode, the instruction is not recommended to be value predicted. The compiler can determine which instructions are inserted with the special directives according to the profile image file and a threshold value supplied by the user. This value determines the prediction accuracy threshold of instructions to be tagged with a directive as value-predictable. For instance, if the user sets the threshold value to 90%, all the instructions in the profile image file that had a prediction accuracy less than 90% are not inserted with directives (marked as unlikely to be correctly predicted) and all those with prediction accuracy greater than or equal to 90% are marked as predictable. When an instruction is marked as value-predictable, the type of the directive (either "stride" or "last-value") still needs to be determined. This can be done by examining the stride efficiency ratio that is provided in the profile image file. A possible heuristic that the compiler can employ is: If the stride efficiency ratio is greater than 50% it indicates that the majority of the correct predictions were non-zero strides and therefore the instruction should be marked as "stride"; otherwise it is tagged with the "last-value" directive. Another way to determine the directive type is to ask the user to supply the threshold value for the stride efficiency ratio. Once this process is completed, the previous hardware-based classification mechanism (the set of saturated counters) becomes unnecessary. Moreover, we can use a hybrid value predictor that consists of two prediction tables: the "last-value" and the "stride" prediction tables (Subsection 2.2). A candidate instruction for value prediction can be allocated to one of these tables according to its opcode directive type. These new capabilities allow us to exploit both value predictability patterns (stride and last-value) and utilize the prediction tables more efficiently. In addition, they allow us to detect in advance the highly predictable instructions, and thus we could reduce the probability that unlikely to be correctly predicted instructions evacuate useful instructions from the prediction table. In order to clarify the principles of our new technique we are assisted by the following sample C program segment: The program sums the values of two vectors, B and C, into vector A. In the first phase, the compilation of the program with the gcc 2.7.2 compiler (using the "-O2" optimization) yields the following assembly code (for a Sun-Sparc machine on SunOS 4.1.3): (1) OG#L#J#O#/RDG#%>L@ (2) OG#L#J#L#/RDG#&>M@ In the second phase we collect the profile image of the program. A sample output file of this process is illustrated by table 3.1. It can be seen that this table includes all the instructions in the program that assign values to a destination register (load and add instructions). For simplicity, we only refer to value prediction where the destination operand is a register. However our methodology is not limited by any means to being applied when the destination operand is a condition code, a program counter, a memory storage location or a special register. Instruction address Prediction accuracy efficiency ratio 3 99.99% 99.99% 7 99.99% 99.99% 9 99.99% 99.99% Table 3.1 - A sample profile image output. In this example the profile image indicates that the prediction accuracy of the instructions that compute the index of the loop was 99.99% and their efficiency ratio was 99.99%. Such an observation is reasonable since the destination value of these instructions can be correctly predicted by the stride predictor. The other instructions in our example accomplished relatively low prediction accuracy and stride efficiency ratio. If the user determines the prediction accuracy threshold to be 90%, then in the third phase the compiler would modify the opcodes of the add operations in addresses 3, 7, and 9 and insert into these opcodes the "stride" directive. All other instructions in the program are unaffected. 4. Examining the potential of profiling through quantitative measurements This section is dedicated to examining the basic question: can program profiling supply the value prediction hardware mechanisms with accurate information about the tendency of instructions to be value-predictable? In order to answer this question, we need to explore whether programs exhibit similar patterns when they are being run with different input parameters. If under different runs of the programs these patterns are correlated, this confirms our claim that profiling can supply accurate information. For our experiments we use different programs, chosen from the Spec95 benchmarks (table 4.1), with different input parameters and input files. In order to collect the profile image we traced the execution of the programs by the SHADE simulator ([12]) on Sun-Sparc processor. In the first phase, all benchmarks were compiled with the gcc 2.7.2 compiler with all available optimizations. Benchmarks Benchmarks Description go Game playing. A simulator for the 88100 processor. gcc A C compiler based on GNU C 2.5.3. compress95 Data compression program using adaptive Lempel-Ziv coding. li Lisp interpreter. ijpeg JPEG encoder. perl Anagram search program. vortex A single-user object-oriented database transaction benchmark. mgrid Multi-grid solver in computing a three dimensional potential field. Table 4.1 - Spec95 benchmarks. For each run of a program we create a profile image containing statistical information that was collected during run-time. The profile image of each run can be regarded as a vector V , where each of its coordinates represents the value prediction accuracy of an individual instruction (the dimension of the vector is determined by the number of different instructions that were traced during the experiment). As a result of running the same program n times, each time with different input parameters and input files, we obtain a set of n vectors { , ,., } the vector ( ) represents the profile image of run j. Note that in each run we may collect statistical information of instructions which may not appear in other runs. Therefore, we only consider the instructions that appear in all the different runs of the program. Instructions which only appear in certain runs are omitted from the vectors (our measurements indicate that the number of these instructions is relatively small). By omitting these instructions we can organize the components of each vector such that corresponding coordinates would refer to the prediction accuracy of same instruction, i.e., the set of coordinates { , ,., } 1,l 2,l n,l refers to the prediction accuracy of the same instruction l under the different runs of the program. Our first goal is to evaluate the correlation between the tendencies of instructions to be value-predictable under different runs of a program with different input files and parameters. Therefore, once the set of vectors { , , ., } is collected, we need to define a certain metric for measuring the similarity (or the correlation) between them. We choose to use two metrics to measure the resemblance between the vectors. We term the first metric the maximum-distance metric. This metric is a vector coordinates are calculated as illustrated by equation 4.1: max{| | , | | , , | | , | | , | | , | | , | |} Equation 4.1 - The Mmax metric. Each coordinate of M(V) max is equal to the maximum distance between the corresponding coordinates of each pair of vectors from the set { , , ., } . The second metric that we use is less strict. We term this metric the average-distance metric. This metric is also a vector, average k where each of its coordinates is equal to the arithmetic-average distance between the corresponding coordinates of each pair of vectors from the set { , , ., } (equation 4.2). average | | , | | , , | | , | | , | | , | | , | |} { 3# Equation 4.2 - The M average metric. Obviously, one can use other metrics in order to measure the similarity between the vectors, e.g., instead of taking the arithmetic average we could take the geometric average. However, we think that these metrics sufficiently satisfy our needs. Once our metrics are calculated out of the profile image, we can illustrate the distribution of its coordinates by building a histogram. For instance, we can count the number of M(V) max coordinates whose values are in each of the intervals: [0,10], (10,20], (30,40], .,(90,100]. If we observe that most of the coordinates are scattered in the lower intervals, we can conclude that our measurements are similar and that the correlation between the vectors is very high. Figures 4.1 and 4.2 illustrate such histograms for our two metrics M(V) max and M(V) average respectively. 0% 20% 40% 80% 100% 126.gcc 129.com press 130.li 132.ijpeg 134.perl 147.vorte x The spread of the coordinates of M(V)max Figure 4.1 - The spread of M(V) max . 126.gcc 129.com press 130.li 132.ijpeg 134.perl 147.vorte x The spread of the coordinates of M(V) average Figure 4.2 - The spread of M(V) average . In these histograms we clearly observe that in all the benchmarks most of the coordinates are spread across the lower intervals. This observation provides the first substantial evidence that confirms one of our main claims - the tendency of instructions in a program to be value predictable is independent of the program's input parameters and data. In addition it confirms our claim that program profiling can supply accurate information about the tendency of instructions to be value predictable. As we have previously indicated, the profile image of the program that is provided to the compiler can be better tuned so that it can indicate which instructions tend to repeat their recently generated value and which tend to exhibit patterns of strides. In order to evaluate the potential of such classification we need to explore whether the set of instructions whose outcome values exhibit tendency of strides is common to the different runs of the program. This can be done by examining the stride efficiency ratio of each instruction in the program from the profile image file. In this case, we obtain from the profile image file a vector S , where each of its coordinates represents the stride efficiency ratio of an individual instruction. When we run the same program n times (each time with different input parameters and input files) we obtain a set of n vectors { , ,., } where the vector represents the profile image of run j. Once these vectors are collected we can use one of the previous metrics either the maximum-distance or the average-distance in order to measure the resemblance between the set of vectors S S S S n { , ,., } . For simplicity we have chosen this time only the average-distance metric to demonstrate the resemblance between the vectors. Once this metric is calculated out of the profile information, we obtain a vector M(S) average . Similar to our previous analysis, we draw a histogram to illustrate the distribution of the coordinates of M(S) average (figure 4.3). 0% 20% 40% 80% 100% 099.go 124.m88ks im 126.gcc 129.compr ess 130.li 132.ijpeg 134.perl 147.vortex107.mgrid9070503010 The spread of the coordinate of M(S) average Figure 4.3 - The spread of M(S) average . Again in this histogram we clearly observe that in all the benchmarks most of the coordinates are spread across the lower intervals. This observation provides evidence that confirms our claim that the set of instructions in the program that tend to exhibit value predictability patterns in form of stride is independent of the program's input parameters and data. Therefore profiling can accurately detect these instructions and provide this information to the compiler. 5. The effect of the profiling-based classification on value-prediction performance In this section we focus on three main aspects: 1. the classification accuracy of our mechanism, 2. its potential to better utilize the prediction table entries and 3. its effect on the extractable ILP when using value prediction. We also compare our new technique versus the hardware only classification mechanism (saturated counters). 5. 1.The classification accuracy The quality of the classification process can be represented by the classification accuracy, i.e., the fraction of correct classifications out of overall prediction attempts. We measured the classification accuracy of our new mechanism and compared it to the hardware-based mechanism. The classification accuracy was measured for the incorrect and correct predictions separately (using the "stride" predictor), as illustrated by figures 5.1 and 5.2 respectively. Note that these two cases represent a fundamental trade-off in the classification operation since improving the classification accuracy of the incorrect predictions can reduce the classification accuracy of the correct predictions and vice versa. Our measurements currently isolate the effect of the prediction table size since in this subsection we wish to focus only on the pure potential of the proposed technique to successfully classify either correct or incorrect value predictions. Hence, we assume that each of the classification mechanisms has an infinite prediction table (a stride predictor), and that the hardware-based classification mechanism also maintains an infinite set of saturated counters. The effect of the finite prediction table is presented in the next subsection.2060100 go m88ksim gcc compress li ijpeg perl vortex mgrid average FSM Prof th=90% Prof th=80% Prof th=70% Prof th=60% Prof th=50% The precentages of the mispredictions which are classified correctly Figure 5.1 - The percentages of the mispredictions which are classified correctly.2060100 go m88ksim gcc compress li ijpeg perl vortex mgrid average FSM Prof th=90% Prof th=80% Prof th=70% Prof th=60% Prof th=50% Thepercentages of the correct predictions which are classified correctly Figure 5.2 - The percentages of the correct predictions which are classified correctly. Our observations indicate that in most cases the profiling-based classification better eliminates mispredictions in comparison with the saturated counters. When the threshold value of our classification mechanism is reduced, the classification accuracy of mispredictions decreases as well, since the classification becomes less strict. Only when the threshold value is less than 60% does the hardware-based classification gain better classification accuracy for the mispredictions than our proposed mechanism (on the average). Decreasing the threshold value of our classification mechanisms improves the detection of the correct predictions at the expense of the detection of mispredictions. Figure 5.2 indicates that in most cases the hardware-based classification achieves slightly better classification accuracy of correct predictions in comparison with the profiling-based classification. Notice that this observation does not imply at all that the hardware-based classification outperforms the profiling-based classification, because the effect of the table size was not included in these measurements. 5. 2.The effect on the prediction table utilization We have already indicated that when using the hardware-based classification mechanism, unpredictable instructions may uselessly occupy entries in the prediction table and can purge out highly predictable instructions. As a result, the efficiency of the predictor can be decreased, as well as the utilization of the table and the prediction accuracy. Our classification mechanism can overcome this drawback, since it is capable of detecting the highly predictable instructions in advance, and hence decreasing the pollution of the table caused by unpredictable instructions. In table 5.1 we show the fraction (in percentages) of potential candidates which are allowed to be allocated in the table by our classification mechanism out of those allocated by the saturated counters. It can be observed that even with a threshold value of 50%, the new mechanism can reduce the number of potential candidates by nearly 50%. Moreover, this number can be reduced even more significantly when the threshold is tightened, e.g., a threshold value of 90% reduces the number of potential candidates by more than 75%. This unique capability of our mechanism allows us to use a smaller prediction table and utilize it more efficiently. Profiling threshold 90% 80% 70% 60% 50% The fraction of potential candidates to be allocated relative to those in the saturated counters Table 5.1 - The fraction of potential candidates to be allocated relative to those in the hardware-based classification. In order to evaluate the performance gain of our classification method in comparison with the hardware-based classification mechanism, we measured both the total number of correct predictions and the total number of mispredictions when the table size is finite. The predictor, used in our experiments, is the "stride predictor", which was organized as a 512-entry, 2-way set associative table. In addition, in the case of the profiling-based classification, instructions were allowed to be allocated to the prediction table only when they were tagged with either the "last-value" or the "stride" directives. Our results, summarized in figures 5.3 and 5.4, illustrate the increase in the number of correct predictions and incorrect predictions respectively gained by the new mechanism (relative to the saturated counters). It can be observed that the profiling threshold plays the main role in the tuning of our new mechanism. By choosing the right threshold, we can tune our mechanism in such way that it outperforms the hardware-based classification mechanism in most benchmarks. In the benchmarks go, gcc, li, perl and vortex, we can accomplish both a significant increase in the number of correct predictions and a reduction in the number of mispredictions. For instance, when using a threshold value in the range of 80-90% in vortex, our mechanism accomplishes both more correct predictions and less incorrect predictions than the hardware-only mechanism. Similar achievements are also obtained in go when the range of threshold values is 60-90%, in gcc when the range is 70-90%, in li when the threshold value is 60% and in perl when the range is 70-90%. In the other benchmarks (m88ksim, compress, ijpeg and mgrid) we cannot find a threshold value that yields both an increase in the total number of correct predictions and a decrease in the number of mispredictions. The explanation of this observation is that these benchmarks employ relatively much smaller working-sets of instructions and therefore they can much less exploit the benefits of our classification mechanism. Also notice that the mispredictions increase, observed for our classification mechanism in m88ksim, is not expected to significantly affect the extractable ILP, since the prediction accuracy of this benchmark is already very high. JR P#NVL JFF FRPSUH OL LMSHJ SHUO YRUWH[ PJULG Figure 5.3 - The increase in the total number of correct predictions. JR P#NVL JFF FRPSU OL LMSHJ SHUO YRUWH[ PJULG Figure 5.4 - The increase in the total number of incorrect predictions. 5. 3.The effect of the classification on the extractable ILP In this subsection we examine the ILP that can be extracted by value prediction under different classification mechanisms. Our experiments consider an abstract machine with a finite instruction window of 40 entries, unlimited number of execution units and a perfect branch prediction mechanism. In addition, the type of value predictor that we use and its table organization are the same as in the previous subsection. In case of value-misprediction, the penalty in our abstract machine is 1 clock cycle. Notice that such a machine model can explore the pure potential of the examined mechanisms without being constrained by individual machine limitations. Our experimental results, summarized in table 5.2, present the increase in ILP gained by using value prediction under different classification mechanisms (relative to the case when value prediction is not used). In most benchmarks we observe that our mechanism can be tuned, by choosing the right threshold value, such that it can achieve better results than those gained by the saturated counters. In addition, we also observe that when decreasing the threshold value from 90% to 50% the ILP gained by our new mechanism increases (in most cases). The explanation of this phenomenon is that in our range of threshold values, the contribution of increasing the correct predictions (as a result of decreasing the threshold) is more significant than the effect of increasing mispredictions. ILP increase Prof. 90% Prof. 80% Prof. 70% Prof. Prof. 50% go 10% 9% 10% 13% 13% 13% gcc 15% 16% 17% 21% 21% 21% compress 11% 7% 7% 8% 8% 8% li 37% 33% 35% 38% 38% 40% ijpeg 16% 14% 14% 15% 16% 15% perl 19% 23% 24% 28% 28% 27% vortex 159% 175% 178% 180% 179% 179% mgrid 24% 7% 10% 11% 11% 11% Notations prediction using saturated counters. X% Value prediction using the profiling-based classification and a threshold value Table 5.2 - The increase in ILP under different classification mechanisms relative to the case when value prediction is not used. 6. Conclusions This paper introduced a profiling-based technique to enhance the efficiency of value prediction mechanisms. The new approach suggests using program profiling in order to classify instructions according to their tendency to be value-predictable. The collected information by the profiler is supplied to the value prediction mechanisms through special directives inserted into the opcode of instructions. We have shown that the profiling information which is extracted from previous runs of a program with one set of input parameters is highly correlated with the future runs under other sets of inputs. This observation is very important, since it reveals various opportunities to involve the compiler in the prediction process and thus to increase the accuracy and the efficiency of the value predictor. Our experiments also indicated that the profiling information can distinguish between different value predictability patterns (such as "last-value" or "stride"). As a result, we can use a hybrid value predictor that consists of two prediction tables: the last-value and the stride prediction tables. A candidate instruction for value prediction can be allocated to one of these tables according to its profiling classification. This capability allows us to exploit both value predictability patterns (stride and last-value) and utilize the prediction tables more efficiently. Our performance analysis showed that the profiling-based mechanism could be tuned by choosing the right threshold value so that it outperformed the hardware-only mechanism in most benchmarks. In many benchmarks we could accomplish both a significant increase in the number of correct predictions and a reduction in the number of mispredictions. The innovation in this paper is very important for future integration of the compiler with value prediction. We are currently working on other properties of the program that can be identified by the profiler to enhance the performance and the effectiveness of value prediction. We are examining the effect of the profiling information on the scheduling of instruction within a basic block and the analysis of the critical path. In addition, we also explore the effect of different programming styles such as object oriented on the value predictability patters. --R Some Experiments in Local Microcode Compaction for Horizontal Machines. A Compiler for VLIW Architecture. The Optimization of Horizontal Microcode Within and Beyond Basic Blocks: An Application of Processor Scheduling with Resources. Speculative Execution based on Value Prediction. An Experimental and Analytical Study of Speculative Execution based on Value Prediction. Computer Architecture a Quantitative Approach. Superscalar Microprocessor Design. Software Pipelining: An Effective Scheduling Technique for VLIW Processors. Value Locality and Load Value Prediction. Exceeding the Dataflow Limit via Value Prediction. Branch Prediction Strategies and Branch-Target Buffer Design A Study of Branch Prediction Techniques. Limits of Instruction-Level Parallelism A Study of Scalar Compilation Techniques for Pipelined Supercomputers. Alternative Implementations of Two-Level Adaptive Branch Prediction --TR Bulldog: a compiler for VLSI architectures A study of scalar compilation techniques for pipelined supercomputers Software pipelining: an effective scheduling technique for VLIW machines Limits of instruction-level parallelism Alternative implementations of two-level adaptive branch prediction Value locality and load value prediction Exceeding the dataflow limit via value prediction Computer architecture (2nd ed.) A study of branch prediction strategies The optimization of horizontal microcode within and beyond basic blocks --CTR Peng Chen , Krishna Kavi , Robert Akl, Performance Enhancement by Eliminating Redundant Function Execution, Proceedings of the 39th annual Symposium on Simulation, p.143-151, April 02-06, 2006 Youtao Zhang , Jun Yang , Rajiv Gupta, Frequent value locality and value-centric data cache design, ACM SIGOPS Operating Systems Review, v.34 n.5, p.150-159, Dec. 2000 Youtao Zhang , Jun Yang , Rajiv Gupta, Frequent value locality and value-centric data cache design, ACM SIGPLAN Notices, v.35 n.11, p.150-159, Nov. 2000 Chao-ying Fu , Jill T. Bodine , Thomas M. Conte, Modeling Value Speculation: An Optimal Edge Selection Problem, IEEE Transactions on Computers, v.52 n.3, p.277-292, March Jun Yang , Rajiv Gupta, Frequent value locality and its applications, ACM Transactions on Embedded Computing Systems (TECS), v.1 n.1, p.79-105, November 2002 Jos Gonzlez , Antonio Gonzlez, The potential of data value speculation to boost ILP, Proceedings of the 12th international conference on Supercomputing, p.21-28, July 1998, Melbourne, Australia Dean M. Tullsen , John S. Seng, Storageless value prediction using prior register values, ACM SIGARCH Computer Architecture News, v.27 n.2, p.270-279, May 1999 Chao-Ying Fu , Matthew D. Jennings , Sergei Y. Larin , Thomas M. Conte, Value speculation scheduling for high performance processors, ACM SIGOPS Operating Systems Review, v.32 n.5, p.262-271, Dec. 1998 Daniel A. Connors , Wen-mei W. Hwu, Compiler-directed dynamic computation reuse: rationale and initial results, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.158-169, November 16-18, 1999, Haifa, Israel Chia-Hung Liao , Jong-Jiann Shieh, Exploiting speculative value reuse using value prediction, Australian Computer Science Communications, v.24 n.3, p.101-108, January-February 2002 Mikio Takeuchi , Hideaki Komatsu , Toshio Nakatani, A new speculation technique to optimize floating-point performance while preserving bit-by-bit reproducibility, Proceedings of the 17th annual international conference on Supercomputing, June 23-26, 2003, San Francisco, CA, USA Tarun Nakra , Rajiv Gupta , Mary Lou Soffa, Value prediction in VLIW machines, ACM SIGARCH Computer Architecture News, v.27 n.2, p.258-269, May 1999 Glenn Reinman , Brad Calder , Dean Tullsen , Gary Tyson , Todd Austin, Classifying load and store instructions for memory renaming, Proceedings of the 13th international conference on Supercomputing, p.399-407, June 20-25, 1999, Rhodes, Greece Huiyang Zhou , Jill Flanagan , Thomas M. Conte, Detecting global stride locality in value streams, ACM SIGARCH Computer Architecture News, v.31 n.2, May Freddy Gabbay , Avi Mendelson, The effect of instruction fetch bandwidth on value prediction, ACM SIGARCH Computer Architecture News, v.26 n.3, p.272-281, June 1998 M. Burrows , U. Erlingson , S-T. A. Leung , M. T. Vandevoorde , C. A. Waldspurger , K. Walker , W. E. Weihl, Efficient and flexible value sampling, ACM SIGOPS Operating Systems Review, v.34 n.5, p.160-167, Dec. 2000 M. Burrows , U. Erlingson , S.-T. A. Leung , M. T. Vandevoorde , C. A. Waldspurger , K. Walker , W. E. Weihl, Efficient and flexible value sampling, ACM SIGPLAN Notices, v.35 n.11, p.160-167, Nov. 2000 Ben-Chung Cheng , Daniel A. Connors , Wen-mei W. Hwu, Compiler-directed early load-address generation, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.138-147, November 1998, Dallas, Texas, United States Brad Calder , Glenn Reinman , Dean M. Tullsen, Selective value prediction, ACM SIGARCH Computer Architecture News, v.27 n.2, p.64-74, May 1999 Martin Burtscher , Amer Diwan , Matthias Hauswirth, Static load classification for improving the value predictability of data-cache misses, ACM SIGPLAN Notices, v.37 n.5, May 2002 Freddy Gabbay , Avi Mendelson, Using value prediction to increase the power of speculative execution hardware, ACM Transactions on Computer Systems (TOCS), v.16 n.3, p.234-270, Aug. 1998 Peng-Sheng Chen , Yuan-Shin Hwang , Roy Dz-Ching Ju , Jenq Kuen Lee, Interprocedural Probabilistic Pointer Analysis, IEEE Transactions on Parallel and Distributed Systems, v.15 n.10, p.893-907, October 2004

value-prediction;speculative execution;instruction-level parallelism

266829

Procedure placement using temporal ordering information.

Instruction cache performance is very important to instruction fetch efficiency and overall processor performance. The layout of an executable has a substantial effect on the cache miss rate during execution. This means that the performance of an executable can be improved significantly by applying a code-placement algorithm that minimizes instruction cache conflicts. We describe an algorithm for procedure placement, one type of code-placement algorithm, that significantly differs from previous approaches in the type of information used to drive the placement algorithm. In particular, we gather temporal ordering information that summarizes the interleaving of procedures in a program trace. Our algorithm uses this information along with cache configuration and procedure size information to better estimate the conflict cost of a potential procedure ordering. We compare the performance of our algorithm with previously published procedure-placement algorithms and show noticeable improvements in the instruction cache behavior.

Introduction The linear ordering of procedures in a program's text segment fixes the addresses of each of these procedures and this in turn determines the cache line(s) that each procedure will occupy in the instruction cache. In the case of a direct-mapped cache, conflict misses result when the execution of the program alternates between two or more procedures whose addresses map to overlapping sets of cache lines. Several compile-time code-placement techniques have been developed that use heuristics and profile information to reduce the number of conflict misses in the instruction cache by a reordering of the program code blocks [5,6,7,8,11]. Though these techniques successfully remove a sizeable number of the conflict misses when compared to the default code layout produced during the typical compilation process, it is possible to do even better if we gather improved profile information and consider the specifics of the hardware configuration. To this end, we propose a method for summarizing the important temporal ordering information related to code placement, and we show how to use this information in a machine-specific manner that often further reduces the number of instruction cache conflict misses. In particular, we apply our new techniques to the problem of procedure placement in direct-mapped caches, where the compiler 2achieves an optimized cache line address for each procedure by specifying the ordering of the procedures and gaps between procedures in an executable. Code-placement techniques may reorganize an application at one or more levels of granularity. Typically, a technique focuses on the placement of whole procedures or individual basic blocks. We use the term code block to refer to the unit of granularity to which a code-placement technique applies. Though we focus on the placement of variable-sized code blocks defined by procedure boundaries, our techniques for capturing temporal information and using this information during placement apply to code blocks of any granularity. The default code layout produced by most compilers places the procedures of an executable in the same order in which they were listed in the source files and preserves the order of object files from the linker command line. Therefore, it is left to chance which code blocks will conflict in the cache. Whenever there are code blocks that are often executed together and happen to overlap in the cache, they can cause a significant number of conflict misses in the instruction cache. Several studies have shown that compile-time optimizations that change the relative placement of code blocks can cause large changes in the instruction cache miss rate [3,4]. When changes in the relative placement of code blocks occurs, the performance of the program will be affected not only by the intended effect of the optimization, but also by the resulting change in instruction cache misses. This makes it difficult to predict the total effect of optimizations which change the code size and use the change in running time of the optimized executable to judge the effectiveness of the optimizations. In summary, code-placement techniques are important because they improve the performance of the instruction fetcher and because they enable the effective use of other compile-time optimizations. To reduce the instruction cache miss rate of an application, a code placement algorithm requires two capabilities: it must be able to assign code blocks to the cache lines; and it must have information on the relative importance of avoiding overlap between different sets of code blocks. There are only a few ways for the compiler to set the addresses of a code block. The compiler can manipulate the order in which procedures appear in the executable, and it can leave gaps between two adjacent procedures to force the alignment of the next procedure at a specific cache line. The more interesting problem is determining how procedures should overlap in the hardware instruction cache. The previous work on procedure placement has almost exclusively been based on summary profile statistics that simply indicate how often a code block was executed. Often this information is organized into a weighted procedure call graph (WCG) that records the number of calls that occurred between pairs of procedures during a profiling run of the program. Figure 1 contains an example of a WCG. This summary information is used to estimate the penalty resulting from the placement of these procedure pairs in the same cache locations. The aim of most existing algorithms is to place procedures such that pairs with high call counts do not conflict in the cache. Counting the number of calls between procedures and summarizing this information in a WCG provides a way of recognizing procedures that are temporally related during the execution of a program. However, a WCG does not give us all the temporal information that we would like to have. In particular, the absence of an edge between two procedures does not necessarily mean that there is no penalty to overlapping the procedures. For example, the WCG in Figure 1 is produced both when the condition cond alternates between true and false (Trace #1 in Figure 1) and when Proc M() loop 20-times loop 4-times if (cond) call else call T; endl call Z; endl Figure 1. Example of a simple program that calls three leaf procedures. The weighted procedure call graph is obtained when the condition cond is true 50% of the time. Notice that this same WCG is obtained from both call traces given on the right of the figure. (a) Example program (b) Weighted procedure call graph (c) Possible traces corresponding to the WCG Trace #1: Trace #2: the condition cond is true 40 times and then false 40 times (Trace #2). Assume for the purposes of this example that all procedures in Figure 1 require only a single cache line and that we have only three locations in our direct-mapped instruction cache. If one cache location is reserved for procedure M, we clearly do not want the same code layout for the last two cache locations in both these execution traces. Trace #1 experiences fewer cache conflict misses when procedures S and T are each given distinct cache line (Z shares a cache line with S or T), while Trace #2 experiences fewer cache conflict misses when procedures S and T share a cache line (Z is given its own cache line). The WCG in Figure 1 does not capture the temporal ordering information that is needed to determine which layout is best. A WCG summarizes only direct call information; no precise information is provided on the importance of conflicts between siblings (as illustrated in Figure 1) or on more distant temporal relationships. To enable better code layout, we want to have a measure of how much the execution of a program alternates between each pair of procedures (not just the pairs connected by an edge in the WCG). We refer to this measure as temporal ordering information. With this and other information concerning procedure sizes and the target cache configuration, we can make a better estimate of the number of conflict misses experienced by any specific layout. We begin in Section 2 with a brief description of a well-known procedure-placement algorithm that sets a framework for understanding our new algorithm. We present the details of our algorithm in Sections 3 and 4. Section 3 describes our method for extracting and summarizing the temporal ordering information in a program trace, while Section 4 presents our procedure-place- ment algorithm that uses the information produced by this method. In Section 5, we explain our experimental methodology and present some empirical results that demonstrate the benefit of our algorithm over previous algorithms for direct-mapped caches. Section 6 describes how to modify our algorithm for set-associative caches. Finally, Section 7 reviews other related work in code lay-out and Section 8 concludes. Procedure placement from Pettis and Hansen Current approaches to procedure placement rely on greedy algorithms. We can summarize the differences between these algorithms by describing how . selects the order in which procedures are considered for placement; and . determines where to place each procedure relative to the already-placed procedures. We begin with a description of the well-known procedure-placement algorithm by Pettis and Hansen [8]. As we will explain, our new algorithm retains much of the structure and many of the important heuristics found in the Pettis and Hansen approach. In addition to procedure placement, Pettis and Hansen also address the issues of basic-block placement and branch alignment. For the purposes of this paper, we use the acronym PH when referring to our implementation of the procedure placement portion of their algorithm. Pettis and Hansen reduce instruction-cache conflicts between procedures by placing the most frequent caller/callee procedure pairs at adjacent addresses. Their approach is based on a WCG summary of the profile information. They use this summary information both to select the next procedure to place and to determine where to place that procedure in relationship to the already- placed procedures. For our implementation of PH, we produce an undirected graph with weighted edges, which contains essentially the same information as a WCG. There is one node in the graph for each procedure in the program. An edge e p,q connects two nodes p and q if p calls q or q calls p. The weight W(e p,q ) given to e p,q is equal to the (2 * (calls p,q + calls q,p )), where calls p,q is the total number of calls made from procedure p to procedure q. We collect this information by scanning an instruc- tion-address trace of the program and noting each transition between procedures, which counts both calls and returns. Therefore, we get an edge weight that is twice the number of calls; the extra factor of two does not change the procedure placement produced by PH. This graph is used to select both the next procedure to place and determine the relative placement for this procedure. PH begins by making a copy of this initial graph; we refer to this copy as the working graph. PH searches this working graph for the edge with the largest weight. Call this edge e u,v . Once this edge is found, the algorithm merges the two nodes u and v into a single node u' in the working graph (more details in a moment). The remaining edges from the original nodes u and v to other nodes become edges of the new node u'. To maintain the invariant of a single edge between any pairs of nodes, PH combines each pair of edges e u,r and e v,r into a single edge e u',r with weight (W(e u,r )). The algorithm then repeats the process, again searching for the edge with the largest weight in the working graph, until the graph is reduced to a single node. In PH, the only way of reducing the chance of a conflict miss between procedures is proximity in the address space: the closer that we place two procedures in the address space, the less likely it is that they will conflict. The procedures within a node are organized as a linear list called a chain [8]. When PH merges two nodes, their chains can be combined into a single chain in four ways. Let A and B represent the chains, and A' and B' the reverse of each chain. The four possibilities are AB, AB', A'B and A'B'. To choose the best one of these, PH queries the original graph to determine the edge e with the largest weight between a procedure p in the first chain and a procedure q in the second chain. Our implementation of PH chooses the merged chain that minimizes the distance (in bytes) between p and q. 3 Summarizing temporal ordering information Any algorithm that aims to optimize the arrangement of code blocks needs a conflict metric which quantifies the importance of avoiding conflicts between sets of code blocks. Ideally, the metric would report the number of cache conflict misses caused by mapping a set of code blocks to overlapping cache lines. We do not expect to find a metric that gives the exact number of resulting cache conflict misses, and we do not need one. We simply need the metric to be a linear function of number of conflict misses. 1 Section 5.3 shows that the metric used in our algorithm exhibits strong correlation with the instruction cache miss rate. 1. Clearly, any difference between the training and testing data sets will also affect the metric's ability to predict cache conflict misses in the testing run. As discussed in the previous section, PH uses the call-graph edge weight W(p,q) between two procedures and q as its conflict metric. This simple metric drives the merging of nodes. Unfortu- nately, this metric has several drawbacks, as illustrated in Section 1. To understand how to build a better conflict metric, it is helpful to review the actions of a cache when processing an instruction stream. Assume for a moment that we are tracking code blocks with a size equal to the size of a cache line. For a direct-mapped cache, a code block b maps to cache line l = (Addr(b) DIV line_size) MOD cache_lines. This code block remains in the cache until another code block maps to the same cache line. In terms of code layout, it is important therefore to note which other code blocks are referenced temporally nearby to a reference to b. Ideally, none of the blocks referenced between consecutive references to b map to the same cache line as b. In this way, we get reuse of the initial fetch of block b and do not experience a conflict miss during the second reference to b. Since the reuse of a code block can be prevented by a single other code block in direct-mapped caches, we construct a data structure that summarizes the frequency of alternating between two code blocks. It is convenient to build this data structure as a weighted graph, where the nodes represent individual code blocks. We refer to this graph as a temporal relationship graph (TRG). As in PH, the conflict metric is simply the edge weight e p,q between two nodes p and q. A TRG is more general than a call graph because it can contain edges connecting any pair of code blocks for which there is some interleaving during the program execution. The rest of this section describes the process by which we build a TRG, and the next section explains how we use the resulting TRG to place procedures. To construct a summary of the temporal locality of the code blocks in a trace, we analyze the set of recently-referenced code blocks at each transition between code blocks. By implementing an ordered set, Q, of code-block identifiers (e.g. procedure names) ordered as they appeared in the trace, we always have access to a history of the recently-referenced code blocks. There is a bound on the maximum size of Q because its entries eventually become irrelevant and can be removed. There are two ways in which a code block identifier p can become irrelevant. First, we need only the latest occurrence of p in Q. Any code blocks that are executed after the most recent occurrence of p can only have an effect on that occurrence, but not on an earlier occurrence of p. Second, p can become irrelevant if a sufficiently large amount of (unique) code has been executed since p's last occurrence and evicted p from the cache. Let T be the set of code block identifiers reached since the last reference to p. Let S(T) be the sum of the sizes of the code blocks referenced in T. Exactly how big S(T) needs to grow before p becomes irrelevant depends on the cache mapping of the code. Assuming that the code layout maximizes the reuse of the members of T, they will be mapped to non-overlapping addresses, and their cache footprint will be equal to S(T). Therefore, p becomes irrelevant when S(T) is greater than the cache size. In summary, we perform the following steps when inserting a new element p into our ordered set Q. First, place p at the most recent end of Q. If there is a previous occurrence of p in Q, remove it. If not, we remove the oldest members of Q until the removal of the next least-recently-used identifier would cause the total size (in bytes) of remaining code blocks in Q to be less than the cache size. To build a TRG, we process the trace one code block identifier at a time. At each processing step, contains a set of code blocks that have temporal locality. The more often that a set of code blocks appears in Q, the more important it is that they not occupy the same cache locations. For each code block identifier p that we remove from the trace, we update the TRG as follows. For every code block q in Q starting from the most-recent end of Q, we increment the weight on the edge e p,q . If node p does not exist, we create it; if the edge e p,q does not exist, we create it with a weight of 1. We continue down Q until we reach previous occurrence of p or the end of Q. We stop when we encounter previous occurrence of p because this indicates a reuse. Any code blocks temporarily referenced before this previous occurrence of p in Q could not have displaced p from the instruction cache. Once we have collected the relationship data for p, we insert p into Q as described above. The process then repeats until we have processed the entire trace. After process- ing, we are left with a TRG whose edge weights W(e p,q ) record the number of times p and q occurred within a sufficiently small temporal distance to be present in Q at the same time, independent of how p and q are related in the program's call graph. 4 Our placement algorithm Given the discussion in Sections 2 and 3, it should be clear that we could use TRG constructed in Section 3 within the procedure-placement algorithm described by Pettis and Hansen [8]. We have found however that extra temporal ordering information alone is not sufficient to guarantee lower instruction cache miss rates. To get consistent improvements, we also make two key changes to the way we determine where to place each procedure relative to the already-placed procedures. The first involves the use of procedure size and cache configuration information that allows us to make a more informed procedure-placement decision. The second involves the gathering of temporal ordering information at a granularity finer than the procedure unit; we use this more detailed information to overcome problems created by procedures that are larger than the cache size. For efficiency reasons, we also consider only popular (i.e. frequently executed) procedures during the building of a relationship graph, as was proposed by Hashemi et al. [5]. The rest of this section outlines our procedure-placement algorithm. Section 4.1 begins with a description of the TRGs required for our algorithm and how we iterate through the procedure list selecting the order in which procedure processed by the main outer loop. Section 4.2 focuses on the portion of our algorithm's main loop that places a procedure relative to the procedures already processed using cache configuration and procedure size information. This placement decision simply specifies a cache-relative alignment among a set of procedures. The determination of each procedure's starting address (i.e. its placement in the linear address space) occurs only after all popular procedures have been processed. Section 4.3 presents the details of this process. 4.1 TRGs and the main outer loop Our algorithm uses two related TRGs. One selects the next procedure to be placed (TRG select ); and other aids in the determination of where to place this selected procedure (TRG place ). In PH, these two graphs are initially the same. In our algorithm, the graphs differ in the granularity of the code blocks processed during TRG build. While a code block in TRG select corresponds to a whole procedure, a code block in TRG place corresponds to a chunk of a procedure. For our benchmarks, we have found that "chunking" procedures into 256-byte pieces works well. TRG place therefore contains nodes for each procedure p in a program. It is straightforward to modify the algorithm in the previous section to generate both TRGs simultaneously. Though we record temporal information concerning the parts of procedures, our procedure-place- ment algorithm places only whole procedures. We use the finer-grain information only to find the best relative alignment of the whole procedures as explained below. Though TRG select contains more edges per node than the relationship graph built in PH (due to the additional temporal ordering information), we process TRG select in exactly the same greedy- merging manner as the relationship graph discussed in Section 2. Though we tried several other methods for creating an order to select procedures for placement, we could not find a more robust heuristic (or one that was as simple and elegant). The only other difference in our "working" relationship graph is that TRG select contains only popular procedures. Section 4.3 discusses how we place the remaining unpopular procedures. 4.2 Determining cache-relative alignments In PH, the data structure for the nodes in the working graph is a linear list (or a chain) of the pro- cedures. The building of a chain is more restrictive in terms of selecting starting addresses for placed procedures than it needs to be however. The only constraint that we need to maintain is that the placed procedures are mapped to addresses that result in a cache layout with a small conflict cost. We explain how exactly we calculate the cost of a placement in a moment. So, instead of chains, we use a data structure for nodes in TRG select that comprises of a set of tuples. Each tuple consists of a procedure identifier and an offset, in cache lines, of the beginning of this procedure from the beginning of the cache. For a node containing only a single procedure, the offset is zero. When two nodes, each containing a single procedure, are merged together, our algorithm modifies the offset of the second procedure to ensure that the cost metric of the placement of these two procedures in the cache is minimized. The algorithm in Figure 2 presents the pseudo-code for the merging of two nodes containing any number of already-placed procedures. Three items are note-worthy concerning the merge_nodes routine in Figure 2. First, when we merge two nodes, we leave the relative alignment of all the procedures within each node Figure 2. Pseudo-code for the merging of two nodes from the temporal relationship graph RG select . Procedure chunks within a node are identified by unique id's. An offset for a chunk id records the cache-line index corresponding to the beginning of that chunk. Offsets are always in units of cache lines. array [#_cache_lines] of {id, .}; merge_nodes (NODE n1, NODE n2) { // Initialize cache array c1 by marking each line with the procedure-chunk id's from node n1 occupying that line. foreach (id, offset) pair p in n1 for { int foreach (id,offset) pair p in p.offset += best_offset; return (n1 - unchanged. We do not backtrack and undo any previous decisions. Though the ability to rearrange the entire set of procedures in the two nodes being merged might lead to a better layout, this flexibility would noticeably increase the computational complexity of the algorithm. We assume that the selection order for procedure placement has guaranteed that we have already avoided the most expensive, potential cache conflicts. As our experimental results show, this greedy heuristic works quite well in practice. It is an open research question if limited amounts of backtracking could improve upon the layouts found by our current approach. Second, merge_nodes calculates a cost metric for each potential alignment of the layout in the first node with respect to the layout in the second node. If we fix the layout of the first node to begin at cache line 0, we can offset the start of the second node's layout by any number between 0 and the number of lines in the cache. We evaluate each of these relative offsets using the fine-grained temporal information in TRG place . For a given offset, we compute, for each procedure piece in the first node, which procedure pieces in the second node overlap with it in the cache. For each pair of overlapping procedure pieces, we compute the estimated number of cache conflicts corresponding to this overlap by accessing the weight on the edge (if any) between these two procedure pieces in TRG place . We then sum all of these estimates to obtain the total estimate for this potential placement. We calculate the estimate for procedure-piece conflicts only between nodes (and not the intra-node conflicts between procedure pieces) because we want the incremental cost of the placement. The cost of the intra-node overlaps are fixed and will not change the ultimate finding. The calculation of this extra cost would only increase the work done by our algorithm. Third, if the cost-metric calculation produces several relative offsets with the same cost, our algorithm selects the first of these offsets. In the simplest case, if we merge two nodes each containing a single procedure (call them p and q) and the total size of these two procedures is less than the cache size, the merging of these nodes will result in a node that is equivalent to the chain created by PH. In other words, merge_nodes selects the first empty cache line after procedure p to begin procedure q since that is the first zero-cost location for q. 4.3 Producing the final linear list The merging phase of our algorithm ends when there are no more edges left in TRG select . 2 The final step in our algorithm produces a linear arrangement of all of the program procedures given the relative alignment decisions contained in remaining TRG select nodes. To begin, we select a procedure p with an cache-line offset of 0. 3 This is the first procedure in our linear layout. To find the next procedure in the linear layout, we search the nodes for a procedure q whose cache-rela- tive offset results in the smallest positive gap in cache lines between the end of p and the start of q. To understand the general case, assume that procedure p is the last procedure in the linear layout. If p ends at the cache-relative offset pEndLine, we choose a procedure q which starts at cache-rel- ative offset qStartLine as the next procedure in the linear layout if q produces the smallest positive value for gap among all unconsidered popular procedures, where Finally, whenever we produce a gap between two popular procedures, we search the unpopular procedures for one that fits in the gap. Once we determine an address for each popular procedures in the linear address space, we simply append any remaining un-placed, unpopular procedures to the end of our linear list. 5 Experimental evaluation In this section, we compare three different procedure-placement algorithms. In addition to PH and our algorithm (GBSC), we present results for a recently published procedure-placement algo- rithm, an algorithm by Hashemi, Kaeli, and Calder [5] which we refer to as HKC. Like our algo- rithm, HKC also extends PH to use knowledge of the procedure sizes, the cache size, and the cache organization. HKC uses a weighted call graph but not any additional temporal information. The key advantage of HKC over PH is that HKC records the set of cache lines occupied by each 2. Unlike PH, our "working" graph, TRG select , is not necessarily reduced to a single node. TRG select contains only popular procedures, and it is possible to have the only connection between two popular procedures be through an unpopular procedure. 3. This assumes that the start of the text segment maps to cache-line 0. If not, it is easy to adjust the algorithm. gap qStartLine pEndLine qStartLine numCacheLines pEndLine procedure during placement, and it tries to prevent overlap between a procedure and any of its immediate neighbors in the call graph. We begin in Section 5.1 with some aspects of the behavior of code placement techniques that need to be addressed in order to make a meaningful comparison of different algorithms. In particular, we introduce an experimental methodology based on randomization techniques. Section 5.2 outlines our experimental methodology while Section 5.3 presents our results. 5.1 Evaluating the performance of code placement algorithms We normally expect code optimizations to behave similar to a continuous function: small changes in the behavior of the optimization cause small changes in the performance of the resulting exe- cutable. With code placement optimizations, this is often not the case: small changes in the layout of a program can cause dramatic changes in the cache miss rate. As an example, we simulated the instruction cache behavior of the SPECint95 perl program for two slightly different layouts. The first layout is the output of our own code layout algorithm, and the second layout is identical to the first except that each procedure is padded by an additional bytes (one cache line) of empty space at its end. The instruction cache miss rate changed from 3.8% for the first layout to 5.4% for the second layout; this is a remarkable change for such a trivial difference between the layouts. In fact, it is possible to introduce a large number of misses by moving one code block by only a single cache line. For greedy code-layout algorithms, we have the additional problem that different layouts, in fact substantially different layouts, often result from small changes in the input profile data. At each step, PH, HKC, and GBSC greedily choose the highest-weight edge in the working graph. If there are two edges, say with weight 1,000,000 and 1,000,001, the (barely) larger edge will always be chosen first, even though such a small difference is unlikely to represent a statistically significant basis for preferring one edge over the other. Worse, ties resulting from identical edge weights are decided arbitrarily. Decisions between two equally good alternatives, which must necessarily be made one way or the other, affect not only the current step of the algorithm, but all future steps. As a result, we find it difficult to draw conclusions about the relative performance of different code layout algorithms from a small number of program traces. Ideally, we would like to have a large enough set of different inputs for each benchmark to get an accurate impression of the distribution of results. Unfortunately, this is very hard to do in practice since common benchmark suites are not distributed with more than a handful of input sets for each benchmark application. We simulate the effect of many slightly different application input sets by first running the application with a single input, and then applying random perturbations to the resulting profile data. For the algorithms in our comparison, we perturb all of our weighted graphs by multiplying each edge weight by a value close to one. Specifically, the initial weight w is replaced by the perturbed weight according to the equation , where X is a random variable, normally distributed with mean 0 and variance 1, and s is a scaling factor which determines the magnitude of the random perturbations. Using multiplicative rather than additive noise is attractive for two reasons. First, additive noise can cause weights to become negative, for which there is no obvious interpretation. Second, the method is inherently self-scaling in the sense that reasonable values for s are independent of the initial edge weights. A large enough value for s will cause the layout to be effectively random, as the perturbed graphs will bear little relationship to the profile data. Low values of s will cause only statistically insignificant differences in edge weights, and we can then observe the range of results produced by these small changes. We use in our experiments. Blackwell [2] shows that for several code placement algorithms, values of s as low as 0.01 elicit most of the range of performance variation from the system, and that values of s as high as 2.0 do not degrade the average performance very much. exp 5.2 Methodology We have implemented the PH, HKC, and GBSC procedure-placement algorithms such that they can be integrated into one of two different environments: a simulation environment based on ATOM [10]; and a compiler environment based on SUIF [9]. The results in Section 5.3 are based on the ATOM environment, but we have used the SUIF environment to verify that our algorithms produce runnable, correct code. Table 1 lists the benchmarks used in our study. Except for ghostscript, they are all from the SPECint95 benchmark suite. We use only five of the eight SPECint95 benchmarks because the other three (compress, ijpeg, and xlisp) are uninteresting in that all have small instruction working sets that do equally well under any reasonable procedure-placement algorithm. We compiled go and perl using the SUIF compiler (version 1.1.2), while all other benchmarks were compiled using gcc 2.7.2 with the -O2 optimization flag. We chose the input data sets to keep the traces to a manageable size. All of the reported miss rates in this and the next section are based on the simulation of an 8 kilobyte direct-mapped cache with a line size of 32 bytes. We use the training input to drive the procedure-placement algorithms, and then simulate the instruction-cache performance of the resulting optimized executable using the testing input. 5.3 Results The graphs in Figure 3 show our experimental results for PH, HKC, and GBSC. Each graph shows the results for a single benchmark. For each of the three algorithms, there is a curve showing the cumulative distribution of results over a set of 20 experiments, all based on the same training and testing traces. As described in Section 5.1, we use randomization to obtain twenty slightly different WCGs or TRGs that result in slightly different placements. For each point along a curve, the X-coordinate is the cache miss rate for one of the placements, and the Y-coordinate gives the percentage of all placements that had an equal or better miss rate. Consequently, if the curve for one algorithm is to the left of the curve for another algorithm, then the first algorithm gives better results. We notice that our algorithm gives clearly better results than the other two for all benchmarks 7except for m88ksim and perl. For these two benchmarks, the ranges of results overlap, though GBSC yields the lowest average miss rate over all placements. In summary, these results demonstrate the benefits of using temporal ordering information as well as an algorithm that considers cache-relative alignments in placing code. In Section 3, we said that a useful conflict metric should be strongly correlated with the number of cache misses. Figure 4 examines this issue by showing the relationship between conflict-metric values and cache miss rates. Each plot in Figure 4 contains 80 points, where each point corresponds to a different placement of the go benchmark. These placements are based on the GBSC algorithm; however we varied the output of this algorithm to produce a placement with a range of different miss rates. We accomplished this by randomly selecting 0-50 procedures in the GBSC placement and randomly changing their cache-relative offsets. The metric value plotted corresponds to the resulting placement. Figure 4a shows that our conflict metric, based on the fine-grained information in TRG place , shows a linear relationship with the actual number of cache misses; all the points in the graph are close to the diagonal. On the other hand, Figure 4b shows that a metric based only on a WCG is not always a good predictor of cache misses. Program Name All procedures Popular procedures Training trace Testing trace Miss rate of default layout Avg. size of procedure history size count size count input description length input description length go 590 K 3221 134 K 112 11x11 board, level 4, no stones level 6, 4 stones ghostscript 1817 K 372 104 K 216 14-page pre- sentation 37 M 3-page paper 38 M 2.63 8.9 limited to 50M BBs limited to 50M BBs 50 M 2.92 14.3 perl 664 K 271 reduced dic- tionary reduced input file vortex 1073 K 923 117 K 156 persons.250, reduced iteration reduced iteration Table 1: Details of our benchmark applications. We report sizes in bytes and trace lengths in basic blocks. A benchmark's ``average size of procedure history'' reports the average number of procedures that were present in our ordered set Q during the building of the TRG. Figure 3. Instruction cache miss rates for our benchmarks. Each graph shows the distribution of miss rates corresponding to the layouts produced by PH, HKC, and our new procedure-placement algorithm (GBSC). Each data point in the graphs represents the result for a single placement. Cache miss rates vary along the x-axis, and the y-axis shows the cumulative distribution of miss rates.0.20.61 "gc.PH" "gc.HKC" "gc.GBSC" (a) gcc0.20.61 "gs.PH" "gs.HKC" "gs.GBSC" (b) ghostscript0.20.61 "go.PH" "go.HKC" "go.GBSC" (c) go (d) m88ksim0.20.61 "m8.PH" "m8.HKC" "m8.GBSC" "pl.PH" "pl.HKC" "pl.GBSC"0.20.61 "vo.PH" "vo.HKC" "vo.GBSC" (f) vortex 6 Extensions for set-associative caches To this point, we have described a technique for collecting and using temporal information that is specific to direct-mapped cache implementations. In other words, we have assumed that a single occurrence of a procedure q between two occurrences of a procedure p is sufficient to displace p. This assumption is not necessarily true for set-associative caches, especially for those that implement a LRU policy. To use our approach for set-associative caches, we construct a slightly different data structure that replaces TRG place , and we slightly modify the cost-metric calculation in merge_nodes. This section focuses on 2-way set-associative caches; the implementation of changes for other associativities follows directly from this explanation. Instead of a graph representation for TRG place , it is now more convenient to think of the temporal- relationship structure as a database D that records the number of times that a code-block pair {r,s} appears between consecutive occurrences of another code block p in a program trace. We can still use our ordered set approach to build this database. However, when we process the temporal associations related to the next code block p in the trace, we associate p with all possible selections of two identifiers from the identifiers currently in Q (up to any previous occurrence of p as before).4812 Figure 4. Correlation between conflict metric and cache misses. Data points are 80 randomized layouts for the go benchmark. The X-coordinate of a point is the cache miss rate for that layout, and its Y-coordinate is the sum of the conflict metrics for the indicated method over the entire placement. cache miss rate conflict estimate (millions) (a) Conflict metric based on a fine-grained TRG. (b) Conflict metric based on a WCG.4812 cache miss rate conflict estimate (millions) We do this because two unique references are required to guarantee no reuse. Thus, the database simply records the frequency of each association between p and the pair {r,s}, accessed as D(p,{r,s}). If r, s, and p all occupy the same set in a two-way set-associative cache, then we estimate that D(p,{r,s}) of the program references to p will result in cache conflicts due to the displacement of p by intervening references to both r and s. We access this information instead of TRG place edge weights during the conflict-metric calculation in merge_nodes. Clearly the inner-loop of this calculation must also change slightly so that we can check the cost of the association between a code block in node n1 against all pairs of code blocks in n2 and vice-versa. Though we change TRG place , we do not change TRG select . TRG select is only a heuristic for selecting the order of code blocks to be placed; it is not obviously affected by a cache's associativity. As we mentioned earlier, other heuristic approaches may work better, but we have not found one. 7 Discussion and related work Much of the prior work in the area of compile-time code placement is related to early work in reducing the frequency of page faults in the virtual memory system and more recent work at reducing the cost of pipeline penalties associated with control transfer instructions. However, we limit our discussion here to studies that directly address the issue of code placement aimed at reducing instruction cache conflict misses. Some of the earliest work in this area was done by Hwu and Chang [6], McFarling [7], and Pettis and Hansen [8]. Hwu and Chang use a WCG and a proximity heuristic to address the problem of basic-block placement. Their approach is unique in that they also perform function inline expansion during code placement to overcome the artificial barriers imposed by procedure call boundaries. McFarling [7] uses an interesting program representation (a DAG of procedures, loops, and condi- tionals) to drive his code-placement algorithm, but the profile information is still summarized in such a way that some of the temporal interleaving of blocks in the trace is lost. In fact, this paper explicitly states that, because he is unable to collect temporal interleaving information, his algorithm assumes and optimizes for a worst-case interleaving of blocks. Like our algorithm, McFarling 1does consider the cache size and its modulo property when evaluating potential layouts, but his cost calculation is obviously different from ours. Finally, his algorithm is unique in its ability to determine which portions of the text segment should be excluded from the instruction cache. Torellas, Xia, and Daigle [11] propose a code-placement technique for kernel-intensive applica- tions. Their algorithm considers the cache address mapping when performing code placement. They define an array of logical caches, equal in size and address alignment to the hardware cache. Code placed within a single logical cache is guaranteed never to conflict with any other code in that logical cache. Though there is sub-area of all logical caches that is reserved for the most frequently-executed basic blocks, there is no general mechanism for calculating the placement costs across different logical caches. Their code placement is guided by execution counts of edges between basic blocks, and therefore does not capture temporal ordering information. The history mechanism we use to analyze the temporal behavior of a trace is similar to the problem of profiling paths in a procedure call graph. Ammons et al. [1] describe a way of implementing efficient path profiling. However, the data structure generated by this technique cannot be used in the place of our TRG, because it does not capture sufficient temporal ordering information. 8 Conclusion We have presented a method for extracting temporal ordering information from a trace. We then described a procedure-placement algorithm that uses this information along with the knowledge of the cache lines each procedure occupies to predict accurately which placements will result in the least number of conflict misses. The results show that these two factors combined allow us to obtain better instruction cache miss rates than previous procedure-placement techniques. Other code-placement techniques, such as "fluff removal" [8] and branch alignment [12], are orthogonal to the problem of placing whole procedures and can therefore be combined with our technique to achieve further improvements. The success of our experiments indicates that it is worthwhile to continue research on the temporal behavior of applications. In particular, we plan to develop similar techniques to optimize the behavior of applications in other layers of the memory hierarchy. 9 --R "Exploiting Hardware Performance Counters with Flow and Context Sensitive Profiling," "Applications of Randomness in System Performance Measurement." "The Effect of Code Expanding Optimizations on Instruction Cache Design," "Performance Issues in Correlated Branch Prediction Schemes," "Efficient Procedure Mapping Using Cache Line Coloring," "Achieving High Instruction Cache Performance with an Optimizing Compiler," "Program Optimization for Instruction Caches," "Profile Guided Code Positioning," "Extending SUIF for Machine-dependent Optimizations," "ATOM: A System for Building Customized Program Analysis Tools," "Optimizing Instruction Cache Performance for Operating System Intensive Workloads," "Near-Optimal Intraprocedural Branch Alignment," --TR Program optimization for instruction caches Achieving high instruction cache performance with an optimizing compiler Profile guided code positioning ATOM Performance issues in correlated branch prediction schemes Exploiting hardware performance counters with flow and context sensitive profiling Efficient procedure mapping using cache line coloring Near-optimal intraprocedural branch alignment Optimizing instruction cache performance for operating system intensive workloads Applications of randomness in system performance measurement --CTR Christophe Guillon , Fabrice Rastello , Thierry Bidault , Florent Bouchez, Procedure placement using temporal-ordering information: Dealing with code size expansion, Journal of Embedded Computing, v.1 n.4, p.437-459, December 2005 Keoncheol Shin , Jungeun Kim , Seonggun Kim , Hwansoo Han, Restructuring field layouts for embedded memory systems, Proceedings of the conference on Design, automation and test in Europe: Proceedings, March 06-10, 2006, Munich, Germany Alex Ramrez , Josep-L. Larriba-Pey , Carlos Navarro , Josep Torrellas , Mateo Valero, Software trace cache, Proceedings of the 13th international conference on Supercomputing, p.119-126, June 20-25, 1999, Rhodes, Greece Alex Ramirez , Josep Ll. Larriba-Pey , Carlos Navarro , Mateo Valero , Josep Torrellas, Software Trace Cache for Commercial Applications, International Journal of Parallel Programming, v.30 n.5, p.373-395, October 2002 John Kalamatianos , Alireza Khalafi , David R. Kaeli , Waleed Meleis, Analysis of Temporal-Based Program Behavior for Improved Instruction Cache Performance, IEEE Transactions on Computers, v.48 n.2, p.168-175, February 1999 Young , Michael D. Smith, Better global scheduling using path profiles, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.115-123, November 1998, Dallas, Texas, United States Rakesh Kumar , Dean M. Tullsen, Compiling for instruction cache performance on a multithreaded architecture, Proceedings of the 35th annual ACM/IEEE international symposium on Microarchitecture, November 18-22, 2002, Istanbul, Turkey S. Bartolini , C. A. Prete, Optimizing instruction cache performance of embedded systems, ACM Transactions on Embedded Computing Systems (TECS), v.4 n.4, p.934-965, November 2005 Architectural and compiler support for effective instruction prefetching: a cooperative approach, ACM Transactions on Computer Systems (TOCS), v.19 n.1, p.71-109, Feb. 2001 Trishul M. Chilimbi , Ran Shaham, Cache-conscious coallocation of hot data streams, ACM SIGPLAN Notices, v.41 n.6, June 2006 Alex Ramirez , Luiz Andr Barroso , Kourosh Gharachorloo , Robert Cohn , Josep Larriba-Pey , P. Geoffrey Lowney , Mateo Valero, Code layout optimizations for transaction processing workloads, ACM SIGARCH Computer Architecture News, v.29 n.2, p.155-164, May 2001 Chandra Krintz , Brad Calder , Han Bok Lee , Benjamin G. Zorn, Overlapping execution with transfer using non-strict execution for mobile programs, ACM SIGOPS Operating Systems Review, v.32 n.5, p.159-169, Dec. 1998 Stephen S. Brown , Jeet Asher , William H. Mangione-Smith, Offline program re-mapping to improve branch prediction efficiency in embedded systems, Proceedings of the 2000 conference on Asia South Pacific design automation, p.111-116, January 2000, Yokohama, Japan Alex Ramirez , Josep L. Larriba-Pey , Mateo Valero, Software Trace Cache, IEEE Transactions on Computers, v.54 n.1, p.22-35, January 2005 Trishul M. Chilimbi, Efficient representations and abstractions for quantifying and exploiting data reference locality, ACM SIGPLAN Notices, v.36 n.5, p.191-202, May 2001 Alex Ramirez , Oliverio J. Santana , Josep L. Larriba-Pey , Mateo Valero, Fetching instruction streams, Proceedings of the 35th annual ACM/IEEE international symposium on Microarchitecture, November 18-22, 2002, Istanbul, Turkey Young , Michael D. Smith, Static correlated branch prediction, ACM Transactions on Programming Languages and Systems (TOPLAS), v.21 n.5, p.1028-1075, Sept. 1999 Ann Gordon-Ross , Frank Vahid , Nikil Dutt, A first look at the interplay of code reordering and configurable caches, Proceedings of the 15th ACM Great Lakes symposium on VLSI, April 17-19, 2005, Chicago, Illinois, USA Brad Calder , Chandra Krintz , Simmi John , Todd Austin, Cache-conscious data placement, ACM SIGPLAN Notices, v.33 n.11, p.139-149, Nov. 1998 Timothy Sherwood , Brad Calder , Joel Emer, Reducing cache misses using hardware and software page placement, Proceedings of the 13th international conference on Supercomputing, p.155-164, June 20-25, 1999, Rhodes, Greece Thomas Kistler , Michael Franz, Automated data-member layout of heap objects to improve memory-hierarchy performance, ACM Transactions on Programming Languages and Systems (TOPLAS), v.22 n.3, p.490-505, May 2000 Murali Annavaram , Jignesh M. Patel , Edward S. Davidson, Call graph prefetching for database applications, ACM Transactions on Computer Systems (TOCS), v.21 n.4, p.412-444, November Rajiv A. Ravindran , Pracheeti D. Nagarkar , Ganesh S. Dasika , Eric D. Marsman , Robert M. Senger , Scott A. Mahlke , Richard B. Brown, Compiler Managed Dynamic Instruction Placement in a Low-Power Code Cache, Proceedings of the international symposium on Code generation and optimization, p.179-190, March 20-23, 2005 Nikolas Gloy , Michael D. Smith, Procedure placement using temporal-ordering information, ACM Transactions on Programming Languages and Systems (TOPLAS), v.21 n.5, p.977-1027, Sept. 1999 Martha Mercaldi , Steven Swanson , Andrew Petersen , Andrew Putnam , Andrew Schwerin , Mark Oskin , Susan J. Eggers, Modeling instruction placement on a spatial architecture, Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures, July 30-August 02, 2006, Cambridge, Massachusetts, USA S. Bartolini , C. A. Prete, A proposal for input-sensitivity analysis of profile-driven optimizations on embedded applications, ACM SIGARCH Computer Architecture News, v.32 n.3, p.70-77, June 2004 Trishul M. Chilimbi , Mark D. Hill , James R. Larus, Cache-conscious structure layout, ACM SIGPLAN Notices, v.34 n.5, p.1-12, May 1999 Sangwook P. Kim , Gary S. Tyson, Analyzing the working set characteristics of branch execution, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.49-58, November 1998, Dallas, Texas, United States Mahmut Kandemir, Compiler-Directed Collective-I/O, IEEE Transactions on Parallel and Distributed Systems, v.12 n.12, p.1318-1331, December 2001 Thomas Kistler , Michael Franz, Continuous program optimization: A case study, ACM Transactions on Programming Languages and Systems (TOPLAS), v.25 n.4, p.500-548, July

profiling;conflict misses;code layout

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Predicting data cache misses in non-numeric applications through correlation profiling.

To maximize the benefit and minimize the overhead of software-based latency tolerance techniques, we would like to apply them precisely to the set of dynamic references that suffer cache misses. Unfortunately, the information provided by the state-of-the-art cache miss profiling technique (summary profiling) is inadequate for references with intermediate miss ratios - it results in either failing to hide latency, or else inserting unnecessary overhead. To overcome this problem, we propose and evaluate a new technique - correlation profiling - which improves predictability by correlating the caching behavior with the associated dynamic context. Our experimental results demonstrate that roughly half of the 22 non-numeric applications we study can potentially enjoy significant reductions in memory stall time by exploiting at least one of the three forms of correlation profiling we consider.

Introduction As the disparity between processor and memory speeds continues to grow, memory latency is becoming an increasingly important performance bottleneck. Cache hierarchies are an essential step toward coping with this problem, but they are not a complete solution. To further tolerate latency, a number of promising software-based techniques have been proposed. For example, the compiler can tolerate modest latencies by scheduling non-blocking loads early relative to when their results are consumed [12], and can tolerate larger latencies by inserting prefetch instructions [7, 9]. While these software-based techniques provide latency-hiding benefits, they also typically incur runtime overheads. For example, aggressive scheduling of non-blocking loads increases register lifetimes which can lead to spilling, and software-controlled prefetching requires additional instructions to compute prefetch addresses and launch the prefetches themselves. While the benefit of a technique typically outweighs its overhead whenever a miss is tolerated, the overhead hurts performance in cases where the reference would have enjoyed a cache hit anyway. Therefore to maximize overall performance, we would like to apply a latency-tolerance technique only to the precise set of dynamic references that would suffer misses. While previous work has addressed this problem for numeric codes [9], this paper focuses on the more difficult but important case of isolating dynamic miss instances in non-numeric applications. 1.1 Predicting Data Cache Misses in Non-Numeric Codes To overcome the compiler's inability to analyze data locality in non-numeric codes, we can instead make use of profiling information. One simple type of profiling information is the precise miss ratios of all static memory references. Throughout the remainder of this paper, we will refer to this approach as summary profiling, since the miss ratio of each memory reference is summarized as a single value. If summary profiling indicates that all significant memory reference instructions (i.e. those which are executed frequently enough to make a non-trivial contribution to execution time) have miss ratios close to 0% or 100%, then isolating dynamic misses is trivial-we simply apply the latency-tolerance technique only to the static references which always suffer misses. In contrast, if the important references have intermediate miss ratios (e.g., 50%), then we do not have sufficient information to distinguish which dynamic instances hit or miss, since this information is lost in the course of summarizing the miss ratio. The current state-of-the-art approach for dealing with intermediate miss ratios is to treat all static memory references with miss ratios above or below a certain threshold as though they always miss or always hit, respectively [2]. However, this all-or-nothing strategy will fail to hide latency when references are predicted to hit but actually miss, and will induce unnecessary overhead when references are predicted to miss but actually hit. Rather than settling for this sub-optimal performance, we would prefer to predict dynamic hits and misses more accurately. 1.1.1 Correlation Profiling By exposing caching behavior directly to the user, informing memory operations [6] enable new classes of lightweight profiling tools which can collect more sophisticated information than simply the per-reference miss ratios. For example, cache misses can be correlated with information such as recent control-flow paths, whether recent memory references hit or missed in the cache, etc., to help predict dynamic cache miss behavior. We will refer to this approach as correlation profiling. Figure 1 illustrates how correlation profiling information might be exploited. The load instruction shown in Figure 1 has an overall miss ratio of 50%. However, depending on the dynamic context of the load, we may see more predictable behavior. In this example, contexts A and B result in a high likelihood of the load missing, whereas contexts C and D do not. Hence we would like to apply a latency tolerance technique within contexts A and B but not C or D. The dynamic contexts shown in Figure 1 should be viewed simply as non-overlapping sets of dynamic instances of the load which can be grouped together because they share a common distinguishable pattern. In this paper, we consider three different types of information which can be used to distinguish these contexts. The first is control-flow information-i.e. the sequence of N basic block numbers preceding the load. The other two are based on sequences of cache access outcomes (i.e. hit or miss) for previous memory references: self correlation considers the cache outcomes of the previous N dynamic instances of the given static reference, and global correlation refers to the previous N dynamic references across the entire program. Note that load M Context C Context D Context A Figure 1: Example of how correlating cache misses with the dynamic context may improve predictability. (X=Y means X misses out of Y dynamic references.) analogous forms of all three types of correlation profiling have been explored previously in the context of branch prediction [4, 10, 15, 16] 1.2 Objectives and Overview The goal of this paper is to determine whether correlation profiling can predict data cache misses more accurately in non-numeric codes than summary profiling, and if so, can we translate this into significant performance improvements by applying software-based latency tolerance techniques with greater precision. We focus specifically on predicting load misses in this paper because load latency is fundamentally more difficult to tolerate (store latency can be hidden through buffering and pipelining). Although we rely on simulation to capture our profiling information in this study, correlation profiling is a practical technique since it could be performed with relatively little overhead using informing memory operations [6]. The remainder of this paper is organized as follows. We begin in Section 2 by discussing the three different types of history information that we use for correlation profiling, and in Section 3 we present a qualitative analysis of the expected performance benefits. In Section 4, we present our experimental results which quantify the performance advantages of correlation profiling in a collection of 22 non-numeric applications. In addition, in Section 5, we report the memory-access behaviors of individual applications which explain when and how correlation profiling is effective. In Section 6, we compare the performance of software prefetching guided by summary and correlation profiling on a modern superscalar processor. Finally, we discuss related work and present conclusions in Sections 7 and 8. Profiling Techniques In this section, we propose and motivate three new correlation profiling techniques for predicting cache outcomes: control-flow correlation, self correlation, and global correlation. 2.1 Control-Flow Correlation Our first profiling technique correlates cache outcomes with the recent control-flow paths. To collect this information, the profiling tool maintains the N most recent basic block numbers in a FIFO buffer, and matches this pattern against the hit/miss outcomes for a given memory reference. Intuitively, control-flow correlation is useful for detecting cases where either data reuse or cache displacement are likely. If we are on a path which leads to data reuse-either temporal or spatial-then the next reference is likely to be a cache hit. Consider the example shown in Figure 2(a)-(b), where a graph is traversed by the recursive procedure walk(). Any cyclic paths (e.g., A!B!C!D!A or P!Q!R!S!P) will result in temporal reuse of p!data. In this example, control-flow correlation can potentially detect that if the last four traversal decisions lead to a cycle (e.g., right, down, left, and up), then there is a high probability that the next p!data reference will enjoy a cache hit. Some control-flow paths may increase the likelihood of a cache miss by displacing a data line before it is reused. For example, if the "x ? 0" condition is true in Figure 2(c), then the subsequent for loop is likely struct node f int data; struct node *left, *right, *up, *down; void walk(node* p) f if (go left(p!data)) elsif (go right(p!data)) elsif (go elsif (go else if (p != NULL) walk(p); R left right up for (a) Code with data reuse (b) Example graph (c) Code with cache displacement Figure 2: Examples of how control-flow correlation can detect data reuse and cache displacement. (Control- flow profiled loads are underlined.) void preorder(treeNode* p) f if (p != NULL) f preorder(p!left); preorder(p!right); preorder traversal self cache outcomes of p->data (a) Example Code (b) Tree constructed and traversed both in preorder Figure 3: Example of using self-correlation profiling to detect spatial locality for p!data. (Consecutively numbered nodes are adjacent in memory.) to displace *p from the primary cache before it can be loaded again. Note that while paths which access large amounts of data are obvious problems, the displacement might also be due to a mapping conflict. 2.2 Self Correlation Under self correlation, we profile a load L by correlating its cache outcome with the N previous cache outcomes of L itself. This approach is particularly useful for detecting forms of spatial locality which are not apparent at compile time. For example, consider the case in Figure 3 where a tree is constructed in preorder, assuming that consecutive calls to the memory allocator return contiguous memory locations, and that a cache line is large enough to hold exactly two treeNodes. Depending on the traversal order (and the extent to which the tree is modified after it is created), we may experience spatial locality when the tree is subsequently traversed. For example, if the tree is also traversed in preorder, we will expect p!data to suffer misses on every-other reference as cache line boundaries are crossed. Therefore despite the fact that the overall miss ratio of p!data is 50% and the compiler would have difficulty recognizing this as a form of spatial locality, self correlation profiling would accurately predict the dynamic cache outcomes for p!data. 2.3 Global Correlation In contrast with self correlation, the idea behind global correlation is to correlate the cache outcome of a load L with the previous N cache outcomes regardless of their positions within the program. The profiling tool maintains this pattern using a single N-deep FIFO which is updated whenever dynamic cache accesses occur. while (1) f register int register listNode* while (curr != NULL) f A G R A->data A->next B->data B->next G->data G->next R->data R->next . A->data A->next B->data B->next . H . H . H . memory global cache outcomes global htab10(a) Example code (b) Hash table accesses Figure 4: Example of using global-correlation profiling to detect bursty cache misses for curr!data. Note that since earlier instances of L itself may appear in this global history pattern, global correlation may capture some of the same behavior as self correlation (particularly in extremely tight loops). Intuitively, global correlation is particularly helpful for detecting bursty patterns of misses across multiple references. One example of this situation is when we move to a new portion of a data structure that has not been accessed in a long time (and hence has been displaced from the cache), in which case the fact that the first access to an object suffers a miss is a good indication that associated references to neighboring objects will also miss. Figure 4 illustrates such a case where a large hash table (too large to fit in the cache) is organized as an array of linked lists. In this case, we might expect a strong correlation between whether htab[i] (the list head pointer) misses and whether subsequent accesses to curr!data (the list elements) also miss. Similarly, if the same entry is accessed twice within a short interval (e.g., htab[10]), the fact that the head pointer hits is a strong indicator that the list elements (e.g., A!data and B!data) will also hit. In summary, by correlating cache outcomes with the context in which the reference occurs-e.g., the surrounding control flow or the cache outcomes of prior references-we can potentially predict the dynamic caching behavior more accurately than what is possible with summarized miss ratios. Qualitative Analysis of Expected Benefits Before presenting our quantitative results in later sections, we begin in this section by providing some intuition on how correlation profiling can improve performance. A key factor which dictates the potential performance gain is the ratio of the latency tolerance overhead (V ) to the cache miss latency (L). In the extreme cases where V there is no point in applying the latency tolerance technique (T ) selectively, since it either has no cost or no benefit. When applying selectively may be important. Figure 5(a) illustrates how the average number of effective stall cycles per load (CPL) varies as a function of V L for various strategies for applying T . (Note that our CPL metric includes any overhead associated with applying T , but does not include the single cycle for executing the load instruction itself.) If T is never applied, then the CPL is simply mL, where m is the average miss ratio. At the other extreme, if we always apply T , then the latency will always be hidden, but all references (even those that normally hit) will suffer the hence the . Note that when V is better to never apply T rather than always applying it. Figure 5(b) shows an alternative view of CPL, where it is plotted as a function of m for a fixed V L . Again, we observe that the choice of whether to always or never apply T depends on the value of m relative to V L . To achieve better performance than this all-or-nothing approach, we apply the same decision-making process (i.e. comparing the miss ratio with V L ) to more refined sets of loads. In the ideal case, we would consider and optimize each dynamic reference individually (the resulting CPL of mV is shown in Figure 5). However, since this is impractical for software-based techniques, we must consider aggregate collections of references. Since summary profiling provides only a single miss ratio per static reference, the finest granularity CPL never CPL single_action_per_load ideal multiple_actions_per_load CPL always CPL CPL single_action_per_load ideal multiple_actions_per_load CPL always CPL never (a) CPL vs. V Figure 5: Illustration of the CPL for different approaches of applying a latency tolerance scheme overall average load miss ratio, latency tolerance overhead, and load miss latency). at which we can decide whether or not to apply T is once for all dynamic instances of a given static reference. Figure 5 illustrates the potential shape of this "single action per load" curve, which is bounded by the cases where T is never, always, and ideally applied. Since correlation profiling distinguishes different sets of dynamic instances of a static load based on path information, it allows us to make decisions at a finer granularity than with summary profiling. Therefore we can potentially achieve even better performance, as illustrated by the "multiple actions per load" curve in Figure 5. (Further details on the actual CPL equations for the summary and correlation profiling cases can be found in the Appendix) Quantitative Evaluation of Performance Gains In this section, we present experimental results to quantify the performance benefits offered by correlation profiling. We begin by measuring and understanding the potential performance advantages for a generic latency tolerance scheme. Later, in Section 6, we will focus on software-controlled prefetching as a specific case study. 4.1 Experimental Methodology We measured the impact of correlation profiling on the following 22 non-numeric applications: the entire SPEC95 integer benchmark suite, the additional integer benchmarks contained in the SPEC92 suite, uniprocessor versions of two graphics applications from SPLASH-2 [14], eight applications from Olden [11] (a suite of pointer-intensive benchmarks), and the standard UNIX utility awk. Table 1 briefly summarizes these applications, including the input data sets that were run to completion in each case, and Table 2 shows some relevant dynamic statistics of these applications. We compiled each application with -O2 optimization using the standard MIPS C compilers under IRIX 5.3. We used the MIPS pixie utility [13] to instrument these binaries, and piped the resulting trace into our detailed performance simulator. To increase simulation speed and to simplify our analysis, we model a perfectly-pipelined single-issue processor (similar to the MIPS R2000) in this section. (Later, in Section 6, we model a modern superscalar processor: the MIPS R10000). To reduce the simulation time, our simulator performs correlation profiling only on a selected subset of load instructions. Our criteria for profiling a load is that it must rank among the top 15 loads in terms of total cache miss count, and its miss ratio must be between 10% and 90%. Using this criteria, we focus only on the most significant loads which have intermediate miss ratios. We will refer to these loads as the correlation-profiled loads. The fraction of dynamic load references in each application that is correlation profiled is shown in Table 2. Table 1: Benchmark characteristics. Suite Name Description Input Data Set Cache Size Integer perl Unix script language Perl train (scrabbl) 128 KB go Computer game "Go" train 8 KB ijpeg Graphic compression and decompression train 8 KB vortex Database program train 8 KB compress Compresses and decompresses file in memory train li LISP interpreter train 8 KB Integer espresso Minimization of boolean functions cps eqntott Translation of boolean equations into truth tables int pri 3.eqn 8KB raytrace Ray-tracing program car 4KB radiosity Light distribution using radiosity method batch 8KB Olden bh Barnes-Hut's N-body force-calculation 4K bodies 16KB mst Finds the minimum spanning tree of a graph 512 nodes 8KB perimeter Computes perimeters of regions in images 4K x 4K image 16KB health Simulation of the Columbian health care system max. level = 5 16KB tsp Traveling salesman problem 100,000 cities 8KB bisort Sorts and merges bitonic sequences 250,000 integers 8KB em3d Simulates the propagation of E.M. waves in a 3D object 2000 H-nodes, 32KB 100 E-nodes voronoi Computes the voronoi diagram of a set of points 20,000 points 8KB UNIX awk Unix script language AWK Extensive test of 32KB Utilities AWK's capabilities We attempt to maintain as much history information as possible for the sake of correlation. For control-flow correlation, we typically maintained a path length of 200 basic blocks-in some cases this resulted in such a large number of distinct paths that we were forced to measure only 50 basic blocks. For the self and global correlation experiments, we maintained a path length of previous cache outcomes (either self or global). We focus on the predictability of a single level of data cache (two levels makes the analysis too compli- cated). The choice of data cache size is important because if it is either too large or too small relative to the problem size, predicting dynamic misses becomes too easy (they either always hit or always miss). Therefore we would like to operate near the "knee" of the miss ratio curve, where predicting dynamic hits and misses presents the greatest challenge. Although we could potentially reach this knee by altering the problem size, we had greater flexibility in adjusting the cache size within a reasonable range. We chose the data cache size as follows. We first used summary profiling to collect the miss ratios of all loads within the application on different cache sizes ranging from 4KB to 128KB. We then chose the cache size which resulted in the largest number of significant loads having intermediate miss ratios-these sizes are shown in Table 1. In all cases, we model a two-way set-associative cache with lines. 4.2 Improvements in Prediction Accuracy and Performance Figure 6 shows how the three correlation profiling schemes-control-flow (C), self (S), and global (G)- improve the prediction accuracy of correlation-profiled loads. Each bar is normalized with respect to the number of mispredicted references in summary profiling (P), and is broken down into two categories. The top section ("Predict HIT / Actual MISS") represents a lost opportunity where we predict that a reference hits (and thus do not attempt to tolerate its latency), but it actually misses. The "Predict MISS / Actual HIT" section accounts for wasted overhead where we apply latency tolerance to a reference that actually hits. As discussed earlier in Section 3, our threshold for deciding whether to apply latency tolerance to a reference is that its miss ratio must exceed V is the latency tolerance overhead and L is the miss latency. For summary profiling, this threshold is applied to the overall miss ratio of an instruction; for correlation profiling, it is applied to groups of dynamic references along individual paths. Figure 6 shows results with two values of V summary profiling tends to apply latency tolerance aggressively, thus resulting in a noticeable amount of wasted overhead. In contrast, for V summary profiling tends to be more conservative, thus resulting in many untolerated misses. Overall, correlation Table 2: Dynamic benchmark statistics (the column "Insts" is the number of dynamic instructions, the column "Loads" is the number of dynamic loads (its percentage out of "Insts" is also given), the column "Load Miss Rate" is the data-cache miss rate of loads, the column "CP Loads" is the fraction of dynamic loads that are correlation profiled, and the column "CP Load Misses" is the fraction of load misses that are correlation profiled). Suite Name Dynamic Statistics Insts Loads Load Miss Rate CP Load Refs CP Load Misses Integer perl 79M 15M (18%) 12.3% 21% 95% go 568M 121M (21%) 7.1% 10% 23% ijpeg 1438M 266M (18%) 2.7% 2% 17% vortex 2838M 830M (29%) 3.3% 7% 48% compress 39M 8M (20%) 3.9% 6% 87% gcc 282M 61M (22%) 1.4% 2% 40% li 228M 54M (24%) 4.0% 8% 73% Integer espresso 560M 112M (20%) 2.2% 6% 70% raytrace 2105M 588M (28%) 4.8% 10% 53% radiosity 996M 236M (24%) 0.4% 1% 32% Olden bh 2326M 667M (29%) 1.0% 3% 82% mst 90M 14M (16%) 6.9% 17% 91% perimeter 123M 17M (14%) 2.3% 5% 88% health 8M 2M (25%) 9.0% 20% 84% tsp 825M 239M (29%) 1.0% 1% 37% bisort 732M 132M (18%) 2.5% 6% 74% em3d 420M 73M (17%) 1.4% 4% 98% voronoi 263M 87M (16%) 1.3% 4% 57% UNIX Utilities awk 70M 9M (7%) 7.6% 16% 90% profiling can significantly reduce both types of misprediction. To quantify the performance impact of this increased prediction accuracy, Figure 7 shows the resulting execution time of the four profiling schemes, assuming a cache miss latency of 50 cycles. Each bar is normalized to the execution time without latency tolerance, and is broken down into four categories. The bottom section is the busy time. The section above it ("Predict MISS / Actual MISS") is the useful overhead paid for tolerating references that normally miss. The top two sections represent the misprediction penalty, including wasted overhead ("Predict MISS / Actual HIT") and untolerated miss latency ("Predict HIT / Actual MISS"). The degree to which improved prediction accuracy translates into reduced execution time 1 depends not only on the relative importance of load stalls but also the fraction of loads that are correlation profiled. When both factors are favorable (e.g., eqntott), we see large performance improvements-when either factor is small (e.g., perimeter and tsp), the performance gains are modest despite large improvements in prediction accuracies. 5 Case Studies To develop a deeper understanding of when and why correlation profiling succeeds, we now examine a number of the applications in greater detail. In addition to discussing the memory access patterns for these applications, we also show the impact of the correlation-profiled loads on three performance metrics: the miss ratio distribution, the stall cycles per load (CPL) due to correlation-profiled loads only, and the overall I. While CPL and CP I measure the impacts on execution time, the miss ratio distribution gives us insight into how effectively correlation profiling has isolated the dynamic hit and miss instances of static load instructions. failing to hide a miss is more expensive than wasting overhead, it is possible to improve performance by replacing more expensive with less expensive mispredictions, even if the total misprediction count increases (e.g., raytrace with control-flow correlation when V Predict MISS / Actual HIT Predict HIT Actual MISS ||||||Normalized Misprediction awk (200 basic blocks) ||||||Normalized Misprediction bh (200 basic blocks) ||||||Normalized Misprediction bisort (200 basic blocks) ||||||Normalized Misprediction compress (200 basic blocks) ||||||Normalized Misprediction em3d (200 basic blocks) ||||||Normalized Misprediction eqntott (50 basic blocks) ||||||Normalized Misprediction espresso (200 basic blocks) ||||||Normalized Misprediction gcc (200 basic blocks) ||||||Normalized Misprediction 100 104 108 100 100 93 go (50 basic blocks) ||||||Normalized Misprediction health (50 basic blocks) ||||||Normalized Misprediction ijpeg (100 basic blocks) ||||||Normalized Misprediction li (100 basic blocks) ||||||Normalized Misprediction ||||||Normalized Misprediction 22 10872 28 2215191001410019 25 mst (200 basic blocks) ||||||Normalized Misprediction perimeter (200 basic blocks) ||||||Normalized Misprediction 71 7078 perl (200 basic blocks) ||||||Normalized Misprediction radiosity (200 basic blocks) ||||||Normalized Misprediction raytrace (50 basic blocks) ||||||Normalized Misprediction sc (200 basic blocks) ||||||Normalized Misprediction tsp (200 basic blocks) ||||||Normalized Misprediction voronoi (200 basic blocks) ||||||Normalized Misprediction vortex (200 basic blocks) Figure Number of mispredicted correlation-profiled loads, normalized to summary profiling summary profiling, control-flow correlation, global correlation). Maximum path lengths used in control-flow correlation are indicated next to the benchmark names. 5.1 li Over half of the total load misses are caused by two pointer dereferences: this!n flags in mark(), and p!n flags in sweep(), as illustrated by the pseudo-code in Figure 8. The access patterns behave as follows. The procedure mark() traverses a binary tree through the three while loops shown in Figure 8(a). Starting at a particular node, the first inner while loop continues descending the tree-choosing either the left or right child as it goes-until it reaches either a marked node or a leaf node. At this point, we then backup to a node where we can continue descending through a search Predict HIT Actual MISS Predict MISS Actual HIT Predict MISS Actual MISS Busy ||||||Normalized Exec Time 88 87 85 85 100 95 95 95 awk (200 basic blocks) ||||||Normalized Exec Time 97 97 97 97 99 99 99 99 bh (200 basic blocks) ||||||Normalized Exec Time 94 92 92 92 100 96 96 96 bisort (200 basic blocks) ||||||Normalized Exec Time 86 84 86 86 95 90 95 95 compress (200 basic blocks) ||||||Normalized Exec Time 96 96 94 95 em3d (200 basic blocks) ||||||Normalized Exec Time 94 eqntott (50 basic blocks) ||||||Normalized Exec Time 96 95 92 93 99 98 96 97 espresso (200 basic blocks) ||||||Normalized Exec Time gcc (200 basic blocks) ||||||Normalized Exec Time 96 94 95 95 99 99 99 99 go (50 basic blocks) ||||||Normalized Exec Time 81 79 78 79 95 94 94 94 health (50 basic blocks) ||||||Normalized Exec Time ijpeg (100 basic blocks) ||||||Normalized Exec Time 92 88 86 88 100 96 94 97 li (100 basic blocks) ||||||Normalized Exec Time ||||||Normalized Exec Time 86 80 79 7988 86 87 mst (200 basic blocks) ||||||Normalized Exec Time 95 94 93 93 100 98 96 97 perimeter (200 basic blocks) ||||||Normalized Exec Time 72 70 69 69 93 perl (200 basic blocks) ||||||Normalized Exec Time radiosity (200 basic blocks) ||||||Normalized Exec Time 90 raytrace (50 basic blocks) ||||||Normalized Exec Time 91 87 87 89 sc (200 basic blocks) ||||||Normalized Exec Time 97 94 93 94 98 97 96 97 tsp (200 basic blocks) ||||||Normalized Exec Time voronoi (200 basic blocks) ||||||Normalized Exec Time 93 vortex (200 basic blocks) Figure 7: Impact of the profiling schemes on execution time, assuming a 50 cycle miss latency (L). summary profiling, control-flow correlation, performed by the second inner while loop. The tree is allocated in preorder, similar to the one shown in Figure 3, except much larger. Therefore we enjoy spatial locality as long as we continue following left branches in the tree, but spatial locality is disrupted whenever we backup in the second inner while loop, as illustrated by Figure 8(c). All three types of correlation profiling provide better cache outcome predictions than summary profiling for the this!n flags reference in mark() for li. Self correlation detects this form of spatial locality effectively. Global correlation is more accurate than summary profiling but less accurate than self correlation in this case because the cache outcomes of other references (which do not help to predict this reference) consume wasted space in the global history pattern. Control-flow correlation also performs well because it void mark(NODE *ptr) f while (TRUE) f /* outer while loop */ while (TRUE) f/* 1st inner while loop */ if (this!n flags & MARK) break; /* a marked node */ else f else if (livecdr(this)) f right */ gelse break; /* a leaf node* / /* ends if-else */ /* ends 1st inner-while */ while (TRUE) f/* 2nd inner while loop */ /* backup to a point where we can continue descending */ /* ends 2nd inner while */ 1st outer while */ (a) Procedure mark() for for if (!(p!n flags & (b) Procedure sweep() tree pointer (c) Tree traversal order in mark() Figure 8: Procedures mark() and sweep() in li, and the memory access patterns of mark(). (Note: consecutively numbered nodes in part (c) correspond to adjacent addresses in memory.) observes that this!n flags is more likely to suffer a miss if we begin iterating in the first inner while loop immediately following a backup performed in the second inner while loop (in the preceding outer while loop iteration). Finally, the reference p!n flags in sweep() (shown in Figure 8(b)) is in fact an array reference written in pointer form. Both self correlation and global correlation detect the spatial locality caused by accessing consecutive elements within the array. (Although the compiler could potentially recognize this spatial locality through static analysis if it can recognize that p!n flags is effectively an array reference, this is not always possible for all such cases.) Figure 9 shows the detailed performance results for li. The miss ratio distribution in Figure 9(a) has ten ranges of miss ratios, each of which contains four bars corresponding to the fraction of total dynamic correlation-profiled load references that fall within this range. The bars for summary profiling represent the inherent miss ratios of these load instructions, and the other three cases represent the degree to which correlation profiling can effectively group together dynamic instances of the loads into separate paths with similar cache outcome behavior. For a correlation scheme to be effective, we would like to see a "U-shaped" distribution where references have been isolated such that they always have very high or very low miss ratios-we refer to such a case as being strongly biased. In contrast, if most of the references are clustered around the middle of the distribution, we say that this is weakly biased. Correlation profiling can outperform summary profiling by increasing the degree of bias, which we do observe in Figure 9(a). With summary profiling, 80% of the loads that we profile 2 have miss ratios in the range of 30-50% (these include the this!n flags and p!n flags references shown earlier in Figure 8). In contrast, with self correlation 2 Recall that we only profile loads with miss ratios between 10% and 90% among the top 15 ranked loads in terms of their contributions to total misses. Therefore the summary profiling case will never have loads outside of this miss ratio range. CPL summary global control-flow self CPI summary global control-flow self ideal (b) CPL due to correlation-profiled loads (c) Overall CP I Figure 9: Detailed performance results for li. profiling only 27% of the isolated loads have miss ratios in the 30-50% range, and over 45% are either below 10% or above 90%. All three correlation schemes increase the degree of bias in this case. This increased degree of bias of correlation-profiled loads translates into a reduction in CPL, as shown in Figure 9(b) where the CPL due to correlation-profiled loads is plotted over a range of overhead-to-latency assuming a miss latency of 50 cycles. As we have discussed in Section 3, correlation profiling partially closes the gap between summary profiling and ideal prediction. The overall CP I is also shown in Figure 9(c). 5.1.1 eqntott Figure shows detailed performance results for eqntott, where we see that all three forms of correlation profiling successfully increase the degree of bias and reduce CPL (and hence CP I). We now focus on the memory access behavior. Most of the load misses are caused by the four loads in cmppt() shown in Figure 11(a), two of which are array references (a ptand[i] and b ptand[i]). Clearly the spatial locality enjoyed by these two array references can be detected through self correlation (and hence global correlation). However, the access patterns of the other two loads (a[0]!ptand and b[0]!ptand) are more complicated. The procedure cmppt() has multiple call sites, and two of them, say S 1 and S 2 , invoke it very frequently. Whenever cmppt() is called at S 1 , a[0] will very likely be unchanged but b[0] will have a new value. In contrast, whenever cmppt() is called at S 2 , b[0] will very likely be unchanged but a[0] will have a new value. Moreover, both S 1 an S 2 repeatedly call cmppt(). This call-site dependent behavior results in the streams of cache outcomes illustrated in Figure 11(b). Self correlation captures these streaming behavior, and control-flow correlation also predicts the cache outcomes accurately by distinguishing the two call sites of cmppt(). The cache outcomes of a[0]!ptand also help predict those of a ptand[i]-if a[0]!ptand is a hit, it implies that the array a ptand[] has been loaded recently, and therefore the a ptand[i] references are likely to also hit. (Similar correlation also exists between b[0]!ptand and b ptand[i]). Hence global correlation is quite effective in this case. Control-flow correlation also predicts the cache outcomes of a ptand[i] and CPL summary control-flow global self ideal 11.21.41.6 CPI summary control-flow global self ideal (b) CPL due to correlation-profiled loads (c) Overall CP I Figure 10: Detailed performance results for eqntott. extern int ninputs, noutputs; int cmppt (a, b) PTERM *a[], *b[]; f register int i, aa, bb; register int* a ptand, *b ptand; a for /* the famous correlated branches */ return (0); a[0]->ptand b[0]->ptand (a) Procedure cmppt() which causes (b) Call-site dependent most load misses cache outcome patterns Figure 11: The memory access behavior in eqntott. To make all loads explicit, we rewrite the two expressions a[0]!ptand[i] and b[0]!ptand[i] in the original cmppt() into the four loads (i.e. a[0]!ptand, a ptand[i], b[0]!ptand, and b ptand[i]) shown in (a). b ptand[i] in an indirect fashion, by virtue of predicting those of a[0]!ptand and b[0]!ptand. CPL summary control-flow global self ideal 0.981.021.061.11.141.18 CPI summary control-flow global self ideal (b) CPL due to correlation-profiled loads (c) Overall CP I Figure 12: Detailed performance results for perimeter.161 middle_right left right middle_leftMore spatial locality found at the bottom void middle first(quadTree* p) f if (p == NULL) return; middle first(p!middle left); middle first(p!middle right); middle middle (a) A quadtree allocated in preorder (b) Code for traversing the quadtree in (a) Figure 13: Example of a case where more spatial locality is found at the bottom of a tree. This example assumes that one cache line can hold three tree nodes and the tree is allocated in preorder. Nodes having consecutive numbers are adjacent in the memory. 5.1.2 perimeter and bisort Figure 12 shows the detailed performance results for perimeter. The main data structures used in both perimeter and bisort are trees: quadtrees in perimeter, and binary trees in bisort. These trees are allocated in preorder, but the orders in which they are traversed are rather arbitrary. As a result, we do not see very regular cache outcome patterns (such as the one illustrated in Figure 3) for these applications. CPL summary control-flow global self CPI summary control-flow global self ideal (b) CPL due to correlation-profiled loads (c) Overall CP I Figure 14: Detailed performance results for mst. Nevertheless, there is still a considerable amount of spatial locality among consecutively accessed nodes while we are traversing around the bottom of a tree that has been allocated in preorder. For example, if we traverse a quadtree using the procedure middle first() shown in Figure 13, we will only miss twice upon accessing nodes 156 through 160 at the tree's bottom, assuming that nodes 156 through 158 are in one cache line and nodes 159 through 161 are in another. In contrast, there is relatively little spatial locality while we are traversing the middle of the tree. Self correlation (and hence global correlation) can discover whether we are currently in a region of the tree that enjoys spatial locality. Control-flow correlation can also potentially detect whether we are close to the bottom of the tree by noticing the number of levels of recursive descent. 5.1.3 mst Most of the misses in mst (see the detailed performance results in Figure 14) are caused by loads in HashLookup() and the tmp!edgehash load in BlueRule(), as illustrated in Figure 15. The mst application consists of two phases: a creation phase and a computation phase. Both phases invoke HashLookup(), but the creation phase causes most of the misses when it calls HashLookup() to check whether a key already exists in the hash table before allocating a new entry for it. During the computation phase, much of the data has already been brought into the cache, and hence there are relatively few misses. Both self correlation and global correlation accurately predict the cache outcomes of these two distinct phases, since they appear as repeated streams of either hits or misses. Control-flow correlation is also effective since it can distinguish the call chains which invoke HashLookup(). The load of tmp!edgehash in BlueRule() accesses a linked lists whose nodes are in fact allocated at contiguous memory locations. Consequently, self correlation detects this spatial locality accurately, but control-flow correlation is not helpful. void *HashLookup(int key, Hash hash) f int j; HashEntry ent; ent && ent!key !=key; if (ent) return ent!entry; return NULL; static BlueReturn BlueRule(.) f for (tmp=vlist!next; tmp; prev=tmp,tmp=tmp!next) f Figure 15: Pseudo codes drawn from mst. ||||||||||||% of Total Correlation-Profiled Miss Ratio control-flow global self (a) Miss ratio distribution of correlation-profiled load references0.050.150.250.350 CPL summary control-flow global self ideal CPI summary control-flow global self ideal (b) CPL due to correlation-profiled loads (c) Overall CP I Figure Detailed performance results for raytrace. 5.1.4 raytrace and tsp In raytrace (refer to Figure 16 for its performance results), over 30% of load misses are caused by the pointer dereference of tmp!bv in prims in box2() (see Figure 17). In subdiv bintree(), the two calls to prims in box2() copy part of the array pe of the current node btn to the arrays btn1!pe and btn2!pe, where btn1 and btn2 are the children of btn. This process of copying pe is performed recursively on the whole tree by create bintree(). As a result, when prims in box2() is called upon a node n, we may have used all values in the array pe (referred to as pepa in prims in box2()) of n before at some antecedent of n and hence hopefully most data loaded by tmp!bv is already in the cache. In this case, most references of tmp!bv will hit in the cache. In contrast, if the values in pepa are new, all tmp!bv references will miss. Hence self correlation captures these streams of hits and streams of misses. In theory, control-flow correlation could also achieve good predictions by observing whether any copying occurred in the parent node-unfortunately, the profiling tool cannot record enough state across the many control-flow changes in subdiv bintree() and prims in box2() to know what decisions were made in the parent node. ELEMENT *prims in box2(pepa, .)f /* computes ovlap */ /* no change in pepa[j] */ if (ovlap == 1) f return (npepa); VOID subdiv bintree(BTNODE* btn, .)f /* btn1 and btn2 are btn's children */ prims in box2(btn!pe, .); prims in box2(btn!pe, .); VOID create bintree(BTNODE* root, .)f if (.) f subdiv bintree(root, .); create bintree(root!btn[0], .); create bintree(root!btn[1], .); Figure 17: Pseudo codes drawn from raytrace. Tree tsp(Tree t,int sz, .) f if (t!size != sz) return conquer(t); return merge(leftval, rightval, t, .); static Tree conquer(Tree t) f for (; l; l=donext) f Figure codes drawn from tsp. Procedure makelist(Tree t) slings t into a list consisting of all nodes of t. Similar to raytrace, tsp also traverses a binary tree recursively, and some data which is read by the current node will be read again by its descendents. As illustrated in Figure 18, the procedure tsp() recursively traverses the tree t and calls conquer(t) if the size of t is not greater than sz. The procedure conquer(t) uses makelist(t) to sling every node of t into a list which is then traversed by the for loop. Therefore since all descendents of t are brought into the cache whenever conquer(t) is called, subsequent recursion down t!left and t!right within tsp() results in many cache hits. Hence the l!data references either mainly hit or mainly miss for a given node t. Self correlation captures this pattern effectively. Control-flow correlation is also quite effective because it can observe the number of times conquer() has been called in a given recursive descent-most misses occur the first time it is invoked. 5.1.5 voronoi and compress Control-flow correlation offers the best prediction accuracy in both of these applications. Most of the misses in voronoi are caused by loading b!next in splice(), which is called from three different places in do merge(), as illustrated in Figure 20(a). When splice() is called from call site 1, b!next will hit since ldi!next loaded this same data into the cache just prior to the call. When splice() is called from the other two call sites, b!next is more likely to miss. Hence control-flow correlation distinguishes the behavior of these different call sites accurately. Self correlation is less effective since b!next does not have regular cache outcome patterns. In compress (see Figure 19 for its performance results), roughly half of the misses are caused by the hash of Total Correlation-Profiled Refs summary4 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Miss Ratio control-flow global self (a) Miss ratio distribution of correlation-profiled load references0.050.150.250.35 CPL summary global self control-flow CPI summary global self control-flow ideal (b) CPL due to correlation-profiled loads (c) Overall CP I Figure 19: Detailed performance results for compress. table access htabof[i] in the procedure compress() (see Figure 20(b)). The index i to the hash table htab is a function of the combination of the prefix code ent and the new character c. If this combination has been seen before, the hash probe test ((htab[i] == fcode)) will be true-if it has been seen recently, the load of htab[i] is likely to hit in the cache. Since the input file we use (provided by SPEC) is generated from a frequency distribution of common English texts, some strings will appear more often than others. Because of this, we expect that the condition (htab[i] == fcode) should be true quite frequently once many common strings have been entered into htab. If the last few tests of (htab[i] == fcode) are false, the probability that the next one is true will be high, which also implies that the next reference of htab[i] is more likely a hit. Therefore, control-flow correlation can make accurate predictions by examining the last several outcomes of this branch. 5.1.6 espresso, vortex, m88ksim, and go For these four applications, correlation profiling mainly improves the cache outcome predictions for array references. In espresso (see Figure 21 for its detailed performance results), many load misses are due to array references, written in pointer form, with variable strides. Figure 22(a) shows one such example. Inside the for loop, p is incremented by BB!wsize, whose value depends on the call chain of setup BB CC() and ranges from 4 to 24 bytes. Different values result in different degrees of spatial locality, but all can be captured by self correlation (and hence global correlation). Control-flow correlation can also make enhanced predictions by exploiting the call-chain information. In vortex, m88ksim, and go, many load misses are caused by array references located inside procedures, where array indices are passed as procedure parameters. See Figure 22(b) for an example drawn from vortex. Each of these procedures have multiple call sites, and the cache outcomes of those array references are mainly call-site dependent. This explains why control-flow correlation offers the highest cache outcome prediction accuracy for these three benchmarks. In vortex, the array index parameter values at a given call are very close or even identical most of the time, but values passed at different call sites are quite different. Consequently, references made through the same call sites will enjoy temporal and/or spatial EDGE PAIR do merge(.) f call site 1*/ /* no dereferences of ldj before */ call site 2*/ /* no dereferences of ldk before */ call site 3*/ splice(QUAD EDGE a, QUAD EDGE b) f while if (htab[i] == fcode) f else f . /* store fcode into htab */ . (a) Code fragment in voronoi (b) Code fragment in compress Figure 20: Pseudo codes drawn from (a) voronoi and (b) compress. locality, but those made through different call sites will not. Since a procedure is usually invoked multiple times by the same call site before being invoked by another call site, this results in a streaming pattern of a miss followed by several hits-hence self correlation also performs well in vortex by capturing these cache outcome patterns. 5.2 Lessons Learned from All Case Studies Although global correlation makes excellent predictions in some cases by correlating behavior across different load instructions (e.g., eqntott), in most cases it essentially assimilates self correlation, but does not perform quite as well since it records less history for a given load. Self correlation is often successful since it recognizes forms of spatial locality which are not recognizable at compile time (e.g, li, perimeter, bisort, and mst), and also long runs of either all hits or all misses (e.g., eqntott, mst, tsp, and raytrace). We often find that as few as four previous cache outcomes per reference are sufficient to achieve good predictability with self correlation. By capturing call chain information, control-flow correlation can distinguish behavior based on call sites (e.g., eqntott, espresso, vortex, m88ksim, go, mst and voronoi) and the depth of the recursion while traversing a tree (e.g., perimeter, bisort, and tsp). Roughly half of the applications enjoy significant improvements from both control-flow and self correlation, and in many of these cases we observe that the same load references can be successfully predicted by both forms of correlation. This is good news, since control-flow correlation profiling is the easiest case to exploit in practice by using procedure cloning [5] to distinguish call-chain dependent behavior. 6 Applying Correlation Profiling to Prefetching To demonstrate the practicality of correlation profiling, we used both summary and correlation profiling to guide the manual insertion of prefetch instructions into three applications: (eqntott, tsp, and raytrace). In the case of correlation profiling, we used procedure cloning [5] to isolate different dynamic instances of a static reference, and adapted the prefetching strategy accordingly with respect to the call sites. We assumed that V deciding whether to insert prefetches, 3 and we performed fully-detailed simulations of a processor similar to the MIPS R10000 [8] (details of the memory hierarchy are shown in Figure 23(a)). 3 We assume an average prefetch overhead (V ) of two cycles, and an average miss latency (L) of 20 cycles. of Total Correlation-Profiled Refs summary26 Miss Ratio control-flow global self (a) Miss ratio distribution of correlation-profiled load references0.020.060.10.140 CPL summary control-flow global self ideal CPI summary control-flow global self ideal (b) CPL due to correlation-profiled loads (c) Overall CP I Figure 21: Detailed performance results for espresso. void setup BB CC(pcover BB, pcover CC)f last=p+BB!count*BB!wsize; boolean ChkGetChunk(numtype ChunkNum, .) f && . (a) Code fragment in espresso (b) Code fragment in vortex Figure 22: Pseudo codes drawn from (a) espresso and (b) vortex. Figure 23(b) shows the resulting execution times, normalized to the case without prefetching. For these applications, summary-profiling directed prefetching actually hurts performance due to the overheads of unnecessary prefetches. In contrast, correlation profiling provides measurable performance improvements by isolating dynamic hits and misses more effectively, thereby achieving similar benefits with significantly less overhead. We would also like to point that these numbers do not represent the limit of what correlation can achieve. For example, with an 8KB primary data cache, correlation profiling offers a 10% speedup over summary profiling in the case of eqntott. 7 Related Work Abraham et al. [2] investigated using summary profiling to associate a single latency tolerance strategy (i.e. either attempt to tolerate the latency or not) with each profiled load. They used this approach to reduce Memory Parameters for the MIPS R10000 Simulator Primary Instr and Data Caches 32KB, 2-way set-assoc. Unified Secondary Cache 2MB, 2-way set-assoc. Line Size 32B Primary-to-Secondary 12 cycles Miss Latency Primary-to-Memory Miss Latency Data Cache Miss 8 Handlers (MSHRs) Data Cache Banks 2 Data Cache Fill Time 4 cycles (Requires Exclusive Access) Main Memory Bandwidth 1 access per 20 cycles ||||||Normalized Exec. Time load stall113 Eqntott Tsp Raytrace store stall inst stall busy (a) Memory Parameters (b) Execution Time Figure 23: Impact of correlation profiling on prefetching performance no prefetching, prefetching directed by summary profiling, prefetching directed by correlation profiling). the cache miss ratios of nine SPEC89 benchmarks, including both integer and floating-point programs. In a follow-up study [1], they also report the improvement in effective cache miss ratio. In contrast with this earlier work, our study has focused on correlation profiling, which is a novel technique that provides superior prediction accuracy relative to summary profiling. Ammons et al.[3] used path profiling techniques to observe that a large fraction of primary data cache misses in the SPEC95 benchmarks occur along a relatively small number of frequently executed paths. The three forms of correlation explored in this study (control-flow, self, and global) were inspired by earlier work on using correlation to enhance branch prediction accuracies [4, 10, 15, 16]. While branch outcomes and cache access outcomes are quite different, it is interesting to observe that correlation-based prediction works well in both cases. Conclusions To achieve the full potential of software-based latency tolerance techniques, we have proposed correlation profiling, which is a technique for isolating which dynamic instances of a static memory reference are likely to suffer cache misses. We have evaluated the potential performance benefits of three different forms of correlation profiling on a wide variety of non-numeric applications. Our experiments demonstrate that correlation profiling techniques always outperform summary profiling by increasing the degree of bias in the miss ratio distribution, and this improved prediction accuracy can translate into significant reductions in the memory stall time for roughly half of the applications we study. Detailed case studies of individual applications show that self correlation works well because the cache outcome patterns of individual references often repeat in predictable ways, and that control-flow correlation works mainly because many cache outcomes are call-chain dependent. Although global correlation offers superior performance in some cases, for the most part it mainly assimilates self correlation. Finally, we observe that correlation profiling offers superior performance over summary profiling when prefetching on a superscalar processor. We believe that these promising results may lead to further innovations in optimizing the memory performance of non-numeric applications. Appendix Derivation of the Stall Cycles Per Load (CPL) under Five Latency-Tolerance Schemes Denote the CPL under a particular tolerance scheme S by CPL S . Let CPL i S be the CPL S of load i in the program and f i be the fraction of references made by load i out of the total references of all loads. Then: S \Theta f i (1) Let L be the cycles stalled upon a load miss, V be the overhead of applying the latency-tolerance technique T to a load reference, m i is miss ratio of load i and m is the overall miss ratio of all loads. load reference is stalled only when it is a cache miss, so: fully tolerates the latencies of all load references but always incurs the overhead, so: CPL single action per load : The miss ratio m i decides whether T should be applied to load i: single action per load = ae L (i.e. not apply T) otherwise (i.e. apply T) (4) CPL single action per load = single action per load \Theta f i single action per load \Theta f i where A is the set of loads with miss ratios ? V L and NA is the set of loads with miss ratios - V L . CPL multiple actions per load : T is only applied to references of load i that belong to contexts with miss L . The formula for CPL i multiple actions per load can be simply obtained adding an extra level to Equation (5) to capture the notion of contexts within load i. That is: multiple actions per load where A i is the set of contexts of load i with miss ratios ? V is the set of contexts of load i of miss ratios - V is the miss ratio of context j of load i, and f i;j is the fraction of references of load i that are on context j. CPL multiple actions per load can be obtained by substituting multiple actions per load into Equation (1). load-miss latencies are fully tolerated and the overhead is only incurred to miss references: --R Predicting load latencies using cache profiling. Predictability of load/store instruction latencies. Exploiting hardware performance counters with flow and context sensitive profiling. Branch classification: a new mechanism for improving branch predictor performance. A methodology for procedure cloning. Informing memory operations: Providing memory performance feedback in modern processors. MIPS Technologies Inc. Design and evaluation of a compiler algorithm for prefetching. Improving the accuracy of dynamic branch prediction using branch correlation. Supporting dynamic data structures on distributed memory machines. Software support for speculative loads. Tracing with pixie. The SPLASH-2 programs: Characterization and methodological considerations A comparison of dynamic branch predictors that use two levels of branch history. Improving the accuracy of static branch prediction using branch correlation. --TR Software support for speculative loads Design and evaluation of a compiler algorithm for prefetching Improving the accuracy of dynamic branch prediction using branch correlation A comparison of dynamic branch predictors that use two levels of branch history Branch classification Improving the accuracy of static branch prediction using branch correlation Supporting dynamic data structures on distributed-memory machines The SPLASH-2 programs Informing memory operations Predictability of load/store instruction latencies Exploiting hardware performance counters with flow and context sensitive profiling --CTR Aleksandar Milenkovic, Achieving High Performance in Bus-Based Shared-Memory Multiprocessors, IEEE Concurrency, v.8 n.3, p.36-44, July 2000 Craig Zilles , Gurindar Sohi, Execution-based prediction using speculative slices, ACM SIGARCH Computer Architecture News, v.29 n.2, p.2-13, May 2001 T. K. Tan , A. K. Raghunathan , G. Lakishminarayana , N. K. Jha, High-level software energy macro-modeling, Proceedings of the 38th conference on Design automation, p.605-610, June 2001, Las Vegas, Nevada, United States Young , Michael D. Smith, Better global scheduling using path profiles, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.115-123, November 1998, Dallas, Texas, United States Jeffrey Dean , James E. Hicks , Carl A. Waldspurger , William E. Weihl , George Chrysos, ProfileMe Craig B. Zilles , Gurindar S. Sohi, Understanding the backward slices of performance degrading instructions, ACM SIGARCH Computer Architecture News, v.28 n.2, p.172-181, May 2000 Adi Yoaz , Mattan Erez , Ronny Ronen , Stephan Jourdan, Speculation techniques for improving load related instruction scheduling, ACM SIGARCH Computer Architecture News, v.27 n.2, p.42-53, May 1999 Chi-Hung Chi , Jun-Li Yuan , Chin-Ming Cheung, Cyclic dependence based data reference prediction, Proceedings of the 13th international conference on Supercomputing, p.127-134, June 20-25, 1999, Rhodes, Greece Abhinav Das , Jiwei Lu , Howard Chen , Jinpyo Kim , Pen-Chung Yew , Wei-Chung Hsu , Dong-Yuan Chen, Performance of Runtime Optimization on BLAST, Proceedings of the international symposium on Code generation and optimization, p.86-96, March 20-23, 2005 Young , Michael D. Smith, Static correlated branch prediction, ACM Transactions on Programming Languages and Systems (TOPLAS), v.21 n.5, p.1028-1075, Sept. 1999 Martin Burtscher , Amer Diwan , Matthias Hauswirth, Static load classification for improving the value predictability of data-cache misses, ACM SIGPLAN Notices, v.37 n.5, May 2002 Jaydeep Marathe , Frank Mueller , Tushar Mohan , Bronis R. de Supinski , Sally A. McKee , Andy Yoo, METRIC: tracking down inefficiencies in the memory hierarchy via binary rewriting, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, March 23-26, 2003, San Francisco, California Chi-Keung Luk, Tolerating memory latency through software-controlled pre-execution in simultaneous multithreading processors, ACM SIGARCH Computer Architecture News, v.29 n.2, p.40-51, May 2001 Tor M. Aamodt , Paul Chow, Optimization of data prefetch helper threads with path-expression based statistical modeling, Proceedings of the 21st annual international conference on Supercomputing, June 17-21, 2007, Seattle, Washington Jiwei Lu , Howard Chen , Rao Fu , Wei-Chung Hsu , Bobbie Othmer , Pen-Chung Yew , Dong-Yuan Chen, The Performance of Runtime Data Cache Prefetching in a Dynamic Optimization System, Proceedings of the 36th annual IEEE/ACM International Symposium on Microarchitecture, p.180, December 03-05, Chi-Keung Luk , Robert Muth , Harish Patil , Richard Weiss , P. Geoffrey Lowney , Robert Cohn, Profile-guided post-link stride prefetching, Proceedings of the 16th international conference on Supercomputing, June 22-26, 2002, New York, New York, USA Jaydeep Marathe , Frank Mueller , Tushar Mohan , Sally A. Mckee , Bronis R. De Supinski , Andy Yoo, METRIC: Memory tracing via dynamic binary rewriting to identify cache inefficiencies, ACM Transactions on Programming Languages and Systems (TOPLAS), v.29 n.2, p.12-es, April 2007 Characterizing the memory behavior of Java workloads: a structured view and opportunities for optimizations, ACM SIGMETRICS Performance Evaluation Review, v.29 n.1, p.194-205, June 2001 Mark Horowitz , Margaret Martonosi , Todd C. Mowry , Michael D. Smith, Informing memory operations: memory performance feedback mechanisms and their applications, ACM Transactions on Computer Systems (TOCS), v.16 n.2, p.170-205, May 1998

non-numeric applications;profiling;latency tolerance;correlation;cache miss prediction

266833

Cache sensitive modulo scheduling.

This paper focuses on the interaction between software prefetching (both binding and nonbinding) and software pipelining for VLIW machines. First, it is shown that evaluating software pipelined schedules without considering memory effects can be rather inaccurate due to stalls caused by dependences with memory instructions (even if a lockup-free cache is considered). It is also shown that the penalty of the stalls is in general higher than the effect of spill code. Second, we show that in general binding schemes are more powerful than nonbinding ones for software pipelined schedules. Finally, the main contribution of this paper is an heuristic scheme that schedules some memory operations according to the locality estimated at compile time and other attributes of the dependence graph. The proposed scheme is shown to outperform other heuristic approaches since it achieves a better trade-off between compute and stall time than the others.

Introduction Software pipelining is a well-known loop scheduling technique that tries to exploit instruction level parallelism by overlapping several consecutive iterations of the loop and executing them in parallel ([14]). Different algorithms can be found in the literature for generating software pipelined sched- ules, but the most popular scheme is called modulo scheduling. The main idea of this scheme is to find a fixed pattern of operations (called kernel or steady state) that consists of operations from distinct iterations. Finding the optimal scheduling for a resource constrained scenario is an NP-complete problem, so practical proposals are based on different heuristic strategies. The key goal of these schemes has been to achieve a high throughput (e.g. [14][11][20][18]), to minimize register pressure (e.g. [9][6]) or both (e.g. [10][15][7][16]), but none of them has evaluated the effect of memory. These schemes assume a fixed latency for all memory operations, which usually corresponds to the cache-hit latency. Lockup-free caches allows the processor not to stall on a cache miss. However, in a VLIW architecture the processor often stalls afterwards due to true dependences with previous memory operations. The alternative of scheduling all loads using the cache-miss latency requires considerable instruction level parallelism and increases register pressure ([1]). Software prefetching is an effective technique to tolerate memory latency ([4]). Software prefetching can be performed through two alternative schemes: binding and nonbinding prefetch- ing. The first alternative, also known as early scheduling of memory operations, moves memory instructions away from those instructions that depend on them. The second alternative introduces in the code special instructions, which are called prefetch instructions. These are nonfaulting instructions that perform a cache lookup but do not modify any register. These alternative prefetching schemes have different drawbacks: . The binding scheme increases the register pressure because the lifetime of the value produced by the memory operation is stretched. It may also increase the initiation interval due to memory operations that belong to recurrences. . The nonbinding scheme increases the memory pressure since it increases the number of memory requests, which may produce an increase in the initiation interval. Besides it may produce an increase in the register pressure since the lifetime of the value used to compute the effective address is stretched. A higher register pressure may require additional spill code, which results in additional memory pressure. In this paper we investigate the interaction between software prefetching and software pipelining in a VLIW machine. First we show that previous schemes that do not consider the effect of memory penalties produce schedules that are far from the optimal when they are evaluated taking into account a realistic cache memory. We evaluate several heuristics to schedule memory operations and to insert prefetch instructions in a software pipelined schedule. The contributions of stalls and spill code is quantified for each case, showing that stall penalties have a much higher impact on performance than spill code. We then propose an heuristic that tries to trade off both initiation interval and stall time in order to minimize the execution time of a software pipelined loop. Finally, we show that schemes based on binding prefetch are more effective than those based on nonbinding prefetch for software pipelined schedules. The use of binding and nonbinding prefetching has been previously studied in [12][1] and [4][8][13][17][3] respectively among others. However, to our knowledge there is no previous work analyzing the interactions of these prefetching schemes with software pipelining techniques. The selective scheduling ([1]) schedules some operations with cache-hit latency and others with cache-miss latency, like the scheme proposed in this paper. However the selective scheduling is based on profiling information whereas our method is based on a static analysis performed at compile-time. In addition, the selective scheduling does not consider the interactions with software pipelining. The rest of this paper is organized as follows. Section 2 motivates the impact that memory latency may have in a software pipelined loop. Section 3 evaluates the performance of simple schemes for scheduling load and stores instructions. Section 4 describes the new algorithm proposed in this paper. Section 5 explains the experimental methodology and presents some performance results. Finally, the main conclusions are summarized in section 6. 2. Motivation A software pipelined loop via modulo scheduling is characterized basically by two terms: the initiation and the stage counter ( ). The former indicates the number of cycles between the initiation of successive iterations. The latter shows how many iterations are over- lapped. In this way, the execution time of the loop can be calculated as: For a given architecture and a given scheduler, the first term of the sum (called compute time in the rest of the paper) is fixed and it is determined at compile time. The stall time is mainly due to dependences with previous memory instructions and it depends on the run-time behavior of the program (e.g. miss ratio, outstanding misses, etc. In order to minimize the execution time, classical methods have tried to minimize the initiation interval with the goal of reduce the fixed part of t exec . The minimum initiation interval is bounded by resources and recurrences: The is the lower bound due to resource constraints of the architecture and assuming that all functional units are pipelined, it is calculated as: where indicates the number of operations of type in the loop body, and indicates the number of functional units of type in the architecture. The is the lower bound due to recurrences in the graph and it is computed as: where represents the sum of all node latencies in the recurrence , and represents the sum of all edge distances in the recurrence . For a particular data flow dependence graph and a given architecture, the resulting II is dependent on the latency that the scheduler assigns to each operation. The latency of operations is usually known by the compiler except for memory operations, which have a variable latency. The II also depends on the , which is affected by the spill code introduced by the scheduler. The other parameters, and , are fixed. Conventional modulo scheduling proposals use a fixed latency (usually the cache-hit time) to schedule memory instructions. Scheduling instructions with its minimum latency minimize the register pressure, and thus, reduces the spill code. On the other hand, this minimum latency scheduling can increase the stall time because of data dependences. In particular, if an operation needs a data that has been loaded in a previous instruction but the memory access has not finished yet, the processor stalls until the data is available. Figure 1 shows a sample scheduling for a data dependence graph and a given architecture. In this case, memory instructions are scheduled with cache-hit latency. If the stall time is ignored, II SC res II rec II res II res max op ARCH NOPS op NFUS op y II rec II rec max rec GRAPH LAT rec DIST rec y as it is usual in studies dealing with software pipeline techniques, the expected optimistic execution time will be (suppose is huge): Obviously this is an optimistic estimation of the actual execution time, which can be rather inaccurate. For instance, suppose that the miss ratio of the N1 load operation is 0.25 (e.g. it has stride 1 and there are 4 elements per cache line). Every cache miss the processor stalls some cycles (called penalty). The penalty for a particular memory instruction depends on the hit latency, the miss latency and the distance in the scheduling between the memory operation and the first instruction that uses the data produced by the memory instruction. For the dependence between N1 and N2 the penalty is 9 cycles, so the stall time assuming that the remaining dependences do not produce any penalty is: and therefore In this case, the actual execution time is near twice the optimistic execution time. If we assume a miss ratio of 1 instead of 0.25, the discrepancy between the optimistic and the actual execution time is even higher. In this case, the stall time is: and therefore load mult load add store1357N1 ALU MEM b) Data flow dependence graph b) Code scheduling c) Kernel Instruction latencies: load/store a) Original code ENDDO Figure 1. A sample scheduling @ If all memory references were considered, the effect of the stall time could be greater, and the discrepancy between the optimistic estimation usually utilized to evaluate the performance of software pipelined schedulers and the actual performance could be much higher. We can also conclude that scheduling schemes that try to minimize the stall time may provide a significant advantage In this paper, the proposed scheduler is evaluated and compared with others using the t exec metric. This requires to consider the run-time behavior of individual memory references, which requires the simulation of the memory system. 3. Basic schemes to schedule memory operations In this section we evaluate the performance of basic schemes to schedule memory operations and point out the drawbacks of them, which motivates the new approach proposed in the next section. We have already mentioned in the previous section that modulo scheduling schemes usually schedule memory operations using the cache-hit latency. This scheme will be called cache-hit latency (CHL). This scheme is expected to produce a significant amount of processor stalls as suggested in the previous section. An approach to reduce the processor stall is to insert a prefetch instruction for every memory operation. Such instructions are scheduled at a distance equal to the cache-miss latency from the actual memory references. This scheme will be called insert prefetch always (IPA). However, this scheme may result in an increase in the number of operations (due to prefetch instructions but also to some additional spill code) and therefore, it may require an II higher than the previous approaches. Finally, an alternative approach is to schedule all memory operations using the cache-miss latency. This scheme will be called early scheduling always (ESA). This scheme prefetches data without requiring additional instructions but it may result in an increase in the II when memory instructions are in recurrences. Besides, it may also require additional spill code. Figure 2 compares the performance of the above three schemes for some SPECfp95 benchmarks and two different architectures (details about the evaluation methodology and the architecture are given in section 5). Each column is split into compute and stall time. In this figure it is also shown a lower bound on the execution time (OPT). This lower bound corresponds to the execution of programs when memory operations are scheduled using the cache-hit latency (which minimizes the spill code) but assuming that they always hit in cache (which results in null stall time). This lower bound was defined as the optimistic execution time in section 2. The main conclusion that can be drawn from Figure 2 is that the performance of the three realistic schemes is far away from the lower bound in general. The CHL scheme results in a significant percentage of stall time (for the aggressive architecture the stall time represents more than 50% of the execution time for most programs). The IPA scheme reduces significantly the stall time but not completely. This is due to the fact that some programs (especially tomcatv and swim) have cache interfering instructions at a very short distance and therefore, the prefetches are not always effective because they may collide and replace some data before being used. Besides, the IPA scheme results in a significant increase in the compute time for some programs (e.g., hydro2d and turb3d among others). The ESA scheme practically eliminates all the stall time. The remaining stall time is basically due to the lack of entries in the outstanding miss table that is used to implement a lockup-free cache. However, this scheme increases significantly the compute time for some programs like the turb3d (by a factor of 3 in the aggressive architecture), mgrid and hydro2d. This is due to the memory references in recurrences that limit the II. 4. The CSMS algorithm In this section we propose a new algorithm, which is called cache sensitive modulo scheduling (CSMS), that tries to minimize both the compute time and the stall time. These terms are not independent and reducing one of them may result in an increase in the other, as we have just shown in the previous section. The proposed algorithm tries to find the best trade-off between the two terms. The CSMS algorithm is based on early scheduling of some selectively chosen memory operations. Scheduling a memory operation using the cache-miss latency can hide almost all memory latency as we have shown in the previous section without increasing much the number of instructions (as opposed to the use of prefetch instructions). However, it can increase the execution time in three ways: . It may increase the register pressure, and therefore, it may increase the due to spill code if the performance of the loop is bounded by memory operations. . It may increase because the latency of memory operations is augmented. . It may increase the because the length of individual loop iterations may be increased. This augments the cost of the prolog and the epilog. Figure 2. Basic schemes performance CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT0.20.61.0 Normalized Loop Execution Time tomcatv swim su2cor hydro2d mgrid turb3d SPECfp95CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT CHL IPA ESA OPT0.20.61.0 Normalized Loop Execution Time tomcatv swim su2cor hydro2d mgrid turb3d 1.257 3.084 a) Simple architecture b) Aggressive architecture II II rec Two of the main issues of the CSMS algorithm is the reduction of the impact of recurrences on the II and the minimization of the stall time. The problem of the cost of the prolog and epilog is handled by computing two alternative schedules. Both focus on minimizing the stall time and the II. However, one of them reduces the impact of the prolog and the epilog at the expense of an increase in the stall time whereas the other does not care about the prolog and epilog cost. Then, depending on the number of iterations of the loop, the most effective one is chosen. The core of the CSMS algorithm is shown in Figure 3. The algorithm makes use of a static locality analysis in addition to other issues in order to determine the latency to be considered when scheduling each individual instruction. The locality analysis is based on the analysis presented in [19]. It is divided into three steps: . Reuse analysis: computes the intrinsic reuse property of each memory instruction as proposed in [21]. The goal is to determine the kind of reuse that is exploited by a reference in each loop. Five types of reuse can be determined: none, self-temporal, self-spatial, group- temporal and group-spatial. . Interference analysis: using the initial address of each reference and the previous reuse analysis, it determines whether two static instructions always conflict in the cache. Besides, self-interferences are also taken into account by considering the stride exhibited by each static instruction. References that interfere with themselves or with other refer- Figure 3. CSMS algorithm function CSMS(InnerLoop IL) return Scheduling is if (RecurrencesInGraph) then else endif if (NITER UpperBound) then return (sch1) else return (sch2) endif endfunction function ComputeSchedMinRecEffect(Graph G) return Scheduling is res foreach (Recurrence R G) do if (II rec (R) II) then endif endforeach return ComputeScheduling(G) endfunction function MinimizeRecurrenceEffect(Rec R, int II) return integer is OrderInstructionsByLocality(R) while (II rec (R) II) do endwhile return endfunction Figure 4. Scheduling a loop with recurrences ences are considered not to have any type of locality even if they exhibit some type of reuse. . Volume analysis: determines which references cannot exploit its reuse because they have been displaced from cache. It is based on computing the amount of data that is used by each reference in each loop. The analysis concludes that a reference is expected to exhibit locality if it has reuse, it does not interfere with any other (including itself) and the volume of data between to consecutive reuses is lower than the cache size. Initially, two data dependence graphs with the same nodes and edges are generated. The difference is just the latency assigned to each node. In grph1, each memory node is tagged according to the locality analysis: it is tagged with the cache-hit latency if it exhibits any type of locality or with the cache-miss latency otherwise. In grph2, all memory nodes are tagged with the cache-miss latency. Then, a schedule that minimizes the impact of recurrences on the II is computed for each graph using the function ComputeSchedMinRecEffect that is shown in Figure 4. The first step of this function is to change the latency of those memory operations inside recurrences that limit the II from cache-miss to cache-hit until the II is limited by resources or by a more constraining recurrence. Nodes to be modified are chosen according to a locality priority order, starting from the ones that exhibit most locality. Then, the second step is to compute the actual scheduling using the modified graph. This step can be performed through any of the software pipelined schedulers proposed in the literature. Finally, the minimum number of iterations (UpperBound) that ensures that sch2 is better than sch1 is computed. A main difference between these two schedules is the cost of the prolog and epilog parts, which is lower for the sch1. This bound depends on the computed schedules and the results of the locality analysis and it is calculated through an estimation of the execution time of each schedule. The sch1 is chosen if . The execution time of a given schedule is estimated as . The stall time is estimated as where penalty is calculated as explained in section 2: and the missratio is estimated by the locality analysis. In this way, sch1 is preferred to sch2 if: We use a scheduling according to the locality a not the CHL (which achieves the minimum SC) in order to take into account the possible poor locality of some loops. 5. Performance evaluation of the CSMS In this section we present a performance evaluation of the CSMS algorithm. We compare its performance to that of the basic schemes evaluated in section 3. It is also compared with some alter- est est t stall est NITER penalty op op MEM penalty LatMiss CycleUse CycleProd op MEM native binding (early scheduling) and nonbinding (inserting prefetch instructions) prefetch schemes. 5.1. Architecture model A VLIW machine has been considered to evaluate the performance of the different scheduling algorithms. We have modeled two architectures in order to evaluate different aspects of the produced schedulings such as execution time, stall time, spill code, etc. The first architecture is called simple and it is composed of four functional units: integer, floating point, branch and memory. The cache-miss latency for the first level cache is 10 cycles. The second architecture is called aggressive and it has two functional units of each type and the cache-miss latency is 20 cycles. All functional units are fully pipelined except divide and square root operations. In both models the first memory level corresponds to a 8Kb lockup-free, direct-mapped cache with lines of 32 bytes and 8 outstanding misses. Other features of the modeled architectures are depicted in Table 1. In the modeled architectures there are two reasons for the processor to stall: (a) when an instruction requires an operand that is not available yet (e.g., it is being read from the second level cache), and (b) when a memory instruction produces a cache miss and there are already 8 outstanding misses. 5.2. Experimental framework The locality analysis and scheduling task has been performed using the ICTINEO toolset [2]. ICTINEO is a source to source translator that produces a code in which each sentence has semantics similar to that of current machine instructions. After translating the code to such low-level representation and applying classical optimizations, the dependence graph of each innermost loop is constructed according the particular prefetching approach. Then, instructions are scheduled Other instructions Latency Machine model Simple Aggressive Integer Branch FUs 1 2 Floating Point Memory Cache Size 8 Kb DIV or SQRT or POW 12 Line Size Outstanding misses 8 Control Memory latency 1/10 1/20 BRANCH 2 Number of registers 32 CALL or RETURN 4 Table 1. Modeled architectures using any software pipelining algorithm. The particular software pipelining algorithm used in the experiments reported here is the HRMS [15], which has been shown to be very effective to minimize both the II and the register pressure. The resulting code is instrumented to generate a trace that feeds a simulator of the architec- ture. Each program was run for the first 100 million of memory references. The performance figures shown in this section refer to the innermost loops contained in this part of the program. We have measured that memory references inside innermost loops represent about 95% of all the memory instructions considered for each benchmark, so the statistics for innermost loops are quite representative of the whole section of the program. The different prefetching algorithms have been evaluated for the following SPECfp95 benchmarks: tomcatv, swim, su2cor, hydro2d, mgrid and turb3d. We have restricted the evaluation to Fortran programs since currently the ICTINEO tool can only process Fortran codes. 5.3. Early scheduling In this section we compare the CSMS algorithm with other schemes based on early scheduling of memory operations. These schemes are: (i) use always cache-hit latency (CHL), (ii) use always cache-miss latency (ESA), and (iii) schedule instructions that have some type of locality using the cache-hit latency and schedule the remaining ones using the cache-miss latency. This later scheme will be called early scheduling according to locality (ESL). The different algorithms have been evaluated in terms of execution time, which is split into compute and stall time. The stall time is due to dependences or to the lack of entries in the outstanding miss table. In Figure 5 we can see the results for both the simple and the aggressive architectures. For each benchmark all columns are normalized to the CHL execution time. It can be seen that the CSMS algorithm achieves a compute time very close to the CHL scheme whereas CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS0.20.61.0 Normalized Loop Execution Time tomcatv swim su2cor hydro2d mgrid turb3d 1.593 1.371 CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS CHL ESA ESL CSMS0.20.61.0 Normalized Loop Execution Time tomcatv swim su2cor hydro2d mgrid turb3d a) Simple architecture b) Aggressive architecture Figure 5. CSMS algorithm compared with early scheduling it has a stall time very close to the ESA scheme. That is, it results in the best trade-off between compute and stall time. In programs where recurrences limit the initiation interval, and therefore the ESA scheme increases the compute time (for instance in hydro2d and turb3d benchmarks) the CSMS method minimize this effect at the expense of a slight increase in the stall time. Table 2 shows the relative speed-up of the different schedulers with respect the CHL scheme. On average all alternative schedulers outperform the CHL scheme (which is usually the one used by software pipelining schedulers). However, for some programs (mainly for turb3d) the ESA and ESL schedulers perform worse than the CHL due to the increase in the II caused by recurrences. The CSMS algorithm achieves the best performance for all benchmarks. For the simple architecture the average speed-up is 1.61, and for the aggressive architecture it is 2.47. Table 3 compares the CSMS algorithm with an optimistic execution time (OPT) as defined in section 3 that is used as a lower bound of the execution time. It also shows the percentage of the execution time that the processor is stalled. It can be seen that for the simple architecture the CSMS algorithm is close to the optimistic bound and it does not cause almost any stall. For the aggressive architecture, the performance of the CSMS is worse than that of OPT and the stall time represents about 10% of the total execution time. Notice however, that the optimistic bound could be quite below the actual minimum execution time. Table 4 compares the different schemes using the CHL algorithm as a reference point. For each schemes it shows the increase in compute time and the decrease in stall time. As we have seen before, scheduling memory operations using the cache-miss latency can affect the initiation interval and the stage counter, which results in an increase in the compute time. The column denoted as DCompute represents the increment in compute time compared with the CHL scheduling. For any scheme s, it is calculated as: The stall time due to dependences can be eliminated by scheduling memory instructions using the cache-miss latency. By default, spill code is scheduled using the cache-hit latency and therefore it may cause some stalls, although it is unlikely because the spill code usually is a store followed by a load to the same address. Since usually they are not close (otherwise the spill code hardly reduces the register pressure), the load will cause a stall only if it interferes with a memory ESA ESL CSMS ESA ESL CSMS tomcatv 2.34 2.28 2.57 3.92 3.41 5.56 su2cor hydro2d 1.13 1.00 1.45 1.13 1.00 2.78 mgrid 1.15 1.00 1.17 1.12 1.00 1.19 turb3d 0.62 0.73 1.18 0.27 0.33 1.42 GEOMETRIC MEAN 1.36 1.22 1.61 1.48 1.15 2.47 Table 2. Relative speed-up DCompute stall s reference in between the store and itself. The column denoted as -Stall represents the percentage of the stall time caused by the CHL algorithm that is avoided. For any scheme s, it is calculated as: We can see in Table 4 that the CSMS algorithm achieves the best trade-off between compute time and stall time, which is the reason for outperforming the others. The ESA scheme is the best one to reduce the stall time but at the expense of a large increment in compute time. 5.4. Inserting prefetch instructions In order to reduce the penalties caused by memory operations, an alternative to early scheduling of memory instructions is inserting prefetch instructions, which are provided by many current instruction set architectures (e.g. the touch instruction of the PowerPC [5]). This new scheme can introduce additional spill code since it increases the register pressure. In particular, the lifetime of values that are used to compute the effective address is increased since they are used by both the OPT/CSMS %Stall OPT/CSMS %Stall su2cor 0.972 1.92 0.873 11.17 hydro2d 0.978 0.18 0.962 1.84 mgrid 0.998 turb3d 0.951 2.54 0.709 19.54 GEOMETRIC Table 3. CSMS compared with OPT scheduling ESA ESL CSMS ESA ESL CSMS DCompute -Stall(%) DCompute -Stall(%) DCompute -Stall(%) DCompute -Stall(%) DCompute -Stall(%) DCompute -Stall(%) su2cor 1.048 100.00 1.015 2.54 1.009 95.90 1.215 97.67 1.060 4.09 1.018 93.27 hydro2d 1.308 99.99 1.008 3.79 1.021 99.62 2.532 99.85 1.067 4.84 1.020 98.98 mgrid 1.023 99.89 0.999 3.59 1.001 99.68 1.469 87.58 1.030 5.19 1.337 87.57 turb3d 1.184 94.35 1.055 85.75 1.184 94.35 7.222 98.21 5.948 87.59 1.134 72.48 GEOMETRIC Table 4. Increment of compute time and decrement of stall time in relation to the CHL t stall CHL stall s prefetch and ordinary memory instructions. It can also increase the initiation interval due to additional memory instructions. We have evaluated three alternative schemes to introduce prefetch instructions: (i) insert prefetch always (IPA), (ii) insert prefetch for those references without temporal locality even if they exhibit spatial locality, according to the static locality analysis (IPT), and (iii) insert prefetch for those instructions without any type of locality (IPL). The first scheme is expected to result in a very few stalls but it requires many additional instructions, which may increase the II. The IPT scheme is more selective when adding prefetch instruction. However, it adds unnecessary prefetch instructions for some references with just spatial locality. Instructions with only spatial locality will cause a cache miss only when a new cache line is accessed if it is not in cache. The IPL scheme is the most conservative in the sense that it adds the less number of prefetch instructions In Figure 6 it is compared the total execution time of the CSMS scheduling against the above-mentioned prefetching schemes. The figures are normalized to the CHL scheduling. The CSMS scheme always performs better than the schemes based on inserting prefetch instructions except for the mgrid benchmark in the aggressive architecture. In this latter case, the IPA scheme is the best one but the performance of the CSMS is very close to it. Among the schemes that insert prefetch instructions, none of them outperforms the others in general. Depending on the particular program and architecture, the best one among them is a different one. The prefetch schemes outperform the CHL scheme in general (i.e. the performance figures in Figure 6 are in general lower than 1) but in some cases they may be even worse than the CHL, which is in general worse than the schemes that are based on early scheduling. Comparing binding (Figure 5) with nonbinding (Figure schemes, it can be observed that binding prefetch is always better for the three first benchmarks. Both schemes have similar per- a) Simple architecture b) Aggressive architecture Figure 6. CSMS algorithm compared with inserting prefetch instructions IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS0.20.61.0 Normalized Loop Execution Time tomcatv swim su2cor hydro2d mgrid turb3d IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS IPA IPT IPL CSMS0.20.61.0 Normalized Loop Execution Time tomcatv swim su2cor hydro2d mgrid turb3d formance for the next two benchmarks and only for the last one, nonbinding prefetch outperforms the binding schemes. To understand the reasons for the behavior of the prefetch schemes, we present below some additional statistics for the aggressive architecture. Table 5 shows the percentage of additional memory instructions that are executed for the CSMS algorithm and for those schemes based on inserting prefetch instructions. In the CSMS algorithm, additional instructions are only due to spill code whereas in the other schemes they are due to spill code and prefetch instructions. We can see in this table that, except for the IPL scheme for the mgrid benchmark, the prefetch schemes require much higher number of additional memory instructions. As expected, the increase in number of memory instructions of the IPA scheme is the highest, followed by IPT, then the IPL and finally the CSMS. Table 6 shows the increase in compute time and the decrease in stall time of the schemes based on inserting prefetch instructions in relation to the CHL scheme. Negative numbers indicate that the stall time is increased instead of decreased. We can see in Table 6 that the compute time is increased by prefetching schemes since the large number of additional instructions may imply a significant increase in the II for those loops that are memory bound. The stall time is in general reduced but the reduction is less than that of the CSMS scheme (see Table 4). The program mgrid is the only one for which there is a prefetch based scheme (IPA) that outperforms the CSMS algorithm. However, the difference is very slight and for the remaining programs the performance of the CSMS scheme is overwhelmingly better than that the IPA scheme. Table 7 shows the miss ratio of the different prefetching schemes compared with the miss ratio of a nonprefetching scheme (CHL). We can see that in general the schemes that insert most memory prefetches produce the highest reductions in miss ratio. However, inserting prefetch instructions do not remove all cache misses, even for the scheme that inserts a prefetch for every memory instruction (IPA). This is due to cache interferences between prefetch instructions before 1.There is spill code, but not in the simulated part of the program. CSMS INSERTING PREFETCH INSTR. IPA IPT IPL su2cor hydro2d 2.12 55.49 39.94 2.85 mgrid 49.90 59.26 56.57 7.50 Table 5. Percentage of additional memory references the prefetched data is used. This is quite common in the programs tomcatv and swim. For instance, if two memory references that interfere in the cache are very close in the code, it is likely that the two prefetches corresponding to them are scheduled before both memory references. In this case, at least one of the two memory references will miss in spite of the prefetch. Besides, if the prefetches and memory instructions are scheduled in reverse order (i.e., instruction A is scheduled before B but the prefetch of B is scheduled before the prefetch of A), both memory instructions will miss. To summarize, there are two main reasons for the bad performance of the schemes based on inserting prefetch instructions when compared with the CSMS scheme: . They increase the compute time due to the additional prefetch instructions and spill code. . They are not always effective in removing stalls caused by cache misses due to interferences between the prefetch instructions. IPA IPT IPL DCompute -Stall(%) DCompute -Stall(%) DCompute -Stall(%) tomcatv 1.396 23.28 1.060 67.14 1.073 19.42 su2cor 1.454 74.48 1.269 82.20 1.090 -3.25 hydro2d 1.608 84.87 1.086 86.81 1.003 4.16 mgrid 1.311 88.32 1.267 35.44 1.030 5.37 turb3d 1.874 68.91 1.787 73.90 1.497 82.60 GEOMETRIC Table 6. Increment of compute time and decrement of stall time for schemes based on inserting prefetch instructions CHL IPA IPT IPL su2cor 25.43 2.35 5.68 21.55 hydro2d 19.57 1.33 5.04 18.80 mgrid 6.46 0.57 2.91 5.35 turb3d 10.68 2.11 2.39 2.64 GEOMETRIC Table 7. Miss ratio for the CHL and the different prefetching schemes 6. Conclusions The interaction between software prefetching and software pipelining techniques for VLIW architectures has been studied. We have shown that modulo scheduling schemes using cache-hit latency produce many stalls due to dependences with memory instructions. For a simple architecture the stall time represents about 32% of the execution time and 63% for an aggressive architec- ture. Thus, ignoring memory effects when evaluating a software pipelined scheduler may be rather inaccurate. We have compared the performance of different prefetching approaches based on either early scheduling of memory instructions (binding prefetch) or inserting prefetch instructions (nonbinding prefetch). We have seen that both provide a significant improvement in general. However, methods based on early scheduling outperform those based on inserting prefetches. The main reasons for the worse performance of the latter methods are the increase in memory pressure due to prefetch instructions and additional spill code, and their limitation to remove short-distance conflict misses. We have proposed an heuristic scheduling algorithm (CSMS), which is based on early scheduling, that tries to minimize both the compute and the stall time. The algorithm makes use of a static locality analysis to schedule instructions in recurrences. We have shown that it outperforms the rest of strategies. For instance, when compared with the approach based on scheduling memory instructions using the cache-hit latency, the produced code is 1.6 times faster for a simple architecture, and 2.5 times faster for an aggressive architecture. In the former case, we have also shown that the execution time is very close to an optimistic lower bound. --R --TR Software pipelining: an effective scheduling technique for VLIW machines Software prefetching A data locality optimizing algorithm Circular scheduling An architecture for software-controlled data prefetching Design and evaluation of a compiler algorithm for prefetching Lifetime-sensitive modulo scheduling Balanced scheduling Evolution of the PowerPC Architecture Iterative modulo scheduling Minimizing register requirements under resource-constrained rate-optimal software pipelining Optimum modulo schedules for minimum register requirements Compiler techniques for data prefetching on the PowerPC Stage scheduling Hypernode reduction modulo scheduling Predictability of load/store instruction latencies Decomposed Software Pipelining Static Locality Analysis for Cache Management Swing Modulo Scheduling --CTR Jess Snchez , Antonio Gonzlez, Instruction scheduling for clustered VLIW architectures, Proceedings of the 13th international symposium on System synthesis, September 20-22, 2000, Madrid, Spain Enric Gibert , Jess Snchez , Antonio Gonzlez, Effective instruction scheduling techniques for an interleaved cache clustered VLIW processor, Proceedings of the 35th annual ACM/IEEE international symposium on Microarchitecture, November 18-22, 2002, Istanbul, Turkey Jess Snchez , Antonio Gonzlez, Modulo scheduling for a fully-distributed clustered VLIW architecture, Proceedings of the 33rd annual ACM/IEEE international symposium on Microarchitecture, p.124-133, December 2000, Monterey, California, United States Enric Gibert , Jess Snchez , Antonio Gonzlez, Local scheduling techniques for memory coherence in a clustered VLIW processor with a distributed data cache, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, March 23-26, 2003, San Francisco, California Javier Zalamea , Josep Llosa , Eduard Ayguad , Mateo Valero, Two-level hierarchical register file organization for VLIW processors, Proceedings of the 33rd annual ACM/IEEE international symposium on Microarchitecture, p.137-146, December 2000, Monterey, California, United States Enric Gibert , Jess Snchez , Antonio Gonzlez, An interleaved cache clustered VLIW processor, Proceedings of the 16th international conference on Supercomputing, June 22-26, 2002, New York, New York, USA Javier Zalamea , Josep Llosa , Eduard Ayguad , Mateo Valero, Modulo scheduling with integrated register spilling for clustered VLIW architectures, Proceedings of the 34th annual ACM/IEEE international symposium on Microarchitecture, December 01-05, 2001, Austin, Texas Alex Alet , Josep M. Codina , Jess Snchez , Antonio Gonzlez, Graph-partitioning based instruction scheduling for clustered processors, Proceedings of the 34th annual ACM/IEEE international symposium on Microarchitecture, December 01-05, 2001, Austin, Texas Enric Gibert , Jesus Sanchez , Antonio Gonzalez, Distributed Data Cache Designs for Clustered VLIW Processors, IEEE Transactions on Computers, v.54 n.10, p.1227-1241, October 2005 Javier Zalamea , Josep Llosa , Eduard Ayguad , Mateo Valero, Software and hardware techniques to optimize register file utilization in VLIW architectures, International Journal of Parallel Programming, v.32 n.6, p.447-474, December 2004

locality analysis;software pipelining;software prefetching;VLIW machines

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Resource-sensitive profile-directed data flow analysis for code optimization.

Instruction schedulers employ code motion as a means of instruction reordering to enable scheduling of instructions at points where the resources required for their execution are available. In addition, driven by the profiling data, schedulers take advantage of predication and speculation for aggressive code motion across conditional branches. Optimization algorithms for partial dead code elimination (PDE) and partial redundancy elimination (PRE) employ code sinking and hoisting to enable optimization. However, unlike instruction scheduling, these optimization algorithms are unaware of resource availability and are incapable of exploiting profiling information, speculation, and predication. In this paper we develop data flow algorithms for performing the above optimizations with the following characteristics: (i) opportunities for PRE and PDE enabled by hoisting and sinking are exploited; (ii) hoisting and sinking of a code statement is driven by availability of functional unit resources; (iii) predication and speculation is incorporated to allow aggressive hoisting and sinking; and (iv) path profile information guides predication and speculation to enable optimization.

Introduction Data flow analysis provides us with facts about a program by statically analyzing the program. Algorithms for partial dead code elimination (PDE) Copyright 1997 IEEE. Published in the Proceedings of Micro-30, December 1-3, 1997 in Research Triangle Park, North Carolina. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. Contact: Manager, Copyrights and Permissions 908-562-3966. y Supported in part by NSF PYI Award CCR-9157371, NSF grant CCR-9402226, Intel Corporation, and Hewlett Packard. [16, 6, 4, 20] and partial redundancy elimination (PRE) [17] solve a series of data flow problems to carry out sinking of assignments and hoisting of expression evaluations. The sinking of an assignment eliminates executions of the assignment that compute values that are dead, i.e., values that are never used. The hoisting of an expression eliminates evaluations of the expression if a prior evaluation of the same expression was performed using the same operands. The existing algorithms for above optimizations suffer from the following drawbacks that limit their usefulness in a realistic compiling environment in which the optimization phase precedes the instruction scheduling phase. ffl The optimization algorithms are insensitive to resource information. Thus, it is possible that instructions that require the same functional unit resource for execution are moved next to each other. Such code motion is not beneficial since the instruction scheduler in separating these instructions from each other may undo the optimization. ffl The data flow analyses used by the algorithms are incapable of exploiting profiling information to drive code sinking and hoisting. Instruction schedulers on the other hand use profiling information to drive code hoisting and sinking. ffl The data flow analyses do not incorporate speculation and predication to enable code hoisting and sinking. Instruction schedulers for modern processor architectures exploit speculation [18] and predication [13, 14] during hoisting and sinking. In this paper we present solutions to the above problems by developing data flow analysis techniques for PRE and PDE that incorporate both resource availability and path profiling [1] information. Fur- thermore, the formulations of hoisting and sinking are generalized to incorporate speculation based hoisting and predication enabled sinking. Our approach performs code motion for PRE and PDE optimizations such that code motion is restricted to situations in which the resulting code placement points are those at which the required functional unit resources are avail- able. Moreover we are able to perform code motion more freely than existing PDE and PRE algorithms [16, 17, 21, 5, 15] through speculative hoisting and predication enabled sinking. Finally, path profiling [1] information is used to guide speculation and pred- ication. In particular, speculation and predication is applied only if their overall benefit in form of increased code optimization along frequently executed program paths is greater than their cost in terms of introducing additional instructions along infrequently executed program paths. (a)8 9614 5 (b) (c) Figure 1: Resource Sensitive Code Sinking. The example in Figure 1 illustrates our approach for code sinking. The flow graph in Figure 1a contains statement node 8 which is partially dead since the value of x computed by the statement is not used along paths 10-8-7-3-1 and 10-8-7-6-4-2-1. Through sinking of the statement to node 5, as shown in Figure 1b, we can completely eliminate the deadness of the statement. Note that in order to enable sinking past node 7 it is necessary to predicate the statement. Furthermore this sinking should only be performed if the paths 10-8-7-3-1 and 10-8-7-6-4-2-1 along which dead code is eliminated are executed more frequently than the path 10-9-7-6-5-2-1 along which an additional instruction has been introduced. If the functional unit for the multiply operation is expected to be busy at node 5 and idle at node 6, then resource sensitive sinking will place the statement at node 6 as shown in Figure 1c. As before, the predication of the statement is required to perform sinking past node 7. However, the sinking past 7 is only performed if the frequency with which the path 10-8-7-3-1 (along which dead code is eliminated) is executed is greater than the sum of the frequencies with which paths 10-9-7-6-4-2-1 and 10-9-7-6-5-2-1 (along which an additional instruction is introduced) are executed. The above solution essentially places the statement at a point where the required resource is available and eliminates as much deadness as possible in the process. It also performs predication enabled sinking whenever it is useful. The example in Figure 2 illustrates our approach for code hoisting. The flow graph in Figure 2a contains partially redundant evaluation of expression x+y in node 8 since the value of x + y is already available at node 8 along paths 1-3-7-8-10 and 1-2-4-6-7-8- 10. Through hoisting of the expression to node 5, as shown in Figure 1b, we can eliminate the redundancy. Note that in order to enable hoisting above node 7, it is necessary to perform speculative motion of x y. Furthermore this hoisting should only be performed if the paths 1-3-7-8-10 and 1-2-4-6-7-8-10 along which redundancy is eliminated are executed more frequently than the path 1-2-5-6-7-9-10 along which an additional instruction is introduced.2 h=x+y h=x+y h=x+y2 h=x+y h=x+y h=x+y (c)2 h=x+y h=x+y .=x+y Figure 2: Resource Sensitive Code Hoisting. If the functional unit for the add operation is expected to be busy at node 5 and idle at node 6, then resource sensitive hoisting will place the statement at node 6 as shown in Figure 1c. As before, speculative execution of the statement is required to perform hoisting above node 7. However, the hoisting past 7 is only performed if the frequency with which path 1-3-7-8-10 (along which redundancy is eliminated) is executed is greater than the frequency with which the paths 1-2-4-6-7-9-10 and 1-2-5-6-7-9-10 (along which an additional instruction is introduced) are executed. The above solution places the expression at a point where the required resource is available and eliminates as much redundancy as possible in the process. It also performs speculative code hoisting whenever it is useful The remainder of this paper is organized as follows. In section 2 we first present extensions to PDE and solutions developed by Knoop et al. [16, 17] to achieve resource sensitive PDE and PRE. In section 3 we present extensions to resource sensitive PDE and PRE algorithms that incorporate path profiling information to drive speculation and predication. Concluding remarks are given in section 4. Resource-Sensitive Code Motion and Optimization The basic approach used by our algorithms is to first perform analysis that determines whether or not resources required by a code statement involved in sinking(hoisting) will be available along paths origi- nating(terminating) at a point. This information is used by the algorithms for PDE(PRE) to inhibit sink- ing(hoisting) along paths where the required resource is not free. Therefore, optimization opportunities are exploited only if they are permitted by resource usage characteristics of a program. In all the algorithms presented in this paper we assume that the program is represented using a flow graph in which each intermediate code statement appears in a distinct node and nodes have been introduced along critical edges to allow code placement along the critical edges. The first assumption simplifies the discussion of our data flow equations and is not essential for our techniques to work. Given this repre- sentation, we next describe how to determine whether a resource is locally free at the exit or entry of a node. This local information will be propagated through the flow graph to determine global resource availability in- formation. In the subsequent sections we present resource analysis which is applicable to acyclic graphs. The extension to loops is straightforward and is based upon the following observation. A resource free with in a loop is not available for hoisting/sinking of instructions from outside the loop since instructions are never propagated from outside the loop to inside the loop. A functional unit resource FU to which an instruction may be issued every c cycles is free at node n if the instructions issued during the c \Gamma 1 cycles prior to n and following n are not issued to FU . The above definition guarantees that if an instruction is placed at n that uses FU , then its issuing will not be blocked due to unavailability of FU . Notice that the above definition can be easily extended to consider situations in which more than one copy of the resource FU is available in the architecture. 2.1 Code Sinking and PDE Partial dead code elimination is performed by sinking partially dead assignments. Through sinking of a partially dead assignment s we migrate s to program points where resources required by s are available and at the same time we remove s from some paths along which it is dead, that is, the value computed by s is not used. In order to ensure that sinking of s is only performed if it can be guaranteed that placement points for s after sinking will be ones at which the resource required for its execution is available, we first develop resource anticipatability analysis. The results of this analysis guide the sinking in a subsequent phase that performs resource sensitive code sinking and PDE. Resource Anticipatability Analysis The resource anticipatability data flow analysis determines the nodes past which the sinking of an assignment s will not be inhibited by the lack of resources. A functional unit resource required to execute an assignment statement s is anticipatable at the entry of a node n if for each path p from n's entry to the terminating node in the flow graph one of the following conditions is true: ffl the value of the variable defined by s at n's entry is dead along path p, that is, the resource will not be needed along this path since the assignment can be removed from it; or ffl there is a node b along p in which the required resource is locally free and statement s is sinkable to b, that is, the required resource will be available after sinking. To perform resource anticipatability analysis for an assignment s, we associate the following data flow variables with each node: PRES s (n) is 1 if given that the resource required by s is anticipatable at n's exit, it is also anticipatable at n's entry; otherwise it is 0. In particular, PRES s (n) is 1 if the statement in n does not define a variable referenced (defined or used) by s. (n) is 1 if the variable defined by s is dead at n's entry (i.e., the value of the variable at n's entry is never used). If the variable is not dead, then the value of DEAD s (n) is 0. (n) is 1 if the resource required by s is free for its use when s is moved to n through sinking; otherwise it is 0. (n)) is 1 if the resource required by s is anticipatable at n's exit(entry); otherwise it is 0. In order to compute resource anticipatability we perform backward data flow analysis with the and confluence operator as shown in the data flow equations given below. The resource used by s is anticipatable at n's exit if it is anticipatable at the entries of all successors of n. The resource is anticipatable at n's entry if the resource is free for use by s in n, or the variable defined by s is dead at n's entry, or the resource is anticipatable at n's exit and preserved through n. N\GammaRANTs (m) (X\GammaRANTs (n) -PRESs(n)) Assignment Sinking The assignment sinking and PDE framework that we use next is an extension of the framework developed by Knoop et al. [16]. PDE is performed in the following steps: assignment sinking followed by assignment elimination. The first step is modified to incorporate resource anticipatability information while the second step remains unchanged. Assignment sinking consists of delayability analysis followed by identification of insertion points for the statement being moved. The data flow equations for delayability analysis and the computation of insertion points are specified below. In this analysis X\GammaDLY s assignment s can be delayed up to the exit(entry) of node n. BLOCK s (n) is 1 for a node that blocks sinking of s due to data dependences; otherwise it is 0. As we can see delayability analysis only allows sinking of s to the entry of node n if the required resource is anticipatable at n's entry and it allows sinking from entry to exit of n if s is not blocked by n. The assignment is removed from its original position and inserted at points that are determined as follows. The assignment s is inserted at n's entry if it is delayed to n's entry but not its exit and s is inserted at n's exit if it is delayed to n's exit but not to the entries of all of n's successors. Assignment deletion eliminates the inserted assignments that are fully dead. X\GammaDLYs (n) N\GammaDLYs (n) X\GammaDLYs (m) owise :N\GammaDLYs (m) The example in Figure 3 illustrates our algorithms by considering the sinking of assignment in node 1. In Figure 3a a control flow graph and the results of resource anticipatability analysis are shown. The node 7 is partially shaded to indicate that the resource is anticipatable at the node's exit but not its entry. Figure 3b shows the outcome of delayability analysis. Notice that the sinking of assignment past node 3 is inhibited since the resource is not anticipatable at 3's suc- cessors. Insertion point computation identifies three insertion points, the exit of 3, entry of 7 and exit of 8. Of these points the assignment is dead at 7's entry and is therefore deleted. Deadness of assignment along path 1-2-4-7-10-11 is removed while along path 1-2-3- 5-9-11 it is not removed. Notice that finally is placed at nodes 3 and 8 where the resource is locally free. 2.2 Code Hoisting and PRE Partial redundancy elimination is performed by hoisting expression evaluations. Through hoisting of a partially redundant expression e we migrate e to program points where resources required by e are available and at the same time we remove e evaluations from some paths along which e is computed multiple times making the later evaluations of e along the path redundant. In order to ensure that hoisting of e is only performed if it can be guaranteed that placement points for e after hoisting will be ones at which the resource required for e's execution is free, we perform resource availability analysis. Its results are used to guide hoisting in a subsequent phase that performs resource sensitive code hoisting and PRE. Resource Availability Analysis The resource availability data flow analysis determines the nodes above which the hoisting of statement s will not be inhibited by the lack of resources. A functional unit resource needed to execute the operation in an expression e is available at the entry of node n if for each path p from the start node to n's entry one the following conditions is true: ffl there is a node b in which resource is locally free and along the path from b to node n's entry, the variables whose values are used in e are not rede- fined. This condition ensures that upon hoisting the expression along the path a point would be found where the resource is free. ffl there is a node b which computes e and after its computation the variables whose values are used in e are not redefined. This condition essentially =. a=. (c) Insertion Point Selection and assignment deletion. if p2 9 10x=. x=. a=. (a) Resource Anticipability Analysis. resource locally free resource globally anticipable x=a*b if p2 9 10x=. x=a*b x=. (b) Delayability Analysis. if p2 9 10x=. x=. x=a*b x=a*b insertion points x=a*b is delayable Figure 3: An Example of Resource-Sensitive PDE. implies that if an earlier evaluation of the expression exists along a path then no additional use of the resource is required during hoisting along that path since the expression being hoisted will be eliminated along that path. To perform resource availability analysis for an expression e, we associate the following data flow variables with each node: (n) is 1 if given that the resource required by e is available at n's entry, it is also available at n's exit; otherwise it is 0. In particular, PRES e (n) is 1 if the statement in n does not define a variable used by e. USED e (n) is 1 if the statement in n evaluates the expression e and this evaluation of e is available at n's exit, that is, the variables used by e are not redefined in n after e's evaluation. (n) is 1 if the required resource is free for use by e in n when e is moved to n through hoisting; otherwise it is 0. (n)) is 1 if the resource required by e is available at n's entry(exit); otherwise it is 0. In order to compute resource availability we perform forward data flow analysis with the and confluence operator. The resource used by expression e is available at n's entry if it is available at the exits of predecessors of n. The resource is available at n's exit if it is available at n's entry and preserved by n, the expression is computed by n and available at n's exit, or resource is locally free at n. X\GammaRAVLe (m) Expression Hoisting The expression hoisting and PRE framework that we use next is a modification of the code motion frame-work developed by Knoop et al. [17]. PRE is performed in two steps: down-safety analysis which determines the points to which expression evaluations can be hoisted and earliestness analysis which locates the earliest points at which expression evaluations are actually placed to achieve PRE. These steps are modified to incorporate resource availability information. The modified equations for down-safety and earliest- ness analysis are given below. In these equations the expression can be hoisted to the entry(exit) of node n; otherwise it is An expression is down-safe at a node as long as it is anticipatable along all paths leading from the node and the required resource is available along all paths leading to the node. The earliestness analysis sets (n)) to 1 up to and including the first down-safe point at which the required resource is free or an expression evaluation exists. Along each path an expression evaluation is placed at the earliest down-safe point. These points are identified by the boolean predicates N\GammaDSafeEarliest e and X\GammaDSafeEarliest e (a) Resource Availability Analysis.2 9 10resource locally free resource globally available a=. (b) Down Safety Analysis. a=. a*b is down safe2 9 10earliest points and PRE transformation. (c) Earliestness Analysis a=. Figure 4: An Example of Resource-Sensitive PDE. N\GammaERLYe (n) USEDe(m)))- X\GammaERLYe (m) The example in Figure 4 illustrates the above al- gorithms. In Figure 4a the results of resource availability analysis are shown. Since the resource is available at node 11, the down-safety analysis will propagate it backwards as shown in Figure 4b. Node 6 is not down-safe because resource is not available at that node. On the other hand node 8 is down-safe because the resource is available at that node. The earliest- ness analysis identifies nodes 5, 9, 7, and 8 as the first nodes which are down-safe and where either resource or expression evaluation exists (node 5 and 7). The final placement of the expression is shown in Figure 4c. Traditional approach would have hoisted the expression above node 9 to node 6. 3 Profile-Directed Resource-Sensitive Code Motion and Optimization In this section we show that additional opportunities for PRE and PDE optimizations can be exploited by enabling more aggressive code hoisting and sink- ing. Speculative code hoisting can be performed to enable additional opportunities for PRE while predication based code sinking can be employed to enable additional opportunities for PDE. However, while speculative hoisting and predication based sinking result in a greater degree of optimization along some program paths, they result in introduction of additional instructions along other program paths. In other words a greater degree of code optimization is achieved for some program paths at the expense of introduction of additional instructions along other program paths. While generating code for VLIW and superscalar architectures, speculation and predication are routinely exploited to generate faster schedules along frequently executed paths at the expense of slower schedules along infrequently executed paths [7, 12, 11]. However, the optimization frameworks today are unable to exploit the same principle. In this section we show how to perform PRE and PDE optimizations by using speculation and predication. Frequently executed paths are optimized to a greater degree at the expense of infrequently executed paths. Path profiling information is used to evaluate the benefits and costs of speculation and predication to program paths. In the subsequent sections we describe code hoisting and sinking frameworks which use path profiling [1] information to enable speculation and predication based hoisting and sinking while inhibiting hoisting and sinking using resource availability and anticipata- bility information. This results in optimization algorithms that are more aggressive than traditional algorithms [16, 17] while at the same time more appropriate for VLIW and superscalar environment as they are resource sensitive and can trade-off the quality of code for frequently executed paths with that of infrequently executed paths. Although the techniques we describe are based upon path profiling information they can also be adapted for edge profiles since estimates of path profiles can be computed from edge profiles [19]. Furthermore we present versions of our algorithms that apply to acyclic graphs. However, the extensions required to handle loops are straight-forward and can be found in [10, 9]. Our algorithms are based upon the following analysis steps. First resource availability and anticipatabil- ity analysis is performed. Next we determine the cost and benefit of enabling speculation and predication at various spilt points and merge points in a flow graph respectively. The benefit is an estimation of increased optimization of some program paths while the cost is an estimate of increase in the number of instructions along other program paths. By selectively enabling hoisting and sinking at program points based upon cost-benefit analysis, we exploit optimization opportunities that the traditional algorithms such as those by Knoop et al. [16, 17] do not exploit while inhibiting optimization opportunities that result in movement of code to program points at which the resource required for an instructions execution is not free. 3.1 Path Profile Directed PDE As mentioned earlier, the resource anticipatability analysis described in section 2.1 remains unchanged and must be performed first. Next cost-benefit analysis that incorporates resource anticipatability information and uses path profiles is performed. The results of this analysis are used to enable predication enabled sinking at selected join points in the next phase. Finally an extension of the sinking framework presented in section 2.1 is used to perform resource-sensitive, profile-guided PDE. The cost-benefit analysis consists of three steps: (a) Availability analysis identifies paths leading to a node along which a statement is available for sinking (i.e., sinking is not blocked by data dependences) at various program points; (b) Optimizability analysis identifies paths originating at a node along which a statement can be optimized because the value computed by it is not live and the sinking required for its removal is not inhibited by the lack of a free resource or presence of data dependences; and (c) Cost-benefit computation identifies the paths through a join point along which additional optimization is achieved or an additional instruction is introduced when predication based sinking is enabled at the join point. By summing the frequencies of the respective paths, provided by path profiles, the values of cost and benefit are obtained. The set of paths identified during availability and optimizability analysis are represented by a bit vector in which each bit corresponds to a unique path from the entry to the exit of the acyclic flow graph. To facilitate the computation of sets of paths, with each node n in the flow graph, we associate a bit vector OnP s(n) where each bit corresponds to a unique path and is set to 1 if the node belongs to that path; otherwise it is set to 0. The steps of the analysis are described next. Availability Analysis In the data flow equations for availability analysis given below (n)) is a one bit variable which is 1 if there is a path through n along which s is available for sinking at n's entry(exit); otherwise its value is is a bit vector which holds the set of paths along which the value of N \Gamma AV L a is 1 at n's entry(exit). Forward data flow analysis with the or confluence operation is used to compute these values. At the entry point of the flow graph the availability value is set to 0, it is changed to 1 when statement a is encountered, and it is set to 0 if a statement that blocks the sinking of a is encountered. In the equations PRES a (n) is a one bit variable which is 1(0) if preserves a, that is, n is not data (anti, output or flow) dependent upon a. At the entry to a node n for which (n) is 0, the set of paths is set to null, that is, to ~ 0. Otherwise the paths in N \Gamma APS a are computed by unioning the sets of paths along which a is available at the exit of one of n's predecessors (i.e., unioning (p), where p is a predecessor of n). In order to ensure that only paths that pass through n are considered, the result is intersected with OnP s(n). The value of X \Gamma APS a (n) is OnP s(n) if n contains a and N \Gamma APS a (n) if n does not block a. OnPs(n)- X\GammaAVL a (m)=1 Optimizability Analysis a (n)) is a one bit variable associated with n's entry(exit) which is 1 if there is a path through n along which a is dead and any sinking of a that may be required to remove this deadness is feasible (i.e., it is not inhibited by lack of resources or presence of data dependences); otherwise its value is 0. Backward data flow analysis with the or confluence operation is used to compute these values. In order to ensure that the sinking of a is feasible, the results of a's availability analysis and resource antici- patability analysis are used. For example, if variable v computed by a is dead at n's exit, then is set to true only if (n) is true because the deadness will only be eliminated if sinking of a to n's exit is not blocked by data dependences. If v is not dead then among other conditions we also check a (n) is true because the sinking of a will only be allowed if the resource required for a's execution along paths where v is not dead is free. In the data flow equations is a one bit variable which is 1 if variable v is fully dead at n's entry(exit), that is, there is no path starting at n along which current value of v is used; otherwise its value is 0. (n)) is a bit vector which holds the set of paths along which the value of OPT a is 1 at n's entry(exit). At the entry(exit) of a node n for which (n)) and N \Gamma AV L a are 1, (n)) is set to OnP s(n). Otherwise the paths in X \Gamma OPS a (n) are computed by unioning the sets of paths along which a is partially dead and removable at the entry of one of n's successors (i.e., by unioning O \Gamma RPS a (p), where p is a successor of n). In order to ensure that only paths that pass through n are considered, the result is intersected with OnP s(n). let v be the variable defined by s; i:e:; OnPs(n)- N\GammaOPT a (m)=1 Computation The cost of enabling predication of a partially dead statement a to allow its movement below a merge point n is determined by identifying paths through the merge point along which in the unoptimized program a is not executed and in the optimized program a predicated version of a is executed. Furthermore resource anticipatability analysis indicates that along paths where predicated version of a is placed, the resource needed by a is available. The sum of the execution frequencies of the above paths, as indicated by path profiles, is the cost. The benefit of enabling predication of a partially dead statement a to allow its movement below a merge point n is determined by identifying paths through the merge point along which in the unoptimized program a is executed while in the optimized program a is not executed. Furthermore the resource anticipatability analysis indicates that the sinking of a required to achieve the above benefit is not inhibited by lack of resources. The sum of the execution frequencies of the above paths, as indicated by path profiles, is the benefit. Code Sinking Framework The results of the cost-benefit analysis are incorporated into a code sinking framework in which predication of a code statement is enabled with respect to the merge points only if resources are available and the benefit of predication enabled sinking is determined to be greater than the cost of predication enabled sink- ing. This framework is an extension of the code sinking framework presented in section 2.1. The data flow equations for enabling predication are presented next. Predication enabled sinking is allowed at join nodes at which the cost of sinking is less than the benefit derived from sinking. In addition, sinking is also enabled at a join node if it has been enabled at an earlier join node. This is to ensure that the benefits of sinking computed for the earlier join node can be fully realized. EPREDa (m) is a join point The delayability analysis of section 2.1 is modified to incorporate the results of enabling predication as shown below. At a join point if predication based sinking is enabled then as long as the assignment is available along some path (as opposed to all paths in section 2.1), it is allowed to propagate below the join node. N\GammaDLYa (n) -PRESa (n) owise N\GammaDLYa (n) if n is a join Consider the paths that contribute to cost and benefit of sinking assignment x = a b in node 8 past the join node 7 in the flow graph of Figure 1a. Availability analysis will determine that the paths that initially contain the subpath 10-8-7 are the ones along which statement is available for sinking at join node 7. Optimizability analysis will determine that the paths that end with subpath 7-3-1 are optimizable while the paths that end with 7-6-4-2-1 and 7-6-5-2-1 are unoptimizable. Although x = a b is dead along the subpath 7-6-4-2-1, the lack of resources inhibits sinking necessary to eliminate this deadness. To eliminate deadness along this path x = a b must be sunk past node 6 to make it fully dead which is prevented by lack of free resource. Based upon the above analysis the path that benefit's from sinking past node 7 is 10-8-7-3-1 while the paths along which cost of an additional instruction is introduced are 10-9-7-6-4-2-1 and 10-9-7-6-5-2-1. Let us assume that the execution frequency of path that benefits is greater than the sum of the frequencies of the two paths that experience additional cost. In this case predication based sinking will be enabled at node 7. The modified sinking frame-work will allow sinking past node 7 resulting in the code placement shown in Figure 1c. 3.2 Path Profile Directed PRE The resource availability analysis described in section 2.2 remains unchanged and must be performed first. Next cost-benefit analysis that incorporates resource availability information and uses path profiles is performed. The results of this analysis are used to enable speculation based hoisting at selected split points in the next phase. Finally an extension of the hoisting framework presented in section 2.2 is used to perform resource-sensitive, profile-guided PRE. The cost-benefit analysis consists of three steps: (a) Anticipatability analysis identifies paths originating at a node along which an expression is anticipatable and thus can be hoisted, i.e., its hoisting is not blocked by data dependences or lack of resources needed to execute the expression; (b) Optimizability analysis identifies paths leading to a node along which an expression can be optimized because a prior evaluation of the expression exists along these paths and the values of the variables used by the expression have not been modified since the computation of the expres- sion; and (c) Cost-benefit computation identifies the paths through a split point along which additional optimization is achieved or an additional instruction is introduced when speculation based hoisting is enabled at the split point. By summing the frequencies of the respective paths, provided by path profiles, the values of cost and benefit are obtained. Due to space limitations we omit the detailed data flow equations of the first two steps of cost-benefit analysis which compute sets of paths OPS e (n)). However, the principles used in their computation analogous to those used in section 3.1. The cost of enabling speculation of a partially redundant expression e to allow its movement above a conditional (split point) n is determined by identifying paths through the conditional along which e is executed in the optimized program but not executed in the unoptimized program. Furthermore, the resource availability analysis indicates that the required resource is available to allow the placement of e along the path. The sum of the execution frequencies of the above paths, as indicated by path profiles, is the cost. The benefit of enabling speculation of a partially redundant expression e to allow its movement above a conditional (split point) n is determined by identifying paths through the conditional along which a redundant execution of e is eliminated. Furthermore the hoisting required to remove the redundant execution of e from these paths is not inhibited due to lack of resources. The sum of the execution frequencies of the above paths, as indicated by path profiles, is the benefit. The incorporation of speculation in the partial redundancy framework of section 2.2 is carried out as follows. The results of cost-benefit analysis are incorporated into a code hoisting framework in which speculation of an expression is enabled with respect to the conditionals only if resources are available and the benefit of speculation enabled hoisting is determined to be greater than the cost of speculation enabled hoisting. The equations for enabling speculation are quite similar to those for enabling predication. The modification of down-safety analysis of section 2.2 as follows. At a split point if the speculative hoisting of an expression is enabled then as long as the expression is anticipatable along some path (as opposed to all paths in section 2.2), it is allowed to propagate above the split point. Consider the paths that contribute to cost and benefit of hoisting expression x+y in node 8 past the split node 7 in the flow graph of Figure 2a. Anticipatabil- ity analysis will determine that the paths ending with the subpath 7-8-10 are the ones along which expression x+y is anticipatable for hoisting at split node 7. Opti- mizability analysis will determine that the paths that start with the subpath 1-3-7 are optimizable while the paths that start with 1-2-4-6-7 and 1-2-5-6-7 are unop- timizable. Although x + y is evaluated along the sub-path 1-2-4-6-7, the lack of resources inhibits hoisting necessary to take advantage of this evaluation in eliminating redundancy. To eliminate redundancy along this path x+ y must be hoisted above node 6 to make it fully redundant which is prevented by lack of free resource. Based upon the above analysis the path that benefits from hoisting above node 7 is 1-3-7-8-10 while the paths along which cost of an additional instruction is introduced are 1-2-4-6-7-9-10 and 1-2-5-6-7-9- 10. Let us assume that the execution frequency of path that benefits is greater than the sum of the frequencies of the two paths that experience additional cost. In this case speculation based hoisting will be enabled at node 7. The modified hoisting framework will allow hoisting above node 7 resulting in the code placement shown in Figure 2c. 3.3 Cost of Profile Guided Optimization An important component of the cost of the analysis described in the preceding sections depends upon the number of paths which are being considered during cost-benefit analysis. In general the number of static paths through a program can be in the millions. How- ever, in practice the number of paths that need to be considered by the cost-benefit analysis is quite small. This is because first only the paths with non-zero execution counts need to be considered. Second only the paths through a given function are considered at any one time. In Figure 5 the characteristics of path profiles for the SPEC95 integer benchmarks are shown. The bar graph shows that in 65% of the functions that were executed no more than 5 paths with non-zero frequency were found and only 1.4% of functions had over 100 paths. Moreover, no function had greater than 1000 paths with non-zero execution count. One approach for reducing the number being considered in the analysis is to include enough paths with non-zero frequency such that these paths account for the majority of the execution time of the program. The first table in Figure 5 shows how the number of functions that contain up to 5, 10, 50, 100, and 1000 paths with non-zero frequency changes as we consider enough paths to account for 100%, 95% and 80% of the program execution time. As we can see the number of functions that require at most 5 paths increases substantially (from 1694 to 2304) while the number of functions that require over hundred paths reduces significantly (from 35 to 1). The second table shows the maximum number of paths considered among all the functions. Again this maximumvalue reduces sharply (from 1000 to 103) as the paths conisdered account for less than 100% of the program execution time. In [10, 9] we illustrate how the solution to cost-benefit analysis that we described earlier can be easily adapted to the situation in which only subset of paths with non-zero frequency are considered. Concluding Remarks In this paper we presented a strategy for PRE and PDE code optimizations that results in synergy between code placements found during optimization and instruction scheduling by considering the presence during selection of code placement points. In addition, the optimization driven by code hoisting and sinking also takes advantage of speculation and predication which till now has only been performed during instruction scheduling. Finally, our data flow algorithms drive the application of speculation and predication based upon path profiling in- formation. This allows us to trade-off the quality of code in favor of frequently executed paths at the cost of sacrificing the code quality along infrequently executed paths. The techniques we have described can also be adapted for application to other optimizations such as elimination of partially redundant loads and partially dead stores from loops [3, 8]. We are extending our algorithms to consider register pressure during optimization. Number of Paths with Non-Zero Execution Frequency Number of Functions 64.9% 8.6% 4% 1.4% Number Number of Functions of Paths 100% 95% 80% 1-5 1694 2022 2304 Total Max. # Exe. Time of Paths 100 1000 Figure 5: Characteristics of Path Profiles for SPEC95 Integer Benchmarks. --R "Efficient Path Profiling," "Partial Dead Code Elimination using Slicing Transformations," "Array Data-Flow Analysis for Load-Store Optimizations in Superscalar Architec- tures," "Using Profile Information to Assist Classic Code Optimiza- tion," "Practical Adaptation of Global Optimization Algorithm of Morel and Renvoise," "VLIW Compilation Techniques in a Superscalar Environment," "Trace Scheduling: A Technique for Global Microcode Compaction," "Code Optimization as a Side Effect of Instruction Scheduling," "Path Profile Guided Partial Dead Code Elimination Using Predi- cation," "Path Profile Guided Partial Redundancy Elimination Using Spec- ulation," "Region Scheduling: An Approach for Detecting and Redistributing Paral- lelism," "The Superblock: An Effective Technique for VLIW and Superscalar Compilation," "Highly Concurrent Scalar Processing," "HPL PlayDoh Architecture Specification: Version 1.0," "Global Optimization by Suppression of Partial Redundancies," "Partial Dead Code Elimination," "Lazy Code Motion," "Sentinel Scheduling for VLIW and Superscalar Pro- cessors," "Data Flow Frequency Analysis," "Critical Path Reduction for Scalar Processors," "Data Flow Analysis as Model Checking," --TR Highly concurrent scalar processing Region Scheduling Using profile information to assist classic code optimizations Lazy code motion Sentinel scheduling The superblock VLIW compilation techniques in a superscalar environment Partial dead code elimination Practical adaption of the global optimization algorithm of Morel and Renvoise Critical path reduction for scalar programs Data flow frequency analysis Efficient path profiling Array data flow analysis for load-store optimizations in fine-grain architectures Partial dead code elimination using slicing transformations Global optimization by suppression of partial redundancies Data Flow Analysis as Model Checking Path Profile Guided Partial Dead Code Elimination Using Predication Code Optimization as a Side Effect of Instruction Scheduling --CTR J. Adam Butts , Guri Sohi, Dynamic dead-instruction detection and elimination, ACM SIGOPS Operating Systems Review, v.36 n.5, December 2002 Sriraman Tallam , Xiangyu Zhang , Rajiv Gupta, Extending Path Profiling across Loop Backedges and Procedure Boundaries, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, p.251, March 20-24, 2004, Palo Alto, California Max Hailperin, Cost-optimal code motion, ACM Transactions on Programming Languages and Systems (TOPLAS), v.20 n.6, p.1297-1322, Nov. 1998 Youtao Zhang , Rajiv Gupta, Timestamped whole program path representation and its applications, ACM SIGPLAN Notices, v.36 n.5, p.180-190, May 2001 Raymond Lo , Fred Chow , Robert Kennedy , Shin-Ming Liu , Peng Tu, Register promotion by sparse partial redundancy elimination of loads and stores, ACM SIGPLAN Notices, v.33 n.5, p.26-37, May 1998 Vikki Tang , Joran Siu , Alexander Vasilevskiy , Marcel Mitran, A framework for reducing instruction scheduling overhead in dynamic compilers, Proceedings of the 2006 conference of the Center for Advanced Studies on Collaborative research, October 16-19, 2006, Toronto, Ontario, Canada Mary Lou Soffa, Complete removal of redundant expressions, ACM SIGPLAN Notices, v.33 n.5, p.1-14, May 1998 John Whaley, Partial method compilation using dynamic profile information, ACM SIGPLAN Notices, v.36 n.11, p.166-179, 11/01/2001 Mary Lou Soffa, Load-reuse analysis: design and evaluation, ACM SIGPLAN Notices, v.34 n.5, p.64-76, May 1999 Mary Lou Soffa, Complete removal of redundant expressions, ACM SIGPLAN Notices, v.39 n.4, April 2004

code optimization;functional unit resources;aggressive code motion;instruction schedulers;data flow algorithms;optimization;data flow analysis;partial dead code elimination;instruction reordering;resource availability;partial redundancy elimination;resource-sensitive profile-directed data flow analysis

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An approach for exploring code improving transformations.

Although code transformations are routinely applied to improve the performance of programs for both scalar and parallel machines, the properties of code-improving transformations are not well understood. In this article we present a framework that enables the exploration, both analytically and experimentally, of properties of code-improving transformations. The major component of the framework is a specification language, Gospel, for expressing the conditions needed to safely apply a transformation and the actions required to change the code to implement the transformation. The framework includes a technique that facilitates an analytical investigation of code-improving transformations using the Gospel specifications. It also contains a tool, Genesis, that automatically produces a transformer that implements the transformations specified in Gospel. We demonstrate the usefulness of the framework by exploring the enabling and disabling properties of transformations. We first present analytical results on the enabling and disabling properties of a set of code transformations, including both traditional and parallelizing transformations, and then describe experimental results showing the types of transformations and the enabling and disabling interactions actually found in a set of programs.

Introduction Although code improving transformations have been applied by compilers for many years, the properties of these transformations are not well understood. It is widely recognized that the place in the program code where a transformation is applied, the order of applying code transformations, and the selection of the particular code transformation to apply can have an impact on the quality of code produced. Although concentrated research efforts have been devoted to the development of particular code improving transformations, the properties of the transformations have not been adequately identified or studied. This is due in part to the informal methods used to describe code improving transformations. Because of the lack of common formal language or notation, it is difficult to identify properties of code transformations, to compare transformations and to determine how transformations interact with one another. By identifying various properties of code improving transformations, such as their interactions, costs, expected benefits, and application frequencies, informed decisions can be made as to what transformation to apply, where to apply them, and in which order to apply them. The order of application is important to the quality of code as transformations can interact with one another by creating or destroying the potential for further code improving transformations. For * This work was partially supported by NSF under grant CCR-9407061 to Slippery Rock University and by CCR- 9109089 to the University of Pittsburgh. * Dr. Whitfield's address is Department of Computer Science, Slippery Rock University, Slippery Rock, PA 16057 example, the quality of code produced would be negatively affected if the potential for applying a beneficial transformation was destroyed by the application of a less beneficial transformation. Certain types of transformations may be beneficial for one architecture but not for another. The benefits of a transformation can also be dependent on the type of scheduler (dynamic or static) that is used. 15 One approach that can be taken to determine the most appropriate transformations and the order of application for a set of programs is to implement a code transformer program (optimizer) that includes a number of code improving transformations, apply the transformations to the programs, and then evaluate the performance of the transformed code. However, actually implementing such a code transforming tool can be a time consuming process, especially when the detection of complex conditions and global control and data dependency information is required. Also, because of the ad hoc manner in which such code transformers are usually developed, the addition of other transformations or even the deletion of transformations may necessitate a substantial effort to change the transformer. Another approach is to modify an existing optimizer. However, optimizing compilers are often quite large (e.g., SUIF 12 is about 300,000 lines of C++ code and the GNU C compiler 8 is over 200,000 lines of code) and complex, making it difficult to use them in experiments that take into account the various factors influencing the performance of the transformed code. In this paper, we present a framework for exploring properties of code improving transformations. The major component of the framework is a code transformation specification language, Gospel. The framework includes a technique that utilizes the specifications to analytically investigate the properties of transformations. Gospel is also used in the design of Genesis, a tool that automatically produces a code transformer program from the specifications, enabling experimentation. A specification for a transformation consists of expressing the conditions in the program code that must exist before the transformation can be safely applied and the actions needed to actually implement the transformation in the program code. The specification uses a variant of first order logic and includes the expression of code patterns and global data and control dependencies required before applying the transformation. The actions are expressed using primitive operations that modify the code. The code improving transformations that can be expressed in Gospel are those that do not require a fix-point computation. This class includes many of the traditional and parallelizing code improving transformations. We demonstrate how the framework can be used to study the phase ordering problem of transformations by exploring the enabling and disabling properties of transformations. Using Gospel, we first show that enabling and disabling properties can be established analytically. We also demonstrate through the use of Genesis that these properties can be studied experimentally. Using Genesis, code transformers were automatically produced for a set of transformations specified in Gospel, and then executed to transform a test suite of programs. We present results on experiments that explored the kinds of transformations found in the test suite and the types and numbers of transformation interactions that were found. A number of benefits accrue from such a framework. Guidelines suggesting an application order for a set of code improving transformations can be derived from both the analytical and experimental exploration of the interactions. Also, a new transformation can be specified in Gospel and its relationship to other transformations analytically and experimentally investigated. From the specifications, a transformer can be generated by Genesis and using sample source programs, the user can experimentally investigate transformations on the system under consideration. The decision as to which transformations to include for a particular architecture and the order in which these transformations should be applied can be easily explored. New transformations that are particularly tailored to an architecture can be specified and used to generate a transformer. The effectiveness of the transformations can be experimentally determined using the architecture. Transformations that are not effective can be removed from consideration, and a new transformation can be added by simply changing the specifications and rerunning Genesis, producing a program (transformer) that implements the new transformation. Transformations that can safely be combined could also be investigated analytically and the need to combine them can be explored experimentally. Another use of Gospel and Genesis is as a teaching tool. Students can write specifications of existing transformations, their own transformations, or can modify and tune transformations. Implementations of these transformations can be generated by Genesis, enabling experimentation with the transformations. Prior research has been reported on tools that assist in the implementation of code improving transformations, including the analysis needed. Research has been performed on automatic code generation useful in the development of peephole transformers. 4,6,7,9 In these works, the transformations considered are localized and require no global data flow information. A number of tools have been designed that can generate analyses. Sharlit 13 and PAG 1 use lattice-based specifications to generate global data-flow analyses. SPARE is another tool that facilitates the development of program analysis algorithms 14 . This tool supports a high level specification language through which analysis algorithms are expressed. The denotational nature of the specifications enables automatic implementation as well as verification of the algorithms. A software architecture useful for the rapid prototyping of data flow analyzers has also recently been presented. 5 Only a few approaches have been developed that integrate analysis and code transformations, which our approach does. A technique to combine specific transformations by creating a transformation template that fully describes the combined operations was developed as part of the framework for iteration-reordering loop transformations. 11 New transformations may be added to the framework by specifying new rules. This work is applied only to iteration-reordering execution order of loop interactions in a perfect (tight) loop nest and does not provide a technique to specify or characterize transformations in general. The next section of this paper discusses the framework developed to specify transformations. Section 3 presents details of the Gospel language. Section 4 shows how Gospel can be used in the analytical investigation of the enabling and disabling conditions of transformations, and in the automatic generation of transformers. Section 5 demonstrates the utility of the specification technique using Genesis and presents experimental results. Conclusions are presented in Section 6. 2. Overview of the Transformation Framework The code improving transformation framework, shown in Figure 1, has three components: Gospel, a code transformation specification language; an analytical technique that uses Gospel specifications to facilitate formal proofs of transformation properties; and Genesis, a tool that uses the Gospel specifications to produce a program that implements the application of transformations. These three components are used to explore transformations and their properties. In this paper, we use the framework to explore disabling and enabling properties. A Gospel specification consists of the preconditions needed in program code in order for a transformation to be applicable, and the code modifications that implement the transformation. Part of the precondition specification is the textual code pattern needed for a transformation. An example includes the existence of a statement that assigns a variable to a constant or the existence of a nested loop. Thus, the code patterns operate on program objects, such as loops, statements, expressions, operators and operands. In order to determine whether it is safe to apply a transformation, certain data and control dependencies may also be needed. Program objects are also used to express these dependence relationships. In describing transformations, Gospel uses dependencies expressed in terms of flow, anti, output, and control dependencies. 21 These dependencies are quantified and combined using logical operators to produce complex data and control conditions. A flow dependence (S i d S j ) is a dependence between a statement that defines a variable and a statement S j that uses the definition in S i . An anti-dependence (S i d - S j ) exists between statement S i that uses a variable that is then defined in statement S j . An output dependence (S i d dependence between a statement S i that defines (or writes) a variable that is later defined (or written) by S j . A control dependence exists between a control statement S i and all of the statements S j under its control. The concept of data direction vectors for both forward and backward loop-carried dependencies of array elements is also needed in transformations for parallelization. 10 Each element of the data dependence vector consists of either a forward, backward, or equivalent direction represented by <, >, or =, respectively. These directions can be combined into >=, <=, and *, with * meaning any direction. The number of elements in the direction vector corresponds to the loop nesting level of the statements involved in the dependence. In some cases, code improving transformations have been traditionally expressed using global data flow information. This information can either be expressed as a combination of the data and control dependencies 21 or can be introduced in Gospel as a relationship that needs to be computed and checked. The underlying assumption of Gospel is that any algorithm needed to compute the data flow or data dependency information is available. Thus, Gospel uses basic control and data dependency information with the possibility of extensions to other types of data flow information. It should be noted that in the more than twenty transformations studied in this research, all data flow information was expressed in terms of combinations of data and control dependencies. 16, 17 A sample of transformation specifications is given in Appendix B. Gospel also includes the specification of the code modifications needed to implement a transformation. Although code improving transformations can produce complex code modifications, the code changes are expressed in Gospel by primitive operations that can be applied in combinations to specify complex actions. These operations are applied to code objects such as statements, expressions, operands and operations. Using primitive operations to express code modifications provides the flexibility to specify a wide range of code modifications easily. Another component of the framework is an analytical technique useful for proving properties of transformations. The technique uses the specification from Gospel to provide a clear, concise description of a transformation useful in analysis. We show how this component was used in establishing the enabling and disabling properties of a set of transformations. The last component of the framework is Genesis, a tool that generates a program that implements transformations from the Gospel specification of those transformations. Thus, the generated program contains code that will check that conditions needed for the safe application of a transformation are satisfied and also contains code that will perform the code modifications as expressed in the Gospel specification. A program to be transformed is then input into the program generated by Genesis and the output produced is the program transformed by the specified transformations. A run-time interface is provided that either permits the user to select the type and place of application of a transformation, or it automatically finds all applicable transformations at all points. We demonstrate the utility of Genesis in determining the kinds and frequencies of transformations occurring in a number of programs, and the types and frequencies of enabling and disabling interactions. Figure 1 presents the code improving framework and uses of the framework. The three components of the framework are shown in the box and some applications of the framework are shown in ovals. Solid lines connect the framework with the applications that are described in this paper. A solid line connects the framework to the interaction prover used to establish enabling and disabling properties of transformations. There is another solid line between the framework and the experimental studies of enabling and disabling properties. The dotted line connecting the framework and the combining transformations represents a potential use of the framework yet to be fully explored. _ Code Improving Transformation Framework Uses Figure 1. Components and Utilization of the Transformation Framework ________________________________________________________________________ 3. Description of the Gospel Language Gospel is a declarative specification language capable of specifying a class of transformations that can be performed without using fix-point computation. We have specified over twenty transformations using Gospel, including specifications for invariant code motion, loop fusion, induction variable elimination, constant propagation, copy propagation and loop unrolling. Transformations that do require fix-point computation such as partial dead code elimination and partial redundancy elimination cannot be specified. Likewise, although Gospel can be used to specify a type of constant propagation and folding, it cannot be used, for example, to specify constant propagation transformations requiring fixed point computation. However, studies have shown that code seldom contains the types of optimizations needing iteration. 3 A BNF grammar for a section of Gospel appears in Appendix A. The grammar is used to construct well-formed specifications and also used in the implementation of the Genesis transformer. In this paper, we assume the general form of statements in a program to be transformed is three address code extended to include loop headers and array references. However, Gospel and Genesis can be adapted to handle other representations including source level representation. We assume that a basic three address code statement has the form: The three address code retains the loop headers and array references from the source program, which enables the user to specify loop level transformations and array transformations. The template for a specification of a transformation consists of a Name that is used to identify the particular code improving transformation followed by three major specification sections identified by keywords: DECLARATION, PRECONDITION and ACTION. The PRECONDITION section is decomposed into two sections, Code_Pattern and Depend. The overall design of a Gospel specification follows. Combining Genesis Proof Technique Enabling & Disabling Interaction Properties Experimental Study of Enabling & Disabling Transformations Gospel DECLARATION PRECONDITION Code_Pattern Depend ACTION The DECLARATION section is used to declare variables whose values are code objects of interest (e.g., loop, statement). Code objects have attributes as appropriate such as a head for a loop and position for an operand. The PRECONDITION section contains a description of the code pattern and data and control dependence conditions, and the ACTION section consists of combinations of primitive operations to perform the transformation. Figure presents a Gospel specification of a Constant Propagation (CTP) transformation (See Section 3.2 for details). The specification uses three variables S i , S j and S l whose values are statements. The Code_Pattern section specifies the code pattern consisting of any statement that defines a constant type (S i .opr 2 ) == const. S i will have as its value such a statement if it exists. In the Depend section, S j is used to determine which statement uses the constant. The pos attribute records the operand position (first, second or third) of the flow dependence between S i and S j . The second statement with S l ensures that there are no other definitions of the constant assignment that might reach S j . Again, the pos attribute records the position of the flow dependence between S j and S l . The S j != S l specification indicates that the two statements are not the same statement and the operand (S j , pos) != operand (S l , pos) specification ensures that the dependence position recorded in S j does not involve the same variables as the dependence found in S 1 . _______________________________________________________________________ DECLARATION PRECONDITION Code_Pattern Find a constant definition any S i Depend Use of S i with no other definitions any (S j , pos): flow_dep (S i , S j , (=)); no (S l , pos): flow_dep (S l , AND operand (S i , pos) != operand (S l , pos); ACTION Change use in S i to be constant modify (operand (S j , pos), S i .opr 2 ); Figure 2. Gospel Specification of Constant Propagation If a S j is found that meets the requirements and no S l 's are found that meet the specified requirements, then the operation expressed in the ACTION section is performed. The action is to modify the use at S j to be the constant found as the second operand of S i . Next consider the specification of the parallelizing transformation Loop Circulation (CRC) found in Figure 3 that defines two statements and three tightly (perfect) nested loops, which are loops without any statements occurring between the headers. In the Code_Pattern section, any specifies an occurrence of tightly nested loops L 1 , L 2 , and L 3 . The data dependence conditions in the Depend section first ensure that the loops are tightly nested by specifying no flow dependences between loop headers. Next, the Depend section expresses that there are no pairs of statements in the loop with a flow dependence and a (<,>) direction vector. If no such statements are found then the Heads and Ends of the loops are interchanged as specified in the ACTION section. The next section provides more details about the Gospel language. 3.1. Gospel Types and Operations Variables, whose values are code elements, are defined in the declaration section and have the DECLARATION: id_list. Variables are declared to be one of the following types: Statement, Loop, Nested loops, Tight loops, or Adjacent loops. Thus, objects of these types have as their value a pointer to a statement, loop, nested loop, tight loop or adjacent loop, respectively. All types have pre-defined attributes denoting relevant properties, such as next (nxt) or previous (prev). The usual numeric constants (integer and real) are available in Gospel specifications. Besides these constants, two classifications of pre-defined constants are also available: operand types and opcode values. These constants ________________________________________________________________________ DECLARATION PRECONDITION Code_Pattern Find Tightly nested loops any (L 1 , Depend Ensure perfect nesting, no flow_dep with<,> no no ACTION Interchange the loops move Figure 3. Gospel Specification of Loop Circulation reflect the constant values of the code elements that are specified in Gospel. Examples of constants include const for a constant operand and var for a variable operand. Typical mathematical opcodes as well as branches and labels can appear in the specification code. Gospel can be extended to include other op codes and variable types by changing the grammar and any tools, such as Genesis, that uses the grammar. A variable of type Statement can have as its value any of the statements in the program and possesses attributes indicating the first, second and third operand (opr 1 , opr 2 , and opr 3 , respectively) and the operation (opcode). Additionally a pos attribute exists to maintain the operand position of a dependence required in the Depend section. ALoop typed variable points to the header of the loop, and has as attributes Body, which identifies all the statements in the loop and Head, which defines Lcv, the loop control variable, Init, the initial value and Final, the last value of the loop control variable. The End of the loop is also an attribute. Thus, a typical loop structure, with its attributes is: Head {L.Head defines L.Init, L.Final, and L.Lcv} Loop_body {L.Body} End_of_Loop {L.End} Nested loops, Tight loops, and Adjacent loops are composite objects whose components are of type Loop. Nested loops are defined as two (or more) loops where the second named loop appears lexically within the first named loop. Tight loops restrict nested loops by ensuring that there are no statements between loop headers. Adjacent loops are nested loops without statements between the end of one loop and the header of the next loop. The id_list after the keyword DECLARATION is either a simple list (e.g., statement and loop identifiers) or a list of pairs (e.g., identifiers for a pair of nested, adjacent or tight loops). For example, Tight: (Loop_One, Loop_Two) defines a loop structure consisting of two tightly nested loops. 3.2. The Gospel Precondition Section In order to specify a code improving transformation and conditions under which it can be safely applied, the pattern of code and the data and control dependence conditions that are needed must be expressed. These two components constitute the precondition section of a specification. The keyword PRECONDITION is followed by the keywords Code_Pattern, which identifies the code pattern specifications, and Depend which identifies the dependence specification. Code Pattern Specification The code pattern section specifies the format of the statements and loops involved in the transformation. The code pattern specification consists of a quantifier followed by the elements needed and the required format of the elements. quantifier element_list: format_of _elements; The quantifier operators can be one of any, all or no with the following meanings: all - returns a set of all the elements of the requested types for a successful match any - returns a set of one element of the requested type if a match is successful no - returns a null set if the requested match is successful For example, the quantifier element list any (S j ) returns a pointer to some statement S j. The second part of the code pattern specification format_of_elements describes the format of the elements required. If Statement is the element type, then format-of-elements restricts the statement's operands and operator. Similarly, if Loop is the element type, format-of-elements restricts the loop attributes. Thus, if constants are required as operands or if loops are required to start at iteration 1, this requirement is specified in the format_of_elements. An example code pattern specification which specifies that the final iteration count is greater than the initial value is: any Loop: Loop.Final - Loop.Init > 0 Expressions can be constructed in format_of_elements using the and and or operators with their usual meaning. Also, restrictions can be placed on either the type of an operand (i.e., const, or var) or the position, pos, of the opcode as seen in the Code_Pattern section of Figure 2. Depend Specification The second component of the PRECONDITION section is the Depend section, which specifies the required data or control dependencies of the transformation. The dependence specification consists of expressions quantified by any, no, or all that return both a Boolean truth value and the set of elements that meet the conditions. If the pos attribute is used, then the operand position of the dependence is also returned. The general form of the dependence specification is: quantifier element: sets_of_elements, dependence_conditions The sets_of_elements component permits specifying set membership of elements; mem(Element, specifies that Element is a member of the defined Set. Set can be described using predefined sets, the name of a specific set, or an expression involving set operations and set functions such as union and intersection. The dependence_conditions clause describes the data and control dependencies of the code elements and takes the form: type_of_dependence (StmtId, StmtId, Direction). In this version of Gospel, the dependence type can be either flow dependent (flow_dep), anti- dependent (anti_dep), output dependent (out_dep), or control dependent (ctrl_dep). Direction is a description of the direction vector, where each element of the vector consists of either a forward, backward or equivalent direction (represented with <, >, =, respectively; also <= and >=, can be used), or any, which allows any direction. Direction vectors are needed to specify loop-carried dependencies of array elements for parallelizing transformations. This direction vector may be omitted if loop-carried dependencies are not relevant. As an example, the following specification is for one element named S i that is an element of Loop 1 such that there is a S j , an element of Loop 2 , and there is either a flow dependence or an anti- dependence between S i and S j . any 3.3. The Gospel Action Section We decompose the code modification effects of applying transformations into a sequence of five primitive operations, the semantics of which are indicated in Table 1. These operations are overloaded in that they can apply to different types of code elements. The five primitive operations, their parameters and semantics are: Table 1. Action Operations An example of a Move operation that moves Loop_1 header after Loop_2 header is: move(Loop_1.Head, Loop_2.Head). An example of a modify action that modifies the end of Loop_2 to jump to the header of Loop_2 is: modify(Loop_2.End, address (Loop_2.Head)). These primitive operations are combined to fully describe the actions of a transformation. It may be necessary to repeat some actions for statements found in the PRECONDITION section. Hence, a list of actions may be preceded by forall and an expression describing the elements to which the actions should be applied. The flow of control in a specification is implicit with the exception of the forall construct available in the action section. In other words, the ACTION keyword acts as a guard that does not permit entrance into this section unless all conditions have been met. 4. Applications of the Gospel Specification The Gospel specifications are useful in a number of ways. In this section, we demonstrate the utilization of the specifications to explore the phase ordering problem of transformations by Operation Parameter Semantics Move (Object, After_Object) move Object and place it following After_Object Add (Obj_Desc, Obj_Name, After_Obj) add Obj_Name with Obj_Desc, place it after_Obj Delete (Object) delete Object Copy (Obj, After_Obj, New_Name) copy Obj into New_Name, place it After_Obj Modify (Object, Object_Description) modify Object with Object_Desc analytically establishing enabling and disabling properties. In Section 4.2, we show how Gospel is used to produce an automatic transformer generator, Genesis, which can be used to explore properties of transformations experimentally. 4.1. Technique to Analyze Specifications The Gospel specifications can be analyzed to determine properties of transformations, and in particular, we use the analysis technique for establishing enabling and disabling properties of transformations. Through the enabling and disabling conditions, the interactions of transformations that can create conditions and those that can destroy conditions for applying other transformations are determined. Knowing the interactions that occur among transformations can be useful in determining when and where to apply transformations. For example, a strategy might be to apply a transformation that does not destroy conditions for applying another transformation in order to exploit the potential of the second transformation, especially if the second transformation is considered to be more beneficial. 4.1.1. Enabling and Disabling Conditions Enabling interactions occur between two transformations when the application of one transformation creates the conditions for the application of another transformation that previously could not be applied. Disabling interactions occur when one transformation invalidates conditions that exist for applying another transformation. In other words, transformation A enables transformation B (denoted A - B) if before A is performed, B is not applicable, but after A is performed, B can now be applied (B's pre-condition is now true). Similarly, transformation A disables transformation B (denoted A B) if the pre-conditions for both transformation A and B are true, but once A is applied, B's pre-condition becomes false. These properties are involved in the phase ordering problem of transformations. Before determining the interactions among transformations, the conditions for enabling and disabling each transformation must be established. The enabling and disabling conditions are found by analyzing the PRECONDITION specifications of the transformations. For each condition in the Code_Pattern and Depend section of a transformation, at least one enabling/disabling condition is produced. For example, if a code pattern includes: any Statement: Statement.opcode == assign then the enabling condition is the creation of a statement with the opcode of assign, or the disabling conditions are the deletion of such a statement or the modification of the statement's opcode. The enabling and disabling conditions of six transformations derived from their specifications (see appendix B for their Gospel specifications) are given in Table 2. 4.1.2 Interactions Among Transformations Using the Gospel specifications, we can prove the non-existence of interactions. We also use the specifications in developing examples that demonstrate the existence of interactions. Such an Transformation Enabling Conditions Disabling Conditions Dead Code Elimination (DCE) 1. Create S i that is not used 2. Non-existence of S l with (S i d= S l ) Delete S l , Path is deleted* 1. Destroy S i that is not used 2. Existence of S l with (S i d= S l ) Introduce S l that uses value computed by S i Constant Propagation (CTP) 1. Create S i 2. Insert S j such that (S i d= 3. Non-existence of S l with (S l d= Modify S l so that S l == S i , Destroy (S l d= S j a) Introduce a definition*, b) Delete S l *, c) Path is deleted* 1. Destroy S i 2. Non-existence of S j with (S i d= 3. Existence of S l with (S l d= Modify S l so that S l - S i , Create (S l d= S j a) Definition is deleted*, b) Introduce S l , where S l - S i c) Path from S l to S j is created Constant Folding (CFO) 1.Create S i of the form CONST opcode CONST 1. Remove or Modify S i Loop Unrolling 1. Create DO Loop, L 1. Destroy DO loop, L Loop Fusion 1. Existence of 2 adjacent loops: Add a loop 2. Two DO loops have identical head- ers: Modify a header 3. The non-existence of S n and S m with a backward dependence before a forward: Remove S n or S m Add definition between S n Delete path between S n and S m 4. Non existence of (S i d= Remove Remove S i * Add def., destroying depend* Delete path between S i and S j * 1.Existence of 2 non-adjacent loops: Add a loop 2.Two DO loops do not have identical headers: Modify a header 3.The existence of S n and S m with a backward dependence before a forward Insert S n or S m Delete definition between S n Create path between S n and S m 4. Existence of (S i d= Delete a def., so dependence holds* Create path between S i and S j Loop Interchanging 1. Existence of 2 nested DO loops: Add a loop 2. Non-existence of S n , S m with a (<,>) dependence: Remove S n * or S m Add definition between S n Delete path between S n and S m 3. Loop headers are invariant: Modify a header 1.Non-Existence of 2 nested DO loops: Remove a loop 2. Existence of S n , S m with a (<,>) dependence Insert S n or S m Remove def. between S n Create path between S n and S m 3.Loop headers vary with respect to each other: Modify a header * denotes condition is not possible in correct specifications (i.e., maintains semantic equivalence) Table 2. Enabling and Disabling Conditions example of an interaction is given in Figure 4, where Loop Fusion (FUS) enables Loop Interchange (INX). The two inner loops on J are fused into one larger loop, which can then be interchanged. Sometimes the interaction between two transformations is more complex in that a transformation can both enable and disable a transformation. Invariant Code Motion (ICM) and Loop Interchange (INX) are two such transformations, as shown in Figure 5. ICM enables INX and also can disable INX. In Figure 5 (a), an example of ICM enabling INX is given and in Figure 5 (b) an example of ICM disabling INX is shown. For ease in proving the non-interaction, we use a formal notation of the Gospel specifications that is directly derived from the specification language by using mathematical symbols in place of the language related words. A comparison of the two styles is exemplified by: Language: no S m , Figure 4. Loop Fusion Enables Loop Interchanging _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ Figure 5: Enabling and Disabling Transformations (a) ICM enables INX (b) ICM disables INX The following claim and proof illustrate the technique to prove non-existence of enabling and disabling interactions between transformations. The claim is that loop interchange (INX) cannot disable the application of constant propagation (CTP). The proof utilizes the disabling conditions for CTP as given previously in Table 2. does not disable constant propagation.) Proof: Assume that INX CTP. For INX to disable CTP, both INX and CTP must be applicable before INX is applied. For INX to be applicable, there must be two tightly nested loops, L 1 and L 2 where the loop limits are invariant and there is no data dependence with a (<,>) direction vector. For CTP to be applicable, there must exist a S i that defines a constant and S j which uses the constant value such that (S i l such that S l Since CTP is applicable, INX must alter the state of the code to disable CTP. The three disabling conditions for CTP given in Table 2, produce the following cases: Case 1: Destroy S i which defines the constant INX does not delete any statements, but does move a header, L 2 . S i defines a variable and a loop header only defines the loop control variable. If the loop control variable and the variable defined in S i were the same, then CTP is not applicable because S i does not define a constant value. \ INX does not destroy S i , the statement defining the constant. Case 2: The non-existence of S j or the removal of the dependence (S i INX does not delete any statements but does move a header, L 2 . However, moving the header to the outside of the loop would not destroy the relationship (S i since the headers must be invariant relative to each other in order for INX to be applicable. \ INX does not destroy S j . Case 3: The creation of S l such that (S l d INX does not create or modify a statement. So there are three ways for INX to create the condition could delete a definition S i , but this is not a legal action for this transformation. could introduce S l . INX does not create any statements, but it does move a header. S l could not be the header because S l defines a constant. creates a path so that S l reaches S j . S j could be the header, but the definition in S l would have reached S j prior to INX since the headers must be invariant. \ INX does not create S l . Thus, we show that INX - CTP.- That is, loop interchange when applied will not destroy any opportunities for constant propagation. By exploring examples of interactions and developing proofs for non-interaction, we derived (by hand) an interaction table that displays the potential occurrence of interactions. Table 3 displays interactions for eight transformations: Dead Code Elimination (DCE), Constant Propagation(CTP), Copy Propagation (CPP), Constant Folding (CFO), Invariant Code Motion (ICM), Loop Unrolling (LUR), Loop Fusion (FUS), and Loop Interchange (INX). Each entry in the table consists of two elements separated by a slash (/). The first element indicates the enabling relationship between the transformation labeling the row and the transformation labeling the column, and the second element is the disabling relationship. A "-" indicates that the interaction does not occur, whereas an "E" or "D" indicates that an enabling or disabling interaction occurs, respectively. As an example, the first row indicates that DCE enables DCE and disables CTP. Notice the high degree of potential interactions among the triples <FUS, INX, and LUR>, and <CTP, CFO, and LUR>. 4.1.3 Impact of the Interactions on Transformation Ordering The disabling and enabling relationships between transformations can be used when transformations are applied automatically or when transformations are applied interactively. When transformations are applied automatically, as is the case for optimizing compilers, the interactions can be used to order the application so as to apply as many transformations as possible. When applying transformations in an interactive mode, knowledge about the interaction can help the user determine which transformation to apply first. Using the interaction properties, two rules are used for a particular ordering, if the goal is to applying as many transformations as possible. 1. If transformation A can enable transformation B, then order A before B - <A,B>. 2. If transformation A can disable transformation B, then order A after B - < B, A>. These rules cannot produce a definite ordering as conflicts arise when: 1. A - B and B - A 2. A B and B A 3. A B and A - B CTP E/-/D E/- E/- E/- E/- E/- LUR E/- E/- E/- E/- E/- E/D E/D FUS -/D -/D -/D E/D E/D Table 3. Theoretical Enabling and Disabling Interactions In these cases, precise orderings cannot be determined from the properties. However, as shown in the next section, experimentation can be performed using Genesis to determine if there is any value in applying one transformation before the other transformation. As an example of using the orderings, consider a scenario where the transformer designer decides that LUR is an extremely beneficial transformation for the target architecture. The transformer designer could benefit from two pieces of information: 1) the transformations that enable LUR, and 2) the transformations that disable LUR. As can be seen in Table 3, CTP, CFO, and LUR all enable LUR. These interactions indicate that CTP and CFO should be applied prior to LUR for this architecture. Additionally, one could infer from the table that since CTP enables CFO and CFO enables CTP, these two transformations should be applied repeatedly before LUR. Of course, there may be other factors to consider when applying loop unrolling. In this paper, we focus on only one, namely transformation interactions. Other factors may include the impact that the unrolled loop has on the cache. When other factors are important in the application of transformations, these factors could be embedded in Genesis experiments (e.g., by adding measures of cache performance). Table 3 also displays the interactions that disable LUR. As FUS is the only transformation that disables LUR, a decision must be made about the importance of applying FUS on the target architecture. If LUR is more important, then either FUS should not be applied at all or only at the end of the transformation process. The information about interactions could also be used in the development of a transformation guidance system that informs the user when a transformation has the potential for disabling another transformation and also informs the user when a transformation has the potential for enabling another transformation. The interactions among the transformations can also be used to determine some pairwise orderings of transformations. For instance, Table 3 indicates that when applying CPP and CTP, CPP should be applied first. Other such information can be gleaned from this table. 4.2 Genesis: An Automatic Transformer Generator Tool Another use of the framework is the construction of a transformer tool that automatically produces transformation code for the specified transformations. The Genesis tool analyzes a Gospel specification and generates code to perform the appropriate pattern matching, check for the required data dependences, and call the necessary primitive routines to apply the specified transformation. 19 Figure 6 presents a pictorial description of the design of Genesis. The value of Genesis is that it greatly reduces the programmer's burden by automatically generating code rather than having the programmer implement the optimizer by hand. In Figure 6, a code transformer is developed from a generator and constructor. The generator produces code for the specified transformations, utilizing pre-defined routines in the transformer library, including routines to compute data and control dependencies. The constructor packages all of the code produced by the generator, the library routines, and adds an interface which prompts interaction with the user. The generator section of Genesis analyzes the Gospel specifications using LEX and YACC, producing the data structures and code for each of the three major sections of a Gospel specification. The generator first establishes the data structures for the code elements in the specifications. Code is then generated to find elements of the required format in the three address code. Code to verify the required data dependences is next generated. Finally, code is generated for the action statements. The Genesis system is about 6,500 lines of C code, which does not include the code to compute data dependencies. A high level representation of the algorithm used in Genesis is given in Figure 7. The generated code relies on a set of predefined routines found in the transformer library. These routines are transformation independent and represent routines typically needed to perform transformations. The library contains pattern matching routines, data dependence computation algorithms, data dependence verification procedures, and code manipulation routines. The pattern matching routines search for loops and statements. Once a possible pattern is found, the generated code is called to verify such items as operands, opcodes, initial and final values of loop control variables. When a possible application point is found in the intermediate code, the data dependences must be verified. Data dependence verification may include a check for the non-existence of a particular data dependence, a search for all dependences, or a search for one dependence within a Generator Constructor Gospel specifications of transformations Library routines transformed by applying User options Transformer Code transforming system Code to perform _______________________________________________________________________ Figure 6. Overview of Genesis loop or set. The generated code may simply be an "if" to ensure a dependence does not exist or may be a more complex integration of tests and loops. For example, if all statements dependent on S i need to be examined, then code is generated to collect the statements. The required direction vectors associated with each dependence in the specification are matched against the direction vectors of the dependences that exist in the source program. If the dependences are verified then the action is executed. Routines consisting of the actions specified in the ACTION section of the specification are generated for the appropriate code elements. The constructor compiles routines from the transformer library and the generated code to produce the transformer for the set of transformations specified. The constructor also generates an interface to execute the various transformations. The interface to the transformer reads the source code, generates the intermediate code and computes the data dependences. The interface also queries the user for interactive options. This interactive capability permits the user to execute any ________________________________________________________________________ GENESIS() For {iterate through transformation list} Read(Gospel specification for Transformation t i ) {Analyze the Gospel specifications using LEX and YACC} {Gen code to setup data structures} {Gen code to search for patterns} gen_code_depend_verify(data_dependences) {Gen code to verify data dependences} {Gen code to perform primitive actions} End for {Create the interface from a template} Construct_optimizer(generated_code, library_routines) Read(source_code) Convert( source, intermediate representation) While (user_interaction_desired) Select_transformations Select_application_points Compute_data_dependences Perform_optimization (user's_direction) EndWhile Figure 7. The Genesis Algorithm number of transformations in any order. The user may elect to perform a transformation at one application point (possibly overriding dependence constraints) or at all possible points in the program. 4.2.1. Prototype Implementation In order to test the viability and robustness of this approach, we implemented a prototype for Genesis and produced a number of transformers. For ease of experimentation, our prototype produces a transformer for every transformation specified. For any transformation specified, the generator produces four procedures tailored to a transformation: set_up_Trans, match_Trans, pre_Trans, and act_Trans. These procedures correspond to the DECLARATION, Code_Pattern, Depend and ACTION sections in the specifications. In our implementation, a transformer consists of a driver that calls the routines that have been generated specifically for that transformation. Code for the driver is given in Figure 8. The format of the driver is the same for any transformer generated. The driver calls procedures in the generated call interface for the specific transformation (set_up_Trans, match_Trans, pre_Trans, and act_Trans). The call interface in turn calls the generated procedures that implement the transformation (the generated transformation specific code). For CTP, as given in Figure 9, the set_up_Trans procedure consists of a single call to set_up_CTP. The driver requires a successful pattern match from match_CTP and pre_CTP in order to continue. Thus, the match_Trans and pre_Trans of the call interface procedures return a boolean value. _______________________________________________________________________ Done := False; match_success := match(); /* Match the code patterns */ IF (match_success) THEN DO pre_success := pre_condition(); /* Verify the dependences */ IF (pre_success) THEN DO Perform actions of the optimization */ Done := True; END Figure 8. The Driver Algorithm Any generated set_up procedure consists of code that initializes data structures for each element specified using any or all in the PRECONDITION section. A type table data structure, TypeTable, contains identifying information about each statement or loop variable specified in the DECLARATION section. The TypeTable holds the identifier string, creates an entry for a quantifier that may be used with this identifier in the PRECOND section, and maintains the type of the identifier (e.g., statement, loop, adjacent loop or nested loop). For type Statement, an entry is initialized with the type and corresponding identifier. If a loop-typed variable is specified, additional flags for nested or adjacent loops are set in the type table entry. These entries are filled in as the information relevant to the element is found when the transformations are performed. For each statement in the DECLARATION section, a call to TypeTable_Insert is generated with the identifier and the type of the identifier and placed in the set_up procedure. During execution of CTP, shown in Figure 9, a type table entry is initialized with type "Statement" and identifier S i when the transformer executes procedure set_up_CTP. After the set_up_CTP procedure terminates, the driver indirectly initiates an exhaustive search for the statement recorded in the type table by calling match_CTP. If the source program's statement does not match, then the transformer driver re-starts the search for a new statement. The match procedure is generated from the statements in the Code_Pattern section of the Gospel specification. For each quantified statement in the Code_Pattern section, a call to SetTable_Insert is made with the identifier, type of identifier, and quantifier. SetTable_Insert searches for the requested type and initializes the Set_Table data structure with the appropriate attributes for the type (e.g., for a statement, the opcode and operands are set). Next the restrictions in the Code_Pattern section are directly translated into conditions of IF statements to determine if the requested restrictions are met. If the current quantifier is an "all", then a loop is generated to check all of the objects found by Set_Table. In the CTP example in Figure 9, code is generated that searches for an assignment statement with a constant on the right hand side. The next routine is the pre procedure, which is generated from the statements in the Depend section. For each quantified statement, a call to SetTable_Insert is generated (however, the pattern matching will not be performed again at run-time.) For the CTP example, the pre_CTP procedure inserts an element into the Set_Table structure for each dependence condition statement. S j is inserted into Set_Table and the dependence library routine is called to find the first statement that is flow dependent on S i ; if no statement is found then the condition fails. S l is also inserted into the Set_Table and the dependence routine is called again. Each S l such that S l is flow dependent on S j is examined to determine if the operand of S l causing the dependence is the same variable involved in the dependence from S i to S j . If such an S l is found then the condition fails. Next, an assignment statement is generated to assign the "hits" field of the Set_Table data structure with the result of the requested dependence or membership procedure call. For example, by setting the "hits" field to a result of a flow_dependence call, the hits field will contain either 1 (for the any quantifier) or many (for the all quantifier) statement numbers that are flow dependent with the required direction vector. Next, IF statements are directly generated from any relational conditions that exist in the specification. The last procedure to be called is the action procedure. The action procedure is generated from the statements in the Action section of the Gospel specification. For each individual action, a call to the primitive transformation is made with the required parameters (e.g., modify requires the object being modified and the new value). If the Gospel forall construct is used, then a for loop is ________________________________________________________________________ Type Table _Insert(Statement, Si); set up type table for statement S i { Table _Insert(Statement, Si, any); classify S i as a set of statements if (Set Table [Si].opcode.kind != ASSGN) then return (failure); if S j 's opcode is not ASSGN, fail if (set Table [Si].operand_a.kind!= CONST) then match successful for S i pre_CTP() { Table _Insert(Statement, Sj, all); classify S j as a set of statements Table _Insert(Statement, Sl, no); classify S l as a set of statements Table find and assign flow dep S j If(Set Table [Si].hits ==NULL) then if flow dep S j does not exist try again return (failure); Table foreach(SetTable[Sl].hits { quad_numbers and involved in dependencies { modify one of S j 's operands if(Si.oprc==Sj.orpa) then modify (Sj.opra, Si.oprc); else endif Figure 9. The Generated Code for CTP ________________________________________________________________________ return (failure); generated and the calls to the primitive transformations are placed within the loop. In the example in Figure simply modifies the operand collected in S j . This modification occurs in either the first or second modify statement depending on the operand that carries the dependence. Thus, the first call to modify considers "operand a" of S j for replacement and the second call considers "operand b" for replacement, effectively implementing the pattern matching needed for determining the operand position of a dependence. The procedure act_CTP is called by the driver only if match_CTP and pre_CTP have terminated successfully. For more implementation details, the reader is referred to another paper. 5. Experimentation Using our prototype implementation of Genesis, we performed experiments to demonstrate that Genesis can be used to explore the properties of transformations including 1) the frequency of applying transformations, and 2) the interactions that occur among the transformations. Using Genesis, transformers were produced for ten of the twenty transformations specified: LUR, and FUS. Experimentation was performed using programs found in the HOMPACK test suite and in a numerical analysis test suite. 2 A short description and the Gospel specifications of these transformations are given in Appendix B. HOMPACK consists of FORTRAN programs to solve non-linear equations by the homotopy method. The numerical analysis test suite included programs such as the Fast Fourier Transform and programs to solve non-linear equations using Newton's method. A total of ten programs were used in the experimentation. The benchmark programs were coded in Fortran, which was the language accepted by our front end. They ranged in size from 110 to 900 lines of intermediate code statements. The programs were numerical in nature and had a mixture of loop structures, including nested, adjacent and single loops. Both traditional optimizations and parallelizing transformations could be applied in the programs, as we were interested in the interaction between these types of transformations. Longer programs would more likely show more opportunities for transformations and thus more opportunities for interactions. In order to verify Genesis' capability to find application points, four transformations were specified in Gospel and run on the HOMPACK test suite. The number of application points for each of the transformations was recorded and compared to the number of application points found by Tiny. 20 The comparison revealed that the Genesis found the same number of applications points that Tiny found. Furthermore, seven optimizations were specified in Gospel and optimizers were generated by Genesis. The generated optimizers were compared to a hand-coded optimizer to further verify Genesis' ability to find application points. Again, the optimizers generated by Genesis found the same application points for optimizations. In the test programs, CTP was the most frequently applicable transformation (often enabled) while no application points for ICM were found. It should be noted that the intermediate code did not include address calculations for array accesses, which may introduce opportunities for ICM. CTP was also found to create opportunities to apply a number of other transformations, which is to be expected. Of the total 97 application points for CTP, 13 of these enabled DCE, 5 enabled CFO and 41 enabled LUR (assuming that constant bounds are needed to unroll the loop). CPP occurred in only two programs and did not create opportunities for further transformation. These results are shown in Table 4 where a "-" entry indicates that no interaction is theoretically possible and a number gives the number of interactions that occurred. For example, the entry for INX/FUS indicates that 5 enabling interactions were found and 4 disabling interactions were found in the 13 application points. To investigate the ordering of transformations, we considered the transformations FUS, INX and LUR which we showed in Section 4 to theoretically enable and disable one another. In one program, FUS, INX, and LUR were all applicable and heavily interacted with one another by creating and destroying opportunities for further transformations. For example, applying FUS disabled INX and applying LUR disabled FUS. Different orderings produced different transformed programs. The transformations also interacted when all three transformations were applied; when applying only FUS and INX, one instance of FUS in the program destroyed an opportunity to apply INX. However, when LUR was applied before FUS and INX, INX was not disabled. Thus, users should be aware that applying a transformation at some point in the program may prevent another transformation from being applicable. To further complicate the process of determining the most beneficial ordering, different parts of the program responded differently to the orderings. In one segment of the program, INX disabled FUS, while in another segment INX enabled FUS. Thus, there is not a "right" order of application. The context of the application point is needed. Using the theoretical results of interactions from the formal specifications of transformations as a guide, the user may need multiple passes to discover the series of transformations that would be most fruitful for a given system. The framework could also be used to explore the value of combining transformations. Freq DCE CTP CPP CFO ICM LUR FUS INX LUR FUS 11 -/5 -/0 -/1 1/0 0/6 Table 4. Enabling and Disabling Interactions Blocking is a transformation that combines Strip Mining and Interchange. 11 We performed a preliminary experiment in which we applied various orders of Loop Interchange (INX), Loop Unrolling (LUR) and Loop Fusion (FUS). In the experiments, LUR when followed by INX produced more opportunities for transformations than other orders. Thus, after performing experimentation to examine what happens when a series of transformations are applied, it might be beneficial to combine certain transformations and apply them as a pair. In our example, we would consider combining LUR and INX. 6. Concluding Remarks The code improving transformation framework presented in this paper permits the uniform specification of code improving transformations. The specifications developed can be used for analysis and to automatically generate a transformer. The analysis of transformations enables the examination of properties such as how transformations interact to determine if a transformation creates or destroys conditions for another transformation. These relationships offer one approach for determining an order in which to apply transformations to maximize their effects. The implementation of the Gospel specifications permits the automatic generation of a transformer. Such an automated method enables the user to experimentally investigate properties by rapidly creating prototypes of transformers to test their feasibility on a particular machine. Genesis also permits the user to specify new transformations and quickly implement them. Future work in this research includes examining the possibility of automatically proving interactions by expanding the specifications to a more detailed level. Such a transformation interaction proving tool would enable the user to determine properties of the transformations. Also, the design of a transformation guidance system prototype is being examined for its feasibility. This type of system would aide the user in applying transformations by interactively providing interaction information. The Gospel specifications are also being explored to determine if they can easily be combined to create more useful transformations. Acknowledgment We are especially grateful to TOPLAS Associate Editor Jack Davidson for his insightful criticisms and advice on earlier drafts of this paper. We also thank the anonymous referees for their helpful comments and suggestions, which resulted in an improved presentation of the paper. Appendix A PRECONDITION Grammar for the Gospel Prototype Precon_list Precon_list - Quantifier Code_list : Mem_list Condition_list ; Precon_list | e Quantifier - ANY | NO | ALL Code_list - StmtId StmtId_list Mem_list - Mem_list OR Mem_list Mem_list AND Mem_list Mem - MEM | NO_MEM Condition_list - NOT Condition_list Condition_list AND Condition_list Condition_list Condition_list - Type (StmtId, StmtId Dir_Vect) Type - FLOW_DEP | OUT_DEP | ANTI_DEP | CTRL_DEP Dir_Vect - ( Dir Dir_List ) | e Gospel Specification of Transformations Bumping (BMP): Modify the loop iterations by bumping the index by a preset amount (e.g., 2). DECLARATION PRECONDITION Code_Pattern any L; Depend all S: flow_dep (L.Lcv, S, (any)); ACTION add (S.Prev, ( -, 2, S.opr 1 , S.opr 1 modify (L.Initial, eval(L.Initial, +, 2)); modify (L.Final, eval(L.Final, +, 2)); Constant Folding (CFO): Replace mathematical expressions involving constants with their equivalent value. DECLARATION PRECONDITION Code_Pattern Find a constant expression any const const AND S i .opcode != assign; checks ACTION Fold the constants into an expression modify modify (S i .opcode, assign); Copy Propagation (CPP): Replace the copy of a variable with the original. DECLARATION PRECONDITION Code_Pattern find a copy statement any S i Depend all uses do not have other defs along the path all no no ACTION propagate and delete the copy modify (operand (S j , pos), S i .opr 2 ); delete (S i ); Loop Circulation (CRC):Interchange perfectly nested loops (more than two) DECLARATION PRECONDITION Code_Pattern Find Tightly nested loops any (L 1 , Depend Ensure perfect nesting, no flow_dep with <,> no no ACTION Interchange the loops move move Common Sub-Expression Elimination (CSE): Replace duplicate expressions so that calculations are perfomed only once. DECLARATION PRECONDITION Code_Pattern Find binary operation any S n Depend Find common sub-expression no all ACTION add modify (S n , (assign, S n .opr 1 , temp) modify Dead Code Elimination (DCE): Remove statements that define values for variables that are not used. DECLARATION PRECONDITION Code_Pattern find statement assigning variable, value or expression any S i Depend statement may not be used no ACTION delete the dead code delete (S i ); Loop Fusion Combine loops with the same headers. DECLARATION PRECONDITION Code_Pattern find adjacent loops with equivalent Heads any L 1 , Depend no dependence with backward direction first; no def reaching prior to loops no no ACTION Fuse the loops modify modify delete (L 1 .End); delete (L 2 .Head); Invariant Code Motion (ICM): Remove statements from within loops where the values computed do not change. DECLARATION PRECONDITION Code_Pattern any loop any L; Depend any statement without dependence within the loop any S k : mem (S k , L) AND mem (S m , L), ACTION move statement to within header move (S k , L.Start.Prev); Loop Unrolling (LUR): Duplicate the body of a loop. DECLARATION PRECONDITION Code_Pattern any loop iterated at least once any const AND type (L 1 .Final) == const checks ACTION unroll one iteration, update original loop's Initial modify modify (L 1 .Initial, eval(L 1 .Initial, +, 1)); delete (L 2 .End); delete (L 2 .Head.Label); Parallelization (PAR): Modify loop type for parallelization. DECLARATION PRECONDITION Code_Pattern any Depend no ACTION modify (L 1 .opcode, PAR); Mining (SMI): Modify loop to utilize vector architecture. DECLARATION PRECONDITION Code_Pattern any L: L.Final - L.Initial > SZ; Depend ACTION copy (L.Head, L.Head.Prev, L 2 .Head); modify (L 2 .Lcv, temp(T)); modify (L 2 .step, SZ); modify (L 1 .Initial, T); modify copy (L.End, L.End, L 2 .End); Loop Unswitching (UNS): Modify a loop that contains an IF to an IF that contains a loop. DECLARATION PRECONDITION Code_Pattern any L; Depend any Find the Else any S k : mem (S k , L) AND NOT ctrl_dep(S i , S k ); ACTION copy (L.Head, S k , L 2 .Head); copy (L.End, L.End.Prev.Prev, L 2 .End); modify (L 2 .End, address(L 2 .Head)); move (L.Head, S i ); move (L.End, S k .Prev); --R "Generation of Efficient Interprocedural Analyzers with PAG," Faires, in Numerical Analysis "Global Code Motion Global Value Numbering," "Automatic Generation of Peephole Transfor- mations," "A Flexible Architecture for Building Data Flow Analyzers," "Automatic Generation of Fast Optimizing Code Generators," "Automatic Generation of Machine Specific Code Transformer," GNU C Compiler Manual (V. "Peep - An Architectural Description Driven Peephole Transformer," "Advanced Compiler Transformations for Supercom- puters," "A General Framework for Iteration-Reordering Loop Transformations," Stanford SUIF Compiler Group. "Sharlit - A tool for building transformers," "SPARE: A Development Environment for Program Analysis Algorithms," "Techniques for Integrating Parallelizing Transformations and Compiler Based Scheduling Methods," "An Approach to Ordering Optimizing Transforma- tions," "Investigation of Properties of Code Transformations," "The Design and Implementation of Genesis," "Automatic Generation of Global Optimizers," Tiny: A Loop Restructuring Research Tool in High Performance Compilers for Parallel Computing --TR Advanced compiler optimizations for supercomputers Automatic generation of fast optimizing code generators An approach to ordering optimizing transformations Automatic generation of global optimizers SharlitMYAMPERSANDmdash;a tool for building optimizers A general framework for iteration-reordering loop transformations Techniques for integrating parallelizing transformations and compiler-based scheduling methods The design and implementation of Genesis Global code motion/global value numbering A flexible architecture for building data flow analyzers Peep Automatic generation of peephole optimizations Automatic generation of machine specific code optimizers Generation of Efficient Interprocedural Analyzers with PAG --CTR Prasad A. 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code-improving transformations;enabling and disabling of optimizations;automatic generation of optimizers;specification of program optimizations;parallelizing transformations

268184

The T-Ruby Design System.

This paper describes the T-Ruby system for designing VLSI circuits, starting from formal specifications in which they are described in terms of relational abstractions of their behaviour. The design process involves correctness-preserving transformations based on proved equivalences between relations, together with the addition of constraints. A class of implementable relations is defined. The tool enables such relations to be simulated or translated into a circuit description in VHDL. The design process is illustrated by the derivation of a circuit for 2-dimensional convolution.

Introduction This paper describes a computer-based system, known as T-Ruby [12], for designing VLSI circuits starting from a high-level, mathematical specification of their behaviour: A circuit is described by a binary relation between appropriate, possibly complex domains of values, and simple relations can be composed into more complex ones by the use of a variety of combining forms which are higher-order functions. The basic relations and their combining forms generate an algebra, which defines equivalences (which may take the form of equalities or conditional equalities) between relational expressions. In terms of circuits, each such equivalence describes a general correctness preserving transformation for a whole family of circuits of a particular form. In the design process, these equivalences are exploited to transform a "specification" in the form of one Ruby expression to an "implementation" in the form of another Ruby expression, in a calculation-oriented style [4, 9, 13]. T-Ruby is based on a formalisation of Ruby, originally introduced by Jones and Sheeran [3], as a language of functions and relations, which we refer to as the T-Ruby language. The purpose of the paper is to demonstrate how such a general language can be used to bridge the gap between a purely mathematical specification and the implementable circuit. The design of a circuit for 2-dimensional convolution is used to illustrate some of the features of the method, in particular that the step from a given mathematical specification to the initial Ruby description is small and obvious, and that the method allows us to derive generic circuits where the choice of details can be postponed until the final actual synthesis. The T-Ruby system enables the user to perform the desired transformations in the course of a design, to simulate the behaviour of the resulting relation and to translate the final Ruby description of the relation into a VHDL description of the corresponding circuit for subsequent synthesis by a high-level synthesis tool. The transformational style of design ensures the correctness of the final circuit with respect to the initial specification, assuming that the equivalences used are correct. Proofs of correctness are performed with the help of a separate theorem prover, which has a simple interface to T-Ruby, so that proof burdens can be passed to the prover and proved equivalences passed back for inclusion in T-Ruby's database. The division of the system into the main T-Ruby-system, a theorem prover and a VHDL translator has followed a "divide and conquer" philosophy. Theorem proving can be very tedious and often needs specialists. In our system the designer can use the proved transformation rules in the computationally relatively cheap T-Ruby system, leaving proofs of specific rules and conditions to the theorem prover. When a certain level of concretisaion is reached efficient tools already exist to synthesise circuits. Therefore we have chosen to translate our relational descriptions into VHDL. Ruby The work described in this paper is based on the so-called Pure Ruby subset of Ruby, as introduced by Rossen [10]. This makes use of the observation that a very large class of the relations which are useful for describing VLSI circuits can be expressed in terms of four basic elements: two relations and two combining forms. These are usually defined in terms of synchronous streams of data as shown in Figure 1. In the Figure 1: The basic elements of Pure Ruby figure, the type sig(T ) is the type of streams of values of type T , usually represented as a function of type Z ! T , where we identify Z with the time. The notation aRb means that a is related to b by R, and is synonymous with (a; b) 2 R. The relation spread f is the lifting to streams of the pointwise relation R of type ff - fi, whose characteristic function is f (of type ff such that (f a b) is true R. The type of spread f is then sig(ff) - sig(fi), the type of relations between streams of type ff and streams of type fi. For notational convenience, and to stress the idea that it describes the lifting to streams of a pointwise relation of type ff - fi, this type will be denoted ff sig Thus spread f , for suitable f , describes (any) synchronously clocked combinational circuit, while the relation D - the so-called delay element - describes the basic sequential circuit. F ; G (the backward relational composition of F with G) describes the serial composition of the circuit described by F with that described by G . If F is of type ff sig - fl and G is of type fl sig fi, then this is of type ff sig - fi. Finally, [F ; G ] (the relational product of F and G) describes the parallel composition of F and G . For F of type sig sig this is of type (ff 1 \Theta ff 2 ) sig The types of the relations describe the types of the signals passing through the interface between the circuit and its environment. However, it is important to note that the relational description does not specify the direction in which data passes through the interface. "Input" and "output" can be mixed in both the domain and the range. R ff fi (a) (b) (c) F G (d) Figure 2: Graphical interpretations of: A feature of Ruby is that relations and combinators not only have an interpretation in terms of circuit elements, but also have a natural graphical interpretation, corresponding to an abstract floorplan for the circuits which they describe. The conventional graphical interpretation of spread (or, in fact, of any other circuit whose internal details we do not wish to show) is as a labelled rectangular box. The components of the domain and range are drawn as wire stubs, whose number reflects the types of the relations in an obvious manner: a simple type gives a single stub, a pair type two and so on. The components of the domain are drawn up the left hand side and the components of the range up the right. The conventional graphical interpretation for D is as a D-shaped figure, with the domain up the flat side and the range up the rounded side, while F ; G is drawn with the range of F "plugged into" the domain of G , and [F with the two circuits F and G in parallel (unconnected). These conventions are illustrated in Figure 2. For further details, see [3]. 3 The T-Ruby Language In T-Ruby all circuits and combinators are defined in terms of the four Pure Ruby elements using a syntax in the style of the typed lambda calculus. Definitions of some circuits and combinators with their types are given in Figure 3. ff, fi and so on denote type variables and can thus stand for any type. The first five definitions are of non-parameterised stream relations, which correspond to circuits. +, defined using the spread element applied to a function which evaluates to true when z equals the sum of x and y , pointwise relates two integers to their sum. ' is the (polymorphic) identity relation, dub pointwise relates a value to a pair of copies of that value and reorg pointwise relates two ways of grouping three values into pairs. These all describe combinational circuits; all except just describe patterns of wiring, and are known as wiring relations. The fifth, SUMspec, describes a simple sequential circuit : an adding machine with an accumulator register. The remaining definitions are examples of combinators, which always have one or more parameters, typically describing the circuits to be combined. Applying a combinator to suitable arguments gives a circuit. Thus (Fst R) is the circuit described by [R; '], R l S (where the combinator l is written as an infix operator) is the circuit where R is below S and the second component in the domain of R is connected to the first component of the range of S . In the definition of l (and elsewhere), R -1 denotes the inverse relation to R. The graphical interpretations of Fst and l are shown in Figure 4. The dialect of Ruby used in the T-Ruby system is essentially that given by Rossen in [8]. This differs from the standard version of Ruby given by Jones and Sheeran in [3] in that "repetitive" combinators and wiring relations are parameterised in the number of repetitions. This is reflected in the type system which includes dependent product types [5], a generalisation of normal function types. This enable us explicitly to express the size of repetive structures in the type system. For example, the combinator map (which "maps" a relation over all elements in a list of streams) has the polymorphic dependent type: where nlist[n]T is the type of lists of exactly n elements of type T . Thus map is a function which takes an integer, n, and a relation of type ff sig - fi as arguments and int \Theta int sig int ': ff sig dub: ff sig dub 4 reorg: ((ff \Theta fi) \Theta fl) sig (int \Theta control) sig (bool \Theta int) loop4 (Fst (Snd D) ; ALU a ; (Snd dub)) Fst sig Fst 4 sig l: (ff \Theta fi) sig sig sig l 4 Fst (R) sig sig sig sig sig else Fst (apr n-1 sig sig sig Figure 3: Examples of circuit and combinator definitions in T-Ruby. Fst (R) R l S map 3 R tri 3 R colf 3 R rdrf 3 R R R R R R R Figure 4: Graphical interpretation of some combinators returns a relation whose type, nlist[n]ff sig - nlist[n]fi, is dependent on n, the so-called \Pi -bound variable. A full description of the T-Ruby type system can be found in [11]. The relation apr n , used in the definition of map, pointwise relates an n-list of values and a single value to the (n 1)-list where the single value is appended "on the right" of the n-list 1 . The combinator mapf is similar to map but the second parameter is a function from integers to relations, so that the relation used can depend on its position in the structure. creates a triangular circuit structure, colf a column structure where each relation is parametrised in its position in the column. Finally rdrf , called "reduce right", is a kind of column structure. It has its name from functional programming and will, if for example used with the relation + as argument, calculate the sum of a list of integers. The graphical interpretations of some of these repetitive combinators are shown in Figure 4. Note that the definitions are all given in a point-free notation, reflecting the fact that they are all expressed in terms of the elements of Pure Ruby. It is easy to show that they are equivalent to the expected definitions using data values; for example, However, defining circuits in terms of Pure Ruby elements offers several advantages: it greatly simplifies the definition and use of general rewrite rules; it simplifies reasoning about circuits in a theorem prover; and it eases the task of translating the language into a more traditional VLSI specification language such as VHDL. 4 The Transformational Phase of T-Ruby Design The design process in T-Ruby involves three main activities, reflecting the overall design of the system: (1) Transformation, (2) Proof and (3) Translation to VHDL. In this section we consider the first phase, which involves transforming an initial specification by rewriting, possibly with the addition of typing or timing constraints, so as to approach an implementable design described as a Ruby relation. 1 Note that the size argument n here, as elsewhere, is written as a subscript to improve readibility. 4.1 Rewriting Rewriting is an essential feature of the calculational style of design which is used in Ruby. The T-Ruby system allows the user to rewrite Ruby expressions according to pre-defined rewrite rules. Rewriting takes place in an interactive manner directed by the user, using basic rewrite functions, known as tactics, which can be combined by the use of higher order functions known as tacticals. This style of system is often called a transformation system to distinguish it from a conventional rewrite system. T-Ruby is implemented in the functional programming language Standard ML (SML), which offers an interactive user environment, and the tactics and tacticals are all SML functions, applied in this environment. In the T-Ruby system, a rewrite rule is an expression with the form of an equality or an implication between two equalities, with explicit, typed universal quantification over term variables and in most cases implicit universal quantification over types via the use of type variables. Apart from this there are no restrictions on the forms of the rules which may be used. In practice, however, the commonly used rules are equalities between relational expressions, corresponding to equivalences between circuits, which can be used to manipulate a circuit description in Ruby to another, equivalent form. Rules for manipulating integer or Boolean expressions could, of course, also be introduced, but most such manipulations are performed automatically by a built-in expression simplifier based on traditional rewriting to a normal form. Some examples of rules can be seen in Figure 5. The first rules express simple facts about the combinators, such as the commutativity of Fst and Snd (fstsndcomm), the fact that the inverse of a serial composition is the backward composition of the inverses (inversecomp), and the distributivity of Fst over serial composition (fstcompdist). The fourth rule, maptricomm, is an example of a conditional rule: the precondition that R and S commute over serial composition must be fulfilled in order for tri n R and map n S to commute. Similarly, forkmap states that if R is a functional relation then a single copy on the domain side of an n-way fork is equivalent to n copies on the range side of the fork. Finally, rules such as retimecol, are used in Ruby synthesis to express timing features, such as the input-output equivalence of a circuit to a systolic version of the same circuit. Note that since all these rules contain universal quantifications over relations of particular types, they essentially express general properties of whole families of circuits. In the T-Ruby system, the directed rules used for rewriting come from three sources. They may be explicit rewrite rule definitions, implicit definitions derived from circuit or combinator definitions (which permit the named circuit or combinator to be replaced by its definition or vice-versa), or lemmata derived from previous rewrite processes which established the equality of two expressions, say t and t 0 . In T-Ruby, the correctness of the explicit rules is proved by the use of a tool [7] based on the Isabelle theorem prover [6], using an axiomatisation of Ruby within ZF set theory. To make life easier for the user, conjectured rewrite rules can, however, be entered without having been proved. When rewriting is finished, all such unproved rewrite rules are printed out. Together with any instantiated conditions from the conditional rules, they form a proof obligation which the user must transfer to the theorem prover, in order to ensure the soundness of the rewriting process. fstsndcomm= sig - sig sig - sig (R fstcompdist 4 sig - sig Fst (R sig - sig (R sig (R retimecol= sig (col Figure 5: Rewrite rules in T-Ruby. 4.2 Constraints The transformation process in the T-Ruby system primarily involves rewriting expressions as described above. However, rewriting can only produce relations which are exactly equivalent to the original, abstract specification. These relations are often too large and have to be restricted to obtain an implementable circuit. As a trivial exam- ple, the relation described above is defined as being of type (int \Theta int) sig - int. For implementation purposes we want to restrict the integers to values representable by a finite number of bits. From a mathematical point of view this means restricting our relation to the subtype given by: - int n+1 where int n is a subtype of integers representable by n bits. In T-Ruby we describe this subtyping by adding relational constraints [13] to the expression. The initial specifi- Bint n int n O n-bit n-bit (n+1)-bit int n+1 int int int int n int n+1 (b) (a) id n+1 int n (c) Bits Figure Adding constraints to a relation cation of + is depicted in Figure 6(a). We narrow the type of + by adding relational constraints to the domain and the range, in this case instantantions of id n , the identity relation for integers representable by n bits. This can be defined by: id n= (Bits n ; Bits n where Bits n relates an integer to a list of n bits which represent it. The constrained relation is: as shown in Figure 6(b). The definition of id n can then be expanded, giving: [(Bits as shown in Figure 6(c), and the relations Bits n can be manipulated into the original relation by using the general rewrite rules. Another style of constraint is to add delay elements, D, to the domain or range of the relation. Since Ruby relations relate streams of data, where each element in the stream corresponds to a specific time instant, this changes the timing properties of the circuit. As a simple example, the relation + defined in Figure 3 describes a purely combinational circuit. Adding n delay elements on the range side would give a specification of an adder with a total delay of n time units. Now we can use the same general relational rewrite rules to "push" the delay elements into the combinational part, thus obtaining a description of an adder with the same external timing properties, but with a different internal arrangement of the registers. This style of manipulation is illustrated in more detail by the convolution example in the following section. Note that the same relational framework is used to describe and manipulate both type and timing constraints. An interesting variation is to mix the two methods above. For example, by adding a relational specification of a bit-serial to integer converter as a constraint on the domain side of + and its inverse on the range side, we obtain a specification of a bit-serial adder and can manipulate it to get an implementable circuit as above. Finally, the specification can be constrained by instantiation of free type or term variables. Free term variables are typically used to describe otherwise unspecified circuit elements or to give the size of regular structures in a generic manner. By instantiation we obtain a description of a more specialised circuit with a particular circuit element or particular dimensions. In general the transformation process starts from a relational specification, spec, of a circuit, at some suitably high level of abstraction. spec is then rewritten by a number of equality rewrites in order to reach a more implementable description. During the rewrite process the relation can be narrowed by adding relational constraints. The process can be illustrated by a series of transformations: where the primes denote the added constraints. The original specification is changed accordingly from spec to spec 00:::0 , reflecting the addition of the constraints and ensuring equality between impl and the final constrained specification. From a logical point of view [15], the constraints can be regarded as the assumptions under which the implementation fulfills the original specification: constraints ' (impl , spec) 4.3 An Example: 2-dimensional Convolution As an example of the tranformation process, we present part of the design of a VLSI circuit for 2-dimensional discrete convolution. The mathematical definition is that from a known as the convolution kernel, and a stream of values a, a new stream of values c should be evaluated, such that: +r +r The intuition behind this is that the stream a represents a sequence of rows of length w , and that each value in c is a weighted sum over the corresponding value in the a-stream and its "neighbours" out to distance \Sigmar in two dimensions, using the weights given by the matrix K. This is commonly used in image processing, where a is a stream of pixel values scanned row-wise from a sequence of images, and K describes some kind of smoothing or weighting function. Note that for each i , the summation over j is equal to the 1-dimensional convolution of a with the i 'th row of K with a time offset of w 1-dimensional convolution is defined by: +r 4.3.1 Formulating the problem in Ruby The first step in the design process is to formulate the mathematical definitions in Ruby. Following the style of design used for a correlator in [3], we now divide the relation between a and c into a combinational part, which relates c-values at a given time to a 0 -values at the same time (for convenience we let the summation run from 1 applying the substitution i new and a temporal part which relates the a 0 values at time t to the original a-values: a 0 The temporal part, the matrix a 0 , can be further split into parts which can easily be specified directly in Ruby. First we for a given i find a relation which relates b i to a 1)-list of a 0 1. An offset dependent on the position j , such that a 0 which in Ruby can be specified by stating that (a 00 ; a 0 ) are related by (tri 2r+1 D 2. A (2r 1)-way fork, such that a 00 specified by (a 000 ; a 00 3. A fixed offset, such that a 000 specified by (b 4. Assembling 1-3 we get: (b 0 Next we find a relation relating a to a (2r + 1)-list of b i 's: 1. An offset dependent on position i , such that b 0 specified by (b 2. A (2r 1)-way fork, such that b 00 specified by (b 3. Another fixed offset, such that b 000 specified by (a; b 000 4. Assembling 1-3 we get: (a; [b 0 It is convenient to rewrite the two relations above (the two (4)) as follows: (b 0 butterfly 2r+1 D) (5) (a; [b 0 butterfly 2r+1 D w where the combinator butterfly is defined by: sig (app n+1;n Fst (irt n+1 R The combinational part of the convolution relation is easily expressed in Ruby in terms of a combinator Q , of type int ((int \Theta int) sig - int), such that (Q relates (a; x ) to (K ij a + x ), which expresses the convolution kernel as a function of position within the matrix K. If we then define c i (t), it is easy to demonstrate that, for all t and arbitrary x , ([a 0 (t)]; x (t)) is related to c i (t) by the Ruby relation (rdrf 2r+1 (Q i )), where rdrf defined in Figure 3. Combining this with the temporal relations given in definitions 5 and 6 we find that the entire 2-dimensional convolution relation CR 2 , which relates (a; x ) and a given w , r , x and Q can be expressed in terms of the one-dimensional convolution relation (CR 1 i ), which relates (b 0 ((int \Theta int) sig butterfly r D) ; rdrf 2r+1 (Q i (int \Theta int) sig int CR 2= Fst (fork butterfly r D w CR 1 corresponds to the inner summation over j in the specification. The graphical interpretation of CR 2 for is shown on the left in Figure 7, and the interpretation of (CR on the right. The butterflies contain increasing numbers of delay elements, D, above the mid-line and increasing numbers of "anti-delay" elements, D -1 below the mid-line. As follows from the definitions, the small butterflies use single delay elements, corresponding to the time difference between consecutive elements in the data stream, while the large butterflies use groups of w delay elements, corresponding to the time difference between consecutive lines in the data stream. To define these relations in T-Ruby, it is convenient to parameterise them, so that they become combinators dependent on r , w and Q . The final definitions are: (int \Theta int) sig (Fst (fork butterfly r D) ; rdrf 2r+1 Q) (int \Theta int) sig (Fst (fork butterfly r (D w With these definitions, the actual circuit for 2-dimensional convolution is described by the relation (conv2 r w Q) for suitable values of r , w and Q . Figure 7: Two dimensional convolution for 4.3.2 Transformation to an implementable relation Unfortunately, the relation given above does not describe a physically implementable circuit, if we assume (as we implicitly have done until now) that the inputs appear in the domain of the relation (as x and a) and the outputs in the range (as c). This is because of the "anti-delays", D -1 , in the butterflies. So instead of trying to implement the relation (conv2 r w Q) as it stands, we implement a retimed version of it, formed by adding a constraint on the domain side which delays all the input signals: Fst (D r ; (D w This will result in the anti-delays being cancelled out, as the delay elements in the constraint are moved "inwards" into the original relation. The resulting circuit will, of course, produce its outputs r units later than the original circuit, but this is the best we can achieve in the physical world we live in! From here on we use a series of rewrite rules to manipulate the relation into a more obviously implementable form. The output from the T-Ruby system during this derivation is shown, in an annotated and somewhat abbreviated form, in Figure 8. In the concrete syntax produced by the T-Ruby prettyprinter, free variables are preceded by a %-sign, repeated composition R n is denoted by R-n, and relational inverse R -1 by R~, while "x:t.b denotes a -expression variable x of type t and body b. The derivation finishes with the relational expression: Fst (fork 2r+1 This is a generic description of a convolution circuit, expressed in terms of three free variables: r and w , corresponding respectively to the kernel size for the convolution and the line size for the 2-dimensional array of points to be convoluted, and Q , which gives the kernel function. To obtain a description of a particular concrete circuit, we can then use the Ruby system's facilities for instantiating such free variables to particular values. For definition of conv2.g (Fst (D-%r;D-%w-%r));((Fst ((fork (2*%r+1));(butterfly %r D-%w))); 2. fstcompdist (used from right to left).g (Fst ((D-%r;D-%w-%r);((fork (2*%r+1));(butterfly %r D-%w)))); use rule forkmap.g (Fst (((fork (2*%r+1));(map (2*%r+1) (D-%r;D-%w-%r)));(butterfly %r D-%w))); butterfly (Fst ((fork (2*%r+1));((map (2*%r+1) D-%r);(tri (2*%r+1) D-%w)))); fRule maptricomm, and then rule fstcompdist.g ((Fst (fork (2*%r+1)));((Fst (tri (2*%r+1) D-%w));(Fst (map (2*%r+1) D-%r)))); (Fst (fork (2*%r+1)));((Fst (tri (2*%r+1) D-%w)); "k:int.(((Fst D-%r); "i:int.(conv1 %r (%Q i)) k)))) definition of conv1.g (Fst (fork (2*%r+1)));((Fst (tri (2*%r+1) D-%w)); "k:int.(((Fst D-%r);((Fst ((fork (2*%r+1));(butterfly %r D))); fNow use a similar procedure to the above to remove the remaining butterfly.g (Fst (fork (2*%r+1)));((Fst (tri (2*%r+1) D-%w)); "k:int.((((Fst (fork (2*%r+1)));(Fst (tri (2*%r+1) D))); using definition of Fst , then use [tri n D; (Fst (fork (2*%r+1)));((Fst (tri (2*%r+1) D-%w)); "k:int.((((Fst (fork (2*%r+1)));((Snd D~-(2*%r+1)); (Fst (fork (2*%r+1)));((Fst (tri (2*%r+1) D-%w)); "k:int.(((Fst (fork (2*%r+1)));((Snd D~-(2*%r+1)); another constraint on the domain side: Snd (D w (Snd D-%w-(2*%r+1));((Fst (fork (2*%r+1)));((Fst (tri (2*%r+1) D-%w)); "k:int.(((Fst (fork (2*%r+1)));((Snd D~-(2*%r+1)); fRule fstsndcomm, and Fst R ; ((Fst (fork (2*%r+1)));[(tri (2*%r+1) D-%w),D-%w-(2*%r+1)]); "k:int.((((Fst (fork (2*%r+1)));(Snd D~-(2*%r+1))); (Fst (fork (2*%r+1))); (rdrf (2*%r+1) "k:int.(((Snd D-%w);((Fst (fork (2*%r+1)));((Snd D~-(2*%r+1)); (Fst (fork (2*%r+1))); "k:int.((((Fst (fork (2*%r+1)));(Snd (D-%w;D~-(2*%r+1)))); \Gamman .g (Fst (fork (2*%r+1))); "k:int.((((Fst (fork (2*%r+1)));(Snd D-(%w -(2*%r+1)))); Figure 8: Derivation of the 2-dimensional convolution relation example, we might instantiate r to 2, w to 64 and Q to - acc 4 describes a multiply-and-add circuit with multiplication factor (i and the kernel element described by (Q use this factor as the weight in accumulating the weighted sum. After suitable reduction of the integer expressions, this would give us the relational description: Fst (fork 5 with no free variables. The graphical interpretation of this final version of the circuit is shown in Figure 9. As can be seen in the figure, the circuit is semi-systolic, with a latchc a z s Figure 9: Semi-systolic version of two dimensional convolution for 2. The left-hand structure depicts the entire circuit. The basic building element shown on the right corresponds to the relation Snd D ; (Q k p) with Q instantiated as described in the text. The middle structure depicts Snd D Only 3 of the 59 delay elements in D 59 are shown. Arrows in the figure indicate the input/output partitioning determined by the causality analysis. (described by a delay element, D) associated with each combinational element, but with a global distribution of the input stream a to all of the combinational elements. 4.4 Selection and Extraction The rewriting system of T-Ruby includes facilities for selection of subterms from the target expression by matching against a pattern with free variables. This can be used to restrict rewriting temporarily to a particular subterm, or, more importantly, for extraction of part of the target expression for implementation. In the latter case, the remainder of the target expression gives a context describing a set of implementation conditions that must be fulfilled for the extracted part to work Extraction is in many respects the converse of adding relational constraints to the specification, and the context specifies the same sorts of requirement. Firstly, it may give representation rules which must be obeyed at the interface to the extracted subterm, and secondly (if the context contains delay elements, D), it may give timing requirements for the implementation of the subterm. For the trivial example of the adder above the extracted part will typically, after some rewriting, be the circuit inside the dashed box on Figure 6(c). The implementation conditions in this case express the fact that integers must be represented by n bits, as specified by Bits n . 5 VLSI Implementation The relational approach to describing VLSI circuits offers a greater degree of abstraction than descriptions using functions alone, since the direction of data flow is not specified. However, real circuits offer particular patterns of data flow, and this means that the interpretation of a relation may in general be 0, 1 or many different circuits. In the case of zero circuits, we say the relation is unimplementable. The widest class of relations which are generally implementable is believed to be the causal relations, as defined by Hutton [2]. These generalise functional relations in the sense that inputs are not restricted to the domain nor outputs to the range. In T-Ruby, causality analysis is performed at the end of the rewriting process, when the user has extracted the part of the relation which is to be implemented. In most cases, in fact, the context from which the relation is extracted is non-implementable: for example, it may specify timing requirements which (if they could be implemented) would correspond to foreseeing the future. 5.1 Causality analysis More exactly, a relation is causal if the elements in each tuple of values in the relation can be partitioned into two classes, such that the first class (the outputs) are functionally determined by the second class (the inputs), and such that the same partitioning and functional dependency are used for all tuples in the relation. For example, the previously defined relation + is causal, in the sense that the three elements of each tuple of values in the relation can be partitioned as described, in fact in three different ways: 1. With x and y as inputs and z as output, so that the relation describes an adder . 2. With x and z as inputs and y as output, so that the relation describes a subtractor . 3. With y and z as inputs and x as output, so that the relation describes another subtractor . Note that the relation + -1 is also causal, although it is not functional. Essentially, causality means that the relation can be viewed under the partitioning as a deterministic function of its inputs. In T-Ruby, the relation to be analysed is first expanded, using the definitions of its component relations, to a form where it is expressed entirely in terms of the four elements of Pure Ruby and relational inverse. The expanded relation is then analysed with a simple bottom-up analysis heuristic. For combinational elements described by spread relations, causality is determined by analysing the body of the spread , which must have the form of a body part which is: ffl an equality with a single variable on the left-hand side, ffl a conjunction of body parts, or ffl a conditional choice between two body parts. In each equality, the result of the analysis depends on the form of the right-hand side. If this is a single variable, no conclusions are drawn, as the equality then just implies a wire in the abstract floorplan. If the right-hand side is an expression, all values in it are taken to be inputs, and the left-hand side is taken to be an output. In choices, all values in the condition are taken to be inputs. If these rules result in conflicts, no causal partitioning can be found. When there are several possible causal partitionings, as in the case of +, on the other hand, the rules enable us to choose a unique one. For delay elements, D, values in the domain are inputs and those in the range are outputs. Parallel composition preserves causality, and so in fact does inversion, but serial compositions in general require further analysis, to determine whether the input/output partitionings for the component relations are compatible with an implementable (unidirectional) data flow between the components. Essentially, checks are made as to whether two or more outputs are used to assign a new signal value to the same wire, whether some wires are not assigned signal values at all or whether there are loops containing purely combinational components. This additional analysis is exploited in order to determine the network of the circuit in the form of a netlist with named wires between active components. At present there is no backtracking, so if the arbitrary choice of partitioning when there are several possibilities is the "wrong" one, then it will not be possible to find a complete causal partitioning for the entire circuit. As an example, let us consider the analysis of parts of the relation for 2-dimensional convolution. The central element in this is the relation given by acc(p describes the combinational multiply-and-add circuit for kernel element (p; k ). Using the definition of acc, and substituting (p reduces to: spread The body of the spread has the form of an equality with a single left-hand side, and thus the causal partitioning will make z an output and m and s both inputs. In this case, the relation is functional from domain to range, but in general this need not be so. Since delay elements, D, can only have inputs on the domain side and outputs on the range side, the serial composition (Snd D compatible with this analysis of acc(p as the range of the delay element corresponds to component s in the domain of acc(p Further analysis proceeds in a similar manner, leading to the final data flow pattern shown by arrows in Figure 9. 5.2 Translation to VHDL Since causality analysis gives both the network of the circuit and the direction of data flow along the individual wires between components, the actual translation to VHDL is comparatively simple. Each translated "top level" Ruby relation is declared as a single design unit, incorporating a single entity with a name specified by the user. In rough terms, each combinational relation C which is not a wiring relation within the expanded Ruby relation is translated into one or more possibly conditional signal assignments, where the outputs of C are assigned new values based on the inputs. For example, the above gives rise to a single concurrent signal assignment of the form: where sig z, sig s and sig m are the names of the VHDL signals corresponding to z , s and m respectively, and W is a constant equal to the value of (p for the circuit element in question. Since the operators available for use with operands of integer, Boolean, bit and character types in Ruby are (with one simple exception: logical implication) a subset of those available in VHDL, this direct style of translation is problem-free. In a similar manner, any conditional (if-then-else) expressions in the body of a spread are directly translated into conditional assignment statements, possibly with extra signal assignments to evaluate a single signal giving the condition. The VHDL types for the signals involved are derived from the Ruby types used in the domain and range of C in an obvious way. Thus for the elementary types, the Ruby type bit is translated to the VHDL type rubybit, bool to rubybool, int to rubyint, and char to rubychar, where the VHDL definitions of rubybit, rubybool, rubyint and rubychar are predefined in a package RUBYDEF, which is referred to by all generated VHDL units. Composed types give rise to groups of signals, generated by (possibly recursive) flattening of the Ruby type, such that a pair is flattened into its two components, a list into its n components and so on until elementary types are reached. If the Ruby relation refers to elementary types other than these pre-defined ones, a package declaration containing suitable type definitions is generated by the translator. For example, if an enumerated type etyp is used, a definition of a VHDL enumerated type rubyetyp with the same named elements is generated. Free variables of relational type and all non-combinational relations in the Ruby relation are translated into instantiations of one or more VHDL components. For example, a Delay relation, D. of Ruby type t sig is a simple type, will be translated into an instantiation of the component dff rubyt, where rubyt is the type corresponding to t , as above. For composed types, such as pairs and lists, two or more components, each of the appropriate simple type, are used. Standard definitions of these components for all standard simple Ruby types are available in a library. Other components (in particular those generated from free relational variables) are assumed to be defined by the user. The final result of translating the fully instantiated 2-dimensional convolution relation into VHDL is shown in Figure 10. The figure does not show the entire VHDL code (which of course is very repetitive owing to the regular nature of the circuit), but illustrates the style. Signal identifiers starting with input and output correspond generated code. Do not edit. Compiled 950201, 11:58:28 from Ruby relation: -%% ((Fst (fork 5)); -%% (rdrf 5 "k:int. -%% ((((Fst (fork 5));(Snd D-59)); ENTITY conv264 IS PORT END conv264; ARCHITECTURE ruby OF conv264 IS COMPONENT dff rubyint PORT END COMPONENT; SIGNAL wire4546,wire4548,wire4550,wire4552,wire4554,wire4556,wire4558, wire19943,wire20128,wire20273,wire20378,wire20478,wire20596, wire20746,wire20928,wire21142: rubyint; BEGIN - Input assignments: - output1 != wire1983; Calculations: - Registers: - D3: dff rubyint PORT MAP (wire20128,wire20746,clk); D4: dff rubyint PORT MAP (wire20273,wire20596,clk); D5: dff rubyint PORT MAP (wire20378,wire20478,clk); D7: dff rubyint PORT MAP (wire19730,wire19732,clk); D319: dff rubyint PORT MAP (wire4546,wire4548,clk); D320: dff rubyint PORT MAP (wire2540,wire4546,clk); Figure 10: VHDL translation of the instantiated 2-dimensional convolution circuit. to the external inputs and outputs mentioned in the formal port clause of the entity, while names starting with wire identify internal signals. A clock input is generated if any of the underlying entities are sequential. The assignments marked Calculations describe the combinational components, and those marked Registers describe the component instantiations corresponding to the Delay elements. (Instantiations of any other user-defined components follow in a separate section if required.) The correctness of the translation relies heavily on two facts: 1. There is a simple mapping between Ruby types and operators and types and operators which are available in VHDL. 2. Relations are only considered translatable if an (internally consistent) causal partitioning can be found. These facts also imply that the VHDL code which is generated can be synthesised into VLSI. At present, we use the Synopsys VHDL Compiler [14] for performing this synthesis automatically. 5.3 Other Components of the System The complete system is illustrated in Figure 11. A similar style of analysis to that used for generating VHDL code is used for controlling simulation of the behaviour of the extracted relation. The user must supply a stream of values for the inputs of the circuit and, if required, initial values for the latches, and the simulation then uses exactly the same assignments of new values to signals as appear in the VHDL description. Obviously, only fully instantiated causal relations can be simulated. SYSTEM Data flow Proof burdens SILICON EQUIVALENCES PROVED PARSER/PRINTER RUBY expressions (internal representation) RUBY terms SYSTEM ANALYSIS Dynamic behaviour Figure 11: The complete Ruby Design System. 6 CONCLUSION 19 6 Conclusion In this paper, we have presented the T-Ruby Design System and outlined a general design method for VLSI circuits based on transformation of formal specifications using equality rewriting, constraints and extraction. The simple mathematical basis of the specification language in terms of functions and relations enables us to prove general transformation rules, and minimises the step from the mathematical description of the problem to the initial specification in our system. The use of the system has been illustrated by the non-trivial example of a circuit for 2-dimensional convolution. This example shows how T-Ruby can be used to describe complex repetitive structures which are useful in VLSI design, and demonstrates how the system can be used to derive descriptions of highly generic circuits, from which concrete circuit descriptions can be obtained by instantiation of free parameters. Circuits described by so-called causal relations can be implemented and their behaviour simulated. In the T-Ruby system, a simple mapping from T-Ruby to VHDL for such relations is used to produce a VHDL description for final synthesis. The design system basically relies on the existence of a large database of pre- proved transformation rules. However, during the design process, conjectured rules can be introduced at any time, and rewrite rules with pre-conditions may be used. In T-Ruby, proofs of conjectures and pre-conditions can be postponed without any loss of formality, as the system keeps track of the relevant proof burdens and these can be transferred later to a separate theorem prover. Our belief is that this "divide and ocnquer" philosophy helps to make the use of formal methods more feasible for practical designs. Acknowledgements The work described in this paper has been partially supported by the Danish Technical Research Council. The authors would like to thank Lars Rossen for many interesting discussions about constructing tools for Ruby, and Ole Sandum for his work on the design of the Ruby to VHDL translator. --R A framework for defining logics. Between Functions and Relations in Calculating Programs. Circuit design in Ruby. Relations and refinement in circuit design. A Generic Theorem Prover A Ruby proof system. Formal Ruby. Ruby algebra. The Ruby framework. T-Ruby: A tool for handling Ruby expressions. Transformational rewriting with Ruby. Constraints, abstraction and verification. --TR

hardware description languages;relational specification;synchronous circuit design;correctness-preserving transformations

268186

Bounded Delay Timing Analysis of a Class of CSP Programs.

We describe an algebraic technique for performing timing analysis of a class of asynchronous circuits described as CSP programs (including Martins probe operator) with the restrictions that there is no OR-causality and that guard selection is either completely free or mutually exclusive. Such a description is transformed into a safe Petri net with interval time delays specified on the places of the net. The timing analysis we perform determines the extreme separation in time between two communication actions of the CSP program for all possible timed executions of the system. We formally define this problem, propose an algorithm for its solution, and demonstrate polynomial running time on a non-trivial parameterized example. Petri nets with 3000 nodes and 10^16 reachable states have been analyzed using these techniques.

Introduction There has been much work in the past decade on the synthesis of speed-independent (quasi-delay- insensitive) circuits. What we develop in this paper are basic results that allow designers to reason about, and thus synthesize, non-speed-independent or timed circuits. Whether designing timed asynchronous circuits is a good idea can be debated ad infinitum. In any event, designers have been applying "seat of the pants" techniques to design timed circuits for years. Our work can be used to verify such designs. Our vision, however, is much more broad, and includes a complete synthesis methodology for developing robust and high-performance timed designs. A description of such a methodology is beyond the scope of this paper but is a major motivation to addressing this difficult timing analysis problem. An asynchronous circuit is specified using CSP as a set of concurrent processes. This description is transformed into a safe Petri net which is the input to the timing analysis algorithm. The analysis determines the extreme case separation in time between two communication actions in the CSP specification over all timed executions. Determining tight bounds on separation times between communication actions (system events) provides information which can be used to answer many different temporal questions. For example, we may wish to know bounds on the cycle period of an asynchronous component so we can use it to drive the clock signal of a synchronous component. Similar information can be used to generate worst-case and amortized performance bounds. We may also perform minimum separation analyses in order to determine if it is feasible to remove circuitry from a speed-independent implementation [16]. Our algorithm performs these sorts of analyses and is useful in many contexts and at many levels of abstraction. Separation analyses at the high-level can be used to help a designer choose among potential designs to perform a given computation. At a lower level they can be used to determine the correctness of the implementation, e.g., whether isochronic fork assumptions are valid [12]. Related work in timing analysis and verification of concurrent systems comes from a variety of different research communities including: real-time systems, VLSI CAD, and operations research. Timed automata [1] is one of the more powerful models for which automated verification methods exists. A timed automaton has a number of clocks (timers) whose values can be used in guards of the transitions of the automaton. Such models have been extensively studied and several algorithms exist for determining timing properties for timed automata [8, 9]. As in the untimed case, timed automata suffer from the state explosion problem when constructing the cross product of component specifications. Furthermore, the verification time is proportional to the product of the maximum value of the clocks and also proportional to the number of permutations of the clocks. To improve the run-time complexity, Burch [7] extends trace theory with discrete time but still uses automata-based methods for verification. This approach also suffers from exponential runtime in the size of the delay values but avoids the factorial associated with the permutations of the clocks. Orbits [19] uses convex regions to represent sets of timed states and thus avoids the explicit enumeration of each individual discrete timed state. Orbits is based on a Petri net model augmented with timing information. Other Petri net based approaches include Timed Petri nets [18] and Time Petri nets [3]. In Timed Petri nets a fixed delay is associated with each transitions while Time Petri nets use a more general model with delay ranges associated with the transitions. This paper is composed of seven sections. We follow this introduction with a description of the CSP specification language and its translation to Petri nets. Timed and untimed execution semantics of Petri nets are introduced in Section 3. The algorithm for performing timing analysis on Petri nets is describe in Section 4 and 5. Section 6 presents a parameterizable example which is used to benchmark the performance of the algorithm. Finally, Section 7 summarizes the contributions of this paper. Specification We now describe the specification language and show how to translate a specification into an intermediate form that is more suitable for timing analysis. 2.1 CSP Programs We specify computations using CSP (Martin style [14]). To simplify the timing analysis, we restrict the expressive power of the specification language. First, we exclude disjunctions in the guards because they correspond to OR-causality, which is known to be difficult [11, 15]. Second, we require the semantic property that during all untimed executions, either choice between guards is completely free (all the guards are identically true), or at most one guard evaluates to true. As we shall see in Section 3.2, this allows the use of all untimed executions in determining the possible timed behaviors, which is an important simplification to the timing analysis problem. These restrictions still allow the analysis of a large and interesting class of CSP programs, including many programs specifying implementations and abstractions of asynchronous control circuits. The syntax for a restricted CSP program P is shown in Table 1. Figure 1(a) shows a simple CSP program with three processes. Table 1: Syntax for the restricted CSP. P is a program, S a statement, C a communication action, E an expression, B a guard, and T a term. The terminal symbols x and X represent a variable identifier and a communication channel, respectively. 2.2 Petri Nets The CSP specification is translated into a safe Petri net which is the direct input for the timing analysis. A net N is a tuple (S; are finite, disjoint, nonempty sets of respectively places and transitions, and F ' (S \Theta T ) [ (T \Theta S) is a flow relation. A Petri net \Sigma is a pair (N; M 0 ), where N is a net and is the initial marking. See [17] for further details on the Petri net model. Graphically, a Petri net is represented as a bipartite graph whose nodes are S and T and whose edges represent the flow relation F . Circles represent places and straight lines represent transitions. The initial marking is shown with dots (tokens). Figure 1(b) shows a simple Petri net. For an element x the preset and postset of x are defined as fy fy A marking represents the global state of the system. A transition t is enabled at a marking M if each input place of t is marked, i.e., 8s Firing the enabled transition t at M produces a new marking M 0 constructed by removing a token from each of the places in the preset of t and adding a token to each of the places in the postset of t. The transformation of M into firing t is denoted by M [tiM 0 . We let [Mi denote the set of markings reachable from the marking M . 2.3 Translation of CSP into Petri Nets The CSP specification is translated into a safe Petri net. Petri net transitions are used to model communication synchronizations and places are used to model control choice. The Petri net can be constructed by syntax directed translation [4]. The mapping amounts to introducing a single token corresponding to the program counter in each communicating process. A variable x is modeled by two places, x 0 and x 1 . If x is true, x 1 is marked, otherwise x 0 is marked. After constructing nets for each process, transitions are combined corresponding to matching communication actions. Figure 1 shows a simple CSP specification and the corresponding Petri net. (a) a d c e f (b) Figure 1: (a) CSP specification and (b) the corresponding Petri net \Sigma. There are two complications in the translation. One is how to consistently label Petri net transitions corresponding to communication actions that occur different numbers of times in connected processes. This labeling problem is illustrated in its simplest form by the following CSP program composed of a divide by two counter connected to a trivial environment: The X communication in P 1 has to connect up to two X communications in P 2 . We solve the labeling problem by introducing a separate label for each possible pairing of communication actions [4]. In the example, we introduce two labels, X (0) and X (1) , and a choice for each of the possible communications, obtaining the nets shown in Figure 2. Y Figure 2: Petri nets for the individual processes in the divide by two counter. The second complication is the translation of the probe construct, X. If the probed communication action X is not completed immediately, we split it in two, ' . The first half implements the guard of the selection statement, and the second half implements the actual communication action. Figure 3 illustrates this translation. 2.4 Properties of Petri Net The Petri net obtained from a CSP specification has the following properties: A A Figure 3: The Petri net for the incomplete CSP program ffl The Petri net is safe, i.e., there is never more than one token at a place: be a choice place, i.e., jsfflj ? 1. Then s is either extended free choice or unique choice. The place s is extended free choice if The place s is unique choice if at most one of the successor transitions ever becomes enabled. denote the number of transitions in sffl that are enabled at the marking M . A place s is unique choice if 8M 2 [M 1. The place s 2 in the net in Figure 1(b) 3 Execution Semantics To represent the set of all (untimed) executions we introduce the notion of a process. Intuitively, a process is an unfolding of a Petri net that represents one possible (finite) execution of the Petri net. Processes are used to give timing semantics to a Petri net; for each process we define the set of legal assignments of time stamps to the transitions of the process. 3.1 A process for the Petri net \Sigma is a net N and a labeling lab (We subscript S, T , and F to distinguish between the nets of \Sigma and -.) The net N is acyclic and without choice, i.e., without branched places. N and lab must satisfy appropriate properties such that - can be interpreted as an execution of \Sigma [5, 21]. Figure 4 shows a process for the Petri net in Figure 1(b). The only true choice in the net is at the place s 2 where there is a non-deterministic selection of either transition c or d. The process represents the execution where the first time transition c fires and the next time transition d fires. We denote all (untimed) executions of a Petri net \Sigma by the set is a process of \Sigmag : A safe Petri net has only a finite number of reachable markings. Processes have the property that any cut of places corresponds to a reachable marking of \Sigma [5, Lemma 2.7]. Therefore, sufficiently long processes will contain repeated segments of processes. We represent the potentially infinite e a c f a d Figure 4: A process - for the Petri net \Sigma in Figure 1(b). The places and transitions in the process have been labeled (using the lab function) with the names of their corresponding places and transitions in \Sigma. set of processes \Pi(\Sigma) by a finite graph we call the process automaton. The vertices of the process automaton correspond to markings of \Sigma and the edges are annotated with segments of processes. We let v 0 denote the vertex corresponding to the initial marking M 0 . Consider a path p in a process automaton from vertex u to v, denoted u p v. Then -(p) is the process obtained by concatenating the process segments annotated on the edges of p. The process automaton has the property that pref (-(p)) v is a path in the process automaton where pref (-) is the set of prefixes (defined on partial orders [21]) of a process -. We can construct the process automaton without first constructing the reachability graph [6, 10]. If there is no concurrency in the net, the size of the process automaton is equal to the size of the reachability graph. However, if there is a high degree of concurrency, the process automaton will be considerably smaller. Figure 5 shows the process automaton and the associated processes for the Petri net in Figure 1(b). The process in Figure 4 is constructed from a f d e Figure 5: To the left, the process automaton for the Petri net in Figure 1(b). The three process segments annotated on the edges are shown to the right (labeled with elements from S \Sigma [ T \Sigma ). 3.2 Timed Execution To incorporate timing into the Petri net model, we associate delay bounds with each place in the net. The lower delay bound, d(s) 2 R 6\Gamma , and the upper delay bound, D(s) 2 R 6\Gamma [f1g, where R 6\Gamma is the set of non-negative real numbers, satisfy 0 - These delay bounds restrict the possible executions of the Petri net. During a timed execution of the net, when a token is added to a place s, the earliest it becomes available for a transition in sffl is later and the latest is D(s) units later. A transition t must fire when there are available tokens at all places in fflt unless the firing of the transition is disabled by firing another transition. The firing of t itself is instantaneous. More formally, a timing assignment for process - , is a function that maps transitions in a process to time values, Definition 3.1 Let \Sigma be a safe Petri net and let - be a process of \Sigma. We consider a cut c ' S - of - and let T enabled ' T \Sigma be the set of transitions enabled at the corresponding marking, M c . For a timing assignment, - , and a transition t 2 T enabled , the earliest and latest global firing time of t is given by and starttime(b) denotes the set of elements of S - which are mapped to s by lab. Note that c " lab \Gamma1 (fflt) is non-empty because t is enabled at marking M c . The function starttime takes a place returns the time when a token entered the place lab(b), i.e., -(e) if . If there is no such transition e, we set starttime(b) to 0. The timing assignment - is consistent at cut c if: and A timing assignment - of a process is consistent if it is consistent at all place cuts c of the process. Let \Pi timed (\Sigma) be the set of all timed executions of \Sigma: timed \Pi(\Sigma) and there exists a consistent timing assignment for - The restrictions on the CSP specification in Section 2 were crafted such that the set of untimed and timed processes of the underlying Petri net are equivalent. This allows us to use the process automaton to enumerate the possible processes without referring to timing information, and then perform timing analysis on each process individually. To prove this, we need two lemmas. The first states that it is always possible to find a timing assignment satisfying (1) in Definition 3.1. The second lemma states a simple structural property of extended free choice places. Lemma 3.2 Let \Sigma be a safe Petri net. For any t 2 T \Sigma , earliest(t) - latest(t) for any process of \Sigma and for any timing assignment (not necessarily consistent). Proof: From the definition of earliest and latest and the fact that for all place s, Lemma 3.3 Let \Sigma be a Petri net and let s 2 S \Sigma be an extended free choice place. Then 8t Proof: By contradiction: assume fflt 1 6= fflt 2 . Then there is a place s 0 in Assume without loss of generality that s 0 is in fflt 1 and not in fflt 2 . From the definition of s being an extended free choice place, it follows that ffl. By the premise of the lemma, t 2 is in sffl, and thus also in s 0 ffl. A simple fact about pre- and post-sets is that if y 2 xffl then x 2 ffly. As t 2 is in s 0 ffl, s 0 is in fflt 2 , contradicting the assumption. 2 Theorem 3.4 Let \Sigma be a safe Petri net where choice is either extended free choice or unique choice. Then \Pi timed Proof: Clearly, \Pi timed (\Sigma) ' \Pi(\Sigma). We show that \Pi(\Sigma) ' \Pi timed (\Sigma), i.e., there exists a consistent timing assignment for all - 2 \Pi(\Sigma). We will prove that for all cuts c of - and any b 2 c, constraint (2) is subsumed by constraint (1). From Lemma 3.2 it follows that for any process there exists a consistent timing assignment. Observe that lab(bffl) ' lab(b)ffl, and thus if lab(bffl) " T enabled is non-empty then so is lab(b)ffl " enabled . Let c ' S - be a cut of - and let b be a place in c. If and (2) is trivially satisfied. For bffl 6= ;, let e be the one element in bffl (all places b in a process has jbfflj - 1) and let s be the place in \Sigma corresponding to b, i.e., lab(b). The observation above then states that lab(e) 2 enabled . Consider two cases: lab(e) is the only element in sffl " T enabled and (2) to reduces to -(e) - latest(lab(e)). 1: The choice place s is either extended free choice or unique choice: s is extended free choice: From Lemma 3.3 we have that from the definition of latest , it follows that 8t As minimization is idempotent and lab(e) 2 reduces to -(e) - latest(lab(e)). s is unique choice: From the definition of unique choice, thus lab(e) is the only element in sffl " T enabled . Condition (2) again reduces to -(e) - latest(lab(e)).4 Timing Analysis Having formally defined the timing semantics of the Petri net, we now state the timing analysis problem and present an algorithm for solving this problem. 4.1 Problem Formulation Given two transitions from a Petri net \Sigma, t from ; t to 2 T \Sigma , we wish to determine the extreme-case separation in time between related firings of t from and t to . We let b \Pi be a set of triples dst i, where - 2 \Pi(\Sigma) and t src ; t dst are transitions in the process - with lab(t src lab(t dst to . The set b \Pi is used to describe all the possible processes where the distinguished transitions t src and t dst have the appropriate relationship. The timing analysis we perform is: for all b \Pi and for all consistent timing assignments - for -, determine the largest ffi and smallest \Delta such that The transitions t src and t dst must be related in order for the timing analysis to yield interesting information. Consider finding the maximum time between consecutive firings of transition a in Figure 1(b), corresponding to the maximum cycle time of a transition. For this separation t from and t to are both a. The occurrences of a, the t src and t dst transitions, must be restricted such that all the elements of b \Pi have the property that no other transition t between t src and t dst has label a. For example, one of the elements in b \Pi is the process in Figure 4 with t src and t dst being the left-most and right-most transitions labeled with a, respectively. The relationship between t src and t dst is defined by backward relative indexing by specifying two numbers fi and fl, and a reference transition, t ref 2 T \Sigma . For a particular -, we find the corresponding transitions t src and t dst by the following procedure: start at the end of the process and move backwards looking for a transition t such that When found, we continue moving backwards, looking for the fith transition t (starting with this is t src . Simultaneously, we find the flth transition having to ; this is t dst . When both are found, we include h-; t src ; t dst i in b \Pi. The specification of a separation analysis on the Petri net \Sigma thus consists of three transitions, t from , t to , and t ref , all in T \Sigma , and two constants fi; fl 2 N. We call fi and fl occurrence indices relative to the transition t ref . We let \Pi(\Sigma; t from ; fi; t to ; fl; t ref ) denote the set of triples h-; t src ; t dst i where - 2 \Pi(\Sigma) and t src and t dst have the relation described above. One communication action in the CSP program may map to many transitions in \Sigma and these transitions are to be considered equivalent when performing the timing analysis. Instead of specifying the separation between individual transitions, we specify it between sets of transitions, i.e., a separation analysis is specified by two occurrence indices fi and fl, and three sets of transitions from \Sigma: From, To, and Ref . Our final formulation is: It is straightforward, given the communication actions, to determine what transitions should be included in these sets. Sometimes we may also want to consider several CSP communication actions as equivalent with respect to the separation analysis (we will see an example of this in Section 6). This can conveniently be achieved by adding the corresponding Petri net transitions to the appropriate From , To, and Ref sets. In the sequel, we will only discuss the maximum separation analysis, i.e., find \Delta, because the separation ffi can be found from a maximum separation analysis of -(t src dst This is accomplished by computing b \Pi using reversed r-oles for From and To, and fi and fl: b 4.2 The CTSE Algorithm Let \Delta( b -) be the maximum separation between t src and t dst for some particular execution b -: \Delta( b is a consistent timing assignment for The maximum separation over all executions is then given by We now show how the elements of b \Pi are constructed to obtain \Delta, and Section 5 describes the algorithm for computing \Delta( b -). The process automaton represents all possible executions. However, whatever topologically dst in a process cannot influence the maximum separation between these two transitions. Any portion of a process following t src and t dst can therefore be ignored and all processes in b \Pi will end with some terminal process segment that includes the two transitions t src and t dst . Let -(p) be a process containing t src and t dst for some path p in the process automaton (starting at We decompose this process into -(p 0 )- T , where p 0 is a path in the process automaton and - T is the minimal process segment containing t src and t dst . The process segment - T is called a terminal segment. We let \Pi T (u) be the finite set of process segments such that for any path v 0 ; u in the process automaton and - T 2 \Pi T (u), the process -(p)- T is in b \Pi. Figure 6 shows the two terminal process segments belonging to \Pi T (v 0 ) for the a-to-a separation analysis in our example. For this example, all processes in b \Pi can be constructed by -(p)- T , where v 0 e a c a f a d a Figure Two terminal processes (labeled using lab) for the separation analysis from a to the next a transition in the Petri net in Figure 1(b). An algorithm for computing \Delta( b -) can be phrased in algebraic terms. For each segment of a process, there is a corresponding element in the algebra. We use [-] to denote this element for the process segment -. The algebra allows us to reuse analysis of shorter processes when computing \Delta( b -) because the operators of the algebra are associative (the details are shown in the next section). There are two operations in the algebra: "choice", j, and "concatenation", fi. Our approach to analyzing the infinite set b \Pi is to enumerate the processes b - of increasing length by unfolding the process automaton using a breadth-first traversal. We traverse the automaton backwards, starting with the terminal segments. An element of the algebra is stored at each node v in the process automaton. Let [v] k denote the algebraic element stored at node v in the process automaton after the k th iteration. Initially, [v] (v)g. When traversing the process automaton backwards, the elements of the algebra are composed (using fi) for two paths in series, and combined (using j ) for two paths in parallel. The choice- operator combines backward paths when they reach the same marking in the process automaton. This is illustrated below by showing a backward traversal with reconvergence corresponding to the process automaton in Figure 5 and the two terminal processes in Figure 6: For this example, [v 0 Whenever the node v 0 is reached in the k th unfolding, [v 0 represents the maximum separation for all executions represented by that unfolding, denoted \Delta k . This value is maximized with the values for the previous unfoldings, \Delta kg. From (3) it follows that \Delta -k is a lower bound on \Delta and that For a given node v in the process automaton, we can compute an upper bound on all further unfoldings; this bound is denoted [v] ?k . Let c be a vertex cut of the process automaton. An upper bound on \Delta for the k th unfolding is \Delta cg. When \Delta ?k is less than or equal to \Delta -k for some k we can stop further unfolding and report the exact maximum separation It is possible that the upper and lower bounds do not converge in which case the bounds may still provide useful information as \Delta is in the range The main loop of the CTSE algorithm is shown in Figure 7. Algorithm: CTSE(G) For each (v)g at v; do f unfold once(G); until return Figure 7: The CTSE algorithm computing \Delta given a process automaton G. The run-time of the algorithm depends on the size of the representation of the algebraic elements. The size of an element may be as large as the number of paths between the two nodes related by the element, i.e., exponential in the number of iterations, k. In practice, pruning drastically reduces the element size. Computing \Delta( b This section describes the algebra used in the CTSE algorithm. This algebra is used to reformulate an algorithm by McMillan and Dill [15] for determining the maximum separation of two events in an acyclic graph. 5.1 Algebras Before presenting the algorithm for computing \Delta( b -) we introduce two algebras. The first is the (min; +)-algebra (R [ f1g; \Phi 0 The elements 1 and 0 are the identity elements for \Phi 0 The second algebra is denoted by Each element in F is a function represented by a set of pairs. The singleton set, fhl; uig, where u is a row-vector of length n, represents the where m is a column-vector of length n and\Omega 0 denotes the inner product in the (min; +)-algebra. In general, the set fhl represents the function We associate two binary operators with functions: function maximization, f \Phi g, and function composition, f\Omega g. It follows from (4) that function maximization is defined as set union: f \Phi g. Function composition, g\Omega h, is defined as f(x; m) = h(g(x; m); m). Notice that we use left-to-right function composition. For m) and Function composition,\Omega , distributes over function maximization, \Phi. The elements are the identity elements for function maximization and composition, respectively. be two pairs in the representation of a function. We can remove since then for all x and m, min(x m) m). Proper application of this observation that can greatly simplify the representation of a function. 5.2 The Acyclic Time Separation of Events Algorithm We can now present the algebraic formulation of McMillan and Dill's algorithm for computing \Delta( b -). For each place and transition in - we compute a pair [f; m] where f 2 F and m 2 R[f1g. The algorithm is shown in Figure 8. Informally, this algorithm works as follows: To maximize the value of -(t dst we need to find a timing assignment that maximizes -(t dst ) and minimizes -(t src ). The first element of [f represents the longest path (using D(s)) from a transition to t dst and the second element represents the shortest path (using \Gammad(s)) to t src . The algebra for the f 0 -part is complicated by the fact that the delay for a given place can not be assigned both d(s) and D(s). The f 0 -part must represent the longest path respecting the delays assigned by the shortest path computation. For details see [13]. To find the maximum separation represented by a [f; m] pair, we evaluate f at m and 0, computing the sum of the longest and shortest paths. To compute \Delta( b -), we maximize over all [f; m] pairs at the initial marking: \Delta( b is the pair at s 2 ffl-g ; where ffl- denotes the set fs similarly, -ffl denotes the set fs 2 S - j ;g. 5.3 Decomposition The algebraic formulation allows for a decomposition of the above computation using matrices. Consider a process segment - having We represent the computation of the algorithm on - by two n \Theta m matrices, F and M. Given a vector of m-values at -ffl, m, we can Algorithm: \Delta(b-) For each element of - in backward topological order: For a place s, compute the pair [f ae 0 if (m)ig\Omega f where [f; m] is the pair stored at ae is the pair stored at For a transition t, compute the pair [f ae dst \Phiff at place s j s 2 tfflg otherwise ae \Phi 0 fm at place s j s 2 tfflg otherwise Figure 8: The algorithm for computing \Delta(b-). find the vector m-values at ffl-, m 0 , from the (\Phi 0 m. This is illustrated using the process segment - 1 from Figure 5, shown in Figure 9. We associate the delay range [1; 2] to each place, i.e., places s. We compute expressions for refers to the m-value computed for element x in the process. In backwards topological order of - 1 we compute: \Gamma1\Omega 0 m(s 5 ) from substitution substitution as\Omega 0 distributes over \Phi 0 and is associative We can represent this computation in matrix form using (\Phi 0 Figure 9: Process segment - 1 from Figure 5. A similar matrix is constructed for the f-part. For the process segment - 1 the computation is: )ig\Omega f(s 5 ) from substitution substitution from substitution )ig\Omega fh2; m(s 5 \Delta\Omega distributes over \Phi by definition of\Omega These expressions depend on the m-values at internal elements of - 1 , e.g., m(t 1 ) in the expression for f(s 0 ). The m-value for these nodes can be computed as a linear expression in the m-values at the places in -ffl. This linear expression is encoded by a vector u of length j-fflj. The vector product computes the m-value for the internal node where u is stored. E.g., the expression used in f(s 0 ) is represented by a the vector: We express the f-computation in matrix form using Given a process segment -, we denote the corresponding function and m-value matrices by F(-) and M(-). The algebraic element [-] is then defined as the singleton set f[F(-); M(-)]g. We can now define the two operators fi and j. The choice operator is defined as set union: The composition operator is more complex. When composing two segments - 1 and - 2 , the functions in - 1 need to refer to the m-values in - 2 ffl rather than those at - 1 ffl. We shift the functions in to make them refer to m-values in - 2 ffl by multiplying the u-vectors in F(- 1 ) with M(- 2 ). For a singleton function fhl; uig, we obtain the function fhl; )ig. Non-singleton functions are shifted by shifting each pair, and a matrix of functions is shifted elementwise. We use the notation F -M to denote a shift of matrix F by matrix M. For singleton sets the composition operator is defined as: \Theta Non-singleton sets are multiplied out by applying the distributive law. 5.4 Pruning Consider the element f[F ]g. We can removed [F from the set if we can show that for any pairs composed to the left and right such that the result is a scalar, this scalar is no greater than the same composition with the [F sufficient condition for eliminating is the following: Let i.e., k is the largest difference between elements in M 2 and M 1 , or 0 if this difference is negative. The where 1 is a row-vector of appropriate length with all entries set to 1. This condition is used to eliminate entire execution paths from further analysis, and is central to obtaining an efficient algo- rithm. More sophisticated conditions, that use more information about the particular computation, are possible and may further increase the efficiency of the algorithm. 5.5 Upper Bound Computation We now consider how to determine an upper bound [v] ?k for node, v, in the process automaton. To determine a non-trivial upper bound, all further backward paths from v to v 0 have to be considered, i.e., we need to bound the infinite set of algebraic elements constructed from backward paths: oe For any simple path p we just compute [-(p)]. If p is not simple, we write p as 3 is a simple path, p 2 is a simple cycle, and p 1 is finite, but may contain cycles. We introduce an upper bound operator, r, with the property that 1. Thus, the expression on the right-hand side is an upper bound on the left-hand side expression. The r operator is recursively applied to the path until this is a simple path. Hence, we can bound the infinite set in (5) by a finite set of algebraic elements constructed from all paths consisting of a simple cycle followed by a simple path ending at v. The r operator is defined as follows: Assume F is a m+ k \Theta n + k matrix of the form 0m where 0 i is a vector of lenght i containing 0 and I k is the identity matrix of size k. The operator r[F; M] is defined as hi z m is a vector of length m containing the function z = fh1; 1ig. The function z is a largest element of F , i.e., z - f for all functions f 2 F . The effect of the r operator is to apply the function z to the part of the F matrix which is not the identity. The upper bound is determined individually for each pair in the set for node v. If the upper bound for a given [F; M] pair is less than or equal to the present global lower bound, \Delta -k , that pair can be removed from the set, further pruning the backward execution paths that must be considered. The order in which [F; M] pairs are multiplied greatly affects the run-time of the algorithm. For example, consider precomputing for each node in the process automaton the algebraic expression for the upper bound, i.e., for each node, compute the algebraic element for the set of simple paths followed by simple cycles (going backwards). Because we don't know what is to be composed with these elements, few pairs can be pruned from the representation. Therefore it may be more efficient to multiply the pairs out in each iteration, even though this doesn't allow the reuse of work from previous iterations. Our experience has been that upper bound expressions become very large when precomputed and we are better off recomputing them at each iteration because effective pruning takes place. We only precompute the r of the simple cycles. This observation was key to achieving polynomial run-time for the example described in the following section. 6 Benchmark Example: The Eager Stack Replicating a single process in a linear array provides an efficient implementation of a last-in, first-out memory which we refer to as an eager stack. The eager stack contains an interesting mixture of choice and concurrency and represents an excellent parameterizable example for explaining analyses that can be performed by our algorithm, and also benchmarking our implementation of the algorithm. 6.1 The Eager Stack A stack capable of storing n elements is constructed from n equivalent processes, arranged in a linear array. Each process has four ports, In, Out , Put , and Get . The ports Put and Get connect to the ports In and Out, respectively, in the stage to the right. Figure 10 shows a block diagram Put Get Environment 3-stage eager stack In Out Put Get In Out Put Get In Out Figure 10: Block diagram of the 3-stage eager stack. The CSP specification of a single stage is: In The Boolean variables b and rb are used to control communication with the adjacent right stage. The value of b indicates whether this stage holds valid data. The value of rb is a mirror of the value of b in the stage to the right. Concurrency occurs when a position must be created or a space must be filled in. The choice of whether to do a Put or Get is made in the environment and is potentially propagated throughout the entire stack. In order to avoid an overflow or underflow condition, the environment interacting with the stack must not attempt a Put if n elements are already stored in the stack and it must not attempt a Get if the stack is empty. The following process represents a suitable environment: E(P ut; Get) j This process is unfolded n times and the actual data (x) is eliminated for simplicity. For get: The construct is repeated if a guard command with a trailing is chosen, and is not repeated otherwise. The number in parenthesis refers to the number of items in the stack at the time when the communication is performed, so after Put (2) the stack is full and only a Get communication is possible 1 . A nice property of this example is that the port names occur the same number of times and along compatible choice paths in adjacent processes. Thus we can identify a (superscripted) number with each occurrence of a port name in the program. We use Petri net transition P i for communications on the Put port in stage i and In port in stage i + 1. Similarly, G i denotes a communications on the Get port in stage i and on the Out port in stage Figure 10). 1 It is possible to have the stack indicate whether it is empty or full and make the environment behave accordingly, but this complicates each stage of the stack. 6.2 Timing Analysis There are numerous interesting time separation analysis we can perform on the eager stacks. We can determine the minimum and maximum separations between consecutive Put communications in the environment process. The maximum separation analysis for a 3-stage stack would correspond to: If we set the delay between communication actions to be the range [1; 2], we get the maximum separation This is obtained by filling an empty stack (three Put operations) and then emptying it again (three Get operation), finally inserting one element (a Put operation). The maximum separation is achieved between the third and the fourth Put operation. For the minimum separation, we exchange the sets To and From and negate the result, in this case A possibly more interesting analysis might be the minimum and maximum separations between consecutive Put or Get communications. This corresponds to the minimum and maximum response time of the stack, or equivalently, the minimum and maximum cycle period of the environment. We must include all Petri net transitions corresponding to Put and Get communications in the environment. Thus The results, again for [1; 2] delay ranges between communication actions, are For fixed delay values, the eager stack has constant response time, i.e., the time between the environment performs either a Put or a Get operation until the next such operation is independent of the size of the stack, n. This is not the case when we introduce uncertainty in the delay values. The maximum response time turns out to be n linear in the stack size. However, if we look at the maximum response time amortized over m Put or Get communication actions we get the following maximum separations \Delta: Dividing \Delta by m, we obtain the amortized separations shown below: \Delta=m 6 5 4.66 4.5 4.4 4.33 4.29 We can predict that for 4. So although the maximum separation between two consecutive operations increases linearly with n, if we amortize over a number of operations, the response time converges to 4. In fact, the maximum response time converges to 4 independently of n. In this sense, the eager stack has constant response time even when the delays are uncertain. 6.3 Run Time Execution times of the CTSE algorithm on eager stacks of various sizes, n, are shown in Table 2 using [1; 2] for all delay ranges. The size of the specification, i.e., number of places, number of transitions, and the size of the flow relation, is given n the table by jS \Sigma j, jT \Sigma j, and jF \Sigma j, respectively. The number of nodes in the reachability graph is shown in the jR:G:j column. Note that the reachability graph is not constructed when performing the timing analysis and is only reported to give an idea of the complexity of the nets. The separation analysis denoted by \Delta 1 is the maximum separation between consecutive Put operations and \Delta 2 is the maximum separation between consecutive Put or Get operations. The CPU times were obtained on a Sparc 10 with 256 MB of memory. .3 4 43 176 268 2.4 2.3 28 1220 813 36 2000 1333 5366 Table 2: Run times of the CTSE algorithm on eager stack of various sizes. Figure 11 shows the CPU times for the two separation analysis plotted as a function the size of the Petri net. Orbits [19] is, to the authors knowledge, the most developed and efficient tool for answering temporal questions about Petri nets specifications. Orbits constructs the timed reachability graph, i.e., the states reachable given the timing information. It should be noted that Orbits is capable of analyzing a larger class of Petri net specifications than the one described here. Partial order techniques are also used in Orbits to reduce the state space explosion [20]. However, the time to construct the timed reachability graph for the eager stack increases exponentially with the size n. For 6 the time is 234 CPU seconds on a Decstation 5000 with 256 MB, i.e., two orders of magnitude slower than the CTSE algorithm. For Orbits ran out of memory. 7 Conclusion We have described an algorithm for solving an important time separation problem on a class of Petri nets that contains both choice and concurrency. In practice, our algorithm is able to analyze nets of considerable size, demonstrated by an example whose Petri net specification consists of more than 3000 nodes and 10 reachable states. While we report a polynomial run-time result for only a single parameterizable example, we expect similar results for other specifications exhibiting limited choice and abundant concurrency. Acknowledgments We thank Chris Myers of Stanford University for many fruitful discussions as well as supplying the Orbits runtimes. This work is supported by an NSF YI Award (MIP-9257987) and by the 110010000 Petri net size, jF \Sigma j Figure 11: Double logarithmic plot of CPU time for the two separation analyses as a function of the Petri net size, jF \Sigma j. DARPA/CSTO Microsystems Program under an ONR monitored contract (N00014-91-J-4041). --R The theory of timed automata. Synchronization and Linearity. Modeling and verification of time dependent systems using time Petri nets. Its relation to nets and to CSP. Partial order behavior and structure of Petri nets. Interleaving and partial orders in concurrency: A formal comparison. Trace Algebra for Automatic Verification of Real-Time Concurrent Systems Minimum and maximum delay problems in real-time systems Computer Aided Verification Using partial orders to improve automatic verification methods. Timing analysis of digital circuits and the theory of min-max functions Practical applications of an efficient time separation of events algorithm. An algorithm for exact bounds on the time separation of events in concurrent systems. Programming in VLSI: From communicating processes to delay-insensitive circuits Algorithms for interface timing verification. Synthesis of timed asynchronous circuits. Petri Net Theory and The Modeling of Systems. Performance evaluation of asynchronous concurrent systems using Petri nets. Automatic verification of timed circuits. Modular Construction and Partial Order Semantics of Petri Nets. --TR --CTR Ken Stevens , Shai Rotem , Steven M. Burns , Jordi Cortadella , Ran Ginosar , Michael Kishinevsky , Marly Roncken, CAD directions for high performance asynchronous circuits, Proceedings of the 36th ACM/IEEE conference on Design automation, p.116-121, June 21-25, 1999, New Orleans, Louisiana, United States

asynchronous systems;concurrent systems;time separation of events;timing analysis;abstract algebra

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Understanding the sources of variation in software inspections.

In a previous experiment, we determined how various changes in three structural elements of the software inspection process (team size and the number and sequencing of sessions) altered effectiveness and interval. Our results showed that such changes did not significantly influence the defect detection rate, but that certain combinations of changes dramatically increased the inspection interval. We also observed a large amount of unexplained variance in the data, indicating that other factors must be affecting inspection performance. The nature and extent of these other factors now have to be determined to ensure that they had not biased our earlier results. Also, identifying these other factors might suggest additional ways to improve the efficiency of inspections. Acting on the hypothesis that the inputs into the inspection process (reviewers, authors, and code units) were significant sources of variation, we modeled their effects on inspection performance. We found that they were responsible for much more variation in detect detection than was process structure. This leads us to conclude that better defect detection techniques, not better process structures, are the key to improving inspection effectiveness. The combined effects of process inputs and process structure on the inspection interval accounted for only a small percentage of the variance in inspection interval. Therefore, there must be other factors which need to be identified.

Introduction Software inspection has long been regarded as a simple, effective, and inexpensive way of detecting and removing defects from software artifacts. Most organizations follow a three-step procedure of Preparation, Collection, and Repair. First, each member of a team of reviewers reads the artifact, detecting as many defects as possible (Preparation). Next, the review team meets, looks for additional defects, and compiles a list of all discovered defects (Collection). Finally, these defects are corrected by the artifact's author (Repair). Several variants of this method have been proposed in order to improve inspection performance. Most involve restructuring the process, e.g., rearranging the steps, changing the number of people working on each step, or the number of times each step is executed. Some of these variants have been evaluated empirically. However, focus has been on their overall performance. Very few investigations attempted to isolate the effects due specifically to structural changes. However, we must know which effect are caused by which changes in order to determine the factors that drive inspection performance, to understand why one method may be better than another, and to focus future research on high-payoff areas. Therefore, we conducted a controlled experiment in which we manipulated the structure of the inspection process[20]. We adjusted the size of the team and the number of sessions. Defects were sometimes repaired in between multiple sessions and sometimes not. Comparing the effects of different structures on inspection effectiveness and interval 1 indicated that none of the structural changes we investigated had a significant impact on effectiveness, but some changes dramatically increased the inspection interval. Regardless of the treatment used, both the effectiveness and interval data seemed to vary widely. To strengthen the credibility of our previous study and to deepen our understanding of the inspection process, we must now study this variation. 1.1 Problem Statement We are asking two questions: (1) Are the effects of process structure obscured by other sources of variation, i.e., is the "signal" swamped by "noise"? (2) Are the effects of other factors more influential than the effects of process structure, i.e., are researchers focusing on the wrong mechanisms? To answer the first question, we will attempt to separate the effects of some external sources of variation from the effects due to changes in the process structure. By eliminating the effects of external variation we will have a more accurate picture of the effects of our experimental treatments. Also, by understanding the external variation we may be able to evaluate how well our experimental design controlled for it, which will aid the design of future experiments. To answer the second question, we will compare the variation due to process structure with that due to other sources. If the other sources are more influential than process structure, then it may Inspections have many different costs and benefits. In this study we restricted our discussion of benefits to the number of defects found, and costs to inspection interval (the time from the start of the inspection to its completion) and person effort. be possible to significantly improve inspections by properly manipulating them. We expect that identifying and understanding these sources will aid the development of better inspection methods. Therefore, we have extended the results of our experiment by identifying some sources of variation and modeling their influence on inspection effectiveness and interval. We will show that our previous results do not change even after these sources of variation are accounted for. This analysis also suggests some improvements for the inspection process and raises some implications about past research and future studies. 1.2 Analysis Philosophy We hope to identify mechanisms that drive the costs and benefits of inspections so that we can engineer better inspections. To do this we will rely heavily on statistical modeling techniques. However, these techniques are not completely automated. Therefore, we must make judgments about which variables or combinations of variables to allow in the models. These choices are guided by our desire to create models that are robust and interpretable. To improve robustness we avoided fitting the data with too many factors. Doing so could result in a model that explains much of the variation in the current data, but has no predictive power when used on a different set of data. To improve interpretability we omitted factors for which we have no readily available measure. We also omitted factors whose effects were known to be confounded with other factors in the model. Finally, we rejected models for which, based on our experience, we could not argue that their variables were causal agents of inspection performance. Specifically, there are four conditions that must be satisfied before factor A can be said to cause response B[12]: 1. A must occur before B. 2. A and B must be correlated. 3. There is no other factor C that accounts for the correlation between A and B. 4. A mechanism exists that explains how A affects B. One implication of all these is that the "best" model for our purpose is not necessarily the one that explains the largest amount of variation. Throughout this research we have chosen certain models over others. Some were rejected because a smaller, but equally effective model could be found, or because one variable was strongly confounded with another, or because a variable failed to show a causal relationship with inspection performance. We will point out these cases as they arise. 2 Summary of Experiment With the cooperation of professional developers working on an actual software project at Lucent Technologies (formerly AT&T Bell Labs), we conducted a controlled experiment to compare the costs and benefits of making several structural changes to the software inspection process. (See et al.[20] for details.) The project was to create a compiler and environment to support developers of Lucent Technologies' 5ESS(TM) telephone switching system. The finished compiler contains over 55K new lines of C++ code, plus 10K which was reused from a prototype. (See Appendix A for a description of the project.) The inspector pool consisted of the 6 developers building the compiler plus 5 developers working on other projects. 2 They had all been with the organization for at least 5 years and had similar development backgrounds. In particular, all had received inspection training at some point in their careers. Data was collected over a period of (June 1994 to December 1995), during which 88 code inspections were performed. 2.1 Experimental Design We hypothesized that (1) inspections with large teams have longer intervals, but find no more defects than smaller teams; (2) multiple-session inspections 3 are more effective than single-session inspections, but at the cost of a significantly longer interval; and (3) although repairing the defects found in each session of a multiple-session inspection before starting the next session will catch even more defects, it will also take significantly longer than multiple sessions meeting in parallel. We manipulated these independent variables: the number of reviewers (1, 2, or 4); the number of sessions (1 or 2); and, for multiple sessions, whether to conduct the sessions in parallel or in sequence. The treatments were arrived at by selecting various combinations of these (e.g., 1 session/4 reviewers, 2 sessions/2 reviewers without repair, etc. Among the dependent variables measured were inspection effectiveness-in terms of observed number of defects, as explained in Appendix B-and inspection interval-in terms of working days from the time the code was made available for inspection up to the collection meeting. 4 2.2 Conducting the Experiment To support the experiment, one of us joined the development team in the role of inspection quality engineer (IQE). He was responsible for tracking the experiment's progress, capturing and validating data, and observing all inspections. He also attended the development team's meetings, but had no development responsibilities. When a code unit was ready for inspection, the IQE randomly assigned a treatment and randomly drew the review team from the inspector pool. In this way, we attempted to control for differences in natural ability, learning rate, and code unit quality. In addition, 6 more developers were called in at one time or another to help inspect 1 or 2 pieces of code, mostly to relieve the regular pool during the peak development periods. It is common practice to get non-project developers to inspect code during peak periods. 3 In this experiment, we used the term "session" to mean one cycle of the preparation-collection-repair process. In multiple-session inspections, different teams inspect the same code unit. 4 For 2-session inspections, the longer interval of the two is selected. The names of the reviewers were then given to the author, who scheduled the collection meeting. If the treatment called for 2 sessions, the author scheduled 2 separate collection meetings. If repair was required between the 2 sessions, then the second collection meeting was not scheduled until the author had repaired all defects found in the first session. The reviewers were expected to prepare sufficiently before the meeting. During preparation, reviewers did not merely acquaint themselves with the code, but carefully examined it for defects. They were not given any specific technical roles (e.g., tester or end-user) nor any checklists. On an individual preparation form, they recorded the time spent on preparation, and the page and line number and the description of each issue (each "suspected" defect). 5 The experiment placed no limit on preparation time. For the collection meeting one reviewer was selected as the moderator and another as the reader. The moderator ran the meeting and recorded administrative data on a moderator report form. This comprised the name of the author, lines of code inspected, hours spent testing the code before inspection, and inspection team members. The reader paraphrased the code. During this activity, reviewers brought up any issues found during preparation or briefly discussed newly discovered issues. On a collection form, the code unit's author recorded the page and line number and description of each issue regarded as valid, as well as the start and end time of the collection meeting. Each valid issue was tagged with a unique Issue ID. If a reviewer had found that particular issue during preparation, he or she recorded that ID next to the issue on his or her preparation form. This enabled us to trace issues back to the reviewers who found them. No limit was placed on meeting duration, although most lasted less than 2 hours. After the collection meeting, the author kept the collection form and resolved all issues. In the process he or she recorded on a repair form the disposition (no change, fixed, deferred), nature (non- issue, optional, requires change not affecting execution, requires change affecting execution), locality (whether repair is isolated to the inspected code), and effort spent (- each issue. Afterwards, the author returned all paperwork to us. We used the information from the repair form and interviews with the author to classify each issue as a true defect (if the author was required to make an execution affecting change to resolve it), soft maintenance issue (any other issue which the author fixed), or false positive (any issue which required no action). In the course of the experiment, several treatments were discontinued because they were either not performing effectively, or were taking too long to complete. These were the 1-session, 1-person treatment and all 2-session treatments which required repair between sessions. After months, we managed to collect data from 88 inspections, with a combined total of 130 collection meetings and 233 individual preparation reports. The entire data set may be examined online at http://www.cs.umd.edu/users/harvey/variance.html. 2.3 Self-Reported Data Self-reported data tend to contain systematic errors. Therefore we minimized the amount of self-reported data by employing direct observation[19] and interviews[2]. The IQE attended 125 of the 5 A sample of this, and all other forms we used may be found at http://www.cs.umd.edu/users/harvey/ variance.html. collection meetings 6 to make sure the meeting data was reported accurately and that reviewers do not mistakenly add to their preparation forms any issues that were not found until collection. We also made detailed field notes to corroborate and supplement some of the data in the meeting forms. The repair information was verified through interviews with the author, who completed the form. Our defect classification was not made available to the reviewers or the authors to avoid biasing them. Among the data that remained self-reported were the amount of preparation time and pre-inspection testing time expended. We had two concerns in dealing with these data: a participant might deliberately fail to tell the truth (e.g., reporting 2 hours preparation time when he or she really did not prepare at all); participants might make errors in recording data (e.g., reporting 2 hours of preparation time when the correct figure was 1.9 hours). During the experiment, the IQE had an office next to those of the compiler development team, and after working with the team for months, a great deal of trust was built up. Also, the development environment routinely collects self-reported data, which is unavailable to management at the individual level. Thus developers are conditioned to answer as reliably as they can. We therefore see no reason to suspect that participants ever deliberately misrepresented their data. As for the element of error, previous observational studies on time usage conducted in this environment have shown that although there are always inaccuracies in self-reported data, the self-reported data is generally within 20% of the observed data[18]. 2.4 Results of the Experiment Our experiment produced three general results: 1. Inspection interval and effectiveness of defect detection were not significantly affected by team size (large vs. small). 2. Inspection interval and effectiveness of defect detection were not significantly affected by number of sessions (single vs. multiple). 3. Effectiveness of defect detection was not improved by performing repairs between sessions of two-session inspections. However, inspection interval was significantly increased. From this we concluded that single-session inspections by small teams were the most efficient, since their defect detection rate was as good as that of other formats, and inspection interval was the same or less. The observed number of defects and the intervals per treatment are shown as boxplots 7 in Figures 1 and 2, respectively. The treatments are denoted [1,or 2] sessions X [1,2, or 4] persons [No- 6 The unattended ones are due to schedule conflicts and illness. 7 We have made extensive use of boxplots to represent data distributions. Each data set is represented by a box whose height spans the central 50% of the data. The upper and lower ends of the box marks the upper and lower quartiles. The data's median is denoted by a bold line within the box. The dashed vertical lines attached to the box indicate the tails of the distribution; they extend to the standard range of the data (1.5 times the inter-quartile range). All other detached points are "outliers."[5] OBSERVED 1sX1p 1sX4p 2sX1pR 2sX2pR Figure 1: Observed Number of Defects by Treatment. The treatment labels are interpreted as follows: the first digit stands for the number of sessions, the second digit stands for the number of reviewers per session, and, for 2-session inspections, the 'R' or `N' suffix indicates "with repair" or "no repair". As seen here, the distributions all seem to be similar except for 1sX1p and 2sX2pR, which were discontinued after 7 and 4 data points, respectively. repair,Repair]. (For example, the label 2sX1pN indicates a two-session, one-person, without-repair inspection.) It can be seen that most of the treatment distributions are similar but that they vary widely within themselves. 3 Sources of Variation 3.1 Process Inputs as Sources of Variation In addition to the process structure, we see that differences in process inputs (e.g., code unit and reviewers) also affects inspection outcomes. Therefore, we will attempt to separate the effects of process inputs from the effects of the process structure. To do this we will estimate the amount of variation contributed by these process inputs. Thus, our first question from Section 1.1 may be refined as, (1) How will our previous results change when we eliminate the contributions due to variability in the process inputs? (2) Did our experimental design spread the variance in process inputs uniformly across treatments? Our second question then becomes, (1) Are the differences due to process inputs significantly larger than the differences in the treatments? (2) If so, what factors or attributes affecting the variability of these process inputs have the greatest influence? Figure 3 is a diagram of the inspection process and associated inputs, e.g., the code unit, the 1sX1p 1sX4p 2sX1pR 2sX2pR INTERVAL (working days) Figure 2: Pre-meeting Interval by Treatment. As seen here, the distributions all seem to be similar except for 2sX2pR, which was significantly higher. Process Inputs Collection Output Inspection Output Reviewers Preparation Collection Issues Code Unit Author Total Defects Repair and Classification Issues Suppressed Issues Meeting Gains Observed Defects Maintenance False Positives Figure 3: A Cause and Effect Diagram of the Inspection Process. The inputs to the process (reviewers, author, and code unit) are shown in grey rectangles on the left, the solid ovals represent process steps, the grouped boxes in between steps show the intermediate outputs of the process. Time flows left to right. reviewers, and the author. It shows how these inputs interact with each process step. This is an example of a cause-and-effect diagram, similar to the ones used in practice[13], but customized here for our use. The number and types of issues raised in the preparation step are influenced by the reviewers selected and by the number of defects originally in the code unit (which in turn may be affected by the author of the code unit). The number and types of issues recorded in the collection step are influenced by the reviewers on the inspection team and the author (who joins the collection meeting), the number of issues raised in preparation, and the number remaining undetected in the Process Inputs Collection Output Inspection Output Reviewers Preparation Collection Issues Prep Time Code Unit Author Total Defects Repair and Classification Discussions Meeting Duration Issues Suppressed Issues Meeting Gains Observed Defects Maintenance False Positives Language Familiarity Application Experience Inspection Experience Type of Change Functionality Code Structure Code Size Pre-inspection Testing Figure 4: The Refined Cause and Effect Diagram. This figure extends the inspection model with some of the factors which we believe to affect reviewer and author performance and code unit quality. code unit. 3.2 Factors Affecting Inspections We considered the factors affecting reviewer and author performance and code unit quality that might systematically influence the outcome of the inspection. (Some of these are shown in Figure 4.) In 3.2.1 through 3.2.3, we examine these factors, explain how they might influence the number of defects, and discuss confounding issues. 8 As we examine them, we caution the reader from making conclusions about the significance of any factor as a source of variation. The goal here is to establish possible mechanisms, not to test significance of correlation. Each plot is meant to be descriptive, showing the relationship of a factor against the number of defects, without eliminating the influence of potentially confounding factors. The actual test of a factor's significance will be carried out when the model is built. (See Section 4.) 3.2.1 Code Unit Factors Some of the possible variables affecting the number of defects in the code unit include: size, author, time period when it was written, and functionality. 8 We did not have any readily available measure of experience nor code complexity, so we did not include them in our analysis. OBSERVED Figure 5: Size vs. Defects Found. This is a scatter plot showing the relation between the size of the code and the number of defects found (cor = 0.40). The line indicates the trend of the data. Note that the plot was "jittered" - a small, random offset was added to each point - to expose overlapping points. (In fact, every scatter plot in this paper that may have overlapping points was jittered.) Code Size. The size of a code unit is given in terms of non-commentary source lines (NCSL). It is natural to think that, as the size of the code increases, the more defects it will contain. From Figure 5 we see that there is some correlation between size and number of defects found 0.4). 9 Author. The author of the code may inadvertently inject defects into the code unit. There were 6 authors in the project. Figure 6 is a boxplot showing the number of defects found, grouped according to the code unit's author. The number of defects could depend on the author's level of understanding and implementation experience. Development Phase. The performance of the reviewers and the number of defects in the code unit at the time of inspection might well depend also on the state of the project when the inspection was held. Figure 7 is a plot of the total defects found in each inspection, in chronological order. Each point was plotted in the order the code unit became available for inspection. There are two distinct distributions in the data. The first calendar quarter of the project (July - September 1994) - which has about a third of the inspections - has a significantly higher mean than the remaining period. This coincided with the project's first integration build. With the front end in place, the development team could incrementally add new code units to the system, possibly with a more precise idea of how the new code is supposed to interact with the integrated system, resulting in fewer misunderstandings and defects. In our data, we tagged each code unit as being from "Phase 1" if they were written in the first quarter and "Phase 2" otherwise. 9 Correlations calculated in this paper are Pearson correlation coefficients. OBSERVED Figure In Authors' Code Units. These boxplots show the total defects found in each inspection, grouped according to the code unit's author. At the end of Phase 1, we met with the developers to evaluate the impact of the experiment on their quality and schedule goals. We decided to discontinue the 2-session treatments with repair because they effectively have twice the inspection interval of 1-session inspections of the same team size. We also dropped the 1-session, 1-person treatment because inspections using it found the lowest number of defects. Figure 8 shows a time series plot of the number of issues raised for each code unit inspection. While the number of true defects being raised dropped as time went by, the total number of issues did not. This might indicate that either the reviewers' defect detection performance were deteriorating in time, or the authors were learning to prevent the true defects but not the other kinds of issues being raised. Functionality. Functionality refers to the compiler component to which the code unit belongs, e.g., parser, symbol table, code generator, etc. Some functionalities may be more straightforward to implement than others, and, hence, will have code units with lower number of defects. Figure 9 is a boxplot showing the number of defects found, grouped according to functionality. Table 1 shows the number of code units each author implemented within each functional area. Because of the way the coding assignments were partitioned among the development team, the effects of functionality are confounded with the author effect. For example, we see in Figure 9 that SymTab has the lowest number of defects found. However, Table 1 shows that almost all the code units in SymbTab were written by author 6, who has the lowest number of reported defects. Nevertheless, we may still be able to speculate about the relative impact of the two factors by examining those functionalities with more than one author (CodeGen) and authors implementing more than one functionality (author 6). In addition, functionality is also confounded with development phase as Phase 1 had most of the INSPECTIONS OBSERVED DEFECTS515Jul 94 Dec 94 Mar 95 Jun 95 Figure 7: Defects Detected Over Time. This is a plot of the inspection results in chronological order showing the trends in number of defects found over time. The vertical lines partition the plot into calendar quarters. Within each quarter, the solid horizontal line marks the mean of that quarter's distribution. The dashed lines mark one standard deviation above and below the mean. Author CodeGen 8 6 8 6 28 Report 3 3 I/O 9 9 Library 12 12 Misc 11 11 Parser 4 4 Table 1: Assignment of Authors to Functionality. Each cell gives the number of code units implemented by an author for a functionality. code for the front end functionalities (input-output, parser, symbol table) while Phase 2 had the back end functionalities (code generation, report generation, libraries). Because author, phase, and functionality are related, they cannot all be considered in the model as they account for much of the same variation. In the end, we selected functionality as it is the easiest to explain. Pre-inspection Testing. The code development process employed by the developers allowed Inspections Total Issues Recorded2060Jul 94 Dec 94 Mar 95 Jun 95 Figure 8: Number of Issues Recorded Over Time. This is a time series plot showing the trends in number of issues being recorded over time. The vertical lines partition the plot into quarters. Within each quarter, the solid horizontal line marks the mean of that quarter's distribution. The dashed lines mark one standard deviation above and below the mean. Note that the scale of the y-axis is different from the previous figure. them to perform some unit testing before the inspection. Performing this would remove some of the defects prior to the inspection. Figure 10 is a scatter plot of pre-inspection testing effort against observed defects in inspection (cor = 0.15). One would suspect that the number of observed defects would go down as the amount of pre-inspection testing goes up, but this pattern is not observed in Figure 10. A possible explanation to this is that testing patterns during code development may have changed across time. As the project progressed and a framework for the rest of the code was set up, it may have become easier to test the code incrementally during coding. This may result in code which has different defect characteristics compared to code that was written straight through. It would be interesting to do a longitudinal study to see if these areas had high maintenance cost. 3.2.2 Reviewer Factors Here we examine how different reviewers affect the number of defects detected. Note that we only look at their effect on the number of defects found in preparation, because their effect as a group is different in the collection meeting's setting. Reviewer. Reviewers differ in their ability to detect defects. Figure 11 shows that some reviewers find more defects than others. 10 Even for the same code unit, different reviewers may find different In addition to the 11 reviewers, 6 more developers were called in at one time or another to help inspect 1 or 2 pieces of code, mostly to relieve the regular pool during the peak development periods. We did not include them in CodeGen Report I/O Library Misc Optimizer Parser SymTab FUNCTIONALITY OBSERVED Figure 9: Defects Found In Code Units Classified by Functionality. These boxplots show the total defects found in each inspection, grouped according to the code unit's functionality. Note that specific authors were assigned to implement specific portions of the project's functionalities, so the effects of functionality is usually not separable from that of authors - the independent factors of author and functionality are confounded. For example, SymTab, which has the lowest number of defects found, was implemented by author 6, who has the lowest number of reported defects. numbers of defects (Figure 12). This may be because they were looking for different kinds of issues. Reviewers may raise several kinds of issues, which may either be suppressed at the meeting, or classified as true defects, soft maintenance issues (issues which required some non-behavior- affecting changes in the code, like adding comments, enforcing coding standards, etc.), or false positives (issues which were not suppressed at the meeting, but which the author later regarded as non-issues). Figure 13 shows the mean number of issues raised by each reviewer as well as the percentage breakdown per classification. We see that some of the reviewers with low numbers of true defects (see Figure 11), like Reviewers H and I, simply do not raise many issues in total. Others, like Reviewers J and K, raise many issues but most of them are suppressed. Still others, like Reviewers E and G, raise many issues but most turn out to be soft maintenance issues. The members of the development team (Reviewers A to F) raise on average more total issues (see left plot in Figure 13), though a very high percentage turn out to be soft maintenance issues (see right plot in Figure 13), possibly because, as authors of the project, they have a higher concern for its long-term maintainability than the rest of the reviewers. An exception is Reviewer F, who found almost as many true defects as soft maintenance issues. Preparation Time. The amount of preparation time is a measure of the amount of effort the reviewer put into studying the code unit. For this experiment, the reviewers were not instructed to follow any prescribed range of preparation time, but to study the code in as much time as they think they need. Figure 14 plots preparation time against defects found, showing a positive trend but little correlation this analysis because they each had too few data points. PRE-INSPECTION TESTING (HOURS) OBSERVED Figure 10: Pre-inspection Testing Effort vs. Defects Found. This is a scatter plot showing how the amount of pre-inspection testing related to the number of defects found in inspection (cor 0.15). Note that the pre-inspection testing data was self-reported by the author. Points cluster at the quarter hours because we asked the authors to only record to that precision. Even if preparation time is found to be a significant contributor, it must be noted that preparation time depends not only on the amount of effort the reviewer is planning to put into the preparation, but also on factors related to the code unit itself. In particular, it is influenced by the number of defects existing in the code, i.e., the more defects he finds, the more time he spends in preparation. Hence, high preparation time may be considered a consequence, as well as a cause, of detecting a large number of defects. Further investigation is needed to quantify the effect of preparation time on defects found as well as the effect of defects found on preparation time. Because there is no way to tell how much of the preparation time was due to reviewer effort or number of defects, we decided not to include it in the model. This is also in keeping with our analysis philosophy to only consider factors that occur strictly before the response. (See Section 1.2.) 3.2.3 Team Factors Team-specific variables also add to the variance in the number of meeting gains. Team Composition. Since different reviewers have different abilities and experiences, and possibly interact differently with each other, different teams also differ in combined abilities and experiences Apparently, this mix tended to form teams with nearly the same performance. This is illustrated in Figure which shows number of defects found by different 2-person teams in each 2sX2pN inspection. Most of the time, the two teams found nearly the same number of defects. This may be due to some interactions going on between team members. However, because teams are formed randomly, there are only a few instances where teams composed of the same people were formed REVIEWER IN PREPARATION Figure 11: Number of Defects in Preparation per Reviewer. This plot shows the number of true defects found in preparation by each reviewer. more than once, not enough to study the interactions. We incorporated the team composition into the model by representing it as a vector of boolean variables, one variable per reviewer in the reviewer pool. When a particular reviewer is in that collection meeting, his corresponding variable is set to "True". Meeting Duration. The meeting duration is the number of hours spent in the meeting. In the one person is appointed the reader, and he reads out the code unit, paraphrasing each chunk of code. The meeting revolves around him. At any time, reviewers may raise issues related to the particular chunk being read and a discussion may ensue. All these contribute toward the pace of the meeting. The meeting duration is positively correlated with the number of meeting gains, as shown in Figure 16 (cor = 0.57). As with the case of preparation time, the meeting duration is partly dependent on the number of defects found, as detection of more defects may trigger more discussions, thus lengthening the duration. It is also dependent on the complexity or readability of the code. Further investigation is needed to determine how much of the meeting duration is due to the team effort independent of the complexity and quality of the code being inspected. For similar reasons as with preparation time (see the previous discussion on preparation time), we did not include this in the model. Combined Number of Defects Found in Preparation. The number of defects already found going into the meeting may also affect the number of defects found at the meeting. Each reviewer gets a chance to raise each issue he found in preparation as a point of discussion, possibly resulting in the detection of more defects. Figure 17 shows some correlation between number of defects found in the preparation and in the meeting (cor = 0.4). INSPECTIONS OBSERVED IN PREPARATION PER Figure 12: Reviewer Performance Per Inspection. This shows the number of defects found in preparation by each reviewer, grouped according to inspection. Each column represents one inspec- tion. The points in that column represent the number of true defects reported during preparation by each reviewer in that inspection. The columns were ordered according to increasing means. 4 A Model of Inspection Effectiveness 4.1 Building the Model To explain the variance in the defect data, we built statistical models of the inspection process, guided by what we knew about it. Model building involves formulating the model, fitting the model, and checking that the model adequately characterizes the process. We built the models in the S programming language[3, 6]. Using the factors described in the previous section, we modeled the number of defects found with a generalized linear model (GLM) from the Poisson family. 11 We started with a model which had all code unit factors, all reviewers, and the original treatment factors, represented by the following RA +RB +RC +RD +RE +R F +RG +RH +R I +R J +RK (1) In this model, Functionality and Author are categorical variables represented in S as sets of dummy 11 The generalized linear model and the rationale for using it are explained in Appendix C. We used S language notation to represent our models[6, pp. 24-31]. For example, the model formula y - is read as, "y is modeled by a, b, and c." SUPPRESSED ISSUES NUMBER OF ISSUES RAISED IN PREPARATION REVIEWER OF ISSUE Figure 13: Classification of Issues Found in Preparation. The bar graph on the left shows the mean number of issues found in preparation by each reviewer, broken down according to issue classification. The bar graph on the right shows the percentage breakdown. PREPARATION TIME (HOURS) OBSERVED IN PREPARATION Figure 14: Preparation Time vs. Defects Found In Preparation. This is a scatter plot showing how the amount of preparation time related to the number of defects found in preparation pp. 20-22,32-36]. They have 7 and 5 degrees of freedom, respectively. Stepwise model selection heuristic 13 selected the following model. 13 Stepwise model selection techniques are a heuristic to find the best-fitting models using as few parameters as possible. To avoid overfitting the data, the number of parameters must always be kept small or the residual degrees of freedom high. To perform stepwise model selection we used the step() function in S[6, pp. 233-238]. INSPECTIONS (2sX2pN only) OBSERVED PER Figure 15: Team Performance Per Inspection (2sX2pN only). This shows the total number of defects found per session in each 2sX2pN inspection. Each column represents one inspection. The points in that column represent the total number of true defects reported in preparation and meeting by each team. "1" and "2" plot the number of defects found by the first and second teams, respectively. The columns are ordered by mean defects found. RB +RC +R F +RG +RH +R I This resulting model is not satisfactory because it retained many factors, making it difficult to interpret. Also, even though these factors were considered important by the stepwise selection criteria, some of them do not explain a lot of the variance. So we increased the selection threshold to produce a smaller model. 14 Increasing the selection threshold did not simplify the model initially, until, at one point, a large number of factors were suddenly dropped. The resulting model then was: It must be noted that the factors left out of the model are not necessarily unimportant. We believe that there are other possible models for our data. In particular, Phase was considered important. Phase is a surrogate variable representing the change in defects being found over time. Figure 7 clearly showed that something had changed over time but it is not clear what caused it. The reason why this change over time explains a significant part of the variability may be attributable to other factors. It is not clear which mechanism explains why Phase affects the number of defects. 14 In S, increase the scale parameter of the step() function. OBSERVED IN Figure Meeting Duration vs. Defects Found in Meeting. This is a scatter plot showing how the amount of time spent in the meeting related to the number of defects found in the meeting We also knew that Phase was confounded with Functionality (e.g., parser was implemented before code generator). Since we knew also that some parts of the compiler are harder to implement than others, the effects due to Functionality are easier to interpret than the effects due to Phase. Thus we replaced Phase by Functionality in our final model: The analysis of variance for this model is in Table 2. For comparison, the treatment factors were added to the model. See Appendix C for details on calculating the significance values. The resulting model explains - 50% of the variance using just 10 degrees of freedom. In this model, Defects is the number of defects found in each of the 88 inspections. Note that the presence of certain reviewers (Reviewers B and F) in the inspection team strongly affects the outcome of the inspection. (See Table 2.) Note also the log transformation on the Code Size factor. We do not really know what the actual underlying functional relationship is between Code Size and Defects and so we applied square root, logarithmic, and linear transformations. Code Size explained more variance under the log transformation than under other transformations. Figure diagnostic plots of the model's goodness-of-fit. The left plot shows the values estimated by the model compared to the original values. It shows that the model reasonably estimates the number of defects. The right plot shows the values estimated by the model compared to the residuals. The residuals appear to be independent of the fitted values, suggesting that the residuals are random. DEFECTS OBSERVED IN PREPARATION OBSERVED IN Figure 17: Defects Found in Preparation vs. Defects Found in Meeting. This is a scatter plot showing how the combined amount of defects found in the preparation related to the number of defects found in the meeting (cor = 0.4). 4.2 Lower Level Models The inspection model is a high level description of the inspection defect detection process. The effects of the process input and of the process structure can be compared using this model. But we also know that defect detection in inspections is performed in two steps: preparation and collection. These two steps may be considered as independent processes which can be modeled separately. Doing so has several advantages. We can understand the resulting models of the simpler separate processes better than the model for the composite inspection process. In addition, there are more data points to fit - 233 individual preparations and 130 collection meetings, as opposed to 88 inspections. 4.2.1 A Model for Defect Detection During Preparation To build the preparation model, we started with the same variables as in inspection model 1. Since the same code unit was inspected several times, we added a categorical variable, CodeUnit, to the regression model. CodeUnit is a unique ID for each code unit inspected. Using stepwise model selection, we selected the variables that significantly affect the variance in the preparation data. These were Functionality, Size, and Reviewers B, E, F, and J. This is represented by the model formula: repDefects In this model, P repDefects is the number of defects found in each of the 233 preparation reports. Factor Degrees of Sum of F Value Pr(F) Effect Freedom Squares Treatment Team Size 2 2.65 0.50 0.6062 factors Sessions 1 1.12 0.43 0.5146 Input log(Code Size) 1 59.63 22.66 factors Functionality 7 43.76 2.38 0.0303 Residuals 73 192.11 Table 2: Factors Affecting Inspection Effectiveness. The sum of squares measure the relative contribution of each factor to the variance of the defect data. The probabilities indicate the significance of the contribution. The last column for each significant scalar factor indicates whether the factor was a positive or negative contributor to the number of defects. (Functionality had 7 degrees of freedom and different functionalities had different effects.) SQRT(FITTED VALUE) Figure 18: Examining the fit of the model. The left plot compares the values estimated by the model with the original values (a perfect fit would imply that everything is on the line There is a substantial correlation between the two 0.69). The right plot shows the relation of the fitted values to the residuals. The residuals appear to be independent of the fitted values. The presence of all the significant factors from the overall model at this level gives us more confidence on the validity of the overall model. 4.2.2 A Model for Defect Detection During Collection We started with the same variables as in preparation model. (See previous section.) Using stepwise model selection to select the variables that significantly affect the meeting data we ended up with Functionality, Size, and the presence of Reviewers B, F, H, J, and K. This is represented by the RESIDUAL (a) NUMBER OF SESSIONS RESIDUAL (b) REPAIR POLICY RESIDUAL (c) Figure 19: Examining the Significance of the Experimental Treatment Factors. These three panels depict the distribution of the residual data grouped according to Team Size, Sessions, and Repair. model formula: MeetingGains In this model, MeetingGains is the number of defects found in each of the 130 collection meetings. This is again consistent with the previous two models. 4.3 Answering the Questions We are now in a position to answer the questions raised in Section 3.1 with respect to inspection effectiveness. 4.3.1 Will previous results change when process inputs are accounted for? In this analysis, we build a GLM composed of the significant process input factors plus the treatment factors and check if their contributions to the model would be significant. The effect of increasing team size is suggested by plotting the residuals of the overall inspection model, grouped according to Team Size (Figure 19(a)). We observe no significant difference in the distributions. When we included the Team Size factor into the model, we saw that its contribution was not significant (p = 0:6, see Table 2). 15 Appendix C Section C.3.1 describes how Tables 2 and 3 were constructed. The effect of increasing sessions is suggested by plotting the residuals of the overall inspection model, grouped according to Session (Figure 19(b)). We observe no significant difference in the distributions. When we included the Session factor into the model, we saw that its contribution was not significant (p = 0:5). The effect of adding repair is suggested by plotting the residuals of the overall inspection model (for those inspections that had 2 sessions), grouped according to Repair policy (Figure 19(c)). We observe no significant difference in the distributions. When we included the Repair factor into the model, we saw that its contribution was not significant (p = 0:2). 4.3.2 Did design spread process inputs uniformly across treatments? We want to determine if the factors of the process inputs which significantly affect the variance are spread uniformly across treatments. This is useful in evaluating our experimental design. Although randomization guarantees that the long run distribution of the factors will be independent of the treatments, we had a single set of 88 data points. Thus we felt it is important to know of any imbalances in this particular randomization. As an informal sanity check we took each of the significant factors in the overall inspection model and tested if they are independent of the treatments. For each factor, we built a contingency table, showing the frequency of occurrence of each value of that factor within each treatment. We then used Pearson's - 2 -test for independence[4, pp. 145-150]. If the result is significant, then the factor is not independently distributed across the treatments. Although the counts in the table cells are too low for this - 2 -test to be valid, we use it as informal means to indicate gross nonuniformities in the assignment of treatments. Results show that the distribution of Reviewer B is independent of treatment Functionality may be unevenly assigned to treatments. Examining further shows us that Reviewer F never got to do any 1sX1p inspections, and that Functionality was not distributed evenly because some functionalities were implemented earlier than others, when there were more treatments. Contingency tables only work with data which have discrete values. To test the independence of log(Size) to treatment, we modeled it instead with a linear model, log(Size) - T reatment, to determine if treatment contribution to log(Size) is significant. The ANOVA result that it is not, indicating that there is no dependence between code sizes and treatment. 4.3.3 Are differences due to process inputs larger than differences due to process structure? Table 2 shows the analysis of variance for our model. The significance of the treatment factors' contribution were included for comparison. The table shows that differences in code units and reviewers drive inspection performance more than differences in any of our treatment variables. This suggests that relatively little improvement in effectiveness can be expected of additional work on manipulating the process structure. 4.3.4 What factors affecting process inputs have the greatest influence? The dominance of process inputs over process structure in explaining the variance also suggests that more improvements in effectiveness can be expected by studying the factors associated with reviewers and code units that drive inspection effectiveness. Differences in code units strongly affect defect detection effectiveness. Therefore, it is important to study the attributes that influence the number of defects in the code unit. Of the code unit factors we studied, code size was the most important in all the models. This is consistent with the accepted practice of normalizing the defects found by the size of the code. The next most important factor is functionality. This may indicate that code functionalities have different levels of implementation difficulty, i.e., some functionalities are more complex than others. Because functionality is confounded with authors, it may also be explained by differences in authors. And because it is also confounded with development phase, another possible explanation is that code functionalities implemented later in the project may have less defects due to improved understanding of requirements and familiarity with implementation environment. The choice of people to use as reviewers strongly affects the defect detection effectiveness of the inspection. The presence of certain reviewers (in particular, Reviewer F) is a major factor in all the models. It suggests that improvements in effectiveness may be expected by selecting the right reviewers or by studying the characteristics and background of the best reviewers and the implicit techniques by which they study code and detect defects. 5 A Model of Inspection Interval Using the same set of factors, we also built a statistical model for the interval data. We measured the interval from submission of the code unit for inspection up to the holding of the collection meeting. Unlike defect detection, we do not see any further decomposition of the inspection process that drives the interval. The author schedules the collection meeting with the reviewers and the reviewers spend some time before the meeting to do their preparation. So instead of splitting the inspection process into preparation and collection, we just modeled the interval from submission to meeting. A linear model was constructed from the factors described in the previous section. 16 We started by modeling interval with the same initial set of factors as in the previous section. Using stepwise model selection heuristic we arrived at the following model. Even though we ended up with a small set of factors, the model was hard to interpret. It did not make sense for Functionality to be an important factor influencing the length of the inspection interval. In addition Functionality and Phase were confounded so they may be explaining part of the same variance. Our belief was that they were masking the effect of the other confounded The linear model was used here rather than the generalized linear model because the original interval data approximates the normal distribution. Factor Degrees of Sum of F Value Pr(F) Effect Freedom Squares Treatment Team Size 2 206.6 0.85 0.4308 factors Sessions 1 161.6 1.28 0.2619 Input Author 5 2195.0 3.62 0.0054 factors R I 1 242.1 2.00 Residuals 77 9340.86 Table 3: Factors Affecting Interval. The sum of squares measure the deviation contributed by each factor to the mean of the interval data. The probabilities indicate the significance of the contribution. The last column for each scalar factor in the model indicates whether the factor was a positive or negative contributor to the interval. (Author had 5 degrees of freedom and different authors had different effects.) factor, Author. It makes more sense for Author to be in the model since he is the central person coordinating the inspection. So we re-ran the stepwise model selection heuristic, instructing it to always retain the Author factor. The result was: Interval - Author +R I +Repair In this model, Interval is the number of days from availability of code unit for inspection up to the last collection meeting. The analysis of variance for this model is in Table 3. For comparison, all the treatment factors were added to the model. The model explains - 25% of the variance using just 7 degrees of freedom. The low explanatory power of the model indicates the limited extent to which structure and inputs affect interval and suggests that other factors (that were not observed in this study) are more important in determining the interval. The presence of Repair confirms our earlier experimental result stating that adding repair in between inspections increases the interval. 5.1 Model Checking Figure 20 gives diagnostic plots of the model's goodness-of-fit. The left plot shows the values estimated by the model compared to the original values. Because the model only explains 25% of the variance, it has limited predictive capabilities. The right plot shows the values estimated by the model compared to the residuals. The residuals appear to be independent of the fitted values. 5.2 Answering the Questions We are now in a position to answer the questions raised in Section 3.1, with respect to inspection FITTED VALUE FITTED VALUE Figure 20: Examining the fit of the model. The left plot compares values estimated by the model with the original values (a perfect fit would imply that everything is on the line There is some correlation between the two 0.48). The right plot shows the relation of the fitted values to the residuals. The residuals appear to be independent of the fitted values. 5.2.1 Will previous results change when process inputs are accounted for? In this analysis, we build a linear model, composed of the significant process input factors plus the treatment factors and check if their contributions to the model are significant. The effect of increasing team size is suggested by plotting the residuals of the interval model consisting only of input factors, grouping them according to Team Size (Figure 21(a)). We observe no significant difference in the distributions. When we included the Team Size factor into the model, we saw that its contribution was not significant (p = 0:4, see Table 3). The effect of increasing sessions is suggested by plotting the residuals of the interval model consisting only of input factors, grouping them according to Session (Figure 21(b)). We observe no significant difference in the distributions. When we included the Session factor into the model, we saw that its contribution was not significant (p = 0:3). The effect of adding repair is suggested by plotting the residuals of the interval model consisting only of input factors (for those inspections that had 2 sessions), grouping them according to Repair policy (Figure 21(c)). We have already seen that Repair has a significant contribution to the model in the previous section and this is supported by the plot. 5.2.2 Are differences due to process inputs larger than differences due to process structure? Table 3 shows the factors affecting inspection interval and the amount of variance in the interval that they explain. We can see that some treatment factors and some process input factors contribute significantly to the interval. Among treatment factors Repair contributes significantly to the interval. This shows that while changes in process structure do not seem to affect defect detection, it does affect interval. RESIDUAL INTERVAL (a) NUMBER OF SESSIONS RESIDUAL INTERVAL (b) REPAIR POLICY RESIDUAL INTERVAL (c) Figure 21: Examining the Significance of the Experimental Treatment Factors. These three panels depict the distribution of the residual data grouped according to Team Size, Sessions, and Repair. 5.2.3 What factors affecting process inputs have the greatest influence? The results of modeling interval show that process inputs explain only - 25% of the variance in inspection interval even after accounting for process structure factors. Clearly, other factors, apart from the process structure and inputs affect the inspection interval. Some of these factors may stem from interactions between multiple inspections, developer and reviewer calendars, and project schedule and may reveal a whole new class of external variation which we will call the process environment. These are beyond the scope of the data we observed for this study but they deserve further investigation. 6 Conclusions 6.1 Intentions and Cautions Our intention has been to empirically determine the influence upon defect detection effectiveness and inspection interval resulting from changes in the structure of the software inspection process (team size, number of sessions, and repair between multiple sessions). We have extended the analysis to study as well the influence of process inputs. All our results were obtained from one project, in one application domain, using one language and environment, within one software organization. Therefore we cannot claim that our conclusions have general applicability until our work has been replicated. We encourage anyone interested to do so, and to facilitate their efforts we have described the experimental conditions as carefully and thoroughly as possible and have provided the instrumentation online. (See http://www.cs.umd.edu/users/harvey/variance.html.) 6.2 The Ratio of Signal to Noise in the Experimental Data Our proposed models of the inspection process proved useful in explaining the variance in the data gathered from our previous experiment. From them we could show that the variance was caused mainly by factors other than the treatment variables. When the effects of these other factors were removed, the result was a data set with significantly reduced variance across all of the treatments, which improved the resolution of our experiment. After accounting for the variance (noise) caused by the process inputs, we showed that the results of our previous experiment do not change (we see the same signal). This has several implications for the design and analysis of industrial experiments. Past studies have cautioned that wide variation in the abilities of individual developers may mask effects due to experimental treatments[8]. However, even with our relatively crude models, we managed to devise a suitable means of accounting for individual variation when analyzing the experimental results. But ultimately, we will get better results only if we can identify and control for factors affecting reviewer and author performance. Note also that the overall drop in defect data over time (see Figure 7) underscores the fact that researchers doing long term studies must be aware that some characteristics of the processes they are examining may change during the study. 6.3 The Need for a New Approach to Software Inspection When process inputs are accounted for, the results of the experiment show that differences in process structure have little effect on defect detection. This reinforces the results of our previous experiment. That work showed that single session inspection by a small team is the most efficient structure for the software inspection process (fewest personnel and shortest interval, with no loss of effectiveness-see summary in Section 2.4 above). If this is the case, and we believe that it is, then further efforts to increase defect detection rates by modifying the structure of the software inspection process will produce little improvement. Researchers should therefore concentrate on improving the small-team-single-session process by finding better techniques for reviewers to carry it out (e.g., systematic reading techniques[1] for the preparation step, meetingless techniques[9, 17, 11] for the collection step, etc. 7 Future Work 7.1 Framework For Further Study Our study revealed a number of influences affecting variation in the data, some internal and some external to the inspection process. Internal sources included factors from the process structure (the manner in which the steps are organized into a process, e.g., team sizes, number of sessions, etc.), and from the process techniques (the manner in which each step is carried out, the amount of effort expended, and the methods used, e.g., reading techniques, computer support, etc. External sources included factors from the process inputs (differences in reviewers' abilities and in code unit quality) and from the process environment (changes in schedules, priorities, workload, etc. 7.2 Premise for Improving Inspection Effectiveness We believe that to develop better inspection methods we no longer need to work on the way the steps in the inspection process are organized (structure), but must now investigate and improve the way they are carried out by reviewers (technique). 7.3 Need for Continued Study of Inspection Interval We have not yet adequately studied the factors affecting interval data. Some of the factors are found in process structure (specifically repairing in between sessions) and process inputs, but much of its variance is still unaccounted for. To address this, we must examine the process environment, including workloads, deadlines, and priorities. Acknowledgments We would like to recognize Stephen Eick and Graham Wills for their contributions to the statistical analysis. Art Caso's editing is greatly appreciated. --R IEEE Trans. A methodology for collecting valid software engineering data. The New S Language. Graphical Methods For Data Analysis. Hastie, editors. Statistical Models in S. Model uncertainty Substantiating programmer variability. Computer brainstorms: More heads are better than one. Estimating software fault content before coding. An instrumented approach to improving software quality through formal technical review. Correlation and Causality. Successful Industrial Experimentation Software research and switch software. Two application languages in software produc- tion Generalized Linear Models. Electronic meeting systems to support group work. Experimental software engineer- ing: A report on the state of the art Understanding and improving time usage in software development. An experiment to assess the cost-benefits of code inspections in large scale software development Assessing software designs using capture-recapture methods --TR A Two-Person Inspection Method to Improve Programming Productivity Electronic meeting systems An experimental study of fault detection in user requirements documents Estimating software fault content before coding An improved inspection technique Assessing Software Designs Using Capture-Recapture Methods An experiment to assess the cost-benefits of code inspections in large scale software development Experimental software engineering An instrumented approach to improving software quality through formal technical review Active design reviews Statistical Models in S Comparing Detection Methods for Software Requirements Inspections --CTR Frank Padberg, Empirical interval estimates for the defect content after an inspection, Proceedings of the 24th International Conference on Software Engineering, May 19-25, 2002, Orlando, Florida Miyoung Shin , Amrit L. Goel, Empirical Data Modeling in Software Engineering Using Radial Basis Functions, IEEE Transactions on Software Engineering, v.26 n.6, p.567-576, June 2000 Dewayne E. Perry , Adam Porter , Michael W. Wade , Lawrence G. Votta , James Perpich, Reducing inspection interval in large-scale software development, IEEE Transactions on Software Engineering, v.28 n.7, p.695-705, July 2002 Trevor Cockram, Gaining Confidence in Software Inspection Using a Bayesian Belief Model, Software Quality Control, v.9 n.1, p.31-42, January 2001 Stefan Biffl , Michael Halling, Investigating the Defect Detection Effectiveness and Cost Benefit of Nominal Inspection Teams, IEEE Transactions on Software Engineering, v.29 n.5, p.385-397, May Oliver Laitenberger , Colin Atkinson, Generalizing perspective-based inspection to handle object-oriented development artifacts, Proceedings of the 21st international conference on Software engineering, p.494-503, May 16-22, 1999, Los Angeles, California, United States Oliver Laitenberger , Thomas Beil , Thilo Schwinn, An Industrial Case Study to Examine a Non-Traditional Inspection Implementation for Requirements Specifications, Empirical Software Engineering, v.7 n.4, p.345-374, December 2002 James Miller , Fraser Macdonald , John Ferguson, ASSISTing Management Decisions in the Software Inspection Process, Information Technology and Management, v.3 n.1-2, p.67-83, January 2002 Lionel C. Briand , Khaled El Emam , Bernd G. Freimut , Oliver Laitenberger, A Comprehensive Evaluation of Capture-Recapture Models for Estimating Software Defect Content, IEEE Transactions on Software Engineering, v.26 n.6, p.518-540, June 2000 Bruce C. Hungerford , Alan R. Hevner , Rosann W. Collins, Reviewing Software Diagrams: A Cognitive Study, IEEE Transactions on Software Engineering, v.30 n.2, p.82-96, February 2004 Andreas Zendler, A Preliminary Software Engineering Theory as Investigated by Published Experiments, Empirical Software Engineering, v.6 n.2, p.161-180, June 2001

software process;software inspection;empirical studies;statistical models

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Asynchronous parallel algorithms for test set partitioned fault simulation.

We propose two new asynchronous parallel algorithms for test set partitioned fault simulation. The algorithms are based on a new two-stage approach to parallelizing fault simulation for sequential VLSI circuits in which the test set is partitioned among the available processors. These algorithms provide the same result as the previous synchronous two stage approach. However, due to the dynamic characteristics of these algorithms and due to the fact that there is very minimal redundant work, they run faster than the previous synchronous approach. A theoretical analysis comparing the various algorithms is also given to provide an insight into these algorithms. The implementations were done in MPI and are therefore portable to many parallel platforms. Results are shown for a shared memory multiprocessor.

Introduction Fault simulation is an important step in the electronic design process and is used to identify faults that cause erroneous responses at the outputs of a circuit for a given test set. The objective of a fault simulation algorithm is to find the fraction of total faults in a sequential circuit that is detected by a given set of input vectors (also referred to as fault coverage). In its simplest form, a fault is injected into a logic circuit by setting a line or a gate to a faulty value (1 or 0), and then the effects of the fault are simulated using zero-delay logic simula- tion. Most fault simulation algorithms are typically of O(n 2 ) time complexity, where n is the number of lines in the circuit. Studies have shown that there is little hope of finding a linear-time fault simulation algorithm [1]. In a typical fault simulator, the good circuit (fault-free cir- cuit) and the faulty circuits are simulated for each test vec- tor. If the output responses of a faulty circuit differ from those of the good circuit, then the corresponding fault is detected, and the fault can be dropped from the fault list, speeding up simulation of subsequent test vectors. A fault simulator can This research was supported in part by the Semiconductor Research Corporation under Contract SRC 95-DP-109 and the Advanced Research Projects Agency under contract DAA-H04-94-G-0273 and DABT63-95-C-0069 administered by the Army Research Office. be run in stand-alone mode to grade an existing test set, or it can be interfaced with a test generator to reduce the number of faults that must be explicitly targeted by the test generator. In a random pattern environment, the fault simulator helps in evaluating the fault coverage of a set of random patterns. In either of the two environments, fault simulation can consume a significant amount of time, especially in random pattern test- ing, for which millions of vectors may have to be simulated. Thus, parallel processing can be used to reduce the fault simulation time significantly. We propose in this paper two scalable asynchronous parallel fault simulation algorithms with the test vector set partitioned across processors. This paper is organized as follows. In Section 2, we describe the various existing approaches to parallel fault simulation and we motivate the need for a test set partitioned approach to parallel fault simulation. In Section 3, we discuss our approach to test sequence partitioning. In Section 4, we present the various algorithms that have been implemented including the two proposed asynchronous algorithms. A theoretical analysis of the sequential and parallel algorithms proposed is given in Section 5 to provide a deeper insight into the algorithms. The results are presented in Section 6, and all algorithms are compared. Section 7 is the conclusion. Parallel Fault Simulation Due to the long execution times for large circuits, several algorithms have been proposed for parallelizing sequential circuit fault simulation [2]. A circuit partitioning approach to parallel sequential circuit fault simulation is described in [3]. The algorithm was implemented on a shared-memory multi- processor. The circuit is partitioned among the processors, and since the circuit is evaluated level-by-level with barrier synchronization at each level, the gates at each level should be evenly distributed among the processors to balance the work- loads. An average speedup of 2.16 was obtained for 8 proces- sors, and the speedup for the ISCAS89 circuit s5378 was 3.29. This approach is most suitable for a shared-memory architecture for circuits with many levels of logic. Algorithmic partitioning was proposed for concurrent fault simulation in [4][5]. A pipelined algorithm was developed, and specific functions were assigned to each processor. An estimated speedup of 4 to 5 was reported for 14 processors, based on software emulation of a message-passing multicomputer [5]. The limitation of this approach is that it cannot take advantage of a larger number of processors. Fault partitioning is a more straightforward approach to parallelizing fault simulation. With this approach [6][7], the fault list is statically partitioned among all processors, and each processor must simulate the good circuit and the faulty circuits in its partition. Good circuit simulation on more than one processor is obviously redundant computation. Alterna- tively, if a shared-memory multiprocessor is used, the good circuit may be simulated by just one processor, but the remaining processors will lie idle while this processing is performed, at least for the first time frame. Fault partitioning may also be performed dynamically during fault simulation to even out the workloads of the processors, at the expense of extra inter-processor communication [6]. Speedups in the range 2.4-3.8 were obtained for static fault partitioning over 8 processors for the larger ISCAS89 circuits having reasonably high fault coverages (e.g., s5378 improvements were obtained for these circuits with dynamic fault partitioning due to the overheads of load redistribution [6]. However, in both the static and dynamic fault partitioning approaches, the shortest execution time will be bounded by the time to perform good circuit logic simulation on a single processor. One observation that can be made about the fault partitioning experiments is that larger speedups are obtained for circuits having lower fault coverages [6][7]. These results highlight the fact that the potential speedup drops as the number of faults simulated drops, since the good circuit evaluation takes up a larger fraction of the computation time. The good circuit evaluation is not parallelized in the fault partitioning ap- proach, and therefore, speedups are limited. For example, if good circuit logic simulation takes about 20 percent of the total fault simulation time on a single processor, then by Am- dahl's law, one cannot expect a speedup of more than 5 on any number of processors. Parallelization of good circuit logic simulation, or simply logic simulation, is therefore very important and it is known to be a difficult problem. Most implementations have not shown an appreciable speedup. Parallelizing logic simulation based on partitioning the circuit has been suggested but has not been successful due to the high level of communication required between parallel processors. Recently, a new algorithm was proposed, where the test vector set was partitioned among the processors [8]. We will call this algorithm, SPITFIRE1. Fault simulation proceeds in two stages. In the first stage, the fault list is partitioned among the processors, and each processor performs fault simulation using the fault list and test vectors in its partition. In the second stage, the undetected fault lists from the first stage are combined, and each processor simulates all faults in this list using test vectors in its partition. Obviously, the test set partitioning strategy provides a more scalable implementa- tion, since the good circuit logic simulation is also distributed over the processors. Test set partitioning is also used in the parallel fault simulator Zamlog [9], but Zamlog assumes that independent test sequences are provided which form the par- tition. If only one test sequence is given, Zamlog does not partition it. If, for example, only 4 independent sequences are given, it cannot use more that 4 processors. Our work does not make any assumption on the independence of test sequences and hence is scalable to any number of processors. It was shown in [8] [10] that the synchronous two-stage al- gorithm, SPITFIRE1, performs better than fault partitioned parallel approaches. Other synchronous algorithms, SPIT- FIRE2 and SPITFIRE3, which are extensions of the SPIT- FIRE1 algorithm, were presented in [10]. SPITFIRE3, in particular, is a synchronous pipelined approach which helps in overcoming any pessimism that may exist in a single or two stage approach. We propose in this paper, two new asynchronous algorithms, based on the test set partitioning strategy for parallel fault simulation. We will demonstrate that the asynchronous algorithms perform better than their synchronous counterparts and shall provide reasons for the same. The first algorithm, SPITFIRE4, is a two stage algorithm, and it is a modification of the SPITFIRE1 algorithm described above. It leaves the first stage unchanged, but the second stage is implemented with asynchronous communication between processors. The second algorithm, SPITFIRE5, obviates the need for two stages. The entire parallel fault simulation strategy is accomplished in one stage with asynchronous communication between processors. 3 Test Sequence Partitioning Parallel fault simulation through test sequence partitioning is illustrated in Figure 1. We use the terms test set and test sequence interchangeably here, and both are assumed to be an ordered set of test vectors. The test set is partitioned Example: A Test Sequence of 5n vectors on 5 Processors Test Sequence Processors 3n Figure 1. Test Sequence Partitioning among the available processors, and each processor performs the good and faulty circuit simulations for vectors in its partition only, starting from an all-unknown (X) state. Of course, the state would not really be unknown for segments other than the first if we did not partition the vectors. Since the unknown state is a superset of the known state, the simulation will be correct but may have more X values at the outputs than the serial simulation. This is considered pessimistic simulation in the sense that the parallel implementation produces an X at some outputs which in fact are known 0 or 1. From a pure logic simulation perspective, this pessimism may or may not be acceptable. However, in the context of fault simulation, the effect of the unknown values is that a few faults which are detected in the serial simulation are not detected in the parallel simulation. Rather than accept this small degree of pes- simism, the test set partioning algorithm tries to correct it as much as possible. To compute the starting state for each test segment, a few vectors are prepended to the segment from the preceding seg- ment. This process creates an overlap of vectors between successive segments, as shown in Figure 1. Our hypothesis is that a few vectors can act as initializing vectors to bring the machine to a state very close to the correct state, if not exactly the same state. Even if the computed state is not close to the actual state, it still has far fewer unknown values than exist when starting from an all-unknown state. Results in [8] showed that this approach indeed reduces the pessimism in the number of fault detections. The number of initializing vectors required depends on the circuit and how easy it is to initialize. If the overlap is larger than necessary, redundant computations will be performed in adjacent processors, and efficiency will be lost. However, if the overlap is too small, some faults that are detected by the test set may not be identified, and thus the fault coverage reported may be overly pessimistic. 4 Parallel Test Partitioned Algorithms We now describe four different algorithms for test set partitioned parallel fault simulation. The first two algorithms are parallel single-stage and two-stage synchronous approaches which have been proposed earlier[8][10]. The third and fourth algorithms are parallel two-stage and single-stage asynchronous approaches. 4.1 SPITFIRE0: Single Stage Synchronous Algorith In this approach, the test set is partitioned across the processors as described in the previous section. This algorithm is presented as a base of reference for the various test set partitioning approaches to be described later. The entire fault list is allocated to each processor. Thus, each processor targets the entire list of faults using a subset of the test vectors. Each processor proceeds independently and drops the faults that it can detect. The results are merged in the end. 4.2 SPITFIRE1: Synchronous Two Stage Algorithm The simple algorithm described above is somewhat inefficient in that many faults are very testable and are detected by most if not all of the test segments. Simulating these faults on all processors is a waste of time. Therefore, one can filter out these easy-to-detect faults in an initial stage in which both the set and the test set are partitioned among the processors. This results in a two stage algorithm. In the first stage, each processor targets a subset of the faults using a subset of the test vectors, as illustrated in Figure 2. A large fraction of the F F F F F 5224Partitioning in Stage 1 U Partitioning in Stage 2 U U U U Figure 2. Partitioning in SPITFIRE1 detected faults are identified in this initial stage, and only the remaining faults have to be simulated by all processors in the second stage. This algorithm was proposed in [8]. The overall algorithm is outlined below. 1. Partition test set T among p processors: g. 2. Partition fault list F among p processors: g. 3. Each processor P i performs the first stage of fault simulation by applying T i to F i . Let the list of detected faults and undetected faults in processor P i after fault simulation be C i and U i respectively. 4. Each processor P i sends the detected fault list C i to processor 5. Processor P 1 combines the detected fault lists from other processors by computing now broadcasts the total detected fault list C to all other processors. 7. Each processor P i finds the list of faults it needs to target in the second stage G 8. Reset the circuit. 9. Each processor P i performs the second stage of fault simulation by applying test segment T i to fault list G i . 10. Each processor P i sends the detected fault list D i to processor 11. Processor P 1 combines the detected fault lists from other processors by computing . The result after parallel fault simulation is the list of detected faults C S D, and it is now available in processor P 1 . Note that G equivalent expression for G i . The reason that a second stage is necessary is because every test vector must eventually target every undetected fault if it has not already been detected on some other processor. Thus, the initial fault partitioning phase is used to reduce redundant work that may arise in detecting easy-to- detect faults. It can be observed though that one has to perform two stages of good circuit simulation with the test segment on any processor. However, the first stage eliminates a lot of redundant work that might have been otherwise per- formed. Hence, the two-stage approach is preferred. The test set partitioning approach for parallel fault simulation is subject to inaccuracies in the fault coverages reported only when the circuit cannot be initialized quickly from an unknown state at the beginning of each test segment. This problem can be avoided if the test set is partitioned such that each segment starts with an initialization sequence. The definitive redundant computation in the above approach is the overlap of test segments for good circuit simu- lation. However, if the overlap is small compared to the size of the test segment assigned to a processor, then this redundant computation will be negligible. Another source of redundant computation is in the second stage when each processor has to target the entire list of faults that remains (excluding the faults that were left undetected in that processor). In this situ- ation, when one of the processors detects a fault, it may drop the fault from its fault list, but the other processors may continue targeting the fault until they detect the fault or until they complete the simulation (i.e., until the second stage of fault simulation ends). This redundantcomputationoverheadcould be reduced by broadcasting the fault identifier, corresponding to a fault, to other processors as soon as the fault is detected. However, the savings in computation might be offset by the overhead in communication costs. 4.3 SPITFIRE4: A Two Stage Asynchronous Algorith We will now describe an asynchronousversion of the Algorithm SPITFIRE1. Consider the second stage of fault simulation in Algorithm SPITFIRE1. All processors have to work on almost the same list of undetected faults that was available at the end of the first stage (except faults that it could not detect in Stage 1). It would therefore be advantageous for each processor to periodically communicate to all other processors a list of any faults that it detects. Thus, each processor asynchronously sends a list of new detected faults to all other processors provided that it has detected at least MinFaultLimit new faults. Each processor periodically probes for messages from other processors and drops any faults that may be received through messages. This helps in reducing the load on a processor if it has not detected these faults yet. Thus, by allowing each processor to asynchronously communicate detected faults to all other processors, we dynamically reduce the load on each processor. It should be observed that in the first stage of Algorithm SPITFIRE1, all processors are working on different sets of faults. Hence, there is no need to communicate detected faults during Stage 1, since this will not have any effect on the work-load on each processor. It would make sense therefore to communicate all detected faults only at the end of Stage 1. The asynchronous algorithm used for fault simulation in Stage 2 by any processor P i is outlined below. For each vector k in the test set T i FaultSimulate vector k (NumberOfNewFaultsDetected ? MinFaultLimit) then Send the list of newly detected faults to all Processors using a buffered asynchronous send while (CheckForAnyMessages()) Receive new message using a blocking receive Drop newly received faults (if not dropped earlier) while end for The routine CheckForAnyMessages() is a non-blocking probe which returns a 1 only if there is a message pending to be received. 4.4 SPITFIRE5: A Single Stage Asynchronous Algorith It is possible to employ the same asynchronous communication strategy used in the algorithm SPITFIRE4 for the algorithm SPITFIRE0. In the latter algorithm, all processors start with the same list of undetected faults, which is the entire list of faults F . Only faults which each processor may detect get dropped, and each processor continues to work on a large set of undetected faults. Once again, it would make sense for each processor to communicate detected faults periodically to other processors provided that it has detected at least MinFaultLimit new faults. The value of MinFaultLimit is circuit dependent. It also depends on the parallel platform that may be used for parallel fault simulation. For a very small circuit with mostly easy to detect faults, it may not make sense to set MinFaultLimit too small, as this may result in too many messages being commu- nicated. On the other hand, if the circuit is reasonably large, or if faults are hard to detect, the granularity of computation between two successive communication steps will be large. Therefore, it may make sense to have a small value of Min- FaultLimit. Similarly, it may be more expensive to communicate often on a distributed parallel platform such as a network of workstations. However, this factor may not matter as much on a shared memory machine. Our results were obtained on a shared memory multiprocessor where the value of MinFault- Limit was empirically chosen to be 5 as we will show. This means that whenever any processor detects at least 5 faults, it will communicate the new faults detected over to other processors to possibly reduce the load on other processors that may still be working on these faults. It is therefore important to ensure that the computation to communication ratio be kept high and hence depending on the parallel platform used, one needs to arrive at a compromise at the frequency at which faults are communicated between processors. One may also use the number of vectors in the test set that have been simulated, say MinVectorLimit, as a control parameter to regulate the frequency of synchro- nization. This may be useful towards the end of fault simulation when there faults are detected very slowly. One can also use both parameters, MinFaultLimit and MinVectorLimit, sim- ulaneously and communicate faults if either control parameter is exceeded. As long as the granularity of the computation is large enough compared to the communication costs involved, one can expect a good performance with an asynchronous ap- proach. If we assume that communication costs are zero, then one would ideally communicate faults as soon as they are detected to other processors. If the frequency of communication is reduced, then one may have to perform more redundant computation. There is a tradeoff between algorithms SPITFIRE4 and SPITFIRE5. As we can see in SPITFIRE4, we have a completely communication independent phase in Stage 1 followed by an asynchronous communication intensive phase. However in SPITFIRE5, we have only one stage of fault simula- tion. This means that the good circuit simulation with test set T i on processor P i needs to be performed only once. Thus, although we may have continuous communication in algorithm SPITFIRE5, we may obtain substantial savings by performing only one stage of fault simulation. We will see in the next section that this is indeed the case. The same approach for asynchronous communication that was discussed in the previous section is used for this algo- rithm. However, the asynchronous communication is applied to the first and only stage of fault simulation that is used for this algorithm. 5 Analysis of Algorithms A theoretical analysis of the various algorithms is now pre- sented. We first provide an analysis of serial fault simulation and then extend the analysis for various test set partitioning approaches and for a fault partitioning approach. 5.1 Analysis of Sequential Fault Simulation We first provide an analysis for a uniprocessor and then proceed to an analysis for a multiprocessor situation. Let us assume that there are N test vectors in the test set f g. Usually in fault simulation, many faults are detected early, and then the remaining faults are detected more slowly. Let us assume that the fraction of faults detected by vector k in the test set is given by ffe \Gamma-(k\Gamma1) , i.e. the fraction of faults detected at each step falls exponentially. (Tradition- ally one assumes that the fraction of faults left undetected after the k'th vector has been simulated is given by ff 1 e \Gamma-k [3] [11]. Hence, the fraction of faults detected by vector k is given by which is of the form ffe \Gamma-(k\Gamma1) . ) Then the fraction of faults detected at this stage after (n-1) vectors have been simulated is given by 1\Gammae \Gamma- . Hence, the number of undetected faults remaining, U(n) when the n'th vector has to be simulated is given by the total number of faults in the circuit. Let us assume that fl is the unit of cost for execution in seconds per gate evalu- ation. Assume that a fraction ffi of the total number of gates G in the circuit is being simulated for each fault. Then the cost for simulating all the faulty circuits left with the n'th vector is fl ffiGU(n). Assume that a fraction fi of the gates G are simulated for the good circuit logic simulation for each vector. (Usually ffi !! fi. This is because, for fault simulation, only the events that are triggered in the fanout cone starting from the node where the fault was inserted need to be processed.) Then the fault simulation cost for simulation of the n'th vector is given by fl(fiG ffiGU(n)). Thus the total fault simulation cost on a uniprocessor, T 1 (N; F; G), for simulating N vectors is given by Since N is large, we may approximate 1\Gammae \Gamma-N by 1. We then find that T 1 (N; F; Neglecting r in relation to N , we obtain 5.2 Analysis of Algorithm SPITFIRE0 In a single stage test set partitioned parallel algorithm, each processor simulates N vectors where o is the vector overlap between processors. Each processor also starts with the same number of faults F . In a single stage synchronous algo- rithm, communication occurs only in the end where all processors exchange all detected faults. This can be neglected in comparison to the total execution time. Therefore, the total execution cost T p;sync;1stage (N; F; G; o) can be approximated as, G; The above formula shows that this approach is scalable but that one has to pay for the redundant computation performed with the vector overlap factor o. 5.3 Analysis of a Fault Partitioning Algorithm In a fault partitioning algorithm, each processor simulates N vectors and targets F faults. Hence, the execution cost in this case is of the form F This formula demonstrates that the fault partitioned approach is not scalable since the first term, which corresponds to the good circuit logic simulation performed, does not scale across the processors. Eventually, this factor will bound the speedup as the number of processors is increased. Table 1. Uniprocessor Execution Times Pri- Pri- Random Test Set Actual Test mary mary Size 10000 from an ATPG tool Inp- Out- Flip Time Faults Test Set Time Faults Circuit Faults Gates uts puts Flops (secs) Detected Size (secs) Detected Table 2. Execution Time on 8 processors of SUN-SparcCenter1000E shared memory multiprocessor Random Test Test Faults Execution Time (secs) Faults Execution Time (secs) Circuit Faults Detected SPF0 SPF1 SPF4 SPF5 Detected SPF0 SPF1 SPF4 SPF5 s526 555 52 16.7 10.4 13.3 8.2 445 8.5 5.1 3.0 2.3 div16 2141 1640 19.2 26.0 24.8 13.4 1801 53.7 28.3 21.1 13.2 pcont2 11300 6829 113.8 137.7 140.7 104.4 6837 116.3 110.3 75.5 52.6 piir8 19920 15004 285.8 271.3 265.0 230.9 15069 51.1 48.4 44.3 33.6 5.4 Analysis of Algorithm SPITFIRE5 For a single stage asynchronous algorithm, we may assume that, due to communication, all processors are aware of all detected faults before a test vector is input. We assume for the purposes of analysis that MinF aultLimit = 1. This means that a processor broadcasts the new faults that it has dropped every time it detects at least 1 fault. If the faults detected by all processors at each stage are all different, then the number of faults detected after (n-1) vectors have been simulated is given by In this case, G; where C p;async;1stage (N; F communication cost involved. In reality, some faults are multiply detected by more than one processor, and the factor pff in the above formula will be smaller. Also, since there is some delay in each processor obtaining information about faults dropped on other processors, this factor may be even smaller. Also, if MinF aultLimit is large, this delay may be even longer, and the factor pff would have to be scaled down further. A smaller value of pff indicates a longer execution time. Let us assume that the communication cost is of the form- 1 +- 2 l where - 1 is the startup cost in seconds, - 2 is the cost in seconds per computer word, and l is the length of the message. Since F ffr(1 \Gamma are detected after the n'th vector has been simulated by each of the p processors, the total cost of communication is given by C p;async;1stage (N; F Note that if MinF aultLimit is large, then the number of messages is smaller, and the term (- 1 (N=p would be scaled down. However, the term (- would remain unchanged, since that is the total amount of data that is communicated. Clearly, there is a tradeoff involved in increasing the value of MinF aultLimit. 5.5 Analysis of SPITFIRE1 and SPITFIRE4 It is easy to show, using similar analysis, that the total execution cost for the two stage algorithms, SPITFIRE1 and SPITFIRE4, can be obtained by replacing F by F ( 1 e \Gamma-( N by counting the good circuit simulation cost twice, in the formulas for the execution cost for the algorithms SPITFIRE0 and SPITFIRE5 respectively. It is apparent that in the two stage algorithms, we have effectively reduced the faulty circuit simulation term but could pay a small price in having to perform two stages of good circuit logic simulation. In addition, the \Gammapff term helps in reducing the execution time for the asynchronous two stage algorithm. We see from the above discussion that the asynchronous algorithms would possibly have the lowest execution time over Table 3. Execution Time and Speedups on SUN-SparcCenter1000E with Algorithm SPITFIRE5 on Random Test Random Test Set Execution Times(seconds) and Speedups Uniprocessor 2 Processors 4 Processors 8 Processors Circuit Time Time Speedup Time Speedup Time Speedup s526 57.39 28.50 2.01 14.76 3.88 8.25 6.95 mult16 143.3 64.36 2.23 33.21 4.31 20.37 7.03 div16 110.6 48.95 2.25 24.76 4.46 13.38 8.26 Table 4. Execution Time and Speedups on SUN-SparcCenter1000E with Algorithm SPITFIRE5 on ATPG Test ATPG Test Set Execution Times(seconds) and Speedups Uniprocessor 2 Processors 4 Processors 8 Processors Circuit Time Time Speedup Time Speedup Time Speedup s526 10.92 5.70 1.91 4.03 2.70 2.26 4.83 pcont2 324.4 168.27 1.93 96.15 3.37 52.56 6.17 piir8 127.4 74.10 1.72 45.1 2.82 33.6 3.79 their synchronous counterparts. There are two factors contributing to this. The first is the N term which corresponds to the partitioning of the test set across processors. The second is the \Gammapff term corresponding to the fact that a processor now has information about the faults that the other processors have dropped due to asynchronous communication. Between the two asynchronous algorithms, the single stage asynchronous algorithm may win over the two stage algorithm simply because only one stage of good circuit logic simulation is performed in this case. The communication cost factor will depend on the platform being used and may have a different impact on different parallel platforms. However, as long as the overall communication cost is small compared to the overall execution cost, the single stage asynchronous algorithm should run faster than all other approaches. 6 Experimental Results The four algorithms described in the paper were implemented using the MPI [12] library. The implementation is portable to any parallel platform which provides support for the MPI communication library. Results were obtained on a SUN-SparcCenter 1000E shared memory multiprocessor with 8 processors and 512 MB of memory. Results are provided for 8 circuits, viz., s5378, s526, s1423, am2910, pcont2, piir8o, mult16, and div16. The circuits were chosen because the test sets available for them were reasonably large. The mult16 circuit is a 16-bit two's complement multiplier; div16 is a 16-bit divider; am2910 is a 12-bit microprogram sequencer; pcont2 is an 8-bit parallel controller used in DSP applica- tions; and piir8 is an 8-point infinite impulse response filter. s5378, s526, and s1423 are circuits taken from the ISCAS89 benchmark suite. Parallel fault simulation was done with a random test set of size 10,000 (i.e. a sequence of 10,000 randomly generated input test vectors) and with actual test sets obtained from an Automatic Test Pattern Generation (ATPG) tool [13]. This shows the performance of the parallel fault simulator both in a random test pattern environment and in a grading environment. Table 1 shows the characteristics of the circuits used, and the timings on a single processor for both types of input test sets. The number of faults detected are also shown. Table 2 shows the execution times in seconds on 8 proces- sors, on the SUNSparcCenter 1000E, obtained using all four algorithms discussed in the previous section. SPF0, SPF1, SPF4 and SPF5 refer to the algorithms SPITFIRE0, SPIT- FIRE1, SPITFIRE4, and SPITFIRE5, respectively. The same number of faults are detected by all algorithms. How- ever, there may be a pessimism is the number of fault detected which can be eliminated by using the pipelined approach in the algorithm SPITFIRE3 [10]. A value of 5 was used for MinFaultLimit in SPF4 and SPF5, MinFaultLimit Execution Times(secs) s526 2.43 2.26 2.33 div16 12.88 13.20 13.32 Table 5. Execution Times with SPITFIRE5 on 8 processors for varying MinFaultLimit as will be explained in the next paragraph. It can be seen from the table that the execution times get progressively smaller, in general, as we proceed from SPF0 to SPF1 to SPF4 to SPF5. Sometimes SPF0 is better than SPF1, and sometimes SPF1 is better than SPF4. The lowest execution time is shown in bold. It can be seen that SPF5 always has the lowest execution time for all algorithms. This shows that the algorithm SPITFIRE5 provides the best performance. We thus see that performing the single stage of fault simulation and simultaneously allowing asynchronous communication between processors provides substantial savings in terms of execution time. Tables 3 and 4 show the execution times and speedups obtained with the random test sets and the ATPG test sets, respectively on 1, 2, 4, and 8 processors for the algorithm SPIT- FIRE5. It can be seen that the algorithm is highly scalable and provides excellent speedups. The combined effect of test set partitioning coupled with asynchronous communication of detected faults has resulted in superlinear speedups in some cases. The results from an experiment to determine the value of MinFaultLimit are shown in Table 5. The experiment was performed with 3 values of MinFaultLimit, viz., 2, 5, and 10, using the actual test sets obtained from a test generator. It was observed that the best performance was obtained with a value of 5 for most circuits on 8 processors. 7 Conclusion Parallel fault simulation has been a difficult problem due to the limited scalability and parallelism that previous algorithms could extract. Parallelization in fault simulation is limited by the serial logic simulation of the fault-free ma- chine. By partitioning the test sets across processors, we have achieved a scalable parallel implementation and have thus avoided the serial logic simulation bottleneck. By performing asynchronous communication, we have dynamically reduced the load on all processors, and the redundant computation that may have otherwise occurred. We have thus presented two asynchronous algorithms for parallel test set partitioned fault simulation. Both asynchronous algorithms provide better performance than their synchronous counterparts on a shared memory multiprocessor. The single stage asynchronous test set partitioned parallel algorithm was shown to provide better performance than the two stage test set partitioned asynchronous parallel algorithm, although the two stage synchronous algorithm was better than the single stage synchronous algorithm. --R "Is there hope for linear time fault simulation," Parallel Algorithms for VLSI Computer-Aided De- sign "Parallel test generation for sequential circuits on general-purpose multiprocessors," "Fault simulation in a pipelined multiprocessor system," "Concurrent fault simulation of logic gates and memory blocks on message passing multicomput- ers," "A parallel algorithm for fault simulation based on PROOFS," "ZAMBEZI: A parallel pattern parallel fault sequential circuit fault simulator," "Overcoming the serial logic simulation bottleneck in parallel fault simulation," "Zamlog: A parallel algorithm for fault simulation based on Zambezi," "SPITFIRE: Scalable Parallel Algorithms for Test Partitioned Fault Simulation," "Distributed fault simulation with vector set partitioning," Portable Parallel Programming with the Message Passing Interface. "Automatic test generation using genetically-engineered distinguishing se- quences," --TR Parallel test generation for sequential circuits on general-purpose multiprocessors Concurrent fault simulation of logic gates and memory blocks on message passing multicomputers Parallel algorithms for VLSI computer-aided design Using MPI <italic>Zamlog</italic> A parallel algorithm for fault simulation based on PROOFS Overcoming the Serial Logic Simulation Bottleneck in Parallel Fault Simulation Automatic test generation using genetically-engineered distinguishing sequences ZAMBEZI --CTR Victor Kim , Prithviraj Banerjee, Parallel algorithms for power estimation, Proceedings of the 35th annual conference on Design automation, p.672-677, June 15-19, 1998, San Francisco, California, United States

circuit analysis computing;synchronous two stage approach;test set partitioned fault simulation;MPI;dynamic characteristics;shared memory multiprocessor;redundant work;sequential VLSI circuits;circuit CAD;software portability;Message Passing Interface;asynchronous parallel algorithms

268906

Optimistic distributed simulation based on transitive dependency tracking.

In traditional optimistic distributed simulation protocols, a logical process (LP) receiving a straggler rolls back and sends out anti-messages. The receiver of an anti-message may also roll back and send out more anti-messages. So a single straggler may result in a large number of anti-messages and multiple rollbacks of some LPs. In the authors' protocol, an LP receiving a straggler broadcasts its rollback. On receiving this announcement, other LPs may roll back but they do not announce their rollbacks. So each LP rolls back at most once in response to each straggler. Anti-messages are not used. This eliminates the need for output queues and results in simple memory management. It also eliminates the problem of cascading rollbacks and echoing, and results in faster simulation. All this is achieved by a scheme for maintaining transitive dependency information. The cost incurred includes the tagging of each message with extra dependency information and the increased processing time upon receiving a message. They also present the similarities between the two areas of distributed simulation and distributed recovery. They show how the solutions for one area can be applied to the other area.

Introduction We modify the time warp algorithm to quickly stop the spread of erroneous computation. Our scheme does not require output queues and anti-messages. This results in less memory overhead and simple memory management algorithms. It also eliminates the problem of cascading rollbacks and echoing [15], resulting in faster simulation. We use aggressive cancellation Our protocol is an adaptation of a similar protocol for the problem of distributed recovery [4, 21]. We supported in part by the NSF Grants CCR-9520540 and ECS-9414780, a TRW faculty assistantship award, a General Motors Fellowship, and an IBM grant. illustrate the main concept behind this scheme with the help of Figure 1. In the figure, horizontal arrows show the direction of the simulation time. Messages are shown by the inter-process directed arrows. Circles represent states. State transition is caused by acting on the message associated with the incoming arrow. For example, the state transition of P1 from s10 to happened when P1 acted on m0. In the time warp scheme, when a logical process (LP) P2 receives a straggler (i.e., a message which schedules an event in back the state s20 and sends an anti-message corresponding to message m2. On receiving this anti-message, P1 rolls back state s10 and sends an anti-message corresponding to m1. It then acts on the next message in its message queue, which happens to be m0. On receiving the anti-message for m1, P0 rolls back s00 and sends an anti-message for m0. On receiving this anti-message, P1 rolls back s11. In our scheme, transitive dependency information is maintained with all states and messages. After rolling back s20 due to a straggler, P2 broadcasts that s20 has been rolled back. On receiving this announce- ment, P1 rolls back s10 as it finds that s10 is transitively dependent on s20. P1 also finds that m0 is transitively dependent on s20 and discards it. Similarly P0 rolls back s00 on receiving the broadcast. We see that P1 was able to discard m0 faster compared to the previous scheme. Even P0 would likely receive the broadcast faster than receiving the anti-message for m1 as that can be sent only after P1 has rolled back s10. Therefore, simulation should proceed faster. As explained later, we use incarnation number to distinguish between two states with the same timestamp, one of which is committed and the other is rolled back. We only need the LP that receives a straggler to broadcast the timestamp of the straggler. Every other LP can determine whether they need to roll back or not by comparing their local dependency information with the broadcast timestamp. Other LPs that roll back in response to a rollback announcement do not send any announcement or anti-messages. Hence, each rolls back at most once in response to a strag- gler, and the problem of multiple rollbacks is avoided. Several schemes have been proposed to minimize the Figure 1: A Distributed Simulation. spread of erroneous computations. A survey of these schemes can be found in [7]. The Filter protocol by Prakash and Subramanian [17] is most closely related to our work. It maintain a list of assumptions with each message, which describe the class of straggler events that could cause this message to be canceled. It maintains one assumption per channel, whereas our protocol can be viewed as maintaining one assumption per LP. In the worst case, Filter tags each message with O(n 2 ) integers whereas our protocol tags O(n) integers, where n is the number of LPs in the sys- tem. Since for some applications even O(n)-tagging may not be acceptable, we also describe techniques to further reduce this overhead. If a subset of LPs interact mostly with each other, then, for most of the time, the tag size of their messages will be bounded by the size of the subset. The paper is organized as follows. Section 2 describes the basic model of simulation; Section 3 introduces the happen before relation between states and the simulation vector which serves as the basis of our optimistic simulation protocol; Section 4 describes the protocol and gives a correctness proof; Section 5 presents optimization techniques to reduce the overhead of the protocol; Section 6 compares distributed simulation with distributed recovery. 2 Model of Simulation We consider event-driven optimistic simulation. The execution of an LP consists of a sequence of states where each state transition is caused by the execution of an event. If there are multiple events scheduled at the same time, it can execute those events in an arbitrary order. In addition to causing a state transi- tion, executing an event may also schedule new events for other LPs (or the local LP) by sending messages. When LP P1 acts on a message from P 2, P1 becomes dependent on P 2. This dependency relation is transitive The arrival of a straggler causes an LP to roll back. A state that is rolled back, or is transitively dependent on a rolled back state is called an orphan state. A message sent from an orphan state is called an orphan message. For correctness of a simulation, all orphan states must be rolled back and all orphan messages must be discarded. An example of a distributed simulation is shown in Figure 2. Numbers shown in parentheses are either the virtual times of states or the virtual times of scheduled events carried by messages. Solid lines indicate useful computations, while dashed lines indicate rolled back computations. In Figure 2(a), s00 schedules an event for P1 at time 5 by sending message m0. P1 optimistically executes this event, resulting in a state transition from s10 to s11, and schedules an event for P2 at time 7 by sending message m1. Then receives message m2 which schedules an event at time 2 and is detected as a straggler. Execution after the arrival of this straggler is shown in Figure 2(b). P1 rolls back, restores s10, takes actions needed for maintaining the correctness of the simulation (to be described later) and restarts from state r10. Then it broadcasts a rollback announcement (shown by dotted arrows), acts on m2, and then acts on m0. Upon receiving the rollback announcement from P 1, P2 realizes that it is dependent on a rolled back state and so it also rolls back, restores state s20, takes actions needed, and restarts from state r20. Finally, the orphan message m1 is discarded by P 2. 3 Dependency Tracking From here on, i,j refer to LP numbers; k refers to incarnation refer to states; P i refers to logical process refers to the number associated with the LP to which s belongs, that is, refers to a message and e refers to an event. 3.1 Happen Before Relation Lamport defined the happen before(!) relation between events in a rollback-free distributed computation [12]. To take rollbacks into account, we extend this relation. As in [4, 21], we define it for the states. For any two states s and u, s ! u is the transitive closure of the relation defined by the following three conditions: 1. immediately precedes u. 2. s is the state restored after a roll-back and u is the state after P u:p has taken the s12 (2) s13 (5) m2 (2) (a) (b) Figure 2: Using Simulation Vector for Distributed Simulation. (a) Pre-straggler computation. (b) Post-straggler computation. actions needed to maintain the correctness of simulation despite the rollbacks. For example, in Figure 3. s is the sender of a message m and u is the re- ceiver's state after the event scheduled by m is executed. For example, in Figure 2(a), s10 ! s11 and s00 ! s21, and in Figure 2(b) s11 6! r10. Saying s happened before u is equivalent to saying that u is transitively dependent on s. For our protocol, "actions needed to maintain the correctness of simulation" include broadcasting a roll-back announcement and incrementing the incarnation number. For other protocols, the actions may be dif- ferent. For example, in time warp, these actions include the sending of anti-messages. Our definition of happen before is independent of such actions. The terms "rollback announcements" and "tokens" will be used interchangeably. Tokens do not contribute to the happen before relation. So if u receives a token from s, u does not become transitively dependent on s due to this token. 3.2 Simulation Vector A vector clock is a vector of size n where n is the number of processes in the system [16]. Each vector entry is a timestamp that usually counts the number of send and receive events of a process. In the context of distributed simulation, we modify and extend the notion of vector clock, and define a Simulation Vector (SV) as follows. To maintain dependency in the presence of rollbacks, we extend each entry to contain both a timestamp and an incarnation number [19]. The timestamp in the i th entry of the SV of corresponds to the virtual time of P i . The timestamp in the j th entry corresponds to the virtual time of the latest state of P j on which P i depends. The incarnation number in the i th entry is equal to the number of times P i has rolled back. The incarnation number in the j th entry is equal to the highest incarnation number of P j on which P i depends. Let entry en be a tuple (incarnation v, timestamp t). We define a lexicographical ordering between entries as follows: en Simulation vectors are used to maintain transitive dependency information. Suppose P i schedules an event e for P j at time t by sending a message m. P i attaches its current SV to m. By "virtual time of m", we mean the scheduled time of the event e. If m is neither an orphan nor a straggler, it is kept in the in-coming queue by P j . When the event corresponding to m is executed, P j updates its SV with m's SV by taking the componentwise lexicographical maximum. its virtual time (denoted by the j th timestamp in its SV) to the virtual time of m. A formal description of the SV protocol is given in Figure 3. Examples of SV are shown in Figure 2 where the SV of each state is shown in the box near it. The SV has properties similar to a vector clock. It can be used to detect the transitive dependencies between states. The following theorem shows the relationship between virtual time and SV. Theorem 1 The timestamp in the i th entry of P i 's SV corresponds to the virtual time of P i . /* incarnation, timestamp */ var sv: array [0.n-1] of entry time at which m should be executed ; send (m:data, m:ts, m:sv) ; Execute message (m:data, m:ts, m:sv) : event scheduled by m */ Figure 3: Formal description of the Simulation Vector protocol Proof. By Induction. The above claim is true for the initial state of P i . While executing a message, the virtual time of the P i is correctly set. After a rollback, virtual time of the restored state remains unchanged. Let s:sv denote the SV of P s:p in state s. We define the ordering between two SV's c and d as follows. In P i 's SV, the j th timestamp denotes the maximum virtual time of P j on which P i depends. This timestamp should not be greater than P i 's own virtual time. Lemma 1 formalizes the above notion. Lemma 1 The timestamp in the i th entry of the SV of a state of P i has the highest value among all the timestamps in this SV. Proof. By induction. The lemma is true for the initial state of P i . Assume that state s of P j sent a message m to P i . State u of P i executed m, resulting in state w. By induction hypothesis, s:sv[j]:ts and the u:sv[i]:ts are the highest timestamps in their SV's. So the maximum of these two timestamps is greater than all the timestamps in w:sv after the max operation in Execute message. Now m:ts, the virtual time of message m, is not less than the virtual time of the state s sending the message. It is also not less than the virtual time of the state u acting on the message, otherwise, it would have caused a rollback. So by theorem 1, m:ts is not less than the maximum of s:sv[j]:ts and the u:sv[i]:ts. Hence setting the w:sv[i]:ts to m:ts preserves the above property. All other routines do not change the timestamps. The following two lemmas give the relationship between the SV and the happen before relation. happens before u, then s:sv is less than or equal to u:sv. Proof. By induction. Consider any two states s and u such that s happens before u by applying one of the three rules in the definition of happen before. In case of rule 1, state s is changed to state u by acting on a message m. The update of the SV by taking the maximum in the routine Execute message maintains the above property. Now consider the next action in which u:sv[u:p]:ts is set to m:ts. Since virtual time of m cannot be less than the virtual time of state s executing it, this operation also maintains the above property. In case of rule 2, in routine Rollback, the update of the SV by incrementing the incarnation number preserves the above property. The case of rule 3 is similar to that of the rule 1. Let state w change to state u by acting on the message m sent by state s. By lemma 1, in m's SV, s:p th timestamp is not less than the u:p th timestamp. Also the virtual time of m is not less than the s:p th timestamp in its SV. Hence setting the i th timestamp to the virtual time of m, after taking max, preserves the above property. The following lemma shows that LPs acquire timestamps by becoming dependent on other LPs. This property is later used to detect orphans. This lemma states that if j th timestamp in state w's SV is not minus one (an impossible virtual time) then w must be dependent on a state u of P j , where the virtual time of u is w:sv[j]:ts . Proof. By induction. Initialize trivially satisfies the above property. In Execute message, let x be the state that sends m and let state s change to state w by acting on m. By induction hypothesis, x and s satisfy the lemma. In taking maximum, let the j th entry from x is se- lected. If j is x:p then x itself plays the role of u. Else, by induction hypothesis, (x:sv[j]:ts Hence either w:sv[j]:ts is -1 or by transitivity, u happens before w. The same argument also applies to the case where the th entry comes from s. In case of Rollback, let s be the state restored and let w be the state resulting from s by taking the actions needed for the correct simulation. By induction hypothesis, s satisfies the lemma. Now s:sv and w:sv differ only in w:p th entry and all states that happened before s also happened before w. Hence w satisfies the lemma. simulation vector */ of set of entry; incarnation end table */ announcement */ Receive else if m:ts ! sv[i]:ts then /* m is a straggler */ receives its own broadcast and rolls back. */ Block till all LPs acknowledge broadcast ; Execute message : messages with the lowest value of m:ts ; Act on the event scheduled by m ; Receive token(v; t) from then Rollback(j; (v; Save the iet ; Restore the latest state s such that Discard the states that follow s ; Restore the saved iet ; sv[i]:inc ++ ; Figure 4: Our protocol for distributed simulation 4 The Protocol Our protocol for distributed simulation is shown in Figure 4. To keep the presentation and correctness proof clear, optimization techniques for reducing overhead are not included in this protocol. They are described in the next section. Besides a simulation vector, each LP P i also maintains an incarnation end table (iet). The j th component of iet is a set of entries of the form (k; ts), where ts is the timestamp of the straggler that caused the rollback of the k th incarnation of P j . All states of the k th incarnation of P j with timestamp greater than ts have been rolled back. The iet allows an LP to detect orphan messages. When P i is ready for the next event, it acts on the message with the lowest virtual time. As explained in Section 3, P i updates its SV and the internal state, and possibly schedules events for itself and for the other LPs by sending messages. Upon receiving a message m, P an orphan. This is the case when, for some j, P i 's iet and the j th entry of m's SV indicate that m is dependent on a rolled back state of P j . If P i detects that m is a straggler with virtual time t, it broadcasts a token containing t and its current incarnation number k. It rolls back all states with virtual time greater than t and increments its incarnation number, as shown in Rollback. Thus, the token basically indicates that all states of incarnation k with virtual time greater than t are orphans. States dependent on any of these orphan states are also orphans. When an LP receives a token containing virtual time t from P j , it rolls back all states with the j th timestamp greater than t, discards all orphan messages in its input queue, and increments its incarnation number. It does not broadcast a token, which is an important property of our protocol. This works because transitive dependencies are maintained. Suppose state w of P i is dependent on a rolled back state u of P j . Then any state x dependent on w must also be dependent on u. So x can be detected as an orphan state when the token from P j arrives at P x:p , without the need of an additional token from P i . The argument for the detection of orphan messages is similar. We require an LP to block its execution after broadcasting a token until it receives acknowledgments from all the other LPs. This ensures that a token for a lower incarnation of P j reaches all LPs before they can become dependent on any higher incarnation of This greatly simplifies the design because, when a dependency entry is overwritten by an entry from a higher incarnation in the lexicographical maximum operation, it is guaranteed that no future rollback can occur due to the overwritten entry (as the corresponding token must have arrived). While blocked, an LP acknowledges the received broadcasts. 4.1 Proof of Correctness Suppose state u of P j is rolled back due to the arrival of a straggler. The simulation is correct if all the states that are dependent on u are also rolled back. The following theorem proves that our protocol correctly implements the simulation. Theorem 2 A state is rolled back due to either a straggler or a token. A state is rolled back due to a token if and only if it is dependent on a state that has been rolled back due to a straggler. Proof. The routine Rollback is called from two places: Receive message and Receive token. States that are rolled back in a call from Receive message are rolled back due to a straggler. Suppose P j receives a strag- gler. Let u be one of the states of P j that are rolled back due to this straggler. In the call from routine Receive token, any state w not satisfying condition (C1) is rolled back. Since the virtual time of u is greater than the virtual time of the straggler, by Lemma 2, any state w dependent on u will not satisfy condition (C1). In the future, no state can become dependent on u because any message causing such dependency is discarded: if it arrives after the token, it is discarded by the first test in the routine Receive message; if it arrives before the token, it is discarded by the first test in the routine Receive token. So all orphan states are rolled back. From Lemma 3, for any state w not satisfying condition (C1) and thus rolled back, there exists a state u which is rolled back due to the straggler, and u ! w. That means no state is unnecessarily rolled back. 5 Reducing the Overhead For systems with a large number of LP's, the overhead of SV and the delay due to the blocking can be substantial. In this section, we describe several optimization techniques for reducing the overhead and blocking. 5.1 Reducing the blocking For simplicity, the protocol description in Figure 4 increments the incarnation number upon a rollback due to a token (although it does not broadcast another token). We next argue that the protocol works even if the incarnation number is not incremented. This modification then allows an optimization to reduce the blocking. We use the example in Figure 2(b) to illustrate this modification. Suppose P 2 executes an event and makes a state transition from r20 to s22 with virtual time 7 (not shown in the figure). If P2 does not increment its incarnation number on rolling back due to the token from P 1, then s22 will have (0; 7) as the 3rd entry of its SV, which is the same as s21's 3rd entry in Figure 2(a). Now suppose the 3rd entry of a state w of another LP P3 is (0; 7). How does P3 decide whether w is dependent on s21 which is rolled back or s22 which is not rolled back? The answer is that, if w is dependent on s21, then it is also dependent on s11. Therefore, its orphan status will be identified by its 2nd entry, without relying on the 3rd entry. The above modification ensures that, for every new incarnation, a token is broadcast and so every LP will have an iet entry for it. This allows the following optimization technique for reducing the blocking. Suppose receives a straggler and broadcasts a token. Instead of requiring P i to block until it receives all acknowl- edgements, we allow P i to continue its execution in the new incarnation. One problem that needs to be solved is that dependencies on the new incarnation of may reach an LP P j (through a chain of messages) before the corresponding token does. If P j has a dependency entry on any rolled back state of the old incarnation then it should be identified as an orphan when the token arrives. Overwriting the old entry with the new entry via the lexicographical maximum operation results in undetected orphans and hence incorrect simulation. The solution is to force P j to block for the token before acquiring any dependency on the new incarnation. We conjecture that this blocking at the token receiver's side would be a improvement over the original blocking at the token sender's side if the number of LPs (and hence acknowledgements) is large. 5.2 Reducing the size of simulation vector The Global Virtual Time(GVT) is the virtual time at a given point in simulation such that no state with virtual time less than GVT will ever be rolled back. It is the minimum of the virtual times of all LPs and all the messages in transit at the given instant. Several algorithms have been developed for computing GVT [2, 20]. To reduce the size of simulation vectors, any entry that has a timestamp less than the GVT can be set to NULL, and NULL entries need not be transmitted with the message. This does not affect the correctness of simulation because: (1) the virtual time of any message must be greater than or equal to the GVT, and so timestamps less than the GVT are never useful for detecting stragglers; (2) the virtual time contained in any token must be greater than or equal to the GVT, and so timestamps less than the GVT are never useful for detecting orphans. Since most of the SV entries are initialized to -1 (see Figure 3) which must be less than the GVT, this optimization allows a simulation to start with very small vectors, and is particularly effective if there is high locality in message activities. Following [21], we can also use a K-optimistic pro- tocol. In this scheme, an LP is allowed to act on a message only if that will not result in more than K non-NULL entries in its SV. Otherwise it blocks. This ensures that an LP can be rolled back by at most K other LPs. In this sense optimistic protocols are N - optimistic and pessimistic protocols are 0-optimistic. Another approach to reducing the size of simulation vectors is to divide the LPs into clusters. Several designs are possible. If the interaction inside a cluster is optimistic while inter-cluster messages are sent conservatively [18], independent SV's can be used inside each cluster, involving only the LPs in the clus- ter. If intra-cluster execution is sequential while inter-cluster execution is optimistic [1], SV's can be used for inter-cluster messages with one entry per cluster. Similarly one can devise a scheme where inter-cluster and intra-cluster executions are both optimistic but employ different simulation vectors. This can be further generalized to a hierarchy of clusters and simulation vectors. In general, however, inter-cluster simulation vectors introduce false dependencies [14] which may result in unnecessary rollbacks. So there is a trade-off between the size of simulation vectors and unnecessary rollbacks. But it does not affect the correctness of the simulation. 6 Distributed Simulation and Distributed Recovery The problem of failure recovery in distributed systems [6] is very similar to the problem of distributed simulation. Upon a failure, a process typically restores its last checkpoint and starts execution from there. However, process states that were lost upon the failure may create orphans and cause the system state to become inconsistent. A consistent system state is one where the send of a message must be recorded if its receive is recorded [6]. In pessimistic logging [6], every message is logged before the receiver acts on it. When a process fails, it restores its last checkpoint and replays the logged messages in the original or- der. This ensures that the pre-failure state is recreated and no other process needs to be rolled back. But the synchronization between message logging and message processing reduces the speed of computation. In optimistic logging [19], messages are stored in a volatile memory buffer and logged asynchronously to the stable storage. Since the content of volatile memory is lost upon a failure, some of the messages are no longer available for replay after the failure. Thus, some of the process states are lost in the failure. States in other processes that are dependent on these lost states then become orphan states. Any optimistic logging protocol must roll back all orphan states in order to bring the system back to a consistent state. There are many parallels between the issues in distributed recovery and distributed simulation. A survey of different approaches to distributed recovery can be found in [6]. In Table 1, we list the equivalent terms from these two domains. References are omitted for those terms that are widely used. The equivalence is exact in many cases, but only approximate in other cases. Stragglers trigger rollbacks in distributed simula- tion, while failures trigger rollbacks in distributed re- covery. Conservative simulation [7] ensures that the current state will never need to roll back. Similarly, pessimistic logging [6] ensures that the current state is always recoverable after a failure. In other words, although a rollback does occur, the rolled back states can always be reconstructed. The time warp optimistic approach [10] inspired the seminal work on optimistic message logging [19]. The optimistic protocol presented in this paper is based on the optimistic recovery protocol presented in [4, 21]. In the simulation scheme by Dickens and Reynolds [5], any results of an optimistically processed event are not sent to other processes until they become definite [3]. In the recovery scheme by Jalote [11], any messages originating from an unstable state interval are not sent to other processes until the interval becomes stable [6]. Both schemes confine the loss of computation, either due to a straggler or a failure, to the local process. Distributed Simulation Distributed Recovery Logical Process Recovery Unit [19] Virtual Time State Interval Index Sim. Vector (this paper) Trans. Dep. Vector [19] Straggler Failure Anti-Message Rollback Announcement Fossil Collection [10] Garbage Collection [6] Global Virtual Time [2] Global Recovery Line [6] Conservative Schemes Pessimistic Schemes Optimistic Schemes Optimistic Schemes Causality Error Orphan Detection Cascading Rollback [15] Domino Effect [6] Echoing [15] Livelock [6] Conditional Event [3] Unstable State [6] Event [3] Stable State [6] Table 1: Parallel terms from Distributed Simulation and Recovery Conservative and optimistic simulations are combined in [1, 18] by dividing LPs into clusters and having different schemes for inter-cluster and intra-cluster executions. In distributed recovery, the paper by Lowry et al. [14] describes an idea similar to the conservative time windows in the simulation literature Now we list some of the main differences between the two areas. While the arrival of a straggler can be prevented, the occurrence of a failure cannot. But pessimistic logging can cancel the effect of a failure through message logging and replaying. The arrival of a straggler in optimistic simulation does not cause any loss of information, while the occurrence of a failure in optimistic logging causes volatile message logs to be lost. So some recovery protocols have to deal with "lost in-transit message" problem [6] which is not present in distributed simulation protocols. Incoming messages from different channels can be processed in an arbitrary order, while event messages in distributed simulation must be executed in the order of increasing timestamps. Due to these differences, some of the protocols presented in one area may not be applicable to the other area. Distributed recovery can potentially benefit from the advances in distributed simulation in the areas of memory management [13], analytical modeling to determine checkpoint frequency [8], checkpointing mechanisms [22], and time constrained systems [9]. Simi- larly, research work on coordinated checkpointing, optimal checkpoint garbage collection, and dependency tracking [6] can potentially be applied to distributed simulation. --R Clustered Time Warp and Logic Simulation. Global Virtual Time Algorithms. The Conditional Event Approach to Distributed Simulation. How to Recover Efficiently and Asynchronously when Optimism Fails. A Survey of Rollback-Recovery Protocols in Message-Passing Systems Parallel Discrete Event Simulation. Comparative Analysis of Periodic State Saving Techniques in Time Warp Simulators. Time Warp Simulation in Time Constrained Systems. Virtual Time. Fault Tolerant Processes. Memory Management Algorithms for Optimistic Parallel Simulation. Optimistic Failure Recovery for Very Large Networks. Virtual Time and Global States of Distributed Systems. An Efficient Optimistic Distributed Simulation Scheme Based on Conditional Knowledge. The Local Time Warp Approach to Parallel Simulation. Optimistic Recovery in Distributed Systems. An Algorithm for Minimally Latent Global Virtual Time. Distributed Recovery with K-Optimistic Logging Automatic Incremental State Saving. --TR Optimistic recovery in distributed systems Virtual time Rollback sometimes works...if filtered Parallel discrete event simulation An algorithm for minimally latent global virtual time The local Time Warp approach to parallel simulation Time Warp simulation in time constrained systems Comparative analysis of periodic state saving techniques in time warp simulators time warp and logic simulation Automatic incremental state saving Time, clocks, and the ordering of events in a distributed system How to recover efficiently and asynchronously when optimism fails Distributed Recovery with K-Optimistic Logging --CTR Reuben Pasquini , Vernon Rego, Optimistic parallel simulation over a network of workstations, Proceedings of the 31st conference on Winter simulation: Simulation---a bridge to the future, p.1610-1617, December 05-08, 1999, Phoenix, Arizona, United States Om. P. Damani , Vijay K. Garg, Fault-tolerant distributed simulation, ACM SIGSIM Simulation Digest, v.28 n.1, p.38-45, July 1998

message tagging;optimistic distributed simulation;transitive dependency information;transitive dependency tracking;time warp simulation;process rollback;rollback broadcasting;straggler;memory management;dependency information;distributed recovery;anti-messages;optimistic distributed simulation protocols;logical process

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Breadth-first rollback in spatially explicit simulations.

The efficiency of parallel discrete event simulations that use the optimistic protocol is strongly dependent on the overhead incurred by rollbacks. The paper introduces a novel approach to rollback processing which limits the number of events rolled back as a result of a straggler or antimessage. The method, called breadth-first rollback (BFR), is suitable for spatially explicit problems where the space is discretized and distributed among processes and simulation objects move freely in the space. BFR uses incremental state saving, allowing the recovery of causal relationships between events during rollback. These relationships are then used to determine which events need to be rolled back. The results demonstrate an almost linear speedup-a dramatic improvement over the traditional approach to rollback processing.

Introduction One of the major challenges of Parallel Discrete Event Simulation (PDES) is to achieve good perfor- mance. This goal is difficult to attain, because, by its very nature, discrete event simulation organizes events in a priority queue based on the timestamp of events, and processes them in that order. When porting a simulation to a parallel platform, this priority queue is distributed among logical processes (LPs) that correspond to the physical processes that are being mod- eled. Because the LPs interact with each other by sending event messages, it is costly to maintain the causality between events. Two basic protocols have been developed to ensure that causality constraints are satisfied [9]: conservative [5] and optimistic. In Time Warp (TW) [11], the best known optimistic protocol, causality errors are allowed to occur, but when such an error is detected, the erroneous computation is rolled back. The research described in this paper utilizes the optimistic protocol and focuses on optimizing rollback processing. The method of rollback processing we will present is applicable to simulations that consist of a space with objects moving freely in it; the space is discretized into a multi-dimensional lattice and divided among LPs. We make use of incremental state saving techniques [18] to detect dependencies between events. Typical implementations of a rollback in such a setting (used in our previous implementation [7]) is to roll back the entire area assigned to the LP. In this paper, we present a novel approach, termed Breadth-First Rollback (BFR), in which the rollback is contained to the area that has been directly affected by the straggler (event message with a timestamp smaller than the current simulation time) or antimessage (cancellation of an event). We also present the improved simulation speedup and performance resulting from the use of this approach. The application that motivated this work is a Lyme disease simulation in which the two-dimensional space is discretized into a two-dimensional lattice. The most important characteristics of the simulation are: the mobile objects moving freely in space (mice) and the stationary objects present at the lattice nodes (ticks). The two main groups of events are: (i) local to a node (such as tick bites, mouse deaths, etc.) and (ii) non-local (such as a mouse moving from one node to another-Move Event). The simulation currently runs on an IBM SP2 (we show results for up to 16 processors). The model was designed in an object oriented fashion and implemented in C++. The communications between processes use the MPI [10] message passing library. Related Work There are two inter-related issues that have arisen in optimizing optimistic protocols for PDES. One is the need to reduce the overhead of rollbacks, and the other is to limit the administrative overhead of partitioning a problem into many "small" LPs (as happens, for example, in digital logic simulations). To address both of these issues, clustering of LPs is often used. Lazy re-evaluation [9] has been been used to determine if a straggler or antimessage had any effect on the state of the simulation. If, after processing the straggler or canceling an event, the state of the simulation remains the same as before, than there is no need to re-execute any events from the time of the rollback to the current time. The problem with this approach is that it is hard to compare the state vectors in order to determine if the state has changed. It is also not applicable to the protocols using incremental state saving. The Local Time Warp (LTW) [15] approach combines two simulation protocols by using the optimistic protocol between LPs belonging to the same cluster and by maintaining a conservative protocol between clusters. LTW minimizes the impact of any rollback to the LPs in a given cluster. Clustered Time Warp (CTW) [1, 2] takes the opposite view. It uses conservative synchronization within the clusters and an optimistic protocol between them. The reason given for such a choice is that, since LPs in a cluster share the same memory space, their tight synchronization can be performed efficiently. Two algorithms for rollback are presented: clustered and lo- cal. In the first case, when a rollback reaches a cluster, all the LPs in that cluster are rolled back. This way the memory usage is efficient, because events that are present in input queues and that were scheduled after the time of the rollback, can be removed. In the local algorithm, only the affected LPs are rolled back. Restricting the rollback speeds up the computation, but increases the size of memory needed, because entire input queues have to be kept. The Multi-Cluster Simulator [16], in which digital circuits are modeled, takes a bit of a different look at clustering. First, the cluster is not composed of a set of LPs; rather, it consists of one LP composed of a set of logical gates. These LPs (clusters) are then assigned to a simulation process. In the case of spatially explicit problems, the issue of partitioning the space between LPs is also of importance. Discretizing the space results in a multi-dimensional lattice for which the following question arises: Should one LP be assigned to each lattice node (which results in high simulation overhead) or should the lattice nodes be "clustered" and the resulting clusters be assigned to LPs? Our first implementation of Lyme disease used the latter approach and assigned spatially close nodes to a single LP, with TW used between the LPs. This was similar to the CTW, except that our implementation did not have multiple LPs within a cluster, to simulate space more efficiently. Unfortunately, this approach did not perform as well as we had hoped, especially when the problem size grew larger, because when a rollback occurred in a cluster, the entire cluster had to roll back. To improve performance, the nodes of the lattice belonging to an LP (cluster) are allowed to progress independently in simulation time; however, all the nodes in a cluster are under the supervision of one LP. When a rollback occurs in a LP/cluster, only the affected lattice nodes are rolled back, thanks to a breadth-first rollback strategy, explained in Section 3. This approach can be classified as an inter-cluster and intra-cluster time warp (TW). The main innovation in BFR is that all future information is global, and information about the past is distributed among the nodes of the spatial lattice. The future information is centralized to facilitate scheduling of events, and the past information is distributed to limit the effects of a rollback. One could say that, from the point of view of the future, we treat a partition as a single LP, whereas, from the point of view of the past, we treat the partition as a set of LPs (one LP per lattice node). The performance of the new method yields a speedup which is close to linear. Breadth-First Rollback Approach Breadth-First Rollback is designed for spatially ex- plicit, optimistic PDES. The space is discretized and divided among LPs, so each LP is responsible for a set of interconnected lattice nodes. The speed of the simulation is dictated by the efficiency of two steps: the forward event and the rollback processing. The forward computation is facilitated when the event queue is global to the executing LP, so that the choice of the next event is quick. The impact of a rollback is reduced when the depth of the rollback is kept to a min- imum: the rollback should not reach further into the past than necessary, and the number of events affected at a given time has to be minimized. For the latter, we can rely on a property of spatially explicit prob- lems: if two events are located sufficiently far apart in space, one cannot affect the other (for certain values of the current logical virtual time (lvt) of the LP and the time of the rollback), so at most one of these events needs to be rolled back when a causality error occurs. Events can be classified as local or non-local. A local event affects only the state of one lattice node. A non-local event, for example the Move Event, which moves an object from one location to the next, affects at least two nodes of the lattice. Local events are easy to roll back. Assume that a local event e at location Original impact point of a rollback Potential 1st,2nd and 3rd waves of the rollback location x Figure 1: Waves of Rollback. x and time t triggers an event e 1 at time t 1 and the same location x (by definition of a local event). If a rollback then occurs which impacts event e, only the state of location x has to be restored to the time just prior to time t. While restoring the state, e 1 will be automatically "undone". If, however, the triggering event e is non-local and triggers an event e 1 at location then restoring the state of location x is not sufficient-it is also necessary to restore the state of location x 1 just prior to the occurrence of event e 1 Regardless of whether an event is local or non-local, the state information can be restored on a node-by- node basis. To show the impact of a rollback on an LP, consider a straggler or an antimessage arriving at a location x, marked in the darkest shade in Figure 1. The roll-back will proceed as follows. The events at x will be rolled back to time t r , the time of the straggler or antimessage. Since incremental state saving is used, events have to be undone in decreasing time order to enable the recovery of state information. The rollback involves undoing events that happened at x. Each event e processed at that node will be examined to determine if e caused another event (let's call it e 1 ) to occur at a different location x 1 6= x (non-local event). In such a case, location x 1 has to be rolled back to the time prior to the occurrence of e 1 . Only then is e undone (this breath-first wave gave the name to the new approach). In our simulation, objects can move only from one lattice node to a neighboring one, so that a rollback can spread from one site only to its neighbors. The time of the rollback at the new site must be strictly greater than the one at site x, because there is a non-zero delay between causally-dependent events. In gen- eral, the breadth of the rollback is bounded by the speed with which simulated objects move around in space. Figure shows potential waves of a rollback, from the initial impact point through three more layers of processing. In practice, the size of the affected area is usually smaller than the shaded area in Figure 1, because events at one site will most likely not affect all their neighboring nodes. Obviously, if an event at location x triggered events on a neighboring LP, antimessages have to be sent. It is interesting to note that each location belonging to a given LP can be at a different logical time. In fact, we do not necessarily process events in a given LP in an increasing-timestamp order. If two events are independent, an event with a higher timestamp can be processed ahead of an event with a lower timestamp. A similar type of processing was mentioned briefly in [17] as CO-OP (Conservative-Optimistic) processing. The justification is that the requirement of processing events in timestamp order is not necessary for provably correct simulations. It is only required that the events for each simulation object be processed in a correct time order. Due to this type of processing, when we process an event (in the forward execution), we have to check the logical time of the node where the event is scheduled. If the logical time is greater than the time of the event, the node has to roll back. 4 Comparison With The Traditional Approach To demonstrate improvements in performance, we present below the model used in our initial simula- tion, which did not use the BFR method. The space, as previously mentioned, is discretized into a two-dimensional lattice. Similar discretization is used, for example, in personal communication services [4], where the space is discretized by representing the net-work as hexagonal or square cells. In these simula- tions, each cell is modeled by an LP. In our research, we have developed a simulation system for spatially explicit problems. The particular application we describe in this paper is the simulation of the spread of the Lyme disease. In Lyme disease simulation, it would be prohibitively expensive to assign one LP to each lattice node, so we "cluster" lattice nodes into a single LP. Currently, the space is divided strip-wise among the available processors. Of course, other spatial decompositions can be used. To achieve better performance, the space can also be divided into more LPs than there are available processors [8]. The LPs in this simulation are called Space Man- agers, because they are responsible for all the events that happen in a given region of space. If the Space speedup processors Figure 2: Speedup For Small Data Set (about 2,400 nodes). Manager determines that an object moves out of local space to another partition, the object and all its future events are sent to the appropriate Space Manager. As previously mentioned, the optimistic approach is used to allow concurrent processing of events happening at the same time at different locations. Because the state information is large, we use incremental state saving of information necessary for rollback. When an event is processed, the state information that it changes is placed into its local data structure. The event is then placed on a processed event list. Events that move an object from one LP to another are also placed in a message list (only pointers to the events are actually placed on the lists; the resulting duplication is not costly and speeds up sending of antimessages). If an object moves to another LP, the sending LP saves the object and the corresponding events in a ghost list to be able to restore this information upon rollback. When a rollback occurs, messages on the message list are removed and corresponding antimessages are sent out (we use aggressive cancellation). Then, the events from the processed event list are removed and undone. Undoing an event which involved sending an object to another process entails restoring the objects from the ghost list and restoring future events of the object to the event queue. For other events, the parts of the state that have been changed by the events have to be restored. During fossil collection, the obsolete information is removed and discarded from the three lists: the processed event list, the message list, and the ghost list. Initial results obtained for a small-size simulation were encouraging (Figure 2); however, the speedup number of processors Figure 3: Speedup For Large Data Set (about 32,000 run time in seconds number of processors Run Time with Multiple Logical Processes Figure 4: Running Time for Large Data Set and Multiple LPs per Processor.1.52.53.52 4 speedup number of processors Figure 5: Speedup with Large Data set and 16LPs. was not impressive for larger simulations (Figure 3). The performance degradation is caused by the large space allocation to individual processes resulting from the increased problem size. When a rollback occurs, the entire space allocated to an LP is rolled back. To minimize the impact of the rollback, we divided the space into more LPs, while keeping the same number of processors. Figure 4 shows the runtime improvement achieved with this approach. For the given problem size, the ultimate number of LPs was ure 5), and the best efficiency was achieved with 8 processors. 5 Challenges Of The New Approach In order to implement BFR, some changes had to be made not only to the simulation engine, but also to the model. A major change was made to the Move Event. The question arose: If an object is moving from location (x; y) to location should the object be placed as "processed"? If it is placed in location (x; y), and location ) is rolled back, there would be no way of finding out that the event affected location If it is placed at location location (x; y) is rolled back, a similar difficulty arises. Placing the Move Event in both processed lists is also not a good solution, because, in one case, the object is moving out of the location, and, in the other case, it is moving into the location. This dilemma motivated us to split the Move event into two: the MoveOut and MoveIn Events. Hence, when an object moves from location (x; y) to location ), the MoveOut is placed in the processed event list at (x; y) and the MoveIn at location (x 1 ). The only exception is when location belongs to another LP. In that case, the MoveIn is placed in the processed event list at location (x; y) (it will be placed on top of the corresponding MoveOut event), to indicate that a message was sent out. Upon rollback, if a MoveIn to another LP is en- countered, an antimessage is sent. The result of such a treatment of antimessages, coupled with the breadth-first processing of rollbacks, gives us an effect of lazy cancellation [12]. An antimessage is sent together with a location (x; y) to which the original message was ad- dressed, to avoid searching the lattice nodes for this information. Since the MoveIn Event indicates when a message has been sent, no message list is necessary. Another affected structure is the ghost list. In the original ap- proach, objects and their events were placed on the list in the order that objects left the partition. Now the time order is not preserved, objects are placed on the list in any timestamp order, because the nodes of the lattice can be at different times. The non-ordered aspect of the ghost list poses problems during fossil collection. The list cannot merely be truncated to remove obsolete objects. The solution, again, is to distribute that list among the nodes. This is useful for load balancing, as described in the final section. How- ever, the ghost list is relatively small (compared to the processed event list), so it might not be necessary to distribute the list if no load balancing is performed. It is sufficient to maintain an order in the list based on the virtual time at which the object is removed from the simulation. Additionally, event triggering information must be preserved. In the original implementation, when an event was created, the identity of the event that caused it was saved in one of the tags (the trigger) of the new event. When an event was undone, the dependent future events were removed by their trigger tags from the event queue. In BFR, it is possible that the future event is already processed, and its assigned location has not been rolled back yet. It is prohibitively expensive to traverse the future event list and then each processed event list in the neighborhood in search of the events whose triggers match the given event tag. The solution is to create dependency pointers from the trigger event to the newly created events. This way, a dependent event is easily accessed, and the location where it resides can be rolled back. Pointer tacking has been previously implemented for shared memory [9] to decide whether an event should be canceled or not. In our approach, we also need to know if a dependent event has been processed or not, in order to be able to quickly locate it either in the event queue or in a processed event list. One more change was required for the random number generation. In the original simulation, a single random number stream was used for an LP. These numbers are used, for example, in calculating the time of occurrence of new events. Now, since the sequence of events executed on a single LP can differ from run to run, the same random number sequence can yield two different results! Obviously, result repeatability is important, so we chose to distribute the random number sequence among the nodes of the lattice. Initially, a single random number sequence is used to seed the sequences at each node. From there, each node generates a new sequence. 6 Examples To demonstrate the behavior of the BFR algorithm, let's consider the example in Figure 6. The figure shows processed lists at three different lattice nodes: (0,0),(0,1), and (0,2). The event MO is a MoveOut MI MI could be any event MO- MoveOut MI- MoveIn event causality relation Most Recent Past MI MO MO MI relation Past Figure View of Processed Lists at Three Nodes of the Lattice. event, MI a MoveIn event, and X can be any local event. If we have a rollback for location (0,1) at time T 0 the following will happen: First MI 3 is undone and placed on the event queue. The same is done to X 2 When MO 2 is being considered, the dependence between it and MI 4 is detected, and a rollback for location (0,2) and time T 2 is performed. As a result, X 3 is undone and MI 4 is undone. Both are placed on the event queue. Next MO 2 is undone, which causes MI 4 to be removed from the event queue. MO 1 is exam- ined, and (0,0) is rolled back to time T 1 . MI 2 and are undone and placed on the event queue. MO 1 is undone and MI 1 is removed from the event queue. If the rollback occurred at location (0,0) for time then the three most recent events at location (0,0) will be undone and placed on the event queue, and no other location will be affected during the rollback. It is possible that the other locations will be affected when the simulation progresses forward. If, for exam- ple, an event MO z was scheduled for time T 2 on (0,0) and triggered an event MI z on (0,1) for time T 3 , then location (0,1) would have to roll back to time T 3 Interesting aside: We can have location (x,y) at simulation time t. The next event in the future list is scheduled for time t 1 and location cessed. If an event comes in from another process for we do not necessarily incur a rollback. If the event is to occur at location (x,y), then no rollback will happen. If, however, it is destined for location localized rollback will occur. As a result, comparing the timestamp of an incoming event uninfected larval tick infected nymphal tick unifected mouse mouse infection tick bite infection tick bite infected Figure 7: The Cycle of Lyme Disease to the local virtual time is not enough to determine if a rollback is necessary. 7 Application Description Before we present the results obtained with BFR, it is important to sketch our application-the simulation of Lyme disease. This disease is prevalent in the Northeastern United States [3, 13]. People can acquire the disease by coming in contact with a tick infected with the spirochete, which may transfer into the human's blood, causing an infection. Since the ticks are practically immobile, the spread of the disease is driven by the ticks' mobile hosts, such as mice and deer. Even though the most visible cases of Lyme disease involve humans, the main infection cycle consists of ticks and mice (Fig. 7). If an infected tick bites a mouse, the animal becomes infected. The disease can also be transmitted from an infected mouse to an uninfected, feeding tick. The seasonal cycle of the disease, and the duration of the simulation, is 180 days, starting in the late spring[6]. This time is the most active for the ticks and mice. Mice, during that time, are looking for nesting sites and may carry ticks a considerable distance [14]. The mice are modeled as individuals, and ticks, because of their sheer number (as many as 1200 larvae/400m 2 [14]) are treated as "background". The space is discretized into nodes of size 20x20m 2 , which represent the size of the home range for a mouse. Each node may contain any number of ticks in various stages of development and various infection status. Mice can move around in space in search of empty nesting sites. The initiation of such a search is described by the Disperse Event, and the moves by the Move Event. Mice can die (Kill Event) if they cannot find a nesting site or by other natural causes, such as old age, attacks by run time in sec number of processors Comparison of Run Times Between Approaches breadth-first old approach Figure 8: Results: Comparison of Runs With BFR and the Traditional Approach. predators, and disease. Mice can be bitten by ticks (Tick Bite) or have ticks drop off (Tick Drop). From the above list of events, only the Move Event is non-local Figure 8 shows the performance of BFR and illustrates almost linear speedup. The running time of the BFR is considerably shorter than that of the traditional approach. Looking at the new algorithm, we observe several benefits. The most important benefit is that, when a rollback occurs, we do not need to roll back all the events belonging to a given LP. Only the necessary events are undone. In the traditional ap- proach, the number of events that needed to be rolled back was ultimately proportional to the number of lattice nodes assigned to a given LP. When a rollback occurred, all the events that happened in that space had to be undone. On the other hand, when a rollback occurs in the BFR version, the number of events being affected by a rollback is proportional to the length of the edges of the space that interface with other LPs. In the case of the space divided into strips, the number of events affected by a given rollback is proportional to the length of the two communicating edges. There- fore, when the size of the space assigned to a given LP increases (when the number of LPs for a given problem size decreases), the number of events affected by a rollback in the case of BFR remains roughly con- stant. In the traditional approach, that number increases proportionally to the increased length of the non-communicating edges. Consequently, we observe that the number of events rolled back using BFR is an order of magnitude smaller than that in the traditional speedup number of processors Comparison of Speedup in Balanced and Unbalanced Computations balanced load uneven load Figure 9: Speedup for Balanced and Unbalanced Computations We also get fewer antimessages being sent as a result of the automatic lazy cancellation. In general, having one LP per processor eliminates on-processor communication delays. There are, of course, some drawbacks to the new method. Fossil collection is much more expensive (because lists are distributed); therefore, it is done only when the Global Virtual Time has increased by a certain amount from the last fossil collection. It is harder to maintain dependency pointers than triggers, because, when an event is un- done, its pointers have to be reset. The pointers have to be maintained when events are created, deleted, and undone, whereas triggers are set only once. There must be code to deal with multiple dependents. There is no aggressive cancellation, but, as can be seen from the results, that does not seem to have an adverse impact on performance. 9 Conclusions and Future Work We have described a new algorithm for rollback processing in spatially explicit problems. The algorithm is based on the optimistic protocol and relies on the space being partitioned into a multi-dimensional lattice. Rollbacks are minimized by examining the processed event list of each lattice node during roll- back, in search of causal dependencies between events which span the lattice nodes. The rollback impacts the minimum number of sites, making the simulation very efficient. As a result, an almost linear speedup is achieved. Obviously this performance is attainable thanks to a large amount of parallelism existing in the application. Up to now, we did not address the issue of load distribution. If the simulation's load per LP is uneven (for example, when the odd LPs have more load then the even ones), the performance degrades, as shown in Figure 9. Another advantage of BFR is that it lends itself well to load balancing, since the local (at the node level) history tracking facilitates load balanc- ing. An overloaded LP can "shed" layers of space in order to balance the load. Nothing special needs to happen on the receiving side. If messages were sent to the space that just arrived, they are simply discarded by the sender of the space and reconstructed from the ghost list by the receiver (we assume that load can only be exchanged between neighboring pro- cesses). On the sending side, however, the priority queue has to be filtered in order to extract the future events for the area sent to the new process. In order to decide if there is a need to migrate the load, the event queue can be scanned to determine the event density. Since there is a large number of events in the queue at any given time, this quantity might prove to be a good measure of load. If the density is too high at some process, some of the space can be sent to the neighboring processes. Acknowledgments This work was supported by the National Science Foundation under Grants BIR-9320264 and CCR- 9527151. The content of this paper does not necessarily reflect the position or policy of the U.S. Government-no official endorsement should be inferred or implied. --R The Dynamic Load Balancing of Clustered Time Warp for Logic Sim- ulations time warp and logic simulation. The biological and social phenomenon of Lyme disease. A Case Study in Simulating PCS Networks Using Time Warp. Distributed Simu- lation: A Case Study in Design and Verification of Distributed Programs Parallel Discrete Event Simulation of Lyme Disease. Continuously Monitored Global Virtual Time in Parallel Discrete Event Simulation. Simulating Lyme Disease Using Parallel Discrete Event Simulation. Parallel Discrete Event Simula- tion Anthony Skjel- lum Virtual Time. A Study of Time Warp Rollback Mechanisms. The epidemiology of Lyme disease in the United States 1987-1998 Temporal and Spatial Dynamics of Ixodes scapu- laris (Acari: Ixodidae) in a rural landscape The Local Time Warp Approach to Parallel Simulation. Dynamic Load Balancing of a Multi-Cluster Simulation of a Network of Workstations SPEEDES: A Unified Approach to Parallel Simulation. Incremental State Saving in SPEEDES using C --TR --CTR Jing Lei Zhang , Carl Tropper, The dependence list in time warp, Proceedings of the fifteenth workshop on Parallel and distributed simulation, p.35-45, May 15-18, 2001, Lake Arrowhead, California, United States Malolan Chetlur , Philip A. Wilsey, Causality representation and cancellation mechanism in time warp simulations, Proceedings of the fifteenth workshop on Parallel and distributed simulation, p.165-172, May 15-18, 2001, Lake Arrowhead, California, United States Ewa Deelman , Boleslaw K. Szymanski, Dynamic load balancing in parallel discrete event simulation for spatially explicit problems, ACM SIGSIM Simulation Digest, v.28 n.1, p.46-53, July 1998 M. Rao , Philip A. Wilsey, Accelerating Spatially Explicit Simulations of Spread of Lyme Disease, Proceedings of the 38th annual Symposium on Simulation, p.251-258, April 04-06, 2005 Boleslaw K. Szymanski , Gilbert Chen, Simulation using software agents I: linking spatially explicit parallel continuous and discrete models, Proceedings of the 32nd conference on Winter simulation, December 10-13, 2000, Orlando, Florida Christopher D. Carothers , David Bauer , Shawn Pearce, ROSS: a high-performance, low memory, modular time warp system, Proceedings of the fourteenth workshop on Parallel and distributed simulation, p.53-60, May 28-31, 2000, Bologna, Italy A. Maniatty , Mohammed J. Zaki, Systems support for scalable data mining, ACM SIGKDD Explorations Newsletter, v.2 n.2, p.56-65, Dec. 2000

speedup;simulation objects;spatially explicit simulations;discrete event simulation;causal relationship recovery;rollback overhead;incremental state saving;optimistic protocol;straggler;antimessage;breadth-first rollback;parallel discrete event simulations;rollback processing

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Billiards and related systems on the bulk-synchronous parallel model.

With two examples we show the suitability of the bulk-synchronous parallel (BSP) model for discrete-event simulation of homogeneous large-scale systems. This model provides a unifying approach for general purpose parallel computing which in addition to efficient and scalable computation, ensures portability across different parallel architectures. A valuable feature of this approach is a simple cost model that enables precise performance prediction of BSP algorithms. We show both theoretically and empirically that systems with uniform event occurrence among their components, such as colliding hard-spheres and ising-spin models, can be efficiently simulated in practice on current parallel computers supporting the BSP model.

Introduction Parallel discrete-event simulation of billiards and related systems is considered a non-obvious algorithmic problem, and has deserved attention in the literature [1, 5, 7, 8, 9, 11, 13, 18, 24, 23, 25]. Currently an important class of applications for these simulations is in computational physics [6, 7, 10, 14, 15, 20, 21] (e.g. hard-particle fluids, ising-spin models, disk- packing problems). However this kind of systems viewed through the more general setting of "many moving objects" [3, 16], are present everywhere in real life (e.g. big cities, transport problems, navigation systems, computer games, and combat models!). On the other hand, these systems have been considered sufficiently general and computationally intensive enough to be used as a sort of benchmark for Time Warp simulation [5, 23, 25], whereas different simulation techniques have been shown to be more efficient when dealing with large systems [8, 9, 11, 13, 18]. Similar to most of the parallel software development in the last few decades, the prevalent approach to the simulation of these systems has followed a machine dependent exploitation of the inherent parallelism associated with the problem. Currently, however, one of the greatest challenges in parallel computing is: "to establish a solid foundation to guide the rapid process of convergence observed in the field of parallel computer systems and to enable architecture independent software to be developed for the emerging range of scalable parallel systems" [17]. The bulk synchronous parallel (BSP) model has been proposed to provide such a foundation [22] and, for a wide range of ap- plications, this model has already been shown to be successful in this bridging role (i.e. a bridge between hardware and software in direct analogy with the role played by the von Neumman model in sequential computing over the last fifty years). At present, the BSP model has been implemented in different parallel architectures shared memory multi-processors, distributed memory systems, and networks of workstations enabling portable, efficient and scalable parallel software to be developed for those machines [19, 4]. A first step in the BSP implementation of conservative and optimistic parallel simulation algorithms has so far been given in [12]. In this paper we follow a different approach by using conservative algorithms designed on purely BSP concepts, and evaluating their performance under two examples: an ising-spin model and a hard-particle fluid. Note that these potentially large-scale systems have the property of a very random and even distribution of events among their constituent elements. We believe, however, that these two examples exhibit sufficient generality and complexity as to be representative of a wide range of other related asynchronous systems (e.g. some instances of the multiple-loop networks described in [8] and the systems there mentioned). Note that because of the synchronous nature of the BSP model, our algorithms are reminiscent to those proposed in [8, 18]. 2 The BSP model For a detailed description of the BSP model the reader is referred to [22, 17]. A bulk-synchronous parallel (BSP) computer consists of: (i) a set of processor-memory pairs, (ii) a communication net-work that delivers messages in a point-to-point man- ner, and (iii) a mechanism for the efficient barrier synchronization of all, or a subset, of the processors. There are no specialized broadcasting or combining facilities. If we define a time step to be the time required for a single local operation, i.e. a basic operation such as addition or multiplication on locally held data values, then the performance of any BSP computer can be characterized by the following four parameters: (i) p the number of processors, (ii) s the processor speed, i.e. number of time step per second, (iii) l the synchronization periodicity, i.e. minimal number of time steps Network l g Ring O(p) O(p) 2D Array O( p p) O( p p) Butterfly O(logp) O(logp) Hypercube O(logp) O(1) Table 1: BSP parameters for some parallel computers. elapsed between two successive barrier synchronizations of the processors, (iv) g the ratio total number of local operations performed by all processors in one second to total number of words delivered by the communication network in one second, i.e. g is a normalized measure of the time steps required to send/receive an one-word message in a situation of continuous traffic in the communication network. See Table 1 taken from [17] which shows bounds for the values of g and l for different communication networks. A BSP computer operates in the following way. A computation consists of a sequence of parallel super- steps, where each superstep is a sequence of steps, followed by a barrier synchronization of processors at which point any remote memory accesses takes effect. During a superstep each processor has to carry out a set of programs or threads, and it can do the following: (i) perform a number of computation steps, from its set of threads, on values held locally at the start of the send and receive a number of messages corresponding to non-local read and write requests. The complexity of a superstep S in a BSP algorithm is determined as follows. Let the work w be the maximumnumber of local computation steps executed by any processor during S. Let h s be the maximum number of messages sent by any processor during S, and h r be the maximum number of messages received by any processor during S. Then the cost of S is given by time steps (or alternatively g). The cost of a BSP algorithm is simply the sum of the costs of its supersteps. The architecture independence is achieved in the BSP model by designing algorithm which are parameterized not only by n, the size of the problem, and p, the number of processors, but also by l and g. The resulting algorithms can then efficiently implemented on a range of BSP architectures with widely differing l and g values. For example, on a machine with large g we must provide an algorithm with sufficient parallel slackness (i.e. a v processor algorithm implemented on a p processor machine with v ? p) to ensure that for every non-local memory access at least g operations on local data are performed. 3 Basic BSP simulation algorithms The kind of systems relevant to this paper (i.e. statistically homogeneous steady state systems with event occurrences randomly and evenly distributed among their constituent elements) can be simulated on a BSP computer using two-phase conservative algorithms as follows. On a p-processor BSP computer the whole system is divided into p equal-sized regions that are owned by a unique processor. Events involving elements located on the boundaries are called border zone events (BZ events), and are used to synchronize the parallel operation of the processors. The most conservative (but less efficient) version of this algorithm works doing iterations composed of two phases: (i) the parallel phase where each processor is simultaneously allowed to simulate sequentially and asynchronously its own region, and (ii) the synchronization phase where the occurrence of one border zone event is simulated by only one processor while the other ones remain in an idle state. [We further improve the efficiency of this algorithm by exploiting opportunities to simulate at most p border zone events in parallel during the synchronization phase.] The synchronization phase is used to cause the barrier synchronization of the processors in the simulated time, and to exchange state information among neighboring regions. [In the system examples studied below, this state information refer to the states of particular atoms and particles located in neighboring regions.] During the parallel phase every processor simulates events whose times are less than the current global next BZ event (i.e. the BZ event with the least time among all of the local next BZ events held in each region or processor). Thus, global processor synchronization is issued periodically at variable time intervals which are driven by the chronological occurrence of the BZ events. See pseudo-code in Figure 1. We assume there are n elements evenly distributed throughout the whole system, with regions made up of n=p) \Theta (a = n=p) elements. The goal is to simulate the occurrence of M events which on average are assumed to occur randomly and evenly distributed among the elements (i.e. M=n per ele- ment). This goal is achieved by the BSP algorithm in I iterations, wherein each iteration simulates a total of NPE events in the parallel phase plus one event in the synchronization phase, namely We define f I =M to be the fraction of BZ events that occur during the simulation. As we show below, for the kind of systems we are interested in we have leading to which shows that by choosing regions sufficiently large it is always possible to achieve some degree of parallelism with this strategy. However, the actual gain in running time due to the parallel phase, where each processor simulates about O(a=p) events sequentially, crucially depends on the cost of the communication and synchronization among the processors during the synchronization phase (this cost depends on the particular parallel computer). The parallel prefix operation in Figure 1 is realized as follows. A virtual t-ary tree is constructed among k be the processor that owns the region where the next border zone event (NBZE) is about to take place. This event is scheduled to occur at time T bz . The parallel prefix operation calculates the minimum among a set of p local NBZEs distributed in the p processors (the minimum is stored in each processor).] Parallel Simulation [processor i] Initialisation; while( not end condition ) begin-superstep Simulate events with time less than T bz ; Processor k reads the state of neighboring regions; end-superstep Processor k simulates the occurrence of the NBZE; endwhile Figure 1: Hyper-conservative simulation algorithm. the p processors: from the leaves to the root the partial are calculated, and then the absolute minimum is distributed among the processors going from the root to the leaves. The cost of this operation is where the value of t depends on the parameters g and l (e.g. for a small number of processors p it could be more convenient to set The efficiency of the algorithm in Figure 1 is improved by attempting to simulate in parallel at most border zone events per iteration. We explain this procedure with an example. Let us assume a situation with next BZ events e a regions R a and R b respectively. In our bg is the identifier of the element in region R i which has scheduled the next BZ event e i to occur at time t i . In addition, we define t to be the time at which an element i 0 (i (R i has scheduled a BZ event. We assume that the elements are related due to the topology of the system being simulated (e.g. neighboring atoms in the ising-spin model described below). Note that t i 0 is not necessarily the time of the next BZ event in region However, the simulation of both e i and e i 0 is restricted by the order relation between their respective scheduled times t i and t i 0 . If t must simulate e i before e i 0 , otherwise we first simulate . Thus we simulate in parallel the two next BZ events e a and e b only if t Otherwise, we must process sequentially more BZ events in the region with lesser t i until the above condition is reached. For each new BZ event processed in a region R i the non-BZ events in the time interval between two consecutive BZ events have to Parallel Simulation [region R a Initialisation; while( not end condition ) begin-superstep end-superstep begin-superstep Simulate events e k with time t k so that t k t a and t k t a 0 if (t a t a 0 ) then Simulate next BZ event e a ; endif end-superstep endwhile Figure 2: Conservative BSP simulation algorithm. be simulated as well. This is described in the pseudo-code for region R a shown in Figure 2. The operation reads the value t a 0 of the element a 0 stored in region R b . 4 Ising-spin systems The ising-spin system is modeled as a n \Theta toroidal network. Every node i of the network is an atom with magnetic spin value \Gamma1 or +1. Each atom i attempts to change its spin value at discrete times given by t is the time at which the atom i has been currently simulated, and X is a random variable with negative exponential distri- bution. The new spin value of i is decided considering the current spin values of its four neighbors. The goal of the simulation is to process the occurrence of M spin changes (events). The sequential simulation of this system is trivial since it is only necessary to deal with one type of event and to use an efficient event-list to administer the times t i(k+1) . Then the cost C 1 of processing each event that takes place in the sequential algorithm is O(logn) or even O(1) if a calendar queue were used [in [2] it has been conjectured that the calendar queue has O(1) cost under a work-load very similar to the one produced by the ising-spin system]. The cost of the sequential simulation of the whole system of n atoms is then In the case of the parallel simulation, the toroidal network is divided into p p \Theta p p regions with n=p) \Theta (a = atoms each. For each region there are a total of 4 (a \Gamma 1) atoms in the border zone, i.e., f In each region the same sequential event-list algorithm is applied during the parallel phase, although it is executed on a smaller number of atoms (n=p). The cost C p of processing every event during the parallel phase is event-list is used. For each iteration, the cost of the parallel phase is determined by the maximum number of events simulated in any processor during that period. This number is hard to determine. We optimistically assume that on average a very similar number of events are simulated by each processor. We are going to assume that from the total of M simulated in all of the parallel phases executed during the simu- lation, a total of M are simulated by each processor (this introduces a constant error since the average maximum per iteration should be considered here). Also we assume in our analysis that the M f bz BZ events that take place are simulated sequentially (hyper-conservative algorithm of Figure 1). Thus the cost TP of the parallel simulation is given by where TCS (p; g; l) is the cost in communication (g) and synchronization (l) among the (p) processors generated in each iteration. To predict the performance of a BSP algorithm we need to compare it with the fastest sequential algorithm for the same problem. With this aim define the speed-up S to be shows that there exists a BSP algorithm with total cost smaller than the cost TS of the sequential alternative (i.e. in a BSP algorithm we not only consider its computation cost but also its cost in communication and synchronization among processors). For the case of the ising-spin model we have Since C p C 1 , we can replace C p by C 1 to obtain an upper bound for TCS required to achieve S ? 1 , which expressed as shows that the effect of the cost TCS is essentially absorbed with f bz and C 1 . That is, given a particular machine (characterized by its parameters g and l) we can always achieve a speedup S ? 1 for a sufficiently large problem (characterized by its parameters f bz and For example, in an extreme situation of a system with very low C 1 to be simulated on an inefficient ma- chine, say very high g and l, the only way to achieve increasing the parallel slackness (by increasing a and/or reducing p) enough to reduce the effect of TCS (p; For the hyper-conservative algorithm given in Figure 1 the cost TCS is dominated by the parallel prefix operation, i.e., p). For Ising-spin system f exp Table 2: Results on an 8-processor IBM/SP2. the more efficient algorithm shown in Figure 2 this cost depends on the number q of different processors to which every processor has to communicate with in order to decide whether or not to simulate its next BZ event, namely (for the case of the 2D ising-spin model we have q 2). For the ising-spin model we can estimate bounds for TCS which ensure S ? 1. To this end we substitute in to obtain O( a log p ) for the hyper-conservative algorithm. On the other hand, if for the less conservative algorithm we assume that p BZ events are processed in each iteration, namely and then we obtain the better bound Note that the more restricted case C leads to the bounds O(a) and O(a p) respectively. Given the bounds for g and l shown in Table 1 we can see that the restriction (upper bound) for TCS is possible to satisfy in practice. For example, running the hyper- conservative algorithm on a 2D array computer would require one to adjust a so that a = O( p p). In Table 2 we show empirical results for S. We obtained S using the running time of the O(log n) sequential algorithm and the less conservative parallel algorithm in Figure 2. In the column 4 (t) we show the fraction of BZ events per iteration, where a value 1.0 means that one BZ event is simulated in each processor (the best case). This column (t) shows the average among all the iterations. 5 Hard-disk fluids Our second example is more complex, it consists of a two-dimensional box of size L \Theta L which contains hard-disks evenly distributed in it. After a random assignment of velocities, the (non-obvious) problem consists of simulating a total of NDDC elastic disk-disk collisions (DDC events) in a running time as small as possible. In this section we show that similar bounds for TCS (although with much higher constant factors) are required to simulate these systems on a BSP computer efficiently. To achieve an efficient sequential running time, the whole box is divided into p n c \Theta p n c cells of size oe \Theta oe with d, where d is the diameter of each disk. The box is periodical in the sense that every time a disk runs out of the box through a boundary wall, it re-enters the box at the opposite point. The neighborhood of a disk i whose center is located in the cell c, is composed of the cell c itself and the eight cells immediately (periodical) adjacent to c. We define m to be the average number of disks per cell. Since oe d, a disk i can only collide with the 9 located in the neighborhood of i. This reduces from O(n) to O(logn) the cost associated with the simulation of every DDC event that takes place since regarded as a constant and we use a O(log n) event-list to administer the pending events (collisions). As the disks move freely between DDC events they will eventually cross into neighboring cells. We regard the instant when a disk i crosses from a cell c to a neighboring cell c as a virtual wall collision (VWC) event. Then each time a VWC event takes place it is necessary to consider the possible collisions among i and the disks located in the new cells that become part of the neighborhood of i, i.e., the cells immediately adjacent to c which are not adjacent to c (3 m disks should be considered here). To consider the effect of these events, we define to be the average number of VWC events that take place between two consecutive DDC events. So the goal of simulating NDDC DDC events actually involves processing the occurrence of events. Note that 1 represents the probability that the next event to take place, in a given instant of the simulated time, is a DDC event whereas is the probability that the next event is a VWC event. To perform the simulation it is necessary to maintain for each disk i, updated information of the time t of all the possible collisions between i and the disks j located in the neighborhood of i. It is also necessary to periodically update the time when i crosses to a neighboring cell through a virtual wall w. These computations are done in a pair-wise manner by considering only the positions and velocities of the two objects involved in the event being calculated. The outcome is a dynamic set of event-tuples represents a disk j or a virtual wall w and e indicates a DDC or VWC event. At initialization, the first future events (event-tuples) are predicted for the n disks of the system and then new future events are successively calculated as the simulation advances to the end, namely every time a disk suffer a DDC or event. Notice that only a subset of all the events calculated for each disk i are the ones that really occur during the simulation, and it is not obvious how to identify these events in advance. Different methods to cope with this problem have been proposed in the literature [6, 10, 14, 20]. However the common principle is to use an efficient data structure to maintain an event-list where the future events are stored until they are removed to take place or they are invalidated by earlier events; a DDC event E(t 1 ; stored in the event-list becomes invalidated if another DDC event E(t place during the simulation. We assume here that only one event E is actually maintained for each disk i (the one with minimal time E(t)) in the event-list, and if this event E becomes invalidated a new event E 0 for i is calculated considering the complete neighborhood of i. In other words, after every DDC and VWC, and when an invalid event is retrieved from the event-list, new collisions are calculated considering the 9 located in the neighborhood (this implies a fairly slower sequential simulation but also simplifies its implementation and analysis). Note that the fraction of invalid events which are retrieved as the "next event" is less than 15% [14], so we neglect this effect in our analysis as well. After initialization, the simulation enters a basic cycle essentially composed of the following operations: (i) picking the chronological next event from the event- list, (ii) updating the state(s) of the disk(s) involved in the current event, (iii) calculating new events for this (these) disk(s) (one VWC event and several DDC events) and (iv) inserting one event in the event-list per disk involved in the current event. These operations are cyclically performed until some end condition is reached (i.e., the occurrence of a border zone event in the case of the parallel simulation). The running time of the sequential algorithm can be estimated as follows. Constant factors are neglected by considering that each of the operations of updating the position or velocity of a disk i, and calculating one DDC event for i, are all single operations with cost O(1). Also, for every disk i it is necessary to consider the disks located in the neighborhood of while calculating new DDC events for i. Calculating a VWC takes time O(1) as well. The cost associated with the event-list is log n per event insertion whereas retrieving the next event is negligible. Selecting the event with minimal time for a disk i is also negligible since this can be done as the new events are calcu- lated. This gives the costs 2 (3 log n) and for the simulation of each DDC and VWC event that takes place respectively. Then the overall cost of the simulation of each DDC that takes place is Notice that C 1 includes the cost of the VWC events that take place between two consecutive DDC events. The total running T S of the sequential algorithm is then Using theory of hard-disk fluids, we have derivated the following expression for [13], is the disk-area density of the system. Calculating d TDDC we obtained the ex- pressionm which even for extreme conditions (e.g., ae = 0:01 and On the other hand, the restriction oe d imposes a lower bound for m opt . Replacing where in practice choosing to an efficient simulation in terms of the total running time and space used by the cells. During the parallel phase every processor simulates the evolution of the disks located in its own region. If there are a total of p processors and an average of n=p disks in every region, then by logarithmic property we have which is the average running time spent by each processor computing the occurrence of two consecutive DDC events in its region. We emphasize here that hard-disk systems are by far more difficult to simulate in parallel than ising-spin models. In particular, it is necessary to cope with the problem that an event scheduled for a particular disk may not occur at the predicted time; this disk can be hit by a neighboring disk in an earlier simulated time. This necessarily leads one to deal with the possibility of "rollbacks" where the whole simulation is re-started at some check point passed without error. For example, in the algorithm of Figure 2 after that processor simulating region R a fetches time t a 0 from region R b in superstep s, it might occur that processor simulating region R b changes the value t a 0 in the next invalidating in this way all the work made by processor simulating R a during its parallel phase (i.e., superstep s 1). To cope with this prob- lem, we maintain an additional copy of the whole state of the simulation. This state is an array with one entry per disk. Each entry keeps disk's information such as position, velocity and local simulation time. We also use a single linked list to register each position of the main state array that is modified during a complete iteration (i.e., parallel phase plus synchronization phase). If the above described problem occurs, then we use the linked list to make the two state arrays identical and repeat the iteration (s; s a better estimation of t a 0 . See [13] for specific details on the BSP simulation of hard-disk fluids (e.g., since a BSP computer is a distributed memory system we maintain in each region a copy of the disks located in the border zone of its neighboring regions, thus the synchronization phase also involves the transference of information among neighboring regions to properly updated the states of these disk copies). The regions to be simulated by each processor are made up of a \Theta a cells with Also we define the BZ cells to be the 4 (a \Gamma 1) cells located in the boundaries of every region. By studying the probabilities of all the cases when a BZ event takes place we can calculate f bz , which is a function of a and other parameters of the hard-disk system. The general expression for f bz is given by where PDDCBZ and PVWCBZ are the probabilities of a DDC and VWC event taking place in border zone res- pectively. These probabilities can be calculated considering that a given disk has the same probability of being in an arbitrary cell and that its direction also has the same probability. These calculations are a bit involved because of the many cases to be considered. Briefly, the expressions given below were obtained by studying the two types of BZ events (DDC and VWC) and the positions of the disk E(i) involved in them. With probability disk is located in a BZ cell whereas with probability the disk is located in a cell neighboring to a BZ cell. For a VWC event the disk crosses to any cell with probability 1/4 whereas for a DDC event the disk collides with a disk located in any neighboring cell with probability 1/8. If the disk is located in a cell neighboring to a BZ cell then with probability the disk is located in a corner of the box. In this case the probability of a BZ event is 5/8 for a DDC and 2/4 for a VWC. When the disk is located out of the corner these last probabilities are 3/8 and 1/4 respectively. Similar considerations are used when the disk is located inside a BZ cell for a VWC. For DDC the probability of a BZ event is just bz . By doing the weighted sum of all the cases we have obtained a 2 and a a 2 where Pm represents the probability of a DDC between two disks located in the same cell. This probability depends on the size of the cells oe, but for the purpose of our analysis it is enough to say f bz !a O 'a The running time TP of the parallel algorithm is given by where TPP is the time spent simulating NPE events (DDC and VWC) during the parallel phase, and T SP is the time spent in the synchronization phase simulating one event plus the cost TCS associated with the communication and synchronization among the pro- cessors. Note that from the NPE events a total of (1=(1+)) NPE are DDC events, and these events are evenly distributed among the p processors, namely p processors simultaneously simulate (1=(1 +)) NPE =p DDC events. Then TPP is given by ''f bz and TSP is given by namely and therefore This last expression can be made more exigent for TCS assuming to obtain log n TCS which shows that for a practical simulation with 1=a, the bound for TCS is similar to those of the ising-spin system. On the other hand, if we assume that p border zone events are simulated in each iteration of the less conservative algorithm, then we have f bz and which leads to a bound similar to the one for the ising- spin model as well. It is important to note that in the calculations involved in the derivation of the speedup S, we have been very conservative in the sense that we are mixing BSP cost units with the ones defined Hard-disk fluid f exp Table 3: Empirical results on the IBM/SP2. by ourselves. Our basic unit of cost (updating disk state or calculating a new event for a disk) is much higher than the cost of each time step assumed for g and l in the BSP model. In Table 3 we show empirical results for the hard-disk fluid simulated with the less conservative algorithm running on a IBM/SP2 parallel computer. 6 Final comments In this paper we have derived upper bounds for the cost of communication and synchronization among processors in order to perform the efficient conservative simulation of two system examples. We conclude that it is possible to satisfy such bounds on current parallel computers. Our empirical results confirm this conclusion. We believe that the examples analyzed in this paper exhibit sufficient generality and complexity to be considered as representatives of a wide class of systems where the events takes place randomly and evenly distributed among their constituent elements. The first example is a very simple system where the links among neighboring regions (processors) are maintained fixed during the whole simulation. How- ever, the cost of each event processed in this system is extremely low. This imposes harder requirements on the cost of communication and synchronization (upper bounds with much lower constant factors). The second example is noticeably more complex because of the dynamic nature of the system. Here the links among regions change randomly during the simulation. Thus, even for the hyper-conservative algorithm of Figure 1, it is necessary to cope with the problem of roll-backs since the scheduled events associated with each disk do not necessarily occur at the predicted time. However the simulated time progresses statistically at the same rate in each region and the upper bounds for communication and synchronization are similar to those of the first simpler example (notice that the constant factors are much higher in this second system example which relaxes the requirements of these bounds). All of our empirical results were obtained with the less conservative algorithm shown in Figure 2 running on an IBM/SP2. We could not obtain speedup S ? 1 with the hyper-conservative algorithm of Figure 1 running under similar conditions. We emphasize, how- ever, that these results were obtained for a particular machine with fairly high g and l values. Only by increasing the slackness to O(n 5 ) disks per processor we obtained with the algorithm of Figure 1 under the experiments described in Table 3. Acknowledgements The author has been supported by University of Magallanes (Chile) and a Chilean scholarship. --R "Distributed simulation and time warp Part 1: Design of Colliding Pucks" "Calendar queues: A fast O(1) priority queue implementation for the simulation event set problem" "Discrete event simulation of object movement and interactions" "The green BSP library" "Per- formance of the colliding pucks simulation on the time warp operating system part 2: Asynchronous behavior & sectoring" "An efficient algorithm for the hard-sphere problem" "Efficient parallel simulations of dynamic ising spin systems" "Efficient distributed event-driven simulations of multiple-loop networks" "Simulating colliding rigid disks in parallel using bounded lag without time warp" "How to simulate billiards and similars systems" "Simulating billiards: Serially and in parallel" "Direct BSP algorithms for parallel discrete-event simulation" "Event-driven hard-particle molecular dynamics using bulk synchronous parallelism" "Ef- ficient algorithms for many-body hard particle molecular dynamics" "An empirical assessment of priority queues in event-driven molecular dynamics simulation" "An object oriented C++ approach for discrete event simulation of complex and large systems of many moving ob- jects" "General purpose parallel comput- ing" "Parallel simulation of billiard balls using shared variables" "The event scheduling problem in molecular dynamics simulation" "Reduction of the event-list for molecular dynamic simulation" "A bridging model for parallel com- putation" "Distributed combat simulation and time warp: The model and its performance" "Im- plementing a distributed combat simulation on the time warp operating system" "Case studies in serial and parallel simulation" --TR Efficient parallel simulations of dynamic Ising spin systems Calendar queues: a fast 0(1) priority queue implementation for the simulation event set problem Implementing a distributed combat simulation on the Time Warp operating system Efficient distributed event-driven simulations of multiple-loop networks A bridging model for parallel computation How to simulate billiards and similar systems Discrete event simulation of object movement and interactions General purpose parallel computing Efficient algorithms for many-body hard particle molecular dynamics An efficient algorithm for the hard-sphere problem Parallel simulation of billiard balls using shared variables An object oriented C++ approach for discrete event simulation of complex and large systems of many moving objects --CTR Wentong Cai , Emmanuelle Letertre , Stephen J. Turner, Dag consistent parallel simulation: a predictable and robust conservative algorithm, ACM SIGSIM Simulation Digest, v.27 n.1, p.178-181, July 1997

billiards;general purpose parallel computing;discrete-event simulation;homogeneous large-scale systems;discrete event simulation;ising-spin models;scalable computation;colliding hard-spheres;bulk-synchronous parallel model

269004

Compositional refinement of interactive systems.

We introduce a method to describe systems and their components by functional specification techniques. We define notions of interface and interaction refinement for interactive systems and their components. These notions of refinement allow us to change both the syntactic (the number of channels and sorts of messages at the channels) and the semantic interface (causality flow between messages and interaction granularity) of an interactive system component. We prove that these notions of refinement are compositional with respect to sequential and parallel composition of system components, communication feedback and recursive declarations of system components. According to these proofs, refinements of networks can be accomplished in a modular way by refining their compponents. We generalize the notions of refinement to refining contexts. Finally, full abstraction for specifications is defined, and compositionality with respect to this abstraction is shown, too.

Introduction A distributed interactive system consists of a family of interacting components. For reducing the complexity of the development of distributed interactive systems they are developed by a number of successive development steps. By each step the system is described in more detail and closer to an implementation level. We speak of levels of abstraction and of stepwise refinement in system development. When describing the behavior of system components by logical specification techniques a simple concept of stepwise refinement is logical implication. Then a system component specification is a refinement of a component specification, if it exhibits all specified properties and possibly more. In fact, then refinement allows the replacement of system specifications by more refined ones exhibiting more specific properties. More sophisticated notions of refinement allow to refine a system component to one exhibiting quite different properties than the original one. In this case, however, we need a concept relating the behaviors of the refined system component to behaviors of the original one such that behaviors of the refined system component can be understood to represent behaviors of the original one. The behavior of interactive system components is basically given by their interaction with their environment. Therefore the refinement of system components basically has to deal with the refinement of their interaction. Such a notion of interaction refinement is introduced in the following. Concepts of refinement for software systems have been investigated since the early 1970s. One of the origins of refinement concepts is data structure refinement as treated in Hoare's pioneering paper [Hoare 72]. The ideas of data structure refinement given there were further explored and developed (see, for instance, [Jones 86], [Broy et al. 86], [Sannella 88], see [Coenen et al. 91] for a survey). Also the idea of refining interacting systems has been treated in numerous papers (see, for instance, [Lamport 83], [Abadi, Lamport 90], and [Back 90]). Typically distributed interactive systems are composed of a number of components that interact for instance by exchanging messages or by updating shared memory. Forms of composition allow to compose systems from smaller ones. Basic forms of composition for systems are parallel and sequential composition, communication feedback and recursion. For a set of forms of composition a method for specifying system components is called compositional (sometimes also the word modular is used), if the specification of composed systems can be derived from the specifications of the constituent components. We call a refinement concept compositional, if refinements of a composed system are obtained by giving refinements for the components. Traditionally, compositional notions of specification and refinement for concurrent systems are considered hard to obtain. For instance, the elegant approach of [Chandy, Misra 88] is not compositional with respect to liveness properties and does not provide a compositional notion of refinement. Note, it makes only sense to talk about compositionality with respect to a set of forms of composi- tion. Forms of composition of system components define an algebra of systems, also called a process algebra. Not all approaches to system specifications emphasise forms of composition for systems. For instance, in state machine oriented system specifications systems are modelled by state transitions. No particular forms of composition of system components are used. As a consequence compositionality is rated less significant there. Approaches being in favor of describing systems using forms of composition are called "algebraic". A discussion of the advantages and disadvantages of algebraic versus nonalgebraic approaches can be found, for instance, in [Janssen et al. 91]. Finding compositional specification methods and compositional interaction refinement concepts is considered a difficult issue. Compositional refinement seems especially difficult to achieve for programming languages with tightly coupled parallelism as it is the case in a "rendezvous" concept (like in CCS and CSP). In tightly coupled parallelism the actions are directly used for the synchronization of parallel activities. Therefore the granularity of the actions cannot be refined, in general, without changing the synchronization structure (see, for instance, [Aceto, Hennessy 91] and [Vogler 91]). The presentation of a compositional notion of refinement where the granularity of interaction can be refined is the overall objective of the following sections. We use functional, purely descriptive, "nonoperational" specification techniques. The behavior of distributed systems interacting by communication over channels is represented by functions processing streams of messages. Streams of messages represent communication histories on channels. System component specifications are predicates characterizing sets of stream processing functions. System components described that way can be composed and decomposed using the above mentioned forms of composition such as sequential and parallel composition as well as communication feedback. With these forms of composition all kinds of finite data processing nets can be described. Allowing in addition recursive declarations even infinite data processing nets can be described. In the following concepts of refinement for interactive system components are defined that allow one to change both the number of channels of a component as well as the granularity of the messages sent by it. In particular, basic theorems are proved that show that the introduced notion of refinement is compositional for the basic compositional forms as well as for recursive declara- tions. Accordingly for an arbitrary net of interacting components a refinement is schematically obtained by giving refinements for its components. The correctness of such a refinement follows according to the proved theorems schematically from the correctness proofs for the refinements of the components. We give examples for illustrating the compositionality of refinement. We deliberately have chosen very simple examples to keep their specifications small such that we can concentrate on the refinement aspects. The simplicity of these examples does not mean that much more complex examples cannot be treated. Finally we generalize our notion of refinement to refining contexts. Refining contexts allow refinements of components where the refined presentation of the input history may depend on the output history. This allows in particular to understand unreliable components as refinements of reliable components as long as the refining context takes care of the unreliability. Refining contexts are represented by predicate transformers with special properties. We give examples for refining contexts. In an appendix full abstraction of functional specifications for the considered composing forms is treated. Specification In this section we introduce the basic notions for functional system models and functional system specifications. In the following we study system components that exchange messages asynchronously via channels. A stream represents a communication history for a channel. A stream of messages over a given message set M is a finite or infinite sequence of messages. We define We briefly repeat the basic concepts from the theory of streams that we shall use later. More comprehensive explanations can be found in [Broy 90]. ffl By x - y we denote the result of concatenating two streams x and y. We assume that x - ffl By hi we denote the empty stream. ffl If a stream x is a prefix of a stream y, we write x v y. The relation v is called prefix order. It is formally specified by ffl By (M ! ) n we denote tuples of n streams. The prefix ordering on streams as well as the concatenation of streams is extended to tuples of streams by elementwise application. A tuple of finite streams represents a partial communication history for a tuple of channels. A tuple of infinite streams represents a total communication history for a tuple of channels. The behavior of deterministic interactive systems with n input channels and output channels is modeled by (n; m)-ary stream processing functions A stream processing function determines the output history for a given communication history for the input channels in terms of tuples of streams. Example 1 Stream processing function Let a set D of data elements be given and let the set of messages M be specified by: Here the symbol ? is a signal representing a request. For data elements a stream processing function is specified by The function (c:d) describes the behavior of a simple storage cell that can store exactly one data element. Initially d is stored. The behavior of the component modeled by (c:d) can be illustrated by an example input The function (c:d) is a simple example of a stream processing function where every input message triggers exactly one output message. End of example In the following we use some notions from domain and fixed point theory that are briefly listed: ffl A stream processing function is called prefix monotonic, if for all tuples of streams We denote the function application f(x) by f:x to avoid brackets. ffl By tS we denote a least upper bound of a set S, if it exists. ffl A set S is called directed, if for any pair of elements x and y in S there exists an upper bound of x and y in S. ffl A partially ordered set is called complete, if every directed subset has a least upper bound. ffl A stream processing function f is called prefix continuous, if f is prefix monotonic and for every directed set S ' M ! we have: The set of streams as well as the set of tuples of streams are complete. For every directed set of streams there exists a least upper bound. We model the behavior of interactive system components by sets of continuous (and therefore by definition also monotonic) stream processing functions. Monotonicity models causality between input and output. Continuity models the fact that for every behavior the system's reaction to infinite input can be predicted from the component's reactions to all finite prefixes of this input 1 . Monotonicity takes care of the fact that in an interactive system output already produced cannot be changed when further input arrives. The empty stream is to be seen as representing the information "further communication unspecified". Note, in the example above by the preimposed monotonicity of the function (c:d) we conclude otherwise, we could construct a contradiction. A specification describes a set of stream processing functions that represent the behaviors of the specified systems. If this set is empty, the specification is called inconsistent , otherwise it is called consistent . If the set contains exactly one element, then the specification is called determined. If this set has more then one element, then the specification is called underdetermined and we also speak of underspecification. As we shall see, an underdetermined specification may be refined into a determined one. An underdetermined specification can also be used to describe hardware or software units that are nondeterministic. An executable system is called nondeterministic, if it is underdetermined. Then the underspecification in the description of the behaviors of a nondeterministic system allows nondeterministic choices carried out during the execution of the system. In the descriptive modeling of interactive systems there is no difference in principle between underspecification und the operational notion of nondeter- minism. In particular, it does not make any difference in such a framework, whether these nondeterministic choices are taken before the execution starts or step by step during the execution. The set of all (n,m)-ary prefix continuous stream processing functions is denoted by The number and sorts of input channels as well as output channels of a specification are called the component's syntactic interface. The behavior, represented by the set of functions that fulfill a specification, is called the component's semantic interface. The semantic interface includes in particular the granularity of the interaction and the causality between input and output. For simplicity we do not consider specific sort information for the individual channels of components in the following and just assume M to be a set of messages. However, all our results carry over straightforwardly to stream processing functions where more specific sorts are attached to the individual channels. This does not exclude the specification of more elaborate liveness properties including fairness. Note, fairness is, in general, a property that has to do with "fair" choices between an infinite number of behaviors. Figure 1: Graphical representation of a component Q A specification of a possibly underdetermined interactive system component with n input channels and m output channels is modeled by a predicate characterizing prefix continuous stream processing functions. Q is called an (n; m)-ary system's specification. A graphical representation of an (n; m)-ary system component Q is given in Figure 1. The set of specifications of this form is denoted by Example 2 Specification A component called C (for storage Cell) with just one input channel and one output channel is specified by the predicate C. The component C can be seen as a simple store that can store exactly one data element. C specifies functions f of the functionality: Let the sets D and M be specified as in example 1. If C receives a data element it sends a copy on its output channels. If it receives a request represented by the signal ?, it repeats its last data output followed by the signal ? to indicate that this is repeated output. The signal ? is this way used for indicating a "read storage content request". The signal ? triggers the read operation. A data element in the input stream changes the content of the store. The message d triggers the write operation. Initially the cell carries an arbitrary data element. This behavior is formalized by the following specification for C: where the auxiliary function (c:d) is specified as in example 1. Notice that the data element stored initially is not specified and thus component C is underdetermined End of example For a deterministic specification Q where for exactly one function q the predicate Q is fulfilled, in other words where we have we often write (by misuse of notation) simply q instead of Q. This way we identify determined specifications and their behaviors. m we denote the identity function; that is we assume We shall drop the index m for I m whenever it can be avoided without confusion. m we denote the function that produces for every input just the empty stream as output on all its output channels; that is we define Similarly we write y m for the unique function in SPF m other words the function with m input channels, but with no output channels. By / L n m we denote the logically weakest specification, which is the specification that is fulfilled by all stream processing functions. It is defined by By n \Upsilon we denote the function that produces two copies of its input. We have 2n and \Upsilon By n+m we denote the function that permutes its input streams as Again we shall drop the index n as well as m \Upsilon whenever it can be avoided without confusion. Composition In this section we introduce the basic forms of composition namely sequential composition, parallel composition and feedback. These compositional forms are introduced for functions first and then extended to component specifications. 3.1 Composition of Functions Given functions we write for the sequential composition of the functions f and g which yields a function in SPF n Given functions we write fkg for the parallel composition of the functions f and g which yields a function in We assume that " ; " has higher priority than "k". Given a function we write -f for the feedback of the output streams of function f to its input channels which yields a function in SPF n Here fix denotes the fixed point operator associating with any monotonic function f its least fixed point fix:f . Thus means that y is with respect the prefix ordering the least solution of the equation We assume that "-" has higher priority than the binary operators ";" and "k". A graphical representation for feedback is given in Figure 2. We obtain a number of useful rules by the fixed point definition of -f . As a simple consequence of the fixed point characterization, we get the unfold rules: A graphical representation of the unfold rules for feedback is given in Figure 3. -f Figure 2: Graphical representation of feedback f f -f f f Figure 3: Graphical representation of the unfold rules for feedback f Figure 4: Graphical representation of semiunfold A useful rule for feedback is semiunfold that allows one to move components outside or inside the feedback loop (let g 2 SPF m A graphical representation for semiunfold is given in Figure 4. For reasoning about feedback loops and fixed points the following special case of semiunfold is often useful: The rule is an instance of semiunfold with y. The correctness of this rule can also be seen by the following argument: if y is the least fixed point of and e y is the least fixed point of then e Semiunfold is a powerful rule when reasoning about results of feedback loops. 3.2 Composition of Specifications We want to compose specifications of components to networks. The forms of composition introduced for functions can be extended to component specifications in a straightforward way. Given component specifications we write for the predicate in SPEC n Trivially we have for all specifications Q 2 SPEC n m the following equations: Given specifications we write QkR for the predicate in SPEC n1+n2 m1+m2 where Given specification we write for the predicate in SPEC n For feedback over underdetermined specifications we get the following rules 2 For determined system specifications Q we get the stronger rules and which do not hold for underdetermined systems, in general. The erroneous assumption that these rules are valid also for underdetermined systems is the source for the merge anomaly (see [Brock, Ackermann 81]). A useful rule for feedback is fusion that allows one to move components that are not affected by the feedback outside or inside the feedback operator application. With the help of the basic functions and the forms of composition introduced so far we can represent all kinds of finite networks of systems (data flow nets) 3 . The introduced composing forms lead to an algebra of system descriptions. 4 Refinement, Representation, Abstraction In this section we introduce concepts of refinement for system components both with respect to the properties of their behaviors as well as with respect to their syntactic interface and granularity of interaction. We start by defining a straightforward notion of property refinement for system component specifications. Then we introduce a notion of refinement for communication histories. Based on this notion we define the concept of interaction refinement for interactive components. This notion allows to refine a component by changing the number of input and output channels as well as the granularity of the exchanged messages. 4.1 Property Refinement Specifications are predicates characterizing functions. This leads to a simple notion of refinement of component specifications by adding logical properties. Given specifications e Q is called a (property) refinement of Q if for all f 2 SPF n e Then we write e If e Q is a property refinement for Q, then e Q has all the properties Q has and may be some more. Every behavior that e Q shows is also a possible behavior of Q. 3 Of course, the introduced combinatorial style for defining networks is not always very useful, in practice, since the combinatorial formulas are hard to read. However, we prefer throughout this report to work with these combinatorial formulas, since this puts emphasis on the compositional forms and the structure of composition. For practical purposesa notation with named channels is often more adequate. All considered composing forms are monotonic for the refinement relation as indicated by the following theorem. Theorem 1 (Compositionality of Refinement) Proof: Straightforward, since all operators for specifications are defined point-wise on the sets of functions that are specified. 2 A simple example of a property refinement is obtained for the component C as described in Example 2 on page 8 if we add properties about the data element initially stored in the cell. A property refinement does not allow one to change the syntactic interface of a component, however. 4.2 Interaction Refinement Recall from section 2 that streams model communication histories on channels. In more sophisticated development steps for a component the number of channels and the sorts of messages on channels are changed. Such steps do not represent property refinements. Therefore we introduce a more general notion of refinement. To be able to do this we study concepts of representation of communication histories on n channels modeled by a tuple of n streams by communication histories on m channels modeled by a tuple of m streams. Tuples of streams y can be seen as representations of tuples of streams we introduce a mapping ae 2 SPF n m that associates with every x its representation. ae is called a representation function. If ae is injective then it is called a definite representation function. Note, a mapping ae is injective, if and only if: If a specification R 2 SPEC n m is used for the specification of a set of representation functions, R is called a representation specification. Example 3 Representation Specification We specify a representation specification R allowing the representation of streams of data elements and requests by two separate streams, one of which carries the requests and the other of which carries the data elements. The representation functions are mappings ae of the following functionality: Here p is used as a separator signal. It can be understood as a time tick that separates messages. Given streams x and y let [x; y] denote a pair of streams and the elementwise concatenation of pairs of streams, in other words: Let T icks be defined by the set of pairs of streams of ticks that have equal We specify the representation specification R explicitly as follows: Note, by the monotonicity of the specified functions: The computation of a representation is illustrated by the following example: The example demonstrates how the time ticks are used to indicate in the streams ae(x) the order of the requests relatively to the data messages in the original stream x. End of example The elements in the images of the functions ae with R:ae are called representations. representation specification) A representation specification R is called definite, if In other words R is definite, if different streams x are always differently represented Obviously, if R is a definite representation specification, then all functions ae with R:ae are definite. For definite representation specifications for elements x and x with x 6= x the sets of representation elements are disjoint. Note, the representation specification given in the example above is definite. For every injective function, and thus for every definite representation function ae, there exists a function ff 2 SPF m n such that: The function ff is an inverse to ae on the image of ae. The function ff is called an abstraction for ae. Notice that ff is not uniquely determined as long as ae is not surjective. In other words, as long as not all elements in (M are used as representations of elements in (M ! ) n there may be several functions ff with A:ff. The concept of abstractions for definite representation functions can be extended to definite representation specifications. m be a definite representation specification; a function ff 2 SPF m n with is called an abstraction function for R. The existence of abstractions follows from the definition of definite representation specification. again for definite representation specifications the abstraction functions ff are uniquely determined only on the image of R, that is on the union of the images of functions ae with R:ae. Definition 3 (Abstraction for a definite representation specification) n be the specification with Then A is called the abstraction for R. For consistent definite representation specifications R with abstraction A we have If ae; contains all possible choices of representation functions for the abstraction A. Example 4 Abstraction For the representation specification R described in example 3 the abstraction functions ff are mappings of the functionality: The specification of A reads as follows. It is a straightforward rewriting proof that indeed: The specification A shows a considerable amount of underspecification, since not all pairs of streams in f?; are used as representations. End of example Parallel and sequential composition of definite representations leads to definite representations again. Theorem 2 Let R be definite representation specifications for (assuming in the second formula) are definite representation specifications Proof: Sequential and parallel composition of injective functions leads to injective functions. 2 Trivially we can obtain the abstractions of the composed representations by composing the abstractions. For many applications, representation specifications are neither required to be determined nor even definite. For an indefinite representation specification sets of representation elements for different elements are not necessarily disjoint. Certain representation elements y do occur in several sets of representations for elements. They ambiguously stand for ("represent") different elements. Such an element may represent the streams x as well as x, if ae:x = ae:x for functions ae and ae with R:ae and R:ae. For indefinite representation specifications the represented elements are not uniquely determined by the representation elements. A representation element y stands for the set For a definite representation specification R this set contains exactly one element while for an indefinite representation specification R this set may contain more than one element. In the latter case, of course, abstraction functions ff with I do not exist. However, even for certain indefinite representations we can introduce the concept of an abstraction. Definition 4 (Uniform representation specifications) A consistent specification m is called a uniform representation specification, if there exists a specification A 2 SPEC m n such that for all ae: The specification A is called again the abstraction for R. The formula expresses that (R; A) is a left-neutral element for every representation function in R. Essentially the existence of an abstraction expresses the following property of R: if for different elements x and x the same representations are possible, then every representation function maps these elements onto equal representations. More formally stated, if there exist functions e ae and ae with R:eae and R:ae such that e then for all functions ae with R:ae: Thus if elements are identified by some representation functions, this identification is present in all representation functions. The same amount of information is "forgotten" by all the representations. The representation functions then are indefinite in a uniform way. Definite representations are always uniform. A function is injective, if for all x and x we have: A function that is not injective ae defines a nontrivial partition on its domain. A representation specification is uniform if and only if all functions ae with R:ae define the same partition. For a uniform representation specification R with abstraction A the product (R; reflects the underspecification in the choices of the representations provided by R. If for a function fl with (R; A):fl we have have the same representations. Definition 5 (Adequate representation) A uniform representation specification R with abstraction A is called adequate for a specification Q, if: Adequacy means that the underspecification in (R; A) does not introduce more underspecification into Q; R; A than already present in Q. Note, definite representations are adequate for all specifications Q. Definition 6 (Interaction refinement) Given representations R 2 SPEC n m and specifications b m we say that b Q is an interaction refinement of Q for the representation specifications R and R, if R R Figure 5: Commuting diagram of interaction refinement This definition indicates that we can replace via an interaction refinement a system of the form Q; R by a refined system of the form R; b Q. We may think about the relationship between Q and b Q as follows: the specification Q specifies a component on a more abstract level while Q 0 gives a specification for the component at a more concrete level. Instead of computing at the abstract level with Q and then translating the output via R onto the output representation level, we may translate the input by R onto the input representation level and compute with b Q. We obtain one of these famous commuting diagrams as shown in Figure 5. Definition 7 (Adequate interaction refinement) The interaction refinement of Q for the representation specifications R and R is called adequate for a specification Q, if R is adequate for Q. For adequate interaction refinements using uniform representation specifications R with abstraction A 2 SPEC m since from the interaction refinement property we get and by the adequacy of R for Q which shows that R; b Q; A is a (property) refinement of Q. A graphical illustration of adequate interaction refinement is shown in Figure 6. R Figure Commuting diagram of interaction refinement The following table summarizes the most important definitions introduced so far. Table of definitions e property refinement of Q e R consistent, definite with abstr. A R; R uniform with abstraction A R:ae ) R; R adequate for Q with abs. A Q; R; A ) Q Inter. refinement b Q of Q for R; R R; b Adequate inter. refinement R uniform and adequate for Q The notion of interaction refinement allows one to change both the syntactic and the semantic interface. The syntactic interface is determined by the number and sorts of channels; the semantic interface is determined by the behavior of the component represented by the causality between input and output and by the granularity of the interaction. Example 5 Interaction Refinement We refine the component C as given in Example 2 into a component b C that has instead of one input and one output channel two input and two output channels. The refinement b C uses one of its channels carrying the signal ? as a read channel and one of its channels carrying data as a write channel. Let R and A be given as specified in the examples above We specify the interaction refinement b C of C explicitly. b C specifies functions of functionality: We specify: where the auxiliary function h is specified by: It is a straightforward proof to show: Assume ae with R:ae and h such that there exist f and d with b C:f and we prove by induction on the length of the stream x that there exist e ae with R:eae and c:d as specified in example 1 such that: ae:(c:d):x For we obtain: there exists t 2 T icks such that: e e ae:(c:d):x Now assume the hypothesis holds for x; there exists t 2 T icks: There exists t 2 T icks: This concludes the proof for finite streams x. By the continuity of h and ae the proof is extended to infinite x. End of example Continuing with the system development after an adequate interaction refinement of a component we may decide to leave R and A unchanged and carry on by just further refining b Q. 5 Compositionality of Interaction Refinement Large nets of interacting components can be constructed by the introduced forms of composition. When refining such large nets it is decisive for keeping the work manageable that interaction refinements of the components lead to interaction refinements of the composed system. In the following we prove that interaction refinement is indeed compositional for the introduced composing forms that is sequential and parallel composition, and communication feedback. 5.1 Sequential and Parallel Composition For systems composed by sequential compositions, refinements can be constructed by refining their components. Theorem 3 (Compositionality of refinement, seq. composition) Assume is an interaction refinement of Q i for the representations R i\Gamma1 and R i is an interaction refinement of Q for the representations R 0 and R 2 . Proof: A straightforward derivation shows the theorem: interaction refinement of Q 1 g interaction refinement of Q 2 g Example 6 Compositionality of Refinement for Sequential Composition Let C and b C be specified as in the example above. Of course, we may compose C as well as b C sequentially. We define the components CC and d CC by: Note, CC is a cell that repeats its last input twice on a signal ?. It is a straight-forward application of our theorem of the compositionality of refinement that d CC is a refinement of CC : Of course, since R; A = I we also have that R; d CC;A is a property refinement of CC. End of example Refinement is compositional for parallel composition, too. Theorem 4 (Compositionality of refinement for parallel composition) Assume b is an interaction refinement of Q i for the representations R i and R i is an interaction refinement of Q 1 kQ 2 for the representations Proof: A straightforward derivation shows the theorem: (R sequential and parallel compositiong (R interaction refinement for sequential and parallel compositiong (R 1 kR 2 )For sequential and parallel composition compositionality of refinement is quite straightforward. This can be seen from the simplicity of the proofs. 5.2 Feedback For the feedback operator, refinement is not immediately compositional. We do not obtain, in general, that - b Q is an interaction refinement of -Q for the representations R and R provided b Q is an interaction refinement of Q for the representations RkR and R. This is true, however, if I ) (A; R) (see below). The reason is as follows. In the feedback loops of - b Q we cannot be sure that only representations of streams (i.e. streams in the images of some of the functions characterized by R) occur. Therefore, we have to give a slightly more complicated scheme of refinement for feedback. Theorem 5 (Compositionality of refinement, feedback) Assume b Q is an interaction refinement of Q for the representation specifications RkR and R where R is uniform; then -((IkA; R); b Q) is an interaction refinement of -Q for the representations R and R. Proof: We prove: (R; -((IkA; R); b From (R; -((IkA; R); b we conclude that there exist functions ae, b q, ae, and ff such that R:ae, b Q:bq, R:ae, and A:ff and furthermore Q is an interaction refinement of Q for the representations RkR and R for functions ae with R:ae and ae with R:ae and - q with b Q:q there exist functions q and e ae such that Q:q and R:eae hold and furthermore ae Given x, because of the continuity of ae, b q, ae, and ff, we may define -((Ikff; ae); b q):ae:x by tby i where Moreover, because of the continuity of q, we may define ~ ae:(-q):x by ~ ae: ty i where We prove: e ae: t y by computational induction. We prove by induction on i the following proposition ae:y is the least elementg e fy 0 is the least elementg e fy 0 is the least elementg fdefinition of b Assume now the proposition holds for i; then we obtain: fdefinition of b y finduction hypothesisg e fdefinition of y e ae:y Furthermore we get: e fdefinition of y e finduction hypothesisg fdefinition of b y i+2 g b y i+2 ?From this we conclude by the continuity of e ae that: and thus and finally q))Assuming an adequate refinement allows us to obtain immediately the following corollary. Theorem 6 (Compositionality of adequate refinement, feedback) Assume Q is an adequate interaction refinement of Q for the representations RkR and R with abstraction A then -( b Q; A; R) is an interaction refinement of -Q for the representations R and R. Proof: Let all the definitions be as in the proof of the previous theorem. Since the interaction refinement is assumed to be adequate there exists a function e with Q:q such that Carrying out the proof of the previous theorem with e q instead of q and ae instead of e ae we get: By straightforward computational induction we may prove This concludes the proof. 2 Assuming that A; R contains the identity as a refinement we can simplify the refinement of feedback loops. Theorem 7 Assume b Q is an interaction refinement of Q for the representations RkR and R with abstraction A and assume furthermore I ) A; R Q is an interaction refinement of -Q for the representations R and R. Proof: Straightforward deduction shows: -Q; RNote, even if I is not a refinement of A; R, in other words even if I ) A; R does not hold, other refinements of A; R may be used to simplify and refine the term A; R in -((IkA; R); b Q). By the fusion rule for feedback as introduced in section 3 we obtain: This may allow further refinements for b Example 7 Compositionality of Refinement for Feedback Let us introduce the component F with two input channels and one output channel. It specifies functions of the following functionality: F is specified as follows: where the auxiliary function g is specified by It is a straightforward proof that for the specification C as defined in Example 1: We carry out this proof by induction on the length of the input streams x. We show that -f fulfills the defining equations for functions c:d in the definition of C in Example 2. Let f be a function with F:f and g be a function as specified above in the definition of F . We have to consider just two cases: by the definition of f there exists g as defined above such that: there exists d: Induction on the length of x and the continuity of the function g conclude the proof. The refinement b F of F according to the representation specification R from example 3 specifies functions of the functionality: It reads as follows: where the auxiliary function g is specified by We have (again, this can be proved by a straightforward rewrite proof): Moreover, we have according to Theorem 5: and therefore Note, the refinement is definite and therefore adequate for F . Therefore we may replace -((IkA; R); b F ) by -( b The component -( b can be further refined by refining A; R. Let us, therefore, look for a simplification for A; R. We do not have I ) A; R since by the monotonicity of all ff with A:ff we have: (otherwise we obtain a contradiction, since by monotonicity the first elements of ff(x; d - y) have to coincide for all x and y). Therefore for all ae with R:ae: This indicates that there are no functions ae and ff with R:ae and A:ff such that is valid for all x. We therefore cannot simply refine A; R into I. We continue the refinement by refining p. We take into account properties of b F . A simple rewriting proof shows: Summarizing our refinements we obtain: This concludes our example of refinement for feedback. End of example Recall that every finite network can be represented by an expression that is built by the introduced forms of composition. The theorems show that a network can be refined by defining representation specifications for the channels and by refining all its components. This provides a modular method of refinement for networks. 6 Recursively defined Specifications Often the behavior of interactive components is specified by recursion. Given a function a recursive declaration of a component specification Q is given by a declaration based on - : Recursive specifications are restricted in the following to functions - that exhibit certain properties. 6.1 Semantics of Recursively Defined Specifications A function - where is monotonic with respect to implication, if: A set of specifications is called a chain, if for all i 2 IN and for all functions f 2 SPF n A function - is continuous with respect to implication, if for every chain Note, the set of all specifications forms a complete lattice. Definition 8 (Predicate transformer) A predicate transformer is a func- tion that is monotonic and continuous with respect to implication (refinement). Note, if - is defined by - Net(X) is a finite network composed of basic component specifications by the introduced forms of composition, then - is a predicate transformer. A recursive declaration of a component specification Q is given by a defining equation (often called the fixed point equation) based on a predicate transformer A predicate Q is called a fixed point of - if: In general, for a function - there exist several predicates Q that are fixed points of - . In fixed point theory a partial order on the domain of - is established such that every monotonic function - has a least fixed point. This fixed point is associated with the identifier f by a recursive declaration of the form For defining the semantics of programming languages the choice of the ordering, which determines the notion of the least fixed point, has to take into account operational considerations. There the ordering used in the fixed point construction has to reflect the stepwise approximation of a result by the execution. For specifications such operational constraints are less significant. Therefore we choose a very liberal interpretation for recursive declarations of specifications in the following. For doing so we define the concept of an upper closure of a specification. The upper closure is again a predicate transformer: It is defined by the following equation: Notice that \Xi is a classical closure operator, since it has the following characteristic properties: A predicate Q is called upward closed, if by \Xi the least element\Omega is mapped onto the specification / L that is fulfilled by every function, that is From a methodological point of view it is sufficient to restrict our attention to specifications that are upward closed 4 . This methodological consideration and the considerable simplification of the formal interpretation of recursive declarations are the reasons for considering only upward closed solutions of recursive equations. A predicate transformer - is called upward closed, if for all predicates Q we By the recursive declaration 4 Taking the upper closure for a specification may change its safety properties. However, only safety properties for those behaviors may be changed where the further output, independent of further input, is empty. A system with such a behavior does not produce a specific message on an output channel, even, if we increase the streams of the messages on the input channels. Then what output is produced on that channel obviously is not relevant at all. we associate with Q the predicate that fulfills the following equation: where the predicates Q i are specified by: According to this definition we associate with a recursive declaration the logically weakest 5 predicate Q such that The predicate Q is then denoted by fix:- . 6.2 Refinement of Recursively Specified Components A uniform representation specification R with abstraction A is called adequate for the predicate transformer - , if for all predicates X: Adequacy implies that specifications for which R is adequate are mapped by - onto specifications by for which R is adequate again. Uniform interaction refinement is compositional for recursive definitions based on predicate transformers for which the refinement is adequate. again definite representations are always adequate. Theorem 8 (Compositionality of refinement for recursion) Let representation specifications R and R be given, where R is uniform with abstraction A and adequate for the predicate transformer For a predicate transformer where and for all predicates X; b (R; b we have 5 True is considered weaker than false. Proof: Without loss of generality assume that the predicate transformers - and are upward closed. Define We prove: This proposition is obtained by a straightforward induction proof on i. For we have to show: which is trivially true, since / L holds for all functions. The induction step reads as follows: from we conclude by the adequacy of - : fdefinition of Q induction hypothesisg fdefinition of Q We prove by induction on i: For we have to prove: This is part of our premises. Now assume the induction hypothesis holds for trivially Therefore, with by our premise we have: By the induction hypothesis and by the fact as can be seen by the derivation We obtain: - with representations R the premise is always valid as the following straightforward derivation shows: fdefinition of / Lg We immediately obtain the following theorem as corollary. It can be useful for simplifying the refinement of recursion. Theorem 9 Given the premisses of the theorem above and in addition I ) A; R we have Proof: The theorem is proved by a straightforward deduction: even if I is not a refinement of A; R, that is even if I ) A; R does not hold, other refinements of A; R may be used to simplify the term A; R in the specification. Example 8 Compositionality of Refinement for Recursion Of course, instead of giving a feedback loop as in example 7 above we may also define an infinite network recursively by 6 : where again we obtain (as a straightforward proof along the lines of the proof above for It is also a straightforward proof to show that (R; b where F Therefore we have where by our compositionality results. again A; R can be replaced by its refinement as shown above. End of example Using recursion we may define even infinite nets. The theorem above shows that a refinement of an infinite net that is described by a recursive equation is obtained by refinement of the components of the net. 7 Predicate Transformers as Refinements So far we have considered the refinement of components by refining on one hand their tuples of input and on the other hand their tuples of output streams. A more general notion of refinement is obtained by considering predicate transformers themselves as refinements. Definition 9 (Refining context) A predicate transformer 6 The predicate transformer - is obtained by the unfold rule for feedback is called a refining context, if there exists a mapping called abstracting context such that for all predicates X we have: Refining contexts can be used to define a quite general notion of refinement. (Refinement by refining contexts) Let R be a refining context with abstracting context A. A specification b Q is then called a refinement for the abstracting context A of the specification Q, if: Note, R:Q is a refinement of the specification Q for the abstracting context A. Refining contexts may be defined by the compositional forms introduced in the previous sections. Example 9 Refining Contexts For component specifications Y with one input channel and two output channels we define a predicate transformer 1by the equation: where the component P specifies functions A graphical representation of A:Y is given in Figure 7. Let P be specified by: For a component specification X with one input channel and one output channel we define a predicate transformer: where the component Q specifies functions Y A:Y Figure 7: Graphical representation of A:Y Let Q be specified by: stand for the finite stream of length k containing just copies of the message m. To show that A and R define a refining context we show that: which is equivalent to showing that for all specifications X: This is equivalent to: which is equivalent to the formula: which can be shown by a proof based on the specifications of P and Q. Let . stand for (I 2 ky) and & stand for the function (ykI 1 ). For functions p and q with P:p and Q:q there exists k 2 IN such that 8i 2 IN with i - k: This can be shown by a straightforward proof of induction on i. By this we obtain for Furthermore: We obtain By induction on the length on x and the continuity of the involved functions the proposition above is proved. End of example Context refinement is indeed a generalization of interaction refinement. Given two pairs of definite representation and abstraction specifications R; A and R; A by a refining context and an abstracting context is defined, since Refining contexts lead to a more general notion of refinement than interaction refinement. There are specifications Q and b Q such that there do not exist consistent specifications R and A where but there may exist refining contexts R and A such that Refining contexts may support the usage of sophisticated feedback loops between the refined system and the refining context. This way a dependency between the representation of the input history and the output history can be achieved. QbHe Figure 8: Graphical representation of the master/slave system A very general form of a refining context is obtained by a special operator for forming networks called master/slave systems. For notational convenience we introduce a special notation for master/slave systems. A graphical representation of master/slave systems is given in Figure 8. A master/slave system is denoted by QbHe. It consists of two components Q and H called the master k+n and the slave H 2 SPEC n m . Then QbHe 2 SPEC i . All the input of the slave is comes via the master and all the output of the slave goes to the master. The master/slave system is defined as follows: or in a more readable notation: where 8x; We can define a refining context and an abstracting context based on the mas- ter/slave system concept: we look for predicate transformers with abstracting context and for specifications V 2 SPEC i+m k+n and W 2 SPEC n+k m+i where the refining context and the abstracting context are specified as follows: y z Figure 9: Graphical representation of the cooperator and the following requirement is fulfilled: We give an analysis of this requirement based on further form of composition called a cooperator. The cooperator is denoted by . For specifications m+k the cooperator is defined as follows: m+m where A graphical presentation of the cooperator is given in Figure 9. A straightforward rewriting shows that the cooperator is indeed a generalization of the master/slave. For H 2 SPEC k In particular we obtain: and therefore the condition: reads as follows: The following theorem gives an analysis for the component W Theorem 10 The implication implies Recall, just swaps its input streams. Proof: By the definition of cooperation we may conclude that for every function i and every function - such that W:i and V:- and for every f where X:f there exists a function e f where X: e f such that: f:x this formula is true for all specifications X and therefore also for definite specifications, the formula holds for all functions f where in addition f . We obtain for the constant function f with z = f:x for all x and for all z: The equation above therefore simplifies to Now we prove that from this formula we can conclude: We do the proof by contradiction. Assume there exists x such and x 6= z. Then we can choose a function f such that f:x 6= f:z. This concludes the proof of the theorem. 2 By the concept of refining contexts we then may consider the refined system QbW bV bHeee The refinement of this refined network can then be continued by refining V bHe and leaving its environment QbW b:::ee as it is. There is a remarkable relationship between master/slave systems and the system structures studied in rely/guarantee specification techniques as advocated among others in [Abadi, Lamport 90]. The master can be seen as the environment and the slave as the system. This indicates that the master/slave situation models a very general form of composition. Every net with a subnet H can be understood as a master/slave system QbHe where Q denotes the surrounding net, the environment, of H. This form of networks is generalized by the cooperator as a composing form, where in contrast to master/slave systems the situation is fully symmetric. The cooperating components Q and Q in Q can be seen as their mutual environments. The concept of cooperation is the most general notion of a composing form for components. All composing forms considered so far are just special cases of cooperation; for k we obtain: Let a net N be given with the set \Gamma of components. Every partition of \Gamma into two disjoint sets of components leads to a partition of the net into two disjoint subnets say Q and Q such that the net is equal to Q the number of channels in N leading from Q to Q and k denotes the number of channels leading from Q to Q. Then both subnets can be further refined independently. 8 Conclusion The notion of compositional refinement depends on the operators, the composing forms, considered for composing a system. Compositionality is not a goal per se. It is helpful for performing global refinements by local refinements. Refining contexts, master slave systems and the cooperator are of additional help for structuring and restructuring a system for allowing local refinements. The previous sections have demonstrated that using functional techniques a compositional notion of interaction refinement is achieved. The refinement of the components of a large net can be mechanically transformed into a refinement of the entire net. Throughout this paper only notions of refinement have been treated that can be expressed by continuous representation and abstraction functions. This is very much along the lines of [CIP 84] and [Broy et al. 86] where it is considered as an important methodological simplification, if the abstraction and representation functions can be used at the level of specified functions. There are interesting examples of refinement, however, where the representation functions are not monotonic (see the representation functions obtained by the introduction of time in [Broy 90]). A compositional treatment of the refinement of feedback loops in these cases remains as an open problem. Acknowledgement This work has been carried out during my stay at DIGITAL Equipment Corporation Systems Research Center. The excellent working environment and stimulating discussions with the colleagues at SRC, in particular Jim Horning, Leslie Lamport, and Mart'in Abadi are gratefully acknowledged. I thank Claus Dendorfer, Leslie Lamport, and Cynthia Hibbard for their careful reading of a version of the manuscript and their most useful comments. A Appendix Full Abstraction Looking at functional specifications one may realize that sometimes they specify more properties than one might be interested in and that one may observe under the considered compositional forms. Basically we are interested in two observations for a given specification Q for a function f with Q:f and input streams x. The first one is straightforward: we are interested in the output streams y where But, in addition, for controlling the behavior of components especially within feedback loops we are interested in causality. Given just a finite prefix. e x of the considered input streams x, causality of input with respect to output determines how much output (which by monotonicity of f is a prefix of y) is guaranteed by f . More technically, we may represent the behavior of a system component by all observations about the system represented by pairs of chains of input and corresponding output streams. A set fx is called a chain, if for all i 2 IN we have Given a specification Q 2 SPEC n m , a pair of chains is called an observation about Q, if there exists a function f with Q:f such that for all and The behavior of a system component specified by Q then can be represented by all observations about Q. Unfortunately, there exist functional specifications which show the same set of observations, but, nevertheless, characterize different sets of functions. For an example we refer to [Broy 90]. Fortunately such functional specifications can be mapped easily onto functional specifications where the set of specified functions is exactly the one characterized by its set of observations. For this reason we introduce a predicate transformer that maps a specification on its abstract counterpart. This predicate transformer basically constructs for a given predicate Q a predicate \Delta:Q that is fulfilled exactly for those continuous functions that can be obtained by a combination of the graphs of functions from the set of functions specified by Q. We define where By this definition we obtain immediately the monotonicity and the closure property of the predicate transformer \Delta. Theorem 11 (Closure property of the predicate transformer \Delta) Proof: Straightforward, since Q:f occurs positively in the definition of \Delta:Q, specification Q is called fully abstract, if We may redefine our compositional forms such that the operators deliver always fully abstract specifications: All the results obtained so far carry over to the abstract view by the monotonicity of \Delta, and by the fact that we have Furthermore, given an upward closed predicate transformer - we have: if Q is the least solution of then \Delta:Q is the least solution of The proof is straightforward. Note, by this concept of abstraction we may obtain I in cases where I ) A; R does not hold. This allows additional simplifications of network refinements. Note, full abstraction is a relative notion. It is determined by the basic concept of observability and the composing forms. In the presence of refinement it is unclear whether full abstraction as defined above is appropriate. We have: However, if a component Q is used twice in a network - [Q], then we do not have, in general, that for (determined) refinements e Q of \Delta:Q there exist (determined) refinements b Q of Q such that: Therefore, when using more sophisticated forms of refinement the introduced notion of full abstraction might not always be adequate. --R Adding Action Refinement to a Finite Process Algebra. Composing Specifications. Refinement Calculus Refinement Calculus Stepwise Refinement of Distributed Systems. Scenarios: A Model of Nondeterminate Computation. Algebraic implementations preserve program correctness. Functional Specification of Time Sensitive Communicating Systems. Algebraic methods for program construc- tion: The project CIP Parallel Program Design: A Foundation. Assertional Data Reification Proofs: Survey and Perspective. Action Systems and Action Refinement in the Development of Parallel Systems - An Algebraic Approach Specifying concurrent program modules. Proofs of Correctness of Data Repre- sentations Systematic Program Development Using VDM. A Survey of Formal Software Development Methods. Bisimulation and Action Refinement. --TR Algebraic implementations preserve program correctness Systematic software development using VDM Parallel program design: a foundation Refinement calculus, part I: sequential nondeterministic programs Refinement calculus, part II: parallel and reactive programs Functional specification of time sensitive communicating systems A logical view of composition and refinement Adding action refinement to a finite process algebra Bisimulation and action refinement Composing specifications Specifying Concurrent Program Modules Scenarios --CTR Bernhard Thalheim, Component development and construction for database design, Data & Knowledge Engineering, v.54 n.1, p.77-95, July 2005 Manfred Broy, Object-oriented programming and software development: a critical assessment, Programming methodology, Springer-Verlag New York, Inc., New York, NY, Bernhard Thalheim, Database component ware, Proceedings of the fourteenth Australasian database conference, p.13-26, February 01, 2003, Adelaide, Australia Antje Dsterhft , Bernhard Thalheim, Linguistic based search facilities in snowflake-like database schemes, Data & Knowledge Engineering, v.48 n.2, p.177-198, February 2004 Einar Broch Johnsen , Christoph Lth, Abstracting refinements for transformation, Nordic Journal of Computing, v.10 n.4, p.313-336, December Manfred Broy, Toward a Mathematical Foundation of Software Engineering Methods, IEEE Transactions on Software Engineering, v.27 n.1, p.42-57, January 2001 Rimvydas Rukenas, A rigorous environment for development of concurrent systems, Nordic Journal of Computing, v.11 n.2, p.165-193, Summer 2004 Marco Antonio Barbosa, A refinement calculus for software components and architectures, ACM SIGSOFT Software Engineering Notes, v.30 n.5, September 2005

specification;interactive systems;refinement

269971

Making graphs reducible with controlled node splitting.

Several compiler optimizations, such as data flow analysis, the exploitation of instruction-level parallelism (ILP), loop transformations, and memory disambiguation, require programs with reducible control flow graphs. However, not all programs satisfy this property. A new method for transforming irreducible control flow graphs to reducible control flow graphs, called Controlled Node Splitting (CNS), is presented. CNS duplicates nodes of the control flow graph to obtain reducible control flow graphs. CNS results in a minimum number of splits and a minimum number of duplicates. Since the computation time to find the optimal split sequence is large, a heuristic has been developed. The results of this heuristic are close to the optimum. Straightforward application of node splitting resulted in an average code size increase of 235% per procedure of our benchmark programs. CNS with the heuristic limits this increase to only 3%. The impact on the total code size of the complete programs is 13.6% for a straightforward application of node splitting. However, when CNS is used, with the heuristic the average growth in code size of a complete program dramatically reduces to 0.2%

Introduction In current computer architectures improvements can be obtained by the exploitation of instruction level parallelism (ILP). ILP is made possible due to higher transistor densities which allows the duplication of function units and data paths. Exploitation of ILP consists of mapping the ILP of the application onto the ILP of the target architecture as efficient as possible. This mapping is used for Very Long Instruction Word (VLIW) and superscalar architectures. The latter are used in most workstations. These architectures may execute multiple operations per cycle. Efficient usage requires that the compiler fills the instructions with operations as efficient as possible. This process is called scheduling. Problem statement: In order to find sufficient ILP to justify the cost of multiple function units and data paths, a scheduler should have a larger scope than a single basic block at a time. A basic block is a sequence of consecutive statements in which the flow of control enters at the beginning and leaves always at the end. Several scheduling scopes can be found which go beyond the basic block level [1]. The most general scope currently used is called a region [2]. This is a set of basic blocks that corresponds to the body of a natural loop. Since loops can be nested, regions can also be nested in each other. Like natural loops, regions have a single entry point (the loop header) and may have multiple exits [2]. In [1] a speedup over 40% is reported when extending the scheduling scope to a region, the problem of region scheduling is that it requires loops in the control flow graph with a single entry point. These flow graphs are called reducible flow graphs. Fortunately, most control flow graphs are reducible, nevertheless the problem of irreducible flow graphs cannot be ignored. To exploit the benefits of region scheduling, irreducible control flow graphs should be converted to reducible control flow graphs. Exploiting ILP also requires efficient memory disambiguation. To accomplish this the nesting of loops must be determined. Since in an irreducible flow graph the nesting of loops is not clear, memory disambiguation techniques cannot directly be applied to these loops. To exploit the benefits of memory disambiguation, irreducible control flow graphs should be converted to reducible control flow graphs as well. Another pleasant property of reducible control flow graphs is the fact that data flow analysis, that is an essential part of any compiler, can be done more efficiently [3]. Related work: The problem of converting irreducible flow graphs to reducible flow graphs can be tackled at the front-end or at the back-end of the compiler. In [4] and [5] methods for normalizing the control flow graph of a program at the front-end are given. These methods rewrite an intermediate program in a normalized form. During normalization irreducible flow graphs are converted to reducible ones. To make a graph reducible, code has to be duplicated, which results in a larger code size. Since the front-end is unaware of the precise number of machine instructions needed to translate a piece of code, it is difficult to minimize the growth of the code size. Another approach is to convert irreducible flow graphs at the back-end. The advantage is that when selecting what (machine)code to duplicate one can take the resulting code size into account. Solutions for solving the problem at the back-end are given in [6, 7, 8, 9]. The solution given by Cocke and Miller [6, 9] is very time complex and it does not try to minimize the resulting code size. The method described by Hecht et al. [7, 8] is even more inefficient in the sense of minimizing the code size, but it requires less analysis. In this paper a new method for converting irreducible flow graphs at the back-end is given which is very efficient in terms of the resulting code size. Paper overview: In section 2 reducible and irreducible flow graphs are defined and a method for the detection of irreducible flow graphs is discussed. The principle of node splitting and the conversion method described by Hecht et al., which is a straightforward application of node splitting, are given in section 3. Our approach, Controlled Node Splitting (CNS), is described in section 4. All known conversion methods convert irreducible flow graphs without minimizing the number of copies. With controlled node splitting it is possible to minimize the number of copies. Unfortunately this method requires much CPU time; therefore we developed a heuristic that reduces the CPU time but still performs close to the optimum. This heuristic and the algorithms for controlled node splitting are presented. The results of applying CNS to several benchmarks are given in section 5. Finally the conclusions are given in section 6. Irreducible Flow Graphs The control flow of a program can be described with a control flow graph. A control flow graph consists of nodes and edges. The nodes represent a sequence of operations or a basic block, and the edges represent the flow of control. Definition 2.1 The control flow graph of a program is a triple where (N; E) is a finite directed graph, with N the collection of nodes and E the collection of edges. From the initial node s 2 N there is a path to every node of the graph. Figure 1 shows an example of a control flow graph with nodes and initial node s. s a c d e f s a c d e f (a) (b) Figure 1: a) A reducible control flow graph. b) The graph As stated in the introduction, finding sufficient ILP requires as input a reducible flow graph. Many definitions for reducible flow graphs are proposed. The one we adopt is given in [8] and is based on the partitioning of the edges of a control flow graph G into two disjoint sets: 1. The set of back edges BE consist of all edges whose heads dominate their tails. 2. The set of forward edges FE consists of all edges which are not back edges, thus A node u of a flow graph dominates node v, if every path from the initial node s of the flow graph to v goes through u. The dominance relations of figure 1 are: node s dominates all nodes, node a dominates all nodes except node s, node c dominates nodes c, d, e, f and node d dominates nodes d, e, f . Therefore f)g. The definition of a reducible flow graph is: Definition 2.2 A flow graph G is reducible if and only if the flow graph is acyclic and every node n 2 N can be reached from the initial node s. The control flow graph of figure 1 is reducible since acyclic. The flow graph of figure 2 however is irreducible. The set of back edges is empty, because neither node a nor node b, dominates the other. FE is equal to f(s; a) s) is not acyclic. From definition 2.2 we can derive that if a control flow graph G is irreducible then the graph contains at least one loop. These loops are called irreducible loops. To remove irreducible loops, they must be s a b Figure 2: The basic irreducible control flow graph. detected first. There are several methods for doing this. One of them is to use interval analysis [10, 11]. The method used here is the Hecht-Ullman T1-T2 analysis [12, 3]. This method is based on two transformations and T2. These transformations are illustrated in figure 3 and are defined as: Definition 2.3 Let be a control flow graph and let u 2 N . The removes the edge (u; u) 2 E, which is a self-loop, if this edge exists. The derived graph becomes G In short G T 1(u) T1(u) u Figure 3: The Definition 2.4 Let be a control flow graph and let node v 6= s have a single predecessor u. The transformation is the consumption of node v by node u. The successor edges of node v become successor edges of node u. The original successor edges of node u are preserved except for the edge to node v. If I is the set of successor nodes of v then the derived graph G In short G T 2(v) Definition 2.5 The graph that results when repeatedly applying the transformations in any possible order to a flow graph, until a flow graph results for which no application of T1 or T2 is possible is called the limit flow graph. This transformation is denoted as In [7] it is proven that the limit flow graph is unique and independent of the order in which the transformations are applied. Theorem 2.6 A flow graph is reducible if and only if after repeatedly applying transformations in any particular order the flow graph can be reduced into a single node. The proof of this theorem can be found in [12]. An example of the application of the transformations is given in figure 4. The flow graph from figure 1 is reduced to a single node, so we can conclude that this flow graph is reducible. f s a c a c T2(c) s a s a s T2(a) T1(a) s a c e f e s a c d f T2(d) T2(e) s a c d f e a f c T1(c) Figure 4: An example of application of the If after applying transformations the resulting flow graph consists of multiple nodes, the graph is irre- ducible. The transformations T1 and T2 not only detect irreducibility but they also detect the nodes that causes the irreducibility. Examples of irreducible graphs are given in figure 5. From theorem 2.6 it follows that we can alternatively define irreducibility by: Corollary 2.7 A flow graph is irreducible if and only if the limit flow graph is not a single node 1 . Another definition, which is more intuitive is that a flow graph is irreducible if it has at least one loop with multiple loop entries [12]. s a b c s a b c s a s a b c (a) (b) (c) (d) Figure 5: Examples of extensions of the basic irreducible control flow graph of figure 2. 3 Flow Graph Transformation If a control flow graph occurs to be irreducible, a graph transformation technique can be used to obtain a reducible control flow graph. In the past some methods are given to solve this problem [6, 7, 8]. Most methods for converting an irreducible control flow graph are based on a technique called node splitting. In section 3.1 this technique to reduce an irreducible flow graph is described. Section 3.2 shows how node splitting can be applied straightforwardly to reduce an irreducible graph. 3.1 Node Splitting Node Splitting is a technique that converts a graph G 1 to an equivalent graph G 2 . We assign a label to each node of a graph; the label of node x i is denoted label Duplication of a node creates a new node with the same label. An equivalence relation between two flow graphs is derived from Hecht [7] and given below. Definition 3.1 If path in a flow graph, then define Labels(P ) to be a sequence of labels corresponding to this path; that is, Labels(P Two flow graphs G 1 and G 2 are equivalent if and only if, for each path P in G 1 , there is a path Q in G 2 such that Labels(P conversely. According to this definition the two flow graphs of figure 6 are equivalent. Note that all nodes a have the same label(a). Node splitting is defined as: 3.2 Node splitting is a transformation of a graph G into a graph G that a node n 2 N , having multiple predecessors p i is split; for any incoming edge (p i ; n) a duplicate n i of n is made, having one incoming edge (p and the same outgoing edges as n. N 0 is defined as N is a successor node of n. This transformation is denoted as G 1 is the splitting of node n 2 N . The principle of node splitting is illustrated in figure 6, node a of graph G 1 is split. Note that if a node n is split in the limit graph, then it is the corresponding node n in the original graph that must be split to remove irreducibility. Theorem 3.3 The equivalence relation between two graphs is preserved under the transformation G 1 a b a b a S(a) Figure simple example of applying node splitting to node a. Proof: We show that node splitting transforms any graph G 1 into an equivalent split graph G 2 . Assume graph G 1 has a node v with n!1 predecessors u i and with m 0 successors w k , as shown in figure 7a. The set of Labels(P ) for all paths P of a graph G is denoted with LABELS (G). With the label notation all paths of graph G 1 of figure 7a are described k=0 f(label label (v) ; label (w k ))g (a) (b) s s Figure 7: Two equivalent graphs, (a) before node splitting, (b) after node splitting. If node v is split in n copies named v i the split graph G 2 results. The set of all paths of graph G 2 is: k=0 f(label label (v i This graph is given in figure 7b. Since label (v i label (v) every path in G 2 exists also in G 1 and conversely. This leads to the conclusion that the graphs G 1 and G 2 are equivalent. Since in figure 7 we split a node with an arbitrary number of incoming and outgoing edges we may in general conclude that splitting a node of any graph results in an equivalent graph. Using the same reasoning it will be clear that the equivalence relation is transitive, splitting a finite number of nodes in either the original graph or any of its equivalent graphs results in a graph which is equivalent to the original graph. 2 The name node splitting is deceptive because it suggests that the node is split in different parts but in fact the node is duplicated. 3.2 Uncontrolled Node Splitting The node splitting transformation technique can be used to convert an irreducible control flow graph into a reducible control flow graph. From Hecht [7] we adopt theorem 3.4. Theorem 3.4 Let S denote the splitting of a node, and let T denote some graph reduction transformation (e.g. any control flow graph can be transformed into a single node by the transformation represented by the regular expression T (ST ) . The proof of the theorem is given in [7]. Hecht et al. describe a straightforward application of node splitting to reduce irreducible control flow graphs. This method selects a node for splitting from the limit graph if the node has multiple predecessors. The selected node is split into several identical copies, one for each entering edge. This approach has the advantage that it is rather simple, but it has the disadvantage that it can select nodes that did not have to be split to make a graph reducible. In figure 8a we see that the nodes a, b, c and d are candidate nodes for splitting. In figure 8b node d is split, the number of nodes reduces after the application of two T2 transformations, but the graph is still irreducible. Splitting of node a neither makes the graph reducible, see figure 8c. Only splitting of node b or c converts the graph into a reducible control flow graph, see figure 8d. Although this method does inefficient node splitting, it does transform an irreducible control flow graph eventually in a reducible one. The consequence of this inefficient node splitting is that the number of duplications becomes unnecessarily large. 4 Presentation of Controlled Node Splitting The problem of existing methods is that the resulting code size after converting an irreducible graph can grow uncontrolled. Controlled Node Splitting (CNS) controls the amount of copies which results in a smaller growth of the code size. CNS restricts the set of candidate nodes for splitting. First we introduce the necessary terminology: Definition 4.1 A loop in a flow graph is a path (n is an immediate successor of n k . The set of nodes contained in the loop is called a loop-set. In figure 8a fa; bg ; fb; cg and fa; b; cg are loop-sets. Definition 4.2 An immediate dominator of a node u, ID(u), is the last dominator on any path from the initial node s of a graph to u, excluding node u itself. b' s a d s a d d' c a a a' d d a. Original irreducible graph. b. Splitting node . c. Splitting node . d. Splitting node . d a b Figure 8: Examples of node splitting. In figure 1 node a dominates the nodes a, b, c, d, e and f , but it immediate dominates only the nodes b and c. Definition 4.3 A Shared External Dominator set (SED-set) is a subset of a loop-set L with the properties that it has only elements that share the same immediate dominator and the immediate dominator is not part of the loop-set L. The SED-set of a loop-set L is defined as: Definition 4.4 A Maximal Shared External Dominator set (MSED-set) K is defined as: SED-set K is maximal , 6 9 SED-set M, K ae M The definition says that an MSED-set cannot be a proper subset of another SED-set. In figure 5a multiple SED-sets can be identified like fa; bg, fb; cg and fa; b; cg. But there is only one MSED-set: fa; b; cg. Definition 4.5 Nodes in an SED-set of a flow graph can be classified into three sets: ffl Common Nodes (CN): Nodes that dominate other SED-set(s) and are not reachable from the SED-set(s) they dominate. ffl Reachable Common nodes (RC): Nodes that dominate other SED-set(s) and are reachable from the SED- they dominate. ffl Normal Nodes (NN): Nodes of an SED-set that are not classified in one of the above classes. These nodes dominate no other SED-sets. In the initial graph of figure 9a we can identify the MSED-sets fa; bg and (c; dg. The nodes a, c and d are elements of the set NN and node b is an element of the set RC. If the edge (c; b) was not present then node b would be an element of the set CN. Note that loop (b; c) is not a SED-set. Theorem 4.6 A SED-set(L) has one node , The corresponding loop L has a single header and is reducible. The proof of this theorem can be derived from [7]. An example of an SED-set which has one node is the graph in figure 4 just before the transformation In section 4.1 a description of CNS is given. It treats a method for minimizing the number of nodes to split. Section 4.2 gives a method for minimizing the amount of copies. The number of copies is not equal to the number of splits because a split creates for every entering edge a copy. If a node has n entering edges then one split creates copies. To speed up the process for minimizing the amount of copies a heuristic is given. The algorithms implementing this heuristic are given in section 4.3. 4.1 Controlled Node Splitting All nodes of an irreducible limit graph, except the initial node s of the graph, are possible candidates for node splitting since they have at least two predecessors. However splitting of some nodes is not efficient; see section 3.2. CNS minimizes the number of splits. To accomplish this, two restrictions are made to the set of candidate nodes. These restrictions are: 1. Only nodes that are elements of an SED-set are candidates for splitting. 2. Nodes that are elements of RC are not candidates for splitting. The first restriction prevents the splitting of nodes that are not in an SED-set. Splitting such a node is inefficient and unnecessary. An example of such a split was shown in figure figure 8b (the only SED-set in figure 8b is fb; cg). The second restriction is more complicated. The impact of this restriction is illustrated in figure 9. This figure shows two different sequences of node splitting. The initial graph of the figure is a graph on which T has been applied. In figure 9a there are three splits needed and in figure 9b only two. In figure 9a node b is split, this node however is an element of the set RC. The second restriction prevents a splitting sequence as the one in figure 9a. Node splitting with the above restrictions, alternated with eventually result in a single node. This can be seen easily. Every time a node that is an element of an SED-set is split, it is reduced by the transformation and the number of nodes involved in SED-sets decreases with one. Since we are considering flow graphs with a finite number of nodes, a single node eventually remains. s a b c d s s c d a. Node splitting sequence of three nodes. b. Node splitting sequence of two nodes. s a c d s a b c d s s a b Figure 9: Graph with two different split graphs. Theorem 4.7 The minimum number of splits needed to reduce an MSED-set with k nodes is given by: Proof: Every time a node is split and T is applied, the number of nodes in the MSED-set decreases with one. For every predecessor of the node to split a duplicate is made, this means that every duplicate has only one predecessor and all the duplicates can be reduced by the T2 transformation. This results in an MSED-set with one node less than the original MSED-set. To reduce the complete MSED-set all nodes but one of the MSED-set must be split until there is only one node left. This results in k-1 splits. 2 Theorem 4.8 The minimum number of splits needed to convert an irreducible graph, with n MSED-sets, into a reducible graph is given by: where T splits is the total number of splits, and k i is the number of nodes of MSED-set i. Proof: The proof consists of multiple parts, first some related lemmas are proven. Lemma 4.9 All MSED-sets are disjoint, that is there are no two MSED-sets that share a node. Proof: If a node is shared by two MSED-sets then this node must have two different immediate domina- tors. This conflicts however with the definition of an immediate dominator as given in 4.2. 2 Since the MSED-sets are disjoint the number of splits of the individual MSED-sets can be added. If however splitting nodes results in merging MSED-sets this result does not hold anymore. Therefore we have to prove that CNS does not merge MSED-sets and that merging MSED-sets does not lead to less splits. Lemma 4.10 Splitting a node that is part of an MSED-set and is not in RC does not result in merging MSED-sets. Proof: First we shall prove that splitting a node that is an element of RC merges MSED-sets. Afterwards we prove that splitting of nodes that are elements of CN or NN do not merge MSED-sets. - Splitting a RC node merges two MSED-sets. Consider the graph of figure 10. Suppose that subgraphs G 1 and G 2 are both MSED-sets. The nodes of both subgraphs form a joined loop because it is possible s s x y y Figure 10: Merging of two MSED-sets. to go from G 1 to G 2 and vice-versa. The reason that both subgraphs do not form a single MSED-set is the fact that they have different immediate dominators. By splitting a node that is in RC, in this case node x, and applying T to the complete graph the immediate dominator of subgraph G 1 becomes also the immediate dominator of subgraph G 2 . Since the subgraphs add up to a single loop and share the same immediate dominator the MSED-sets are merged. This holds also in the general case where x dominates and is reachable by n MSED-sets. Splitting nodes that are not in RC do not merge MSED-sets. There are now two types of nodes left that are candidates for splitting, these are the nodes of the sets NN and CN. Splitting nodes that are element of the set NN do not merge MSED-sets. These nodes do not have edges that go to other MSED-sets, therefore splitting of these nodes does not affect the edges from one MSED-set to another and therefore the splitting will never result in merging MSED-sets. Splitting nodes that are element of the set CN do not merge MSED-sets. These nodes do not form a loop with the MSED-set it dominates. By splitting such a node the nodes of both MSED-sets get the same immediate dominator but there is no loop between the MSED-sets and therefore are not merged. Lemma 4.11 Reducing two merged MSED-sets results in more splits to reduce a graph than reducing the MSED-sets separately. Proof: Suppose SED-set 1 consists of x nodes and SED-set 2 has y nodes. Merging them costs one split since the RC node must be split. Reducing the resulting SED-set which has now x splits. The total number of splits is x \Gamma 1+y. Reducing the two SED-sets separately results in splits. This is one split less than the splits needed when merging the SED-sets. 2 The combination of lemmas 4.10 and 4.11 justifies the restriction to prevent the splitting of nodes that are elements of RC. Lemma 4.12 There exists always a node in an irreducible graph that is part of an MSED-set but that is not an element of RC. Proof: If all nodes of all MSED-sets are elements of RC then these nodes must dominate at least two other nodes because a node cannot dominate its own dominators. These nodes are also elements of an MSED-set and of RC. The graph therefore must have an infinite number of nodes. Since we are considering graphs with a finite number of nodes there must be a node that is part of an MSED-set but that is not an element of Since MSED-sets are disjoint and our algorithm can always find a node that can be split without merging MSED- sets the result of equation 1 holds. 2 Example 4.13 In figure 9 the MSED-sets fa; bg and fc; dg can be identified. They have both two nodes. This results in a minimal number of splits needed to reduce the graph. 4.2 Minimizing the amount of copies In the previous section we saw that the algorithm minimizes the number of splits, but this does not result in a minimum number of copied instructions or basic blocks. In the following the quantity to minimize is denoted with Q, Q (n) means the quantity of node n, Q(G) is the quantity of a graph G and is defined as: The purpose of CNS is to minimize Q(G ), where G is the transformation of G into a single node using some sequence of splits, more formally Q (G Two conditions must be satisfied to achieve this minimum: 1. The freedom of selecting nodes to split must be as big as possible. Notice that the number of splits is also minimized if we prevent the splitting of all nodes that dominate another MSED-set, that is prevent splitting of nodes that are elements of RC and CN. But this has the disadvantage that we lose some freedom in selecting nodes. This loss of freedom is illustrated in figure 11. Suppose that the nodes contain a number of s a b c d Figure 11: A graph that has a common node that is not in the set RC. instructions and that we want to minimize the total resulting code size, which means that we would like to copy as less instructions as possible. The number of copied instructions if we prevent splitting nodes that are elements of RC and CN is: Q (a) If we only prevent the splitting of nodes that are element of RC the number of copied instructions is: min (Q (a) ; If the number of instructions in node b is less than in node a then the number of copied instructions is less in the latter case. Thus keeping the set of candidate nodes as big as possible pays off if one would like to minimize the amount of copies. 2. The sequence of splitting nodes must be chosen optimal. There exists multiple split sequences to solve an irreducible graph. A tree can be build to discover them all. Figure 12 shows a flow graph and the tree with all possible split sequences. The nodes of this tree indicate how many copies are introduced by the split. The s a b c a 2b c c b+a c+b a+b a c b+c a a a Figure 12: An irreducible graph with its copy tree. edges give the split sequence. The number of copies can be found by following a path from the root to a leaf and adding the quantities of the nodes. Suppose that the nodes contain a number of instructions and that we want to minimize the total resulting code size, which means that we would like to copy as less instructions as possible, then we can choose from 6 different split sequences with 5 different numbers of copies. The minimum number of copied instructions is: min(Q (a The problem is to pick a split sequence that minimizes the number of copied instructions. Theorem 4.14 Minimizing the resulting Q(G ) of an irreducible graph that is converted to a reducible graph requires a minimum number of splits, where G is a single node; that is the totally reduced graph. In short: splits to produce G is minimal. Proof: Suppose all nodes of a limit flow graph, except the initial node s, are candidates for splitting, then nodes that are not in an MSED-set and nodes that are elements of RC are also candidates. Splitting a node of one of these categories results in a number of splits that is greater than the minimal number of splits. If we can prove that splitting these nodes always result in a Q(G ) that is greater than the one we obtain if we exclude these nodes, then we have proven that a minimum number of splits is required in order to minimize Q(G ). ffl Splitting a node that is not in an MSED-set cannot result in the minimum Q(G ). As seen in the previous section splitting of nodes that are not in an MSED-set do not make a graph more reducible since splitting these nodes does not decrease the number of nodes in an MSED-set. This means that the MSED-set still needs the same number of splits. ffl Splitting nodes that are element of RC cannot result in the minimum Q(G ). s G a s a sag s ag s G a sa ga saga sa Ga a. Splitting nodes that are not in the set RC. b. Splitting a node that is in the set RC. Figure 13: The influence on the number of copies by splitting a RC node. Consider the graph of figure 13. In this figure the subgraph G has at least one MSED-set, otherwise the graph would not be irreducible. Figure 13a shows the reduction of a graph in the case that splitting of a RC node is not allowed and in 13b splitting of such a node is allowed. The node g is the reduced subgraph G and the notation sa in a node means that node s has consumed a copy of node a. The resulting quantity of the node is the sum of the quantities of nodes s and a. As we can see the resulting total quantity of the split sequence of figure 13a is Q(s)+Q(a)+Q(g) and the resulting total quantity of the reduced graph of figure 13b is Q(s)+2Q(a)+Q(g). Without loss of generality we can conclude that splitting a node that is in RC never can lead to the minimum total quantity. 2 As one can easily see, the more nodes in MSED-sets the larger the tree and the number of possible split sequences increases. It takes much computation time to compute all possibilities, therefore a heuristic is constructed which picks a node n i to split with the smallest H (n i ) as defined by: The results of this heuristic, compared to the best possible split sequence, are given in section 5. 4.3 Algorithms The method described in the previous sections detects an irreducible control flow graph and converts it to a reducible control flow graph. In this section the algorithm for this method is given. The algorithm consists of three parts (1) the T1 and T2 transformations, (2) the selection of a candidate node and (3) the splitting of a node. Algorithm 4.1 Controlled Node Splitting Input : The control flow graph of a procedure. : The reducible control flow graph of the procedure. (1) Copy the flow graph of basic blocks to a flow graph G of nodes (2) Apply repeatedly T1-T2 transformations to G (3) while G has more than one node do selection (5) Split candidate node Apply repeatedly T1-T2 transformations to G Algorithm 4.1 expects as input a control flow graph of basic blocks. The structure of this flow graph is copied to a flow graph of nodes (1). Now we have two different flow graphs: a flow graph of basic blocks and a flow graph of nodes. This means that initially every node represents a basic block. Every duplicate introduced by splitting the flow graph of nodes is also performed in the flow graph of basic blocks. After the graph is copied the and T2 transformations are applied till the graph of nodes does not change any more (2). If the graph of nodes is reduced to a single node, the graph is reducible and no splitting is needed. However if there remain multiple nodes, node splitting must be applied. First a node for splitting is selected (4). This is done with algorithm 4.2 that is discussed later. The selected node is then split (5) as defined in definition 3.2. In the graph of basic blocks, the corresponding basic blocks are copied also. After splitting the T1 and T2 transformations are applied again on the graph of nodes (6). When there is still more than one node left the process start over again. The algorithm terminates if the graph of nodes is reduced to a single node and thus the graph of basic blocks is converted to a reducible flow graph. The algorithms for the transformations and for node splitting are quite straightforward and not given here. Algorithm 4.2 selects a node for splitting. Initially every node is a candidate. A node is rejected as a candidate if it does not fulfill the restrictions (3) as discussed in subsection 4.1. For all nodes that fulfill these restrictions the heuristic is calculated (4) with equation 1. The node with the smallest heuristic is selected for splitting. The goal of our experiments is to measure the quality of controlled node splitting in the sense of minimizing the amount of copies. In the experiments four methods for node splitting are used: Optimal Node Splitting, ONS. This method computes the best possible node split sequence with respect to the quantity to minimize; the number of basic blocks or the number of instructions. This algorithm however Algorithm 4.2 Node selection Input : The control flow graph of nodes. node for splitting. (2) for all nodes n do (3) if n in an SED-set and n not in RC then Calculate value H(n) return candidate node requires a lot of computation time (up to several days on a HP735 workstation). Uncontrolled Node Splitting, UCNS. A straightforward application of node splitting, no restrictions are made to the set of nodes that are candidate for splitting. Controlled Node Splitting, CNS. Node splitting with the restrictions discussed in section 4.1. Controlled Node Splitting with Heuristic, CNSH. The same method as CNS but now a heuristic is used to select a node from the set of candidate nodes. The algorithms are applied to a selective group of benchmarks. These benchmarks are procedures with an irreducible control flow graph and are obtained from the real world programs: a68, bison, expand, gawk, gs, gzip, sed, tr. The programs are compiled with the GCC compiler which is ported to a RISC architecture 2 . The amount of copies of two different quantities are considered. In table 1 the number of copies of basic blocks are listed and in table 2 the number of copied instructions. The reported results of the methods UCNS, CNS and CNSH are the averages of all possible split sequences. The first column in the tables 1 and 2 lists the procedure name, with the program name in parentheses. The second column gives the number of basic blocks or instructions of the procedure before an algorithm is applied. The other columns give the number of copies that result from the algorithms. The absolute number of copies is given and a percentage that indicates the growth of the quantity with respect to the original quantity is given. From the results of the ONS method we can conclude that node splitting does not have to lead to an excessive number of copies. Furthermore we can conclude that CNS outperforms UCNS. UCNS can lead to an enormous amount of copies, the average percentage of growth in basic blocks is 241.7% and in code size it is 235.5%. CNS performs better, a growth of 26.2% for basic blocks and 30.1% for the number of instruction, but there is still a big gap with the optimal case. When using the heuristic, controlled node splitting performs very close to the optimum. The average growth in basic blocks for the methods CNSH and ONS is respectively 3.1% and 3.0%. The growth We used a RISC like MOVE architecture. The MOVE project [13, 1] researches for the generation of application specific processors (ASPs) by means of Transport Triggered Architectures (TTA). Table 1: The number of copied basic blocks. atof output program(bison) 14 2 (14%) 9.7 (69%) 9.7 (69%) 2.0 (14%) copy definition(bison) 119 2 (2%) 417.0 (350%) 27.7 (23%) 2.0 (2%) copy guard(bison) 190 4 (2%) 2273.5 (1197%) 133.4 (70%) 4.0 (2%) copy action(bison) next file(expand) 17 1 (6%) 5.0 (29%) 5.0 (29%) 1.0 (6%) re compile pattern(gawk) 787 1 (0%) 1202.7 (153%) 47.5 (6%) 1.0 (0%) gs copy block(gzip) 17 2 (12%) 2.5 (15%) 2.5 (15%) 2.0 (12%) compile program(sed) 145 1 (1%) 80.1 (55%) 60.0 (41%) 1.0 (1%) re search 2(sed) 486 20 (4%) 1328.7 (273%) 50.0 (10%) 21.0 (4%) squeeze filter(tr) 33 8 (2%) 16.3 (49%) 15.5 (47%) 8.0 (24%) total 2692 82 (3.0%) 6505.3 (241.7%) 704.1 (26.2%) 83 (3.1%) in code size is for both methods 2.9%. Comparing the results of ONS and CNSH lead to the conclusion that CNSH performs very close to the optimum. In our experiments there was only one procedure with a very small difference. The results in the tables 1 and 2 show an substantial improvement when using CNSH. But the question is: what is the impact of the code expansion, when using a more simple method like UCNS, on the code size of the complete program. If the impact is small then why bother, except for the theoretical aspects. In the tables 3 and 4 the effects for the complete code expansion are shown. All procedures of the benchmarks that have an irreducible control flow graph are converted to procedures with a reducible control flow graph. In table 3 we show the impact in basic blocks and in table 4 the impact on the code size is listed. The first column of both tables lists the program name, the second column list the total number of basic blocks or instructions. The remaining columns list the increase in basic blocks or in instructions for each method. As can be seen from the tables the impact of node splitting can be substantial in terms of number of basic blocks or instructions. As we can see, for UCNS the average increase in basic blocks is 15.4% and in instructions it is 13.6%. Using UCNS can even result in a code size increase of 80% for the program bison. When using controlled node splitting the increases are smaller and quite acceptable. CNSH results as expected in the smallest increases for both quantities. These results show the importance of a clever transformation of irreducible control flow graphs. 6 Conclusions A method has been given which transforms an irreducible control flow graph to a reducible control flow graph. This gives us the opportunity to exploit ILP over a larger scope than a single basic block for any program. The method is based on node splitting. To achieve the minimum number of splits the set of possible candidate nodes is limited to nodes with specific properties. Since splitting of these nodes can result in a minimum resulting code size the algorithm can be used to prevent uncontrolled growth of the code size. Because the computation time to determine the optimum split sequence is (very) large, a heuristic has been developed. Table 2: The number of copied instructions. instructions ONS UCNS CNS CNSH atof output program(bison) 59 9 (15%) 41.5 (70%) 41.5 (70%) 9.0 (15%) copy copy guard(bison) 880 copy action(bison) 858 9 (1%) 2961.4 (345%) 122.5 (14%) 9.0 (1%) next file(expand) re compile pattern(gawk) 2746 1 (0%) 4106.9 (150%) 218.5 (8%) 1.0 (0%) gs s LZWD read buf(gs) 228 62 (27%) 95.0 (42%) 95.0 (42%) 62.0 (27%) copy block(gzip) 88 4 (5%) 7.5 (9%) 7.5 (9%) 4.0 (5%) compile program(sed) 693 2 (0%) 391.4 (56%) 267.5 (39%) 2.0 (0%) re search 2(sed) 1857 91 (5%) 4803.7 (259%) 227.5 (12%) 93.0 (5%) squeeze filter(tr) 119 22 (18%) 57.0 (48%) 55.5 (47%) 22.0 (18%) total 11588 335 (2.9%) 27291.7 (235.5%) 3484.1 (30.1%) 337 (2.9%) The method with the heuristic is called controlled node splitting with heuristic. This method is applied to a set of procedures which contain irreducible control flow graphs. The results are compared with the results of the other methods; these methods are uncontrolled node splitting and controlled node splitting. From our experiments it follows that uncontrolled node splitting can lead to an enormous number of copies, the average growth in code size per procedure is 235.5%. Controlled node splitting performs better (30.1%) but there is still a big gap with the optimal case. We observed that the average number of copies when using controlled node splitting with heuristic is very close to that of the optimum; the average growth in code size per procedure for both methods is 2.9%. We also looked at the impact on the total code size of the benchmarks containing procedures with irreducible control flow graphs. The same methods used as for the analysis per procedure are used. For CNSH the impact on the total code size is very small, only 0.2% on average. The impact of UCNS is however surprisingly large. An average of code size growth of 13.6% with a maximum for bison of 80%. Table 3: The increase of basic blocks per program. Program # basic blocks ONS UCNS CNS CNSH bison 4441 14 (0%) 3501.7 (79%) 222.8 (5%) 14.0 (0%) expand 1226 1 (0%) 5.0 ( 0%) 5.0 (0%) 1.0 (0%) gs 16514 sed 3823 21 (1%) 1408.8 (37%) 110.0 (3%) 22.0 (1%) tr 1554 8 (1%) 16.3 ( 1%) 15.5 (1%) 8.0 (1%) total 43116 108 (0.3%) 6631.2 (15.4%) 754.1 (1.7%) 109.0 (0.3%) Table 4: The increase of instructions per program. instructions ONS UCNS CNS CNSH bison 19689 63 (0%) 15858.4 (80%) 983.8 (5%) 63.0 (0%) expand gs 85824 210 (0%) 2169.7 (3%) 1804.1 (2%) 210.0 (0%) sed 17489 93 (1%) 5195.1 (30%) 495.0 (3%) 95.0 (1%) tr total 205073 452 (0.2%) 27884.0 (13.6%) 3695.4 (1.8%) 454.0 (0.2%) --R Global instruction scheduling for superscalar machines. Elimination algorithms for data flow analysis. Taming control flow: A structured approach to eliminating goto statements. A control-flow normalization algorithm and its complexity Some analysis techniques for optimizing computer programs. Flow Analysis of Computer Programs. On certain graph-theoretic properties of programs A basis for program optimization. A program data flow analysis procedure. Flow graph reducibility. --TR Compilers: principles, techniques, and tools Elimination algorithms for data flow analysis Global instruction scheduling for superscalar machines A Control-Flow Normalization Algorithm and its Complexity An elimination algorithm for bidirectional data flow problems using edge placement A new framework for exhaustive and incremental data flow analysis using DJ graphs loops using DJ graphs A Fast and Usually Linear Algorithm for Global Flow Analysis Fast Algorithms for Solving Path Problems A program data flow analysis procedure Microprocessor Architectures Flow Analysis of Computer Programs Transport-Triggering versus Operation-Triggering --CTR Han-Saem Yun , Jihong Kim , Soo-Mook Moon, Time optimal software pipelining of loops with control flows, International Journal of Parallel Programming, v.31 n.5, p.339-391, October Sebastian Unger , Frank Mueller, Handling irreducible loops: optimized node splitting versus DJ-graphs, ACM Transactions on Programming Languages and Systems (TOPLAS), v.24 n.4, p.299-333, July 2002 Fubo Zhang , Erik H. D'Hollander, Using Hammock Graphs to Structure Programs, IEEE Transactions on Software Engineering, v.30 n.4, p.231-245, April 2004 Reinhard von Hanxleden , Ken Kennedy, A balanced code placement framework, ACM Transactions on Programming Languages and Systems (TOPLAS), v.22 n.5, p.816-860, Sept. 2000

instruction-level parallelism;control flow graphs;reducibility;irreducibility;compilation;node splitting

270413

Optimal Parallel Routing in Star Networks.

AbstractStar networks have recently been proposed as attractive alternatives to the popular hypercube for interconnecting processors on a parallel computer. In this paper, we present an efficient algorithm that constructs an optimal parallel routing in star networks. Our result improves previous results for the problem.

Introduction The star network [2] has received considerable attention recently by researchers as a graph model for interconnection network. It has been shown that it is an attractive alternative to the widely used hypercube model. Like the hypercube, the star network is vertex- and edge-symmetric, strongly hierarchical, and maximally fault tolerant. Moreover, it has a smaller diameter and degree while comparing with hypercube of comparable number of vertices. The rich structural properties of the star networks have been studied by many researchers. The n-star network S n is a degree 1)-connected, and vertex symmetric Cayley graph [1, 2]. Jwo, Lakshmivarahan, and Dhall [15] showed that the star networks are hamiltonian. Qiu, Akl, and Meijer [22] showed that the n-star network can be decomposed into (n \Gamma 1)! node-disjoint paths of length can be decomposed into (n \Gamma 2)! node-disjoint cycles of length (n \Gamma 1)n. Results in embedding hypercubes into star networks have been obtained by M. Nigam, S. Sahni, and B. Krishnamurthy [20] and by Miller, Pritikin, and Sudborough [18]. Broadcasting on star networks has also been studied recently [3, 4, 14, 23]. Routing on star networks was first studied by Akers and Krishnamurthy [2] who derived a formula for the length of the shortest path between any two nodes in a star network and developed an efficient algorithm for constructing such a path. Recently, parallel routing, i.e., constructing node-disjoint paths, on star networks has received much attention. Sur and Srimni [24] demonstrated that node-disjoint paths can be constructed between any two nodes in S n in polynomial time. Dietzfelbinger, Madhavapeddy, and Sudborough [11] derived an improved algorithm that constructs node-disjoint paths of length bounded by 4 plus the diameter of S n . The algorithm was further improved by Day and Tripathi [10] who developed an efficient algorithm that constructs paths of length bounded by 4 plus the distance from u to v in S n . The problem was also investigated by Jwo, Lakshmivarahan, and Dhall [16]. Misic and Jovanovic [19] derived a general algebraic expression for all (not necessarily node-disjoint) shortest paths between any two nodes in S n . Palis and Rajasekaran [21], Dietzfelbinger, Madhavapeddy, and Sudborough [11], Qiu, Akl, and Meijer [22], and Chen and Chen [9] have considered the problem of node-disjoint paths between two sets of nodes in a star network. In this paper, we will improve the previous results on node-to-node parallel routing in star networks by developing an efficient algorithm that constructs optimal parallel routing between any two nodes in a star network. More specifically, let u and v be any two nodes in the n-star network Department of Computer Science and Engineering, Tatung Institute of Technology, Taipei 10451, Taiwan, R.O.C. Supported in part by Engineering Excellence Award from Texas A&M University. Email: [email protected]. y Corresponding author. Department of Computer Science, Texas A&M University, College Station, 3112. Supported in part by the National Science Foundation under Grant CCR-9110824. Email: [email protected]. S n and let dist(u; v) be the distance from u to v in S n . The bulk length of a group of node-disjoint paths connecting u and v in S n is defined to be the length of the longest path in the group. Define the bulk distance Bdist(u; v) between u and v to be the minimum bulk length over all groups of connecting u and v in S n . We develop an O(n 2 log n) time algorithm that, given two nodes u and v in S n , constructs a group of node-disjoint paths of bulk length Bdist(u; v) that connect the nodes u and v in S n . Our algorithm involves careful analysis on the lower bound on the bulk distance Bdist(u; v) between two nodes u and v in S n , a non-trivial reduction from the parallel routing problem on star networks to a combinatorial problem called Partition Matching, a subtle solution to the Partition Matching problem, and a number of routing algorithms on different kinds of pairs of nodes in a star network. The basic idea of the algorithm can be roughly described as follows. Let u and v be two nodes in the n-star network S n . According to Day and Tripathi [10], the bulk distance Bdist(u; v) is equal to dist(u; v) or 4. We first derive a necessary and sufficient condition for a pair of nodes u and v to have bulk distance dist(u; 4. For a pair u and v whose bulk distance is less than dist(u; v) + 4, we develop an efficient algorithm that constructs a group of node-disjoint paths of bulk length dist(u; v) between u and v. Finally, an efficient algorithm, which is obtained from a reduction of an efficient algorithm solving the Partition Matching problem, is developed that constructs the maximum number of node-disjoint shortest paths (of length dist(u; v)) between u and v. Combining all these analysis and algorithms gives us an efficient algorithm that constructs an optimal parallel routing on star networks. We should also point out that the running time of our algorithm is almost optimal (differs at most by a log n factor) since a lower running time of parallel routing algorithms on star networks can be easily derived. Preliminary A permutation of the symbols can be given by a product of disjoint cycles [5], which is called the cycle structure of the permutation. A cycle is nontrivial if it contains more than one symbol. Otherwise the cycle is trivial. The cycle containing the symbol 1 will be called the primary cycle in the cycle structure. A -[1; i] transposition on the permutation u is to exchange the positions of the first symbol and the ith symbol in u: -[1; It is sometimes more convenient to write the transposition -[1; i](u) on u as -[a i ](u) to indicate that the transposition exchanges the positions of the first symbol and the symbol a i in u. Let us consider how a transposition changes the cycle structure of a permutation. Write u in its cycle structure If a i is not in the primary cycle, then -[1; i] "merges" the cycle containing a i into the primary cycle. More precisely, suppose that a (note that each cycle can be cyclically permuted and the order of the cycles is irrelevant), then the permutation -[1; i](u) will have the following cycle structure: If a i is in the primary cycle, then -[1; i] "splits" the primary cycle into two cycles. More precisely, suppose that a we have let a 1n 1 (note that a we assume i ? 1), then -[1; i](u) will have the following cycle structure: In particular, if a then we say that the transposition -[1; i] "deletes" the symbol a 11 from the primary cycle. Note that if a symbol a i is in a trivial cycle in the cycle structure of the permutation a 1 a 1 \Delta \Delta \Delta a n , then the symbol is in its "correct" position, i.e., a that if a symbol is in a nontrivial cycle, then the symbol is not in its correct position. Denote by " the identity permutation every symbol is in a trivial cycle. The n-dimensional star network (or simply the n-star network) S n is an undirected graph consisting of n! nodes labeled with the n! permutations on symbols There is an edge between two nodes u and v in S n if and only if there is a transposition -[1; i], that v. A path from a node u to a node v in the n-star network S n corresponds to a sequence of applications of the transpositions -[1; i], starting from the permutation u and ending at the permutation v. A (node-to-node) parallel routing in S n is to construct the maximum number of node-disjoint paths between two given nodes in S n . Since the n-star network S n is (n \Gamma 1)-connected [2] and every node in S n is of degree theorem [17], a parallel routing in S n should produce exactly between two given nodes. Moreover, since the n-star network is vertex symmetric [2], a parallel routing in the n-star network any two given nodes can be easily mapped to a parallel routing between a node and the identity node " in S n . We will write dist(u) and Bdist(u) for the distance dist(u; ") and the bulk distance Bdist(u; "), resepctively. By the definitions, we always have Bdist(u) - dist(u). Let u be a node in the n-star network with cycle structure are nontrivial cycles and e j are trivial cycles. If we further let denotes the number of symbols in the cycle c i , then the distance from the node u to the identity node " is given by the following formula [2]. if the primary cycle is a trivial cycle if the primary cycle is a nontrivial cycle Combining this formula with the above discussion on the effect of applying a transposition on a permutation, we derive the following necessary rules for tracing a shortest path from the node u to the identity node " in the n-star network S n . Shortest Path Rules Rule 1. If the primary cycle is a trivial cycle in u, then in the next node on any shortest path from u to ", a nontrivial cycle c i is merged into the primary cycle. This corresponds to applying transposition -[a] to u with a 2 c Rule 2. If the primary cycle c nontrivial cycle in u, where a 1n 1 then in the next node on any shortest path from u to ", either a nontrivial cycle c i 6= c 1 is merged into the primary cycle (this corresponds to applying transposition -[a] to u, where a 2 c i ), or the symbol a 11 is deleted from the primary cycle c 1 (this corresponds to applying transposition -[a 12 ] to u). Fact 2.1. A shortest path from u to " in S n is obtained by a sequence of applications of the Shortest Path Rules, starting from the permutation u. Fact 2.2. If a symbol a 6= 1 is in a trivial cycle in u, then a will stay in a trivial cycle in any node on a shortest path from u to ". Fact 2.3. If an edge [u; v] in S n does not lead to a shortest path from u to ", then Consequently, let P be a path from u to " in which exactly k edges do not follow the Shortest Path Rules, then the length of the path P is equal to dist(u) Fact 2.4. [10] There are no cycles of odd length in an n-star network. Consequently, the length of any path from a node u to the node " is equal to dist(u) plus an even number. Given a node in the n-star network S n , where c i are nontrivial cycles and e j are trivial cycles, a shortest path from u to " can be constructed through the following two-stage process: 1. Merge in an arbitrary order each of the nontrivial cycles into the primary cycle. This will result in a node whose cycle structure has a single nontrivial cycle, which is the primary cycle. For example, suppose that c 1) is the primary cycle in u and that by applying the transpositions -[a 21 ], -[a in this order, we obtain a node with cycle structure (a k1 \Delta \Delta \Delta a kn k a 2. Delete each of the symbols in the primary cycle until the node " is reached. For example, suppose that we start with the node with cycle structure as described in Equation (1), then we apply the transpositions -[a k2 -[1], in this order, to reach the node ". The above process will be called the Merge-Delete process. It is easy to verify, using the Shortest Path Rules, that the Merge-Delete process produces a shortest path from the node u to the node " in the n-star network S n . The most important property of the Merge-Delete process is that in each node (except the node ") of the constructed shortest path, the primary cycle is of form 1), where a 1n 1 is fixed for all nodes on the path. Parallel routing from a node u to " in S n is particularly simple when the primary cycle in u is a trivial cycle [10, 16]. For completeness, we describe the routing process for this case here. Let u be such a node in S n . For each symbol i, 2 - i - n, construct a path P i as follows. The path P i starts with the edge [u; -[i](u)] followed by a shortest path from the node -[i](u) to ", which is obtained by applying the Merge-Delete process starting from the node -[i](u). It is easy to verify that if i is in a nontrivial cycle in u, then the path P i has length dist(u), and that if i is in a trivial cycle in u then the path P i has length dist(u) 2. On the other hand, if any symbol i, 2 - i - n, is in a trivial cycle in u, then any path from u to " via -[i](u) has length at least dist(u) 2. Since the node u has degree any group of node-disjoint paths from u to " contains a path whose first edge is [u; -[i](u)]. This implies that the bulk distance Bdist(u) is at least dist(u) is a symbol i in a trivial cycle in u, for 2 - i - n. Therefore, to show that the constructed paths length Bdist(u), we only need to prove that all these paths are node-disjoint. In fact, since the first edge on P i is via -[i](u) and the subpath from -[i](u) to " on P i follows the shortest path obtained by the Merge-Delete process, it is easy to verify that for each path P i , there is a ng uniquely associated with the path P i such that for each interior node on the path P i , the primary cycle is of form Therefore, no two of the P can share a common interior node, and the paths P i , 2 - i - n, are node-disjoint. This gives a group of node-disjoint paths of bulk length Bdist(u) from u to ". Therefore, throughout the rest of this paper, we discuss the parallel routing problem in star networks based on the following assumption: Assumption 2.1. The node u in the n-star network S n has cycle structure are nontrivial cycles and e j are trivial cycles, and the primary cycle c is a nontrivial cycle. 3 Nodes with bulk distance dist(u) Day and Tripathi [10] have developed a routing algorithm that constructs of bulk length dist(u) + 4 from a node u to " in the n-star S n . The basic idea of Day-Tripathi's algorithm can be described as follows. For each symbol a construct a path P a in which each node has a cycle of form (\Delta \Delta \Delta a1). This cycle distinguishes the path P a from the other constructed paths. Let 1), be the node as described in Assumption 2.1. We describe the path P a in three different cases. Case 1. The symbol a is in a nontrivial cycle c i , for i ? 1. Without loss of generality, let a). In this case, the first four nodes on the path P a are u The rest of the path P a is constructed by applying the Merge- Delete process, starting from the node u 4 . Case 2. The symbol a is in a trivial cycle e (a). Then the first four nodes on the path P a are u the rest of the path P a is obtained by applying the Merge-Delete process starting from the node u 4 . Case 3. The symbol a is in the cycle c 1 . Let a = a j , where 1 d. Then the first four nodes on the path P a are u simply let u discard the nodes u 1 , u 2 , and u 3 ), and the rest of the path P a is obtained by applying the Merge-Delete process starting from the node u 4 . It is easy to verify that in each of the above three cases, the nodes u 2 and u 3 on the path P a are not contained in any other constructed path P a 0 for a 0 6= a. It is also easy to check that the fourth node u 4 on the path P a contains a cycle of form the Merge-Delete process keeps the cycle pattern along the path P a , we conclude that all these P a , a are node-disjoint. Finally, by examining each constructed path P a , we find out that at most two edges on P a do not follow the Shortest Path Rules. By Fact 2.3, the path P a has length at most 4. This completes the description of Day-Tripathi's algorithm. Combining Day-Tripathi's algorithm and Fact 2.4, we conclude that for any node u in the n- star network S n , the bulk distance Bdist(u) is equal to dist(u) or 4. In this section, we derive a necessary and sufficient condition for a node u to have bulk distance dist(u)+4. Let P be a path in S n from u to ". We say that the path P leaves u with symbol a if the second node on P is -[a](u), and we say that the path P enters " with symbol a 0 if the node next to the last node " on P is -[a 0 ]("), which has a single nontrivial cycle (a 0 1). Lemma 3.1 Let 1), be the node in the n-star network S n as described in Assumption 2.1. Suppose that m ? minf2 jg. Then any group of node-disjoint paths from u to " has bulk length at least Proof. Assume that P 1 node-disjoint paths from u to " of bulk length bounded by dist(u) 2. We show that in this case we must have First note that for each symbol b that is in a trivial cycle in u, one of the paths P must leave u with b, and one of the paths P must enter " with b. Suppose that a path P i leaves u with a symbol b in a trivial cycle in u. By the Shortest Path Rules, the node -[b](u) does not lead to a shortest path from u to ". By Fact 2.3, 1. On the other hand, the length of P i is bounded by dist(u) 2. Thus, starting from the node -[b](u), the path P i must strictly follow the Shortest Path Rules. In particular, no node on the path P i (including the node -[b](u)) can contain a cycle of form This implies that the path P i must enter " with a symbol in the set fa d g [ S k Since there are exactly m of the paths P leaving with symbols in trivial cycles in u, and since all these paths are node-disjoint and must enter " with symbols in fa d g that there are at least m different symbols in the set fa d g [ Now again consider the path P i leaving u with a symbol b in a trivial cycle in u. Let u i be the first node on P i in which the primary cycle is a trivial cycle. Note that u i 6= u and u i 6= -[b](u). Moreover, since the nontrivial cycles in the node -[b](u) are c 0 and the path P i strictly follows the Shortest Path Rules after the node -[b](u), every nontrivial cycle in the node u i is a cycle in the set fc g. Therefore, the node u i , hence the path P i , corresponds to a subset of the set fc g. Similarly, suppose that P h is a path among in a trivial cycle in u. Since at node u, the symbol b is in a trivial cycle while at node -[b](") on the path P h , the symbol b is in a nontrivial cycle, we can find two consecutive nodes u h and v h on the path P h such that the symbol b is in a trivial cycle in the node u h while in a nontrivial cycle in the node v h . By Fact 2.2, the edge [u h ; v h ] on the path P h does not follow the Shortest Path Rules. By our assumption, the length of the path P h is bounded by dist(u) 2. We conclude that the subpaths of P h from u to u h and from v h to ", respectively, must strictly follow the Shortest Path Rules. Note that by our assumption on the nodes u h and v h , we must have and the symbols b and 1 are in the same cycle in the node v h . Since the node trivial cycles) is on the subpath of P h from v h to ", in order to let the subpath of P h from v h to " to strictly follow the Shortest Path Rules, the node v h must have a cycle of form (b 1). Consequently, the symbol 1 is in a trivial cycle in the node u h . Now since the subpath of P h from u to u h follows strictly the Shortest Path Rules, every nontrivial cycle in the node u h must be one of the nontrivial cycles in the node u. This proves that the node u h , thus the path P h , corresponds to a subset of the set fc g. As we have shown before, no path in P 1 leave u with a symbol b and enters " with a symbol b 0 , where b and b 0 are symbols in trivial cycles in the node u. Therefore, there are exactly 2m paths among leaves u or enters " with a symbol in a trivial cycle in u. Each of these 2m paths corresponds to a subset in the set fc g. Since all these paths are node-disjoint, we conclude that there are at least 2m different subsets of the set g. That is, 2 equivalently, m - 2 k\Gamma2 . Combined with Day-Tripathi's algorithm, Lemma 3.1 provides a sufficient condition for a node u in the n-star network S n to have bulk distance 4. In the following, we show that this condition is also necessary. For this, we demonstrate that when m - minf2 k\Gamma2 can always construct node-disjoint paths from u to " of bulk length dist(u) 2. We first make some conventions. We assume that n - 3 since the parallel routing on an n-star network S n for n - 2 is trivial. Let c be a cycle, given in any fixed cyclic order. We will denote by [c] the sequence of the symbols in the cycle c. Therefore, ([c] a 1 \Delta \Delta \Delta a d ) is a cycle starting with the symbols in the cycle c followed by the symbols a 1 a d . Recall that a cycle can be arbitrarily cyclically rotated, still resulting in the same cycle. In many cases, we will discard irrelevant trivial cycles in a cycle structure. Let 1), be the node in the n-star network S n as described in Assumption 2.1, and suppose m - minf2 k\Gamma2 jg. We describe our construction in three different cases. Case I. The number k of nontrivial cycles in u is equal to 1. By the condition Therefore, the cycle structure of the node u consists of a single (nontrivial) cycle 1). Note that the unique cycle in u contains at least 3 symbols since n - 3. The node-disjoint paths from u to " are given as follows. A path P 1 leaving u with a 2 and entering " with a n\Gamma1 is given by (a 1 a 2 \Delta \Delta \Delta a stands for a sequence of transpositions that repeatedly deletes a symbol from the primary cycle. A path P 2 leaving u with 1 and entering " with a 1 is given by (a 1 a 2 \Delta \Delta \Delta a For there is a path P j leaving u with a j and entering " with a j \Gamma1 , given by It is easy to verify that each path P j , 1, has at most one edge that does not follow the Shortest Path Rules. Thus, all these constructed paths have length at most dist(u) 2. The path P 1 keeps a cycle of form 1). The path P 2 keeps a cycle of form 1). For the first part of the path P j keeps a distinguished cycle (a 1 the second part of the path P j keeps a cycle of form 1). Therefore, all these constructed paths are node-disjoint. This gives in this case a group of node-disjoint paths of bulk length bounded by from u to ". Case II. The number k of nontrivial cycles in u is at least 2, and the number m of trivial cycles in u is 0. In this case, the node u can be written as A path P 1 leaving u with 1 and entering " with a d is given by stands for a sequence of transpositions that repeatedly merges a nontrivial cycle into the primary cycle. For there is a path P j leaving u with a j and entering " with a j \Gamma1 , given by Note that this group of paths is constructed only when d - 2. For each symbol a 2 [ k there is a path P a leaving u with a and entering " with a. Since each nontrivial cycle can be cyclically rotated and the order of the cycles c 2 we can assume, without loss of generality, that c path P a is given by It is again easy to verify that all these are of length at most dist(u) disjoint. Thus, this gives in this case a group of node-disjoint paths of bulk length bounded by from u to ". Case III. The number k of nontrivial cycles in u is at least 2 and the number m of trivial cycles in u is at least 1. In this case, the node u can be written as are nontrivial cycles and (b 1 are trivial cycles. A path P 1 leaving u with b 1 and entering " with a d is constructed as follows. A path P 0 leaving u with a 2 and entering " with b 1 is given by If then the above constructed path leaves u with the symbol 1 and enters " with b 1 . For each i, we construct two paths P i and P 0 i as follows. First mark all symbols in [ k as "unused", and mark all subsets of the set as "unused". Then for each pick an unused symbol a 0 and an unused subset S from the set such that S i 6= S and a 0 i is contained in a cycle in the subset S i . Let loss of generality, we assume that the cycle c (i) contains the leaves u with a 0 i and enters " with b i , and is given by c (i) (a d 1)c (i) and the path P 0 leaves u with b i and enters " with a 0 , and is given by c (i) (b c (i) Now mark the symbol a 0 i and the subsets fc (i) and fc (i) k g of the set fc as "used". We must justify that the construction of the paths P i and P 0 i is always possible. Since m - there are at least Therefore, for each i, should always be able to find an unused symbol a 0 Now fix the symbol a 0 . There are different subsets S 0 of the set such that S 0 6= S and a cycle in S 0 contains the symbol a 0 at least one of such subsets has not been used for constructing the previous paths P j and P 0 (note that in the two subsets fc (j) and used for the paths P j and P 0 only one of them has a cycle containing the symbol a 0 Also note that if S 0 is an unused subset of S, then S \Gamma S 0 is also an unused subset of S. Therefore, we are always able to find an unused subset S i of S such that S i 6= S and a cycle in S i contains the symbol a 0 i . This ensures the possibility of the construction of the paths P i and P 0 If the cycle c contains more than two symbols, i.e., d - 2, then we also construct the following The path Q 2 leaves u with the symbol 1 and enters " with a and for d, the path Q j leaves u with a j and enters " with a Finally, for each symbol a in [ k that is not used in constructing the paths P i and P 0 we construct a path P a leaving u with a and entering " with a. Without loss of generality, let the symbol a be in the cycle The above process constructs paths from the node u to ". It is easy to verify that each of the constructed paths has at most one edge that does not follow the Shortest Path Rules. By Fact 2.3, all these are of length at most dist(u) 2. What remains is to show that all these paths are node-disjoint. On each of the constructed paths, the nodes have a special cycle pattern to ensure that the path is node-disjoint from the other paths. 1. Path P 1 consists of two parts of different format: each node in the first part has a cycle of and each node in the second part has a single nontrivial cycle of form 1), where the symbol a d is uniquely associated with the path 2. Path P 0 1 consists of two parts of different format: each node in the first part has a cycle structure of form and each node in the second part has a cycle of form where the symbol b 1 is uniquely associated with the path P 0 3. Path P i , consisits of three parts of different format: each node in the first part has a cycle of form 1), each node in the second part has a cycle structure of form , where the subset fc (i) k g is uniquely associated with the path and each node in the third part has a cycle of form where the symbol b i is uniquely associated with the path P i . Note that the assumption S i 6= S is necessary here - otherwise, the path P i would enter " with the symbol a d and would share a common interior node with path Similarly, path P 0 consists of three parts of different format: each node in the first part has a cycle of form 1), each node in the second part has cycle structure of form , where the subset fc (i) g is uniquely associated with the path P 0 and each node in the third part has a cycle of form 1), where the symbol a 0 i is uniquely associated with the path P 0 . Again here the condition S i 6= S is necessary - otherwise, the third node of the path P 0 i would be the node u again and the path P 0 would then leave u with the symbol a 2 and share an interior node with the path P 0 1 . 4. The second node of the path Q 2 has a distinguished cycle (a 1 \Delta \Delta \Delta a d ), and the rest of the nodes in the path Q 2 have a cycle of form (\Delta \Delta \Delta a 1 1), where the symbol a 1 is uniquely associated with the path 5. The path Q d, consists of two parts of different format: each node in the first part has a distinguished cycle (a 1 each node in the second part has a cycle of the symbol a j \Gamma1 is uniquely associated with the path Q 6. For each symbol a that is not used in the paths P i and P 0 m, the second node in the path P a has a cycle (aa 1), the third node in the path P a has a cycle and the rest of the nodes in P a have a cycle of form (\Delta \Delta \Delta a1), where the symbol a is uniquely associated with the path P a . Therefore, all these constructed paths are node-disjoint. Summarizing all the above discussions, we obtain the following lemma. Lemma 3.2 Let 1), be the node in the n-star network S n as described in Assumption 2.1. If m - minf2 k\Gamma2 a group of paths of bulk length dist(u) + 2 from u to " can be constructed in time O(n 2 log n). Proof. The construction of a group of node-disjoint paths of bulk length dist(u) + 2 from u to " has been given in the above discussion. For each P of the paths and P a , where a is a symbol in [ k is not used in the construction of the paths P i and P 0 m, the construction of P can be obviously implemented in time O(n). (Note that according to [2], To construct the paths P i and P 0 i , we need to find an unused symbol a 0 . The symbol a 0 i can be simply found in time O(n) if we keep a record for each symbol to indicate whether it has been used. Once the symbol a 0 i is decided, we need to find an unused subset S i of the set g. For this, suppose that the symbol a 0 i is in cycle c 0 . We pick arbitrarily another r from the set S (note that by our assumption, log m+ 2 - k). Now instead of using the set S, we use the set S r g. Since the set S 0 has and the cycle c 0 is in S 0 , at least one of these subsets has not been used. These subsets of S 0 can be enumerated in a systematic manner in time O(r2 r log n) since each such subset contains at most r = O(log m) cycles. Now an unused subset of S 0 is also an unused subset of S, with which we can construct the paths P i and P 0 i in time O(n). In conclusion, the paths i can be constructed in time O(n log n) for each total time needed to construct the paths P i and P 0 is bounded by O(n 2 log n). Combining Lemma 3.1 and Lemma 3.2, we obtain immediately the following theorem. Theorem 3.3 Let 1), be the node in the n-star network S n as described in Assumption 2.1. The bulk distance Bdist(u) from the node u to the identity node " is 4 The maximum number of node-disjoint shortest paths In this section we diverge to a slightly different problem. Let u be the node in the n-star network S n as described in Assumption 2.1. How many node-disjoint shortest paths (of length dist(u)) can we find from u to "? This problem is closely related to a combinatorial problem, called Maximum Partition Matching, formulated as follows: be a collection of subsets of the universal set ng such that S k partition matching (of order m) of S consists of two ordered subsets of m elements of U (the subsets L and R may not be disjoint), together with a sequence of m distinct partitions of S: such that for all a i is contained in a subset in the collection A i and b i is contained in a subset in the collection B i . The Maximum Partition Matching problem is to construct a partition matching of order m for a given collection S with m maximized. Theorem 4.1 The Maximum Partition Matching problem is solvable in time O(n 2 log n). Proof. An O(n 2 log n) time algorithm has been developed in [7] that, given a collection S of subsets in constructs a maximum partition matching of S. We refer our readers to [7] (and [6]) for details. Now we show how Theorem 4.1 can be used to find the maximum number of node-disjoint shortest paths from a node u to the identity node " in star networks. Lemma 4.2 Let 1), be the node in the n-star network S n as described in Assumption 2.1. Then the number of node-disjoint shortest paths from u to " cannot be larger Proof. According to Rule 2 of the Shortest Path Rules, only the path leaving u with a symbol in the set fa 2 g [ can be a shortest path from u to ". Another upper bound on the number of node-disjoint shortest paths from u to " can be derived in terms of the maximum partition matching of the collection of nontrivial cycles in u, where each nontrivial cycle is regarded as a set of symbols. Lemma 4.3 Let 1), be the node in the n-star network S n as described in Assumption 2.1. Then the number of node-disjoint shortest paths from u to " cannot be larger than 2 plus the number of partitions in a maximum partition matching in the collection g, where the cycles c i are regarded as sets of symbols. Proof. Let shortest paths from u to ". For each path P i , let u i be the first node on P i such that in u i the primary cycle is a trivial cycle. The node u i is obtained by repeatedly applying Rule 2 of the Shortest Path Rules, starting from the node u. It is easy to prove, by induction, that in any node v on the subpath from u to u i on the path P i , the only possible nontrivial cycle that is not in fc is the primary cycle. In particular, the node u i must have a cycle structure of the form are nontrivial cycles, e 0 are trivial cycles and B g is a subcollection of the collection g. Assume that the path P i leaves u with the symbol b i . By Rule 2 in the Shortest Path Rules, b i is either a 2 or one of the symbols in [ k also by this rule, once b i is contained in the primary cycle in a node in the path P i , it will stay in the primary cycle in the nodes along the path P i until it is deleted from the primary cycle, i.e., until b i is contained in a trivial cycle. In particular, the symbol b i is not in the set [ k i . Now suppose that the path P i enters the node " with a symbol d i . Thus, the node w on the path P i must have a cycle structure (d i 1) if we discard trivial cycles in w i . Since the symbol d i is in a nontrivial cycle in w i , by Fact 2.2, d i is also in a nontrivial cycle in the node u i , that is . The only exception is d (in this case u Now we let A . Then we can conclude that except for at most two paths P 1 and P 2 , each of the other paths P 3 must leave the node u with a symbol b i in A i and enter " with a symbol d i in B i . (The two exceptional paths P 1 and P 2 may leave u with the symbol a 2 or enter " with the symbol a d .) Now since the s paths are node-disjoint, the symbols b 3 are all pairwise distinct, and the symbols d 3 are also pairwise distinct. Moreover, since all nodes u are pairwise distinct, the collections B 3 of cycles are also pairwise distinct. Consequently, the partitions of the collection together with the symbol sets partition matching of the collection S. This concludes that s cannot be larger than 2 plus the number of partitions in a maximum partition matching of the collection thus proves the lemma. Now we show how we construct a maximum number of node-disjoint shortest paths from the node 1), as described in Assumption 2.1, to the identity node " in the n-star network S n . We first show how to route a single shortest path from u to ", given a partition (A; B) of the collection g, and a pair of symbols b and d, where b is in a cycle in A and d is in a cycle in B. We also allow b to be a 2 - in this case d must be in [ k S. Similarly, we allow d to be a d - in this case b must be in [ k S. Consider the algorithm Single Routing given in Figure 1. Since the algorithm Single Routing starts with the node u and applies only transpositions described in the Shortest Path Rules, we conclude that the algorithm Single Routing constructs a shortest path from the node u to the node ". Now we are ready for describing the final algorithm. Consider the algorithm Maximum Shortest Routing given in Figure 2. Algorithm. Single Routing input: A partition (A;B) of and two symbols b and d, where b is in a cycle in A and d is in a cycle in B. b can be a2 with A = OE, and d can be ad with output: A shortest path from u to " leaving u with b and entering " with d. 1. if b 6= a2 apply -[b] to u to merge the cycle in A that contains b into the primary cycle c1 ; then merge in an arbitrary order the rest of the cycles in A into the primary cycle; 2. repeatedly delete symbols in the primary cycle until the primary cycle becomes a trivial cycle; 3. if d 6= ad suppose that the cycle c containing d in B is Apply -[d 0 ] to merge c into the primary cycle (1); then merge in an arbitrary order the rest of the cycles in B into the primary cycle; 4. repeatedly delete symbols in the primary cycle until reach the node "; Figure 1: The algorithm Single Routing Algorithm. Maximum Shortest Routing input: The node u in the n-star network Sn , as described in Assumption 2.1. output: A maximum number of node-disjoint shortest paths from u to ". 1. Construct a maximum partition matching M [(b1 ds )] in ck g with the partitions 2. if shortest paths as follows. 2.1. Call the algorithm Single Routing with the partition (OE; S) of S and the symbol 2.2. Call the algorithm Single Routing with the partition (S; OE) of S and the symbol 2.3. For to s, call the algorithm Single Routing with the partition the symbol pair (b 3. if s ! shortest paths as follows. 3.1. Let b0 ; 3.2. Call the algorithm Single Routing with the partition (OE; S) of S and the symbol 3.3. Call the algorithm Single Routing with the partition (S; OE) of S and the symbol 3.4. For to s, call the algorithm Single Routing with the partition the symbol pair (b Figure 2: The algorithm Maximum Shortest Routing Theorem 4.4 The algorithm Maximum Shortest Routing constructs a maximum number of node-disjoint shortest paths from the node u to the identity node " in time O(n 2 log n). Proof. From Lemma 4.2 and Lemma 4.3, we know that the number of shortest paths constructed by the algorithm Maximum Shortest Routing matches the maximum number of node-disjoint shortest paths from u to ". What remains is to show that all these paths are node-disjoint. Suppose that the algorithm Maximum Shortest Routing constructs h shortest paths h from node u to node ", here depending on whether suppose that the path P i is constructed by calling the algorithm Single Routing on partition then we have then we have d Now fix an i and consider the path P i , which is constructed from the partition and the symbol pair (b l g and g, where if A i 6= OE then the cycle c (i) 2 is of form c (i) then the cycle c (i) l+1 is of form c (i) Finally, recall that the primary cycle c 1 has the form The interior nodes of the path P i can be split into three segments I (i) 2 , and I (i) 3 . The first segment I (i) 1 corresponds to nodes constructed in step 1 of the algorithm Single Routing that first merges cycle c (i) 2 into the primary cycle c 1 , obtaining a cycle of form (b merges cycles c (i) l into the primary cycle. Therefore, for all nodes in this segment, the primary cycle is of the form The second segment I (i) 2 corresponds to the nodes constructed by step 2 of the algorithm Single Routing that deletes symbols in the primary cycle. All nodes in this segment are of the form The third segment I (i) 3 corresponds to the nodes constructed by step 3 and step 4 of the algorithm Single Routing, which first merges the cycle c (i) l+1 into the primary cycle (1), obtaining a cycle of merges the cycles c (i) k into the primary cycle, and then deletes symbols in the primary cycle. Therefore, in all nodes in this segment, the primary cycle should have the In case A and the segment I (i) 1 is empty, and in case a d and the segment I (i) 3 is empty. We now show that any two shortest paths P i and P j , i 6= j, constructed by the algorithm Maximum Shortest Routing are node-disjoint. Let v be a node on the path P i . Suppose that is a node on the first segment I (i) 1 of the path P i . The node cannot be on the first segment I (j) 1 of the path P j since all nodes on I (j) 1 are of form Moreover, the node v cannot be on the second or the third segment of P j since the cycle structure of a node on the second or the third segment of P j has more trivial cycles (note that each execution of step 2 of the algorithm Single Routing creates a new trivial cycle in the cycle structure). k is on the second segment I (i) 2 of the path P i , then v cannot be on the second segment I (j) of P j since each node on I (j) 2 is of form k and The node v can neither be on the third segment I (j) 3 of P j since each node on the segment I (j) 3 is of while the primary cycle in the node v is either a trivial cycle or of form (\Delta \Delta \Delta a d 1) where a d is in c 1 . Finally, if is on the third segment of the path P i , then v cannot be on the third segment of P j because d i 6= d j . By symmetry, the above analysis shows that the two shortest paths P i and P j constructed by the algorithm Maximum Shortest Routing must be node-disjoint. The running time of the algorithm Maximum Shortest Routing is dominated by step 1 of the algorithm, which takes time O(n 2 log n) according to Theorem 4.1. Thus, the algorithm Maximum Shortest Routing runs in time O(n 2 log n). Construction of the maximum number of node-disjoint shortest paths between two nodes in star networks was previously studied in [16], which presents an algorithm that runs in exponential time in the worst case. More seriously, the algorithm seems based on an incorrect observation, which claims that when there are more than one nontrivial cycles in a node u, the maximum number of node-disjoint shortest paths from u to " is always an even number. Therefore, the algorithm in [16] always produces an even number of node-disjoint shortest paths from u to ". A counterexample to this observation has been constructed in [6]. 5 Conclusion: an optimal parallel routing Combining all the previous discussion in the present paper, we obtain an O(n 2 log n) time algorithm, Optimal Parallel Routing as shown in Figure 3, that constructs node-disjoint paths of bulk length Bdist(u) from any node u to the identity node " in the n-star network S n . The correctness of the algorithm Optimal Parallel Routing has been proved by Lemma 3.1, Lemma 3.2, Theorem 4.4, and the results in [10]. The running time of the algorithm is O(n 2 log n). We would like to make a few remarks on the complexity of our algorithm. The bulk distance problem on general graphs is NP-hard [13]. Thus, it is very unlikely that the bulk distance problem can be solved in time polynomial in the size of the input graph. On the other hand, our algorithm solves the bulk distance problem in time O(n 2 log n) on the n-star network. Note that the n-star network has n! nodes. Therefore, the running time of our algorithm is actually a polynomial of the logarithm of the size of the input star network. Moreover, our algorithm is almost optimal (differs at most by a log n factor) since the following lower bound can be easily observed - the distance dist(u) from u to " can be as large as \Theta(n). Thus, constructing node-disjoint paths from u to " takes time at least \Theta(n 2 ) in the worst case. --R The star graph: an attractive alternative to the n-cube A group-theoretic model for symmetric interconnection networks A routing and broadcasting scheme on faulty star graphs A Survey of Modern Algebra Combinatorial and algebraic methods in star and de Bruijn networks The maximum partition matching problem with applications Optimal parallel routing in star networks An improved one-to-many routing in star networks A comparative study of topological properties of hypercubes and star graphs Three disjoint path paradigms in star networks Short length versions of Menger's theorem Computers and Intractability: A Guide to the Theory of NP-Completeness The cost of broadcasting on star graphs and k-ary hypercubes Embedding of cycles and grides in star graphs Characterization of node disjoint (par- allel) path in star graphs Near embeddings of hypercubes into Cayley graphs on the symmetric group Routing function and deadlock avoidance in a star graph interconnection network Embedding hamiltonians and hypercubes in star interconnection graphs Packet routing and PRAM emulation on star graphs and leveled networks On some properties and algorithms for the star and pancake interconnection networks An optimal broadcasting algorithm without message redundancy in star graphs Topological properties of star graphs --TR --CTR Cheng-Nan Lai , Gen-Huey Chen , Dyi-Rong Duh, Constructing One-to-Many Disjoint Paths in Folded Hypercubes, IEEE Transactions on Computers, v.51 n.1, p.33-45, January 2002 Eunseuk Oh , Jianer Chen, On strong Menger-connectivity of star graphs, Discrete Applied Mathematics, v.129 n.2-3, p.499-511, 01 August Chi-Chang Chen , Jianer Chen, Nearly Optimal One-to-Many Parallel Routing in Star Networks, IEEE Transactions on Parallel and Distributed Systems, v.8 n.12, p.1196-1202, December 1997 Adele A. Rescigno, Optimally Balanced Spanning Tree of the Star Network, IEEE Transactions on Computers, v.50 n.1, p.88-91, January 2001

shortest path;parallel routing;star network;partition matching;network routing;graph container

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Analysis for Chorin''s Original Fully Discrete Projection Method and Regularizations in Space and Time.

Over twenty-five years ago, Chorin proposed a computationally efficient method for computing viscous incompressible flow which has influenced the development of efficient modern methods and inspired much analytical work. Using asymptotic error analysis techniques, it is now possible to describe precisely the kind of errors that are generated in the discrete solutions from this method and the order at which they occur. While the expected convergence rate is seen for velocity, the pressure accuracy is degraded by two effects: a numerical boundary layer due to the projection step and a global error due to the alternating or parasitic modes present in the discretization of the incompressibility condition. The error analysis of the projection step follows the work of E and Liu and the analysis of the alternating modes is due to the author. The two are combined to show the asymptotic character of the errors in the scheme. Regularization methods in space and time for recovering full accuracy for the computed pressure are discussed.

Introduction In 1968, Chorin [3] proposed a computationally efficient method for computing viscous, incompressible flow. The method was based on the primitive variables, velocity and pressure, with all unknowns at the same grid points. The discretization was centered in space (second order in space step h) and implicit in time (first order in time step with the projection part of the Stokes operator split off for computational efficiency. The discretization of the incompressibility condition allowed for alternating (parasitic or checkerboard) modes. The idea of a projection step has been used in modern efficient methods (e.g. [6]) (since the literature in this field is vast, the references here and in what follows are only intended to be illustrative not exhaustive). Many authors have considered the analysis of the projection step [4, 15, 16, 9] and have proposed higher order corrections [13, 18, 14]. The precise description of the errors from the projection step in [9] will be used in this work. The effect of parasitic modes on the accuracy of the computed pressure has also been an area of much interest, especially in the finite element method [8] and spectal method [2] communities. It is well known that the presence of parasitic modes can lead to a degradation in the convergence rate of the pressure. A precise description of this effect for Chorin's discretization is given in [20]. What the work in [9, 20] does is to characterize the errors from the projection step and the parasitic modes precisely for smooth problems with a smooth discretization in space and time. These analyses will be combined in this paper to give a precise description of the errors from Chorin's original fully discrete scheme. Although there are no significant new difficulties that arise from the interaction of the discretizations in space and time, the author believes it is worthwhile to give the full error analysis for this historically important algorithm. A simplified presentation of the boundary layer effects from the projection step is given. We describe the numerical order reducing effects on the pressure briefly below. If it is assumed that convergence of second order in h might be expected. While this is true for the velocity, it is not true for the pressure. In fact, the discrete pressure from Chorin's method has an O(k 1=2 ) numerical boundary layer due to the projection step and an O(h) global error due to the alternating modes. In order to recover full accuracy for the pressure, we also consider regularization methods in space and time. Computational evidence for all predictions is given. The reader may wish to take a short tour through pressure errors from Chorin's original scheme O(h) alternating terms dominate, to a space regularized scheme only the boundary layer is left, to a fully regularized scheme (Fig.5) where the errors are spatially smooth. Some discussion should be made here about the real and artificial limitations of this work. First of all, the analysis is presented for the two-dimensional (2D) Stokes equations with homogeneous Dirichlet boundary conditions but can be extended in a straightforward way to 3D with nonhomogeneous compatible boundary conditions and to smooth solutions of the nonlinear Navier-Stokes equations. Secondly, the error expansions presented assume a great number of compatibility conditions at When these are not satisfied, convergence is not uniform in some quantities up to (see [12] for computational results and formal analysis of the behaviour near Thirdly, the computational issue of how to efficiently implement the projection step is not addressed in this work. We use a simplified geometry for our numerical tests in which it is easy to implement an exact projection efficiently which allows us to obtain refined solutions to verify the predicted error structure. Fourthly, the temporal regularization discussed in the final section uses unsplit time integration. In the context of split-step projection methods, it would have been more appropriate to present a pressure increment scheme [18, 6] but the error analysis of these schemes is not well understood in the fully discrete case [10]. Finally, because the discrete divergence and gradient operators are not adjoint, a simple stability result based on energy estimates as used in [4, 9] is not possible for Chorin's original method when boundaries are present. Thus, the stability analysis of the method is still an open problem. In the next section, Chorin's original method is described. Then, computational results showing the boundary layer and alternating errors in the pressure are presented. In Section 4 we present the error analysis for the method, describing the alternating and boundary layer errors and at what order they occur. Finally, in Section 5 we present analysis and computation of regularized methods. 2 Description of the Scheme We consider fluid flow in a simplified domain: a two-dimensional (2D) [0; 1] \Theta [0; 1] channel with fixed walls on the top and bottom boundaries and periodic in the horizontal direction. The incompressible Stokes equations are given below where are the velocities, p is the pressure and - is the kinematic viscosity. Boundary conditions are used. Initial data u 0 is given and it is assumed that r \Delta u We note that p can only be determined up to an arbitrary constant. A unique p is recovered if we require Z It is well known that any square integrable vector function can be orthogonally decomposed into a divergence-free part with homogeneous normal boundary conditions and a part that can be represented as the gradient of a scalar (see e.g. [5]). In this framework, we can interpret the pressure gradient rp as a term that projects the right hand side of (1) onto the space of divergence free fields and summarize its action with the projection operator P: To describe the discrete scheme we approximate in space on a regular grid with spacing h and in time with spacing k. It is assumed that 1=h is even for convenience. We use and P n to denote approximations of u(ih; jh; nk) and p(ih; jh; nk) respectively. To proceed, we need to define the approximate "projection", P h , derived by Chorin [3]. We use discrete divergence D h and gradient operators G h based on long, centered differences, i.e. away from boundaries. Near the lower boundary we can use the fact that U derive On the boundary, using second order one sided differencing gives Similar expressions apply on the upper wall. To divide an arbitrary vector W defined in the interior of the domain into a gradient part G h P and a discrete divergence free part U (D h \Delta must solve and then We summarize this process as denotes the scalar corresponding to the gradient part of the vector. This projection approach is convenient because it does not require the specification of any additional "pressure boundary conditions". Such conditions can be considered to be implicitly given by (7). We note that P h is not a projection matrix since D h and G h are not negative adjoint. Also, the matrix D h \Delta G h has four null modes corresponding to the four null modes of G h , constant vectors on the four subgrids shown in Figure 1. However, (7) is solvable [1] up to the four null vectors of G h . The four arbitrary constants are normalized using appropriate trapezoidal or midpoint approximations of (3). From the structure of G h P it is easy to see that the errors on the four subgrids can be different, leading to so-called alternating error expansions. The order that these effects enter the velocity and pressure is described in detail below. We now turn to a discretization in time. Chorin [3] proposed splitting the diffusion step and the projection step in the following scheme: where \Delta h denotes the usual five point approximation of the Laplacian with Dirichlet data. This scheme gives an uncoupled system for P n+1 and W n+1 , an auxiliary quantity computed during the diffusion sub-step. The fact that the system is decoupled is the advantage of using the split-step technique. Table 1: Normalized pointwise pressure errors e p and velocity errors e u (and estimated convergence rates in h) from Chorin's scheme. We note that in Chorin's original work an ADI technique was used to approximate (10) and an iterative technique was used to approximate the projection step (11). Here we analyze the underlying exact discretization for simplicity and because more modern solution techniques are available that can efficiently solve these subproblems. We consider computational results for this method below, showing the numerical boundary layers from the projection step and the alternating errors from the parasitic modes. The detailed analysis of these phenomena is then done in Section 4. Computational Results We demonstrate the types of errors discussed above with computational results for the Stokes equations in the periodic channel. The initial data from [19] is used (a perturbation of Poiseuille flow) with 1=64. Errors are calculated by comparing the solutions from Chorin's method with those from the Marker and Cell (MAC) grid with high order accurate explicit time stepping (the discrete pressures from this scheme have no alternating or boundary layer effects [11]). Comparisons are made at When relatively large and k relatively small), the pressure errors are dominated by the O(h) alternating errors from the parasitic mode effects as shown in Figure 2. Note that the error alternates in sign in the vertical direction only and is not confined to a region near the boundary. If the computation had been done in a box with vertical walls as well as horizontal walls there would also be horizontally alternating components of the error. When relatively small and k relatively large) the pressure errors are dominated by the boundary layer due to the projection step with size and width O(k 1=2 ). This is seen in the top picture of Figure 3. A contour plot of this same data has jagged contour lines, showing the presence of (smaller) alternating terms. When k is reduced to 0:01, the boundary layer is reduced in size and extent as shown in the lower picture of Figure 3. The error plots above verify the qualitative description of the errors. To show their order reducing effects on computed P we perform computations with several with . In Table 1 we see that P converges with first order (in h) and that U converges with second order (in h). We develop a asymptotic error expansion for the pressure and computed velocity for Chorin's original scheme (10)-(12) consisting of regular and alternating errors and numerical boundary layer terms as described in [20]. We will use the asymptotic descriptions of P h from [20] and present a simplified derivation of the errors from the split-step time-stepping first given in [9]. It is convenient to first derive an error expansion for W, the intermediate velocity, and then derive expansions for U and P from (11) and (12). The update equation for W is with boundary conditions convenient scaling for this analysis. We index the errors by powers of h so O(k) errors are listed as O(h 2 ) errors. Part of the errors in W at grid point level n can be described by numerical boundary layers of the form where A 2 (x; t) is a smooth function that depends only the exact solution u and - depends only on -. These errors appear at in the computed velocities W. Here that (14) has a width of a fixed number of grid points in space and so will shrink as the computation is refined (the size is thus similar boundary layer will appear at the upper boundary. From now on, we will consider only the bottom boundary explicitly. It will be shown below (in Lemma 1) that the projection of such a boundary layer (14) is zero at O(1): This allows us to determine -. The boundary layer should satisfy the discrete equations (13) exactly to highest order. Inserting (14) into (13), using Lemma 1 and collecting terms of O(1) (so the differences in the x direction can be neglected) we obtain which reduces to a quadratic equation for -: This equation has two real positive roots for every - ? 0 that occur in reciprocal pairs. The root with magnitude less than one we denote by - (the other root describes the boundary layer at the upper wall). The boundary layer (14) does not satisfy the boundary conditions for \Delta h . In fact, it is generated by a mismatch in the boundary conditions for W from the global error terms. The details of this are seen below. We note that numerical boundary layers are normally associated with finite difference methods with wide stencils that require additional, artificial boundary values to be specified. This is not the case in (13). The boundary layer that arises in the projection method can be shown formally to arise from a singular perturbation of the underlying pressure equations with a mismatch in boundary conditions [9]. We now show the action of the discrete projection on the boundary layer (14). has values at grid point that tend asymptotically to 2 (x), a (2) (x; y) and - a (2) (x; y) are smooth functions determined by A 2 . These functions are also smooth. We discuss the notation and meaning of this lemma before turning to the proof. In general, the superscript in brackets denotes the order a term appears and a subscript denotes the component for a vector quantity. Vectors appear in bold. The term described by a (2) is a smooth, global regular error term and the term described by - a (2) is an alternating error term caused by the decoupled stencil for the pressure. Alternating errors dominate in the pressure errors in Figure 2. What the expansions show is what will be computed when the discrete projection operator acts on a boundary layer to high accuracy (a weak but sufficient stability result for the projection step alone can be derived). Later, we will write as shorthand for (16). In following lemmas, we present expansions for P h acting on boundary layers in the horizontal component and regular and alternating terms. We can then derive an error expansion for W (as noted in the Introduction, a stability argument for the scheme is still missing so convergence cannot be shown). Here, the terms in the error expansion will show the order that the various types of errors appear. Expansions for U and P follow easily. Here and in what follows we retain only the highest order terms of each type except when necessary to explain some more subtle point. We return now to the proof of Lemma 1. Proof of Lemma 1 We denote To satisfy the interior equations (7) (for Q not P ) the following conditions for the boundary conditions apply Where centered differencing of a boundary layer is like multiplication by -h \Gamma1 ) and the primes denote differentiation in x. These equations determine the C functions in terms of A The interior equations for global terms from (7) are \Deltaq since there is no global source term. To determine q (2) and - q (2) we derive Neumann boundary conditions for them. In [20] it was shown that the effect of the reduced stencils near the boundary (5), (6) was equivalent to the following two discrete boundary conditions for Q: ~ where ~ centered differencing in the x and y directions. These can be interpreted as pressure Neumann boundary conditions. Using (17) these relationships are both satisfied at O(1). The action of ~ B on a boundary layer is like multiplication by smooth terms ~ to third order and for alternating terms, ~ f i;0 to first order. We note that D y f actually approximates \Gamma - f y since centered differencing uses adjacent grid points of opposite sign. Putting this together we find that at second order (22) and (23) give the following relationships at q (2) y and q (2) y q (2) y These give solvable Neumann data for (19) and (20) and so determine q (2) and - q (2) . All the listed terms in the expansion for Q in (16) have been determined. Further terms in the expansion can be determined similarly. Having determined Q we can now determine At O(1) the boundary layers cancel (recall centered differencing in y of a boundary layer is like multiplication by -h \Gamma1 and note (17)). We then have q (2) where r due to the effect noted above. 2 We have shown Lemma 1 in some detail so the reader can see the idea of the technical arguments. However, the important features of Lemma 1 are that a vertical boundary layer is removed (to highest order) by the action of P h and that the boundary layer in smaller by a factor of h. Later, we will see that there is a boundary layer of size O(h 2 ) in W. This leads to a boundary layer of size O(h 3 ) in kP and so a boundary layer of O(h) in P n+1 . By taking P h of (15) and bringing all the boundary layers to the left hand side we obtain the following Corollary. The fact that a (2) and - a (2) in Lemma 1 are pure gradients is used with Lemmas 3 and 5 to show that the global errors are suppressed to fourth order, although this is not important. Corollary 2 What we have created is a "pure gradient" boundary layer that has given normal boundary data at highest order. This is implicitly done in the spatially continuous analysis in [9]. Lemmas describing the action of P h on regular and alternating terms (denoted by are stated below. Proofs of Lemmas 3 and 5 can be found in [20]. Lemma 3 When a is a smooth function, P h a has the following error expansion: when a is compatible. For incompatible a the error terms of both types will appear at first order. A compatible function a is one for which the tangential component of Pa also vanishes on the boundary. A solution u of the Stokes equations and \Deltau are compatible as well as pure gradient fields. We need a small refinement of this lemma for the error analysis below. We note that on an alternating term D h \Delta - a approximates r \Delta - a. The modified projection P describes the projection onto divergence-* free fields with zero normal boundary values, which is orthogonal to gradient-* fields. Details are given in [20]. Corollary 4 If a is divergence free then a (1) and a (2) are pure gradients. If a is divergence free and compatible then a (3) is a pure gradient and - a (3) is a pure gradient-* field. Proof: We refer the reader to [20] for the details of the proof of Lemma 3 to make this rigorous, but the idea is simple. We use a. The error a (2) comes from two sources: rq (2) (a pure gradient) and the second order errors from computing G h q instead of rq. When a is divergence so the second source of error is not present and a (2) is a pure gradient. Similar reasoning applies to the other statements. 2 Lemma 5 The discrete projection acting on an alternating term gives the following error expansion: a h- a (1) The discrete projection acting on an alternating gradient-* field has an expansion beginning at second order. We are now in a position to state and prove the main error expansion result for the intermediate computed velocities W: Theorem 6 The intermediate velocities have the following error expansion where u is the exact solution of the Stokes equations. That is, regular errors and vertical boundary layers begin at second order and alternating errors and horizontal boundary layers begin at third order. Proof: For notational simplicity we assume . The divergence-free and gradient parts of the regular errors are determined at different levels in the discrete equations (13) so we divide the error terms explicitly d We similarly divide the alternating terms into divergence-* free and gradient-* fields. We insert (27) into the discrete equations (13), expanding \Delta h in a Taylor series as well as W n about the time level n+ 1. Regular interior terms are collected below: a (2) [u] a (3) [u] (31) d \Deltau (2) +a (2) d [u (2) d a (2) d +a (2) a (2) The terms a represent error terms from the discrete projection operator, with square brackets to denote their source. The interior equations force P h so A (3) using Corollary 4. To determine the equations for alternating terms we use the fact that the discrete Laplacian amplifies alternating terms in the following way (see [20]). The third order alternating terms in (13) give - boundary conditions are written below for normal component and tangential component: +A (2) d We will now discuss all of the terms above in detail. Equations (28), (29) and boundary conditions (36) show that u is indeed the solution of the Stokes equations we seek so W is a consistent approximation. Equation (30) then determines u (2) to be a (2) [u] +rp (38) is the exact pressure gradient for the Stokes equations. We know a (2) [u] is a pure gradient from Corollary 4. Once u (2) is known, A (2) 2 can be determined from (37) and tangential boundary conditions for u (2) d are known and can be used with the equations (32) to determine u (2) d . We note that u is given in (31). Continuing to ignore the alternating terms for the moment, the pattern to determine the regular and boundary layer terms is the following: 1. If u (p) d is known, u (p+2) can be determined from the O(h p+2 ) g expansion (i.e. is determined from (33)). 2. u (p+2) determines the vertical boundary layer at order p and the tangential boundary conditions for u (p+2) d and (through the effect of the tangential boundary layers) u (p+3) d 3. u (p+2) d can now be determined from the O(h p+4 ) d expansion (i.e. u (2) d is determined from (32)). In the discrete setting an important technical detail is that a (2) d d example, in the second line of (32) there is no term a (2) d [u (2) d ]. This is guaranteed by Corollary 4. This separated determination of the gradient and divergence-free components of the error expansion for the space continuous analysis is implicitly present in [9] but not clearly laid out. This technique easily allows for the implicit handling of the convection terms, for instance, which is avoided in [9]. We turn our attention now to the alternating errors. Equation (35) implies - u (3) d d - u (3) In fact, 1- a (3) d u (3) is a pure gradient-*, we can use Lemma 5 to justify the missing error terms from P h - u (3) in (32) and (33). Higher order alternating error terms are determined statically like (39) and (40) from alternating errors from the projection of lower order terms. An alternating divergence-* error appears at fifth order. 2 We now turn to the expansions for U and P . Theorem 7 The computed W has an error expansion with regular errors at second order, alternating errors at third order and no boundary layers. The computed pressure has alternating errors and boundary layers at first order and regular errors at second order. Proof: We take P h of (27). The boundary layers are removed (they were so constructed) and the following results: d d This verifies the first part of the Theorem. Now asymptotically by (27) minus (41) and results in a (2) [u]) with a nonzero fourth order regular error a (4) . When (38), (31) and (40) are used this becomes The corresponding scalar has an expansion (recall how boundary layers scale from Lemma 1). Since we have chosen the convenient scaling for the analysis, we see that This verifies the second claim of the Theorem. 2 Regularizations The alternating errors were generated by the uncoupled stencil for D h and G h . Following [17] we can use higher order regularizing terms (with corrections at the boundary) in D h and G h to eliminate these alternating errors. A projection operator based on this idea is described in [20]. We consider the scheme (10)-(12) with this regularized projection: Theorem 8 The computed velocities W from the spatially regularized scheme have an error expansion with regular errors at second order, numerical boundary layers at third order and no alternating errors. The computed pressure has boundary layers at first order, regular errors at second order and no alternating errors. Here, numerical boundary layers that occur due to the wide stencil of D h \Delta G h do enter the computed velocities U. This theorem can be proven using the asymptotic error description of the regularized projection in [20] following the general framework of the proof of Theorem 6. We omit the technical details. The presence of the dominant boundary layer error in the computed pressure for this scheme and the suppression of the alternating errors can be seen in Fig. 4 (compare to Fig. 2 for Chorin's original scheme with the same h and k). Using the regularized D h and G h as discussed above, we can further eliminate the dominant boundary layer errors in the pressure by using a non-split-step scheme: As shown in the theorem below, this scheme suppresses the numerical boundary layers from the projection step. Table 2: Normalized pointwise pressure errors e p (and estimated convergence rates in h) from the fully regularized scheme. Theorem 9 The computed velocities from the scheme (42),(43) with spatially regularized D h and G h have an error expansion with regular errors at second order, numerical boundary layers at fourth order and no alternating errors. The computed pressure has regular errors at second order, numerical boundary layers at third order and no alternating errors. Again, we omit the technical details. We note that the scheme (42),(43) requires the solution of a coupled system for U n+1 and P n+1 . As mentioned in the introduction, it would be more computationally efficient to use a pressure increment scheme [18, 6] to suppress the numerical boundary layers, but the analysis of this approach is not fully understood in the discrete setting. Second order convergence for the pressure from the fully regularized scheme is shown in Table 2 using the same parameters as the convergence study from Section 3 (compare Table 1). The smooth nature of the errors in computed pressure is shown in Fig. 5. 6 Summary We have presented an error analysis for Chorin's original fully discrete method for computing the incompressible Navier-Stokes equations. The velocities from this scheme converge with full order O(k)+O(h 2 ). The computed pressures have O(h) global alternating errors due to the uncoupled approximation used for the incompressibility condition and layers due to the split-step projection step. These errors can be removed by using a regularized stencil to approximate the incompressibility condition and a non-split-step time integration procedure. --R "Derivation and solution of the discrete pressure equations for the incompressible Navier Stokes equations," Spectral Methods in Fluid Dynamics (Section 11.3) "Numerical solution of the Navier-Stokes equations," "On the convergence of discrete approximations to the Navier-Stokes equations," "A Mathematical Introduction to Fluid Dynamics," "A second-order projection method for the incompressible Navier-Stokes equations," "An efficient second-order projection method for viscous incompressible flow," Finite Element Methods for Navier-Stokes Equations "Projection Method I: Convergence and Numerical boundary Layers," "Projection Method II: Rigorous Godunov-Ryabenki Analysis," "Second Order Convergence of a Projection Scheme for the Incompressible Navier-Stokes Equations with Boundaries," "Discrete Compatibility in Finite Difference Methods for Viscous Incompressible Flow," "Application of a fractional-step method to incompressible Navier-Stokes equations," "boundary conditions for incompressible flows," "On Chorin's projection method for the incompressible Navier-Stokes equations," "On error estimates of projection methods for Navier-Stokes equations: first order schemes," "High-Order Accurate Schemes for Incompressible Viscous Flow," "A second-order accurate pressure-correction scheme for viscous incompressible flow," "Finite Difference Vorticity Methods" "Analysis of the spatial error for a class of finite difference methods for viscous incompressible flow," --TR --CTR Robert D. Guy , Aaron L. Fogelson, Stability of approximate projection methods on cell-centered grids, Journal of Computational Physics, v.203 n.2, p.517-538, 1 March 2005 Weinan , Jian-Guo Liu, Projection method III: spatial discretization on the staggered grid, Mathematics of Computation, v.71 n.237, p.27-47, January 2002

parasitic modes;numerical boundary layers;projection methods

270656

First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity.

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H1 product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are naturally uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity, where we obtain the more substantive result that the estimates are uniform in the Poisson ratio.

Introduction . In earlier work [10], [11], we developed least-squares functionals for a first-order system formulation of general second-order elliptic scalar partial differential equations. The functional developed in [11] was shown to be elliptic in the sense that its homogeneous form applied to the (pressure and velocities) is equivalent to the norm. This means that the individual variables in the functional are essentially decoupled (more precisely, their interactions are essentially subdominant). This important property ensures that standard finite element methods are of H 1 -optimal accuracy in each variable, and that multiplicative and additive multigrid methods applied to the resulting discrete equations are optimally convergent. The purpose of this paper is to extend this methodology to the Stokes equations in two and three dimensions. To this end, we begin by reformulating the Stokes equations as a first-order system derived in terms of an additional vector variable, the velocity flux, defined as the vector of gradients of the Stokes velocities. We first apply a least-squares principle to this system using L 2 and H \Gamma1 norms weighted appropriately by the Reynolds number, Re. We then show that the resulting functional is elliptic in a product norm involving Re and the L 2 and H 1 norms. While of theoretical interest in its own right, we use this result here primarily as a vehicle for establishing that a modified form of this functional is fully elliptic in an H 1 product norm scaled by Re. This appears to be the first general theory of this kind for the Stokes equations in general dimensions with velocity boundary conditions. Bochev and Gunzburger [6] developed least-squares functionals for Stokes equations in norms that include stronger Sobolev terms and mesh weighting, but none are product H 1 elliptic. Chang [13] also used velocity derivative variables to derive a product H 1 elliptic functional for Stokes equations, but it is inherently limited to two dimensions. For general dimensions, a vorticity-velocity-pressure Center for Applied Mathematical Sciences, Department of Mathematics, University of Southern Cali- fornia, 1042 W. 36th Place, DRB 155, Los Angeles, CA 90089-1113. email: [email protected] y Program in Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526. email: [email protected] and [email protected]. This work was sponsored by the Air Force Office of Scientific Research under grant number AFOSR-91-0156, the National Science Foundation under grant number DMS-8704169, and the Department of Energy under grant number DE-FG03-93ER25165. form (cf.[4] and [20]) proved to be product H 1 elliptic, but only for certain nonstandard boundary conditions. For the more practical (cf. [17], [22], and [25]) velocity boundary conditions treated here, the velocity-vorticity-pressure formulation examined by Chang [14] can be shown by counterexample [3] not to be equivalent to any H 1 product norm, even with the added boundary condition on the normal component of vorticity. Moreover, this formulation admits no apparent additional equation, such as the curl and trace constraints introduced below for our formulation, that would enable such an equivalence. The velocity- pressure-stress formulation described in [7] has the same shortcomings. (If the vorticity and deformation stress variables are important, then they can be easily and accurately reconstructed from the velocity-flux variables introduced in our formulation.) While our least-squares form requires several new dependent variables, we believe that the added cost is more than offset by the strengthened accuracy of the discretization and the speed that the attendant multigrid solution process attains. Moreover, while our modified functional requires strong regularity conditions, this is to be expected for obtaining full product H 1 ellipticity in all variables, including velocity fluxes. (We thus obtain optimal estimates for the derivatives of velocity.) In any case, strengthened regularity is not necessary for the first functional we introduce. Our modified Stokes functional is obtained essentially by augmenting the first-order system with a curl constraint and a scalar (trace) equation involving certain derivatives of the velocity flux variable, then appealing to a simple L 2 least-squares principle. As in [11] for the scalar case, the important H 1 ellipticity property that we establish guarantees optimal finite element accuracy and multigrid convergence rates applied to this Stokes least-squares functional that are uniform in Re. One of the more compelling benefits of least squares is the freedom to incorporate additional equations and impose additional boundary conditions as long as the system is consistent. In fact, many problems are perhaps best treated with overdetermined (but first-order systems, as we have here for Stokes. We therefore abandon the so-called ADN theory (cf. [1] [2]), which is restricted to square systems, in favor of more direct tools of analysis. An important aspect of our general formulation is that it applies equally well to the Dirichlet problem for linear elasticity. This is done by posing the Stokes equations in a slightly generalized form that includes a pressure term in the continuity equation. Our development and results then automatically apply to linear elasticity. Most important, our optimal discretization and solver estimates are uniform in the Lam'e constants. We emphasize that the discretization and algebraic convergence properties for the generalized Stokes equations are automatic consequences of the H 1 product norm ellipticity established here and the finite element and multigrid theories established in Sections 3-5 of [11]. We are therefore content with an abbreviated paper that focuses on establishing ellipticity, which we do in Section 3. Section 2 introduces the generalized Stokes equations, the two relevant first-order systems and their functionals, and some preliminary theory. Concluding remarks are made in Section 4. 2. The Stokes Problem, Its First-Order System Formulation, and Other Preliminaries. Let\Omega be a bounded, open, connected domain in ! n Lipschitz boundary @ \Omega\Gamma The pressure-perturbed form of the generalized stationary Stokes equations in dimensionless variables may be written as \Gamma- \Delta u +r r where the symbols \Delta, r, and r\Delta stand for the Laplacian, gradient, and divergence operators, - is the reciprocal of the Reynolds number Re; f is a given vector function; g is a given scalar function; and ffi is some nonnegative constant linear elasticity). Without loss of generality, we may assume that Z \Omega Z \Omega equation (2.1) can have a solution only when g satisfies (2.2), and we are then free to ask that p satisfy (2.2). For ffi ? 0, in general we have only that R\Omega but this can be reduced to (2.2) simply by replacing p by and g by 0 in (2.1).) We consider the (generalized) Stokes equations (2.1) together with the Dirichlet velocity boundary condition The slightly generalized Stokes equations in (2.1) allow our results to apply to linear elasticity. In particular, consider the Dirichlet problem where u now represents displacements and - and - are the (positive) Lam'e constants. By here we mean the n-vector of components \Delta u i , that is, \Delta applies to u componentwise. This is recast in form (2.1)-(2.2) by introducing the pressure variable 1 rescaling f , and by letting is easy to see that this p must satisfy (2.2).) An important consequence of the results we develop below is that standard Rayleigh-Ritz discretization and multigrid solution methods can be applied with optimal estimates that are uniform in h, -, and -. For example, we obtain optimal uniform approximation of the gradients of displacements in the H 1 product norm. This in turn implies analogous H 1 estimates for the stresses, which are easily obtained from the "velocity fluxes". For related results with a different methodology and weaker norm estimates, see [16]. Let curl j r\Theta denote the curl operator. (Here and henceforth, we use notation for the case consider the special case in the natural way by identifying with the if u is two dimensional, then the curl of u means the scalar function r\Theta 1 Perhaps a more physical choice for this artificial pressure would have been r \Delta u, since it then becomes the hydrostatic pressure in the incompressible limit. We chose our particular scaling because it most easily conforms to (2.1). In any case, our results apply to virtually any nonnegative scaling of p, with no effect on the equivalence constants (provided the norms are correspondingly scaled); see Theorems 3.1 and 3.2. where u 1 and u 2 are the components of u.) The following identity is immediate: r\Theta (r\Theta interpreted as r ? is the formal adjoint of r\Theta defined by We will be introducing a new independent variable defined as the n 2 -vector function of gradients of the u i , It will be convenient to view the original n-vector functions as column vectors and the new n 2 -vector functions as either block column vectors or matrices. Thus, given and denoting u then an operator G defined on scalar functions (e.g., r) is extended to n-vectors componentwise: and If U i j Gu i is a n-vector function, then we write the matrix U U n1 U We then define the trace operator tr according to tr If D is an operator on n-vector functions (e.g., its extension to matrices is defined by When each DU i is a scalar function (e.g., then we will want to view the extension as a mapping to column vectors, so we will use the convention We also extend the tangential operator n\Theta componentwise: Finally, inner products and norms on the matrix functions are defined in the natural componentwise way, e.g., Introducing the velocity flux variable then the Stokes system (2.1) and (2.3) may be recast as the following equivalent first-order system: r Note that the definition of U, the "continuity" condition r \Delta in \Omega\Gamma and the Dirichlet condition @\Omega imply the respective properties r\Theta Then an equivalent extended system for (2.6) is r r\Theta Let D(\Omega\Gamma be the linear space of infinitely differentiable functions with compact support on\Omega and let D 0 (\Omega\Gamma denote the dual space of D(\Omega\Gamma0 The duality pairing between D D(\Omega\Gamma is denoted by ! \Delta; \Delta ?. We use the standard notation and definition for the Sobolev spaces H s the standard associated inner products are denoted by (\Delta; \Delta) s;\Omega and (\Delta; \Delta) s; @\Omega , and their respective norms by k \Delta k s;\Omega and k \Delta k s; @\Omega . (We suppress the superscript n because dependence of the vector norms on dimension will be clear by context. We also omit\Omega from the inner product and norm designation when there is no risk of confusion.) For coincides with L . In this case, the norm and inner product will be denoted by k \Delta k and (\Delta; \Delta), respectively. As usual, H s 0(\Omega\Gamma is the closure of with respect to the norm k \Delta k s and H \Gammas(\Omega\Gamma is its dual with norm defined by Define the product spaces H s \Gammas(\Omega\Gamma with standard product norms. Let and which are Hilbert spaces under the respective norms and @\Omega where denote the respective unit vectors normal and tangent to the boundary. Finally, define Z \Omega It is well-known that the (weak form of the) boundary value problem (2.1)-(2.2) has a unique solution (u; n \Theta L 2 (e.g., see [21, 22, 17]). Moreover, if the boundary of the domain\Omega is C or a convex polyhedron, then the following H 2 -regularity result holds: (We use C with or without subscripts in this paper to denote a generic positive constant, possibly different at different occurrences, that is independent of the Reynolds number and other parameters introduced in this paper, but may depend on the domain\Omega or the constant .) Bound (2.9) is established for the case the case for general and the case ffi ? 0 follows from the well-known linear elasticity bound kuk 2 is the (unscaled) source term in (3.19) and oe is the stress tensor. We will need (2.9) to establish full H 1 product ellipticity of one of our reformulations of (2.1)-(2.2); see Theorem 3.2. The following lemma is an immediate consequence of a general functional analysis result due to Ne-cas [24] (see also [17]). Lemma 2.1. For any p in L 2 , we have Proof. See [24] for a general proof. A curl result analogous to Green's theorem for divergence follows from [17] (Theorem 2.11 in Chapter I): (r\Theta z; Z @\Omega ds for z 2 H(curl ; \Omega\Gamma and OE 2 H Finally, we summarize results from [17] that we will need for G 2 in the next section. The first inequality follows from Theorems 3.7-3.9 in [17], while the second inequality follows from Lemmas 3.4 and 3.6 in [17]. Theorem 2.1. Assume that the domain\Omega is a bounded convex polyhedron or has C boundary. Then for any vector function v in either H 0 (div; If, in addition, the domain is simply connected, then kr 3. First-Order System Least Squares. In this section, we consider least-squares functionals based on system (2.6) and its extension (2.8). Our primary objective here is to establish ellipticity of these least-squares functionals in the appropriate Sobolev spaces. Our first least-squares functional is defined in terms of appropriate weights and norms of the residuals for system (2.6): Note the use of the H \Gamma1 norm in the first term here. Our second functional is defined as a weighted sum of the L 2 norms of the residuals for system (2.8): Let n \Theta L 2 n \Theta (H where @\Omega g: Note that V 2 ae V 1 . For 2, the first-order system least-squares variational problem for the Stokes equations is to minimize the quadratic functional G i (U; u; find (U; u; p) 2 V i such that Theorem 3.1. There exists a constant C independent of - such that for any (U; u; p) 2 and Proof. Upper bound (3.5) is straightforward from the triangle and Cauchy-Schwarz inequalities. We proceed to show the validity of (3.4) for (U; u; p) g. Then (3.4) would follow for (U; u; p) 2 V 1 by continuity. For any (U; u; p) n , we have (r Hence, by Lemma 2.1, we have From (3.6) and the Poincar'e-Friedrichs inequality on u we have Using the "-inequality, 2ab - 1 for the first two products yields Again from (3.6) and the Poincar'e-Friedrichs inequality on u we have Using the "-inequality on the first three products and (3.7), we then have Again using the "-inequality, we find that Using (3.8) in (3.6) and (3.7), we now have that The theorem now follows from these bounds, (3.8), and the Poincar'e-Friedrichs inequality on u. The next two lemmas will be useful in the proof of Theorem 3.2. Lemma 3.1. (Poincar'e-Friedrichs-type inequality) Suppose that the assumptions of Theorem 2.1 hold. Let p 2 H where C depends only on\Omega . Further, let q 2 (H 1 kr where C depends only on\Omega . Proof. Equation R at some point in \Omega\Gamma The first result now follows from the standard Poincar'e-Friedrichs inequality. The second result follows from the fact that R Lemma 3.2. Under the assumptions of Theorem 2.1 with simply connected\Omega , for any (\Omega\Gamma we have: 2(\Omega\Gamma and each OE is such that \DeltaOE i 2 L @\Omega , then jr jr 2(\Omega\Gamma and each @\Omega , then jr jr Proof. The assumptions of Theorem 2.1 are sufficient to guarantee H 2 -regularity of the Laplace equation on \Omega\Gamma that is, the second inequality in the equation Note that tr (r ? OE OE. Then, from the above and the triangle inequality, we have jr jr which is (3.11). applied to each column of r\Theta\Phi imply that since each OE i is divergence free. Eqn. (3.12) now follows from the triangle inequality as for the case 2. Theorem 3.2. Assume that the domain\Omega is a bounded convex polyhedron or has C boundary and that regularity bound (2.9) holds. Then, there exists a constant C independent of - such that for any (U; u; and Proof. Upper bound (3.14) is straightforward from the triangle and Cauchy-Schwarz inequalities. To prove (3.13), note that the H \Gamma1 norm of a function is always bounded by its Hence, by Theorem 3.1, we have From Theorem 2.1 and (3.9), we have It thus suffices to show that We will prove (3.17) only for the case because the proof for First we assume that the domain\Omega is simply connected. Since n \Theta U = 0 on @ the following decomposition is admitted (see Theorems 3.4 in [17]): and \Phi is columnwise divergence free with n \Theta (r\Theta @ Here we choose q to satisfy By taking the curl of both sides of this decomposition, it is easy to see that (3. so that k\Delta \Phik is bounded and Lemma 3.2 applies. Hence, kr\Theta Uk 2 (by equation (3.18)) (by Lemma 3.2) assumption (2.9) with (by equation (3.18)) This proves (3.17) and, hence, the theorem for simply connected\Omega\Gamma The proof for general\Omega\Gamma that is, when we assume only that @\Omega is C 1;1 , now follows by an argument similar to the proof of Theorem 3.7 in [17]. We now show that the last two terms in the definition of G 2 are necessary for the bound (3.13) to hold, even with the extra boundary condition n \Theta U = 0. We consider the Stokes equations, so that first that we omit the term kr\ThetaUk 2 but include the term krtr Uk 2 . We offer a two-dimensional counterexample; a three-dimensional counterexample can be constructed in a similar manner. Let Choose any ! 2 D(\Omega\Gamma such that \Deltar! 6= 0 and define Clearly, n \Theta U = 0. It is easy to show that r However, (r\Theta by construction. Thus, which cannot bound kUk 2 1 . That is, since D(\Omega\Gamma is arbitrary, we may choose it so oscillatory that kUk 1 =kUk is as large as we like. This prevents the bound (3.13) from holding. Next suppose we include the kr\Theta Uk 2 term but omit the krtr Uk 2 term. Now set choose q i to satisfy 2. Then We also know that where C is independent of k. Now set 2. Then n \Theta U where C is independent of k. On the other hand, we have which again prevents the bound (3.13) from holding. 4. Concluding Remarks. Full regularity assumption (2.9) is needed in Theorem 3.2 only to obtain full H 1 product ellipticity of augmented functional G 2 in (3.2). This somewhat restrictive assumption is not necessary for functional G 1 in (3.1), which supports an efficient practical algorithm (the H \Gamma1 norm in (3.1) can be replaced by a discrete inverse norm or a simpler mesh weighted norm; see [5] and [8] for analogous inverse norm algorithms) and which has the weaker norm equivalence assured by Theorem 3.1. Nevertheless, the principal result of this paper is Theorem 3.2, which establishes full H 1 product ellipticity of least-squares functional G 2 for the generalized Stokes system. Since we have assumed full H 2 -regularity of the original Stokes (linear elasticity) equations, we may then use this result to establish optimal finite element approximation estimates and optimal multiplicative and additive multigrid convergence rates. This can be done in precisely the same way that these results were established for general second-order elliptic equations (see [11], Sections 3-5). We therefore omit this development here. However, it is important to recognize that the ellipticity property is independent of the Reynolds parameter - (Lam'e constants - and -). This automatically implies that the optimal finite element discretization error estimates and multigrid convergence factor bounds are uniform in - and -). At first glance, it might appear that the scaling of some of the H 1 product norm components might create a scale dependence of our discretization and algebraic convergence estimates. However, the results in [11] are based only on assumptions posed in an unscaled H 1 product norm, in which the individual variables are completely decoupled; and since the constant - appears only as a simple factor in individual terms of the scaled H 1 norm, these assumptions are equally valid in this case. On the other hand, for problems where the necessary H 1 scaling is not (essentially) constant, extension of the theory of Section 3-5 of [11] is not straightforward. Such is the case for convection-diffusion equations, which will be treated in a forthcoming paper. --R Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II Analysis of least-squares finite element methods for the Navier-Stokes equations Accuracy of least-squares methods for the Navier-Stokes equations Analysis of least-squares finite element methods for the Stokes equations A least-squares approach based on a discrete minus one inner product for first order system On the existence Schwarz alternating procedure for elliptic problems discretized by least-squares mixed finite elements A mixed finite element method for the Stokes problem: an acceleration-pressure formulation Appl An error estimate of the least squares finite element methods for the Stokes problem in three dimensions Math. The Finite Element Method for Elliptic Problems analysis of some Galerkin least squares methods for the linear elasticity equationns Finite Element Methods for Elliptic Problems in Nonsmooth Domains Finite Element Methods for Viscous Incompressible Flows Theoretical study of the incompressible Navier-Stokes equations by the least-squares method A regularity result for the Stokes problem in a convex polygon The Mathematical Theory of Viscous Incompressible Flow Variational multigrid theory --TR --CTR Hongxing Rui , Seokchan Kim , Sang Dong Kim, A remark on least-squares mixed element methods for reaction-diffusion problems, Journal of Computational and Applied Mathematics, v.202 n.2, p.230-236, May, 2007 Suh-Yuh Yang, Analysis of piecewise linear approximations to the generalized stokes problem in the velocity-stress-pressure formulation, Journal of Computational and Applied Mathematics, v.147 n.1, p.53-73, 1 October 2002 M. M. J. Proot , M. I. Gerrtisma, Least-squares spectral elements applied to the Stokes problem, Journal of Computational Physics, v.181 n.2, p.454-477, 20 September 2002 On the velocity-vorticity-pressure least-squares finite element method for the stationary incompressible Oseen problem, Journal of Computational and Applied Mathematics, v.182 n.1, p.211-232, 1 October 2005 Yves Tourigny, The Optimisation of the Mesh in First-Order Systems Least-Squares Methods, Journal of Scientific Computing, v.24 n.2, p.219-245, August 2005 J. P. Pontaza , J. N. Reddy, Space-time coupled spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equations, Journal of Computational Physics, v.197 n.2, p.418-459, 1 July 2004 J. P. Pontaza , J. N. Reddy, Spectral/hp least-squares finite element formulation for the Navier-Stokes equations, Journal of Computational Physics, v.190 n.2, p.523-549, 20 September Youngmi Choi , Hyung-Chun Lee , Byeong-Chun Shin, A least-squares/penalty method for distributed optimal control problems for Stokes equations, Computers & Mathematics with Applications, v.53 n.11, p.1672-1685, June, 2007 P. Bolton , R. W. Thatcher, On mass conservation in least-squares methods, Journal of Computational Physics, v.203 n.1, Sang Dong Kim , Byeong Chun Shin, H-1 least-squares method for the velocity-pressure-stress formulation of Stokes equations, Applied Numerical Mathematics, v.40 n.4, p.451-465, March 2002 J. J. Heys , T. A. Manteuffel , S. F. McCormick , J. W. Ruge, First-order system least squares (FOSLS) for coupled fluid-elastic problems, Journal of Computational Physics, v.195 J. A. Sethian , Jon Wilkening, A numerical model of stress driven grain boundary diffusion, Journal of Computational Physics, v.193 n.1, p.275-305, January 2004 J. J. Heys , T. A. Manteuffel , S. F. McCormick , L. N. Olson, Algebraic multigrid for higher-order finite elements, Journal of Computational Physics, v.204

stokes equations;multigrid;least squares

270671

On Krylov Subspace Approximations to the Matrix Exponential Operator.

Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equations. As a side result, we obtain error bounds for Galerkin-type Krylov methods for linear equations, namely, the biconjugate gradient method and the full orthogonalization method. For Krylov approximations to matrix exponentials, we show superlinear error decay from relatively small iteration numbers onwards, depending on the geometry of the numerical range, the spectrum, or the pseudospectrum. The convergence to exp$(\tau A)v$ is faster than that of corresponding Krylov methods for the solution of linear equations $(I-\tau A)x=v$, which usually arise in the numerical solution of stiff ordinary differential equations (ODEs). We therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step.

Introduction . In this article we study Krylov subspace methods for the approximation of exp(-A)v when A is a matrix of large dimension, v is a given vector, scaling factor which may be associated with the step size in a time integration method. Such Krylov approximations were apparently first used in Chemical Physics [20, 22, 17] and were more recently studied by Gallopoulos and Saad [10, 24]; see also their account of related previous work. They present Krylov schemes for exponential propagation, discuss the implementation, report excellent numerical results, and give some theoretical error bounds. As they also mention, these bounds are however too pessimistic to explain the numerically observed error reductions. Moreover, their error bounds do not make evident that - or when and why - Krylov methods perform far better than standard explicit time stepping methods in stiff problems. A further open question concerns the relationship between the convergence properties of Krylov subspace methods for exponential operators and those for the linear systems of equations arising in implicit time integration methods. In the present paper we intend to clear up the error behavior. When we wrote this paper, we were unaware of the important previous work by Druskin and Knizhnerman [3, 4, 14, 15] who use a different approach to the analysis. We will comment on the relationship of some of their results to ours in a note at the end of this paper. Our error analysis is based on a functional calculus of Arnoldi and Lanczos methods which reduces the study of approximations of exp(-A)v to that of the corresponding iterative methods for linear systems of equations. Somewhat oversimplified, it may be said that the error of the mth iterate for exp(-A)v behaves like the minimum, Mathematisches Institut, Universit?t T-ubingen, Auf der Morgenstelle 10, D-72076 T-ubingen, Germany. E-mail: [email protected], [email protected] taken over all ff ? 0, of e ff multiplied with the error of the mth iterate for the solution of by the same Krylov subspace method. This minimum is usually attained far from especially for large iteration numbers. Unless a good preconditioner for I \Gamma -A is available, the iteration for exp(-A)v converges therefore faster than that for v. We do not know, however, of a way to "precondition" the iteration for exp(-A)v . Gallopoulos and Saad showed that the error of the mth iterate in the approximation of exp(-A)v has a bound proportional to k-Ak m =m!, which gives superlinear convergence for m AE k-Ak. In many cases, however, superlinear error decay begins for much smaller iteration numbers. For example, we will show that for symmetric negative definite matrices A this occurs already for m - k-Ak, whereas for skew-Hermitian matrices with uniformly distributed eigenvalues substantial error reduction begins in general only for m near k-Ak. We will obtain rapid error decay for m- k-Ak also for a class of sectorial, non-normal matrices. Convergence within the required tolerance ensures that the methods become superior to standard explicit time stepping methods for large systems. For m - k-Ak, our error bounds improve upon those of [10] and [24] typically by a factor 2 \Gammam e \Gammack- with a c ? 0. The analysis explains how the error depends on the geometry of critical sets in the complex plane, namely the numerical range of A for Arnoldi-based approximations, and the location of the spectra or pseudospectra of A and the Krylov-Galerkin matrix Hm for both Lanczos- and Arnoldi-based approximations. In our framework, it is also easily seen that clustering of eigenvalues has similar beneficial effects in the Krylov subspace approximation of exp(-A)v as in the iterative solution of linear systems of equations. As mentioned above, exp(-A)v can often be computed faster than Krylov subspace methods. This fact has implications in the time integration of very large systems of ordinary differential equations arising, e.g., in many-particle simulations and from spatial discretizations of time-dependent partial differential equations. It justifies renewed interest in ODE methods that use the exponential or related functions of the Jacobian instead of solving linear or nonlinear systems of equations in every time step. Methods of this type in the literature include the exponential Runge-Kutta methods of Lawson [18] and Friedli [8], the adaptive Runge-Kutta methods of Strehmel and Weiner [27] in their non-approximated form, exponentially fitted methods of [5], and the exponential multistep methods of [9]. In the last section of this paper, we propose a promising new class of "exponen- tial" integration methods. With Krylov approximations, substantial savings can be expected for large, moderately stiff systems of ordinary differential equations which are routinely solved by explicit time-stepping methods despite stability restrictions of the step size, or when implicit methods require prohibitively expensive Jacobians and linear algebra. The paper is organized as follows: In Section 2, we describe the general framework and derive a basic error bound for the Arnoldi method. In Section 3, this leads to specific error bounds for the approximation of exp(-A)v for various classes of matrices A. Lanczos methods are studied in Section 4, which contains also error bounds for BiCG and FOM. In Section 5, we introduce a class of time-stepping methods for large systems of ODEs which replace the solution of linear systems of equations by multiplication with '(-A), where whose Krylov subspace approximations converge as fast as those for exp(-A)v. Throughout the paper, k \Delta k is the Euclidean norm or its induced matrix norm. 2. Arnoldi-based approximation of functions of matrices. In the sequel, let A be a complex square matrix of (large) dimension N , and v 2 C N a given vector of unit length, 1. The Arnoldi process generates an orthonormal basis of the Krylov space matrix Hm of dimension m (which is the upper left part of its successor Hm+1 ) such that where e i is the ith unit vector in R m . By induction this clearly implies, as noted in [3, Theorem 2] and [24, Lemma 3.1], A standard use of the Arnoldi process is in the solution of linear equations [23], where one approximates when - is not an eigenvalue of A, and hopefully not of Hm . The latter condition is always satisfied when - is outside the numerical range since (2.1) implies m AVm and therefore We now turn to the approximation of functions of A. Let f be analytic in a neighborhood of F(A). Then Z where \Gamma is a contour that surrounds F(A). In view of (2.3), we are led to replace this Z so that we approximate Such an approximation was proposed previously [22, 33, 3, 10], with different derivations In practice, we are then left with the task of computing the lower-dimensional expression f(Hm )e 1 , which for m - N is usually much easier to compute than f(A)v, e.g., by diagonalization of Hm . The above derivation of (2.7) also indicates how to obtain error bounds: Study the error in the Arnoldi approximation (2.3) of linear systems and integrate their error bounds, multiplied with jf(-)j, for - varying along a suitable contour \Gamma. This will actually be done in the present paper. Our error bounds are based on Lemma 1 below. To prepare its setting, let E be a convex, closed bounded set in the complex plane. Let OE be the conformal mapping that carries the exterior of E onto the exterior of the unit circle fjwj ? 1g, with We note that ae is the logarithmic capacity of E. Finally, let \Gamma be the boundary curve of a piecewise smooth, bounded region G that contains E, and assume that f is analytic in G and continuous on the closure of G. Lemma 1. Under the above assumptions, and if the numerical range of A is contained in E, we have for every polynomial q m\Gamma1 of degree at most Z with is the length of the boundary curve @E of E, and where d(S) is the minimal distance between F(A) and a subset S of the complex plane. If E is a straight line segment or a disk, then (2:8) holds with Remark. It will be useful to choose the integration contour dependent on m, in order to balance the decay of OE \Gammam away from E against the growth of f outside E. For functions f , such as the exponential function studied in detail below, this will ultimately yield superlinear convergence. On the other hand, the liberty in choosing the polynomial q m\Gamma1 will not materialize in the study of the exponential function. Proof. (a) We begin by studying the error of (2.3) and consider a fixed the moment. Our argumentation in this part of the proof is inspired by [25]. Using m v, we rewrite the error as with . By (2.1) and the orthogonality of Hence we have for arbitrary y We note that is a polynomial of degree - m with Conversely, for every such (A)v is of the above form. To bound \Delta m , we recall kVm use the estimates k(-I which follow from [26, Thm.4.1] and (2.4). We thus obtain for every polynomial p m of degree at most m with (b) It remains to bound p m (A). Since Z we have For the special case when E is a line segment, we have A of the form A with a Hermitian B and complex coefficients ff; fi, so that then When E is a disk jz \Gamma -j - ae, then p m (z) will be chosen as a multiple of (z and an inequality of Berger (see [1, p. 3]) then tells us that In all these cases we thus have with M as stated in Lemma 1. (c) To proceed in the proof, we use near-optimality properties of Faber polynomials. These have been employed previously in analyses of iterative methods by Eiermann [6] and Nevanlinna [21]. Let OE m (z) denote the Faber polynomial of degree m associated with the region E. This is defined as the polynomial part of OE(z) m , i.e., 1. We now choose the polynomial p m (z) with the normalization cf. [21, p.76]. A theorem of K-ovari and Pommerenke [16, Thm.2] provides us with the bound This implies max z2E jOE m (d) Combining inequalities (2.10),(2.11), and (2.14) gives us The proof is now completed by inserting this bound into the difference of formulas (2.5) and (2.6) and taking account of (2.2). Remark. Part (c) of the above proof, combined with Cauchy's integral formula, shows that there exists a polynomial \Pi m\Gamma1 (z) of degree at most m \Gamma 1 such that Z where ffi is the minimal distance between \Gamma and E. This holds for \Pi R of (2.12). Polynomial approximation bounds of this type are closely tied to Bernstein's theorem [19, Thm.III.3.19]. 3. Approximation of the matrix exponential operator. In this section we give error bounds for the Arnoldi approximation of e -A v for various classes of matrices A. We may restrict our attention to cases where the numerical range of A is contained in the left half-plane, so that ke -A k and, in view of (2.4), also are bounded by unity. This assumption entails no loss of generality, since a shift from A to A changes both e -A v and its approximation by a factor e - ff . The same bounds as in Theorems 2 to 6 below (even slightly more favorable are valid also for Krylov subspace approximations of '(-A)v, with Theorem 2. Let A be a Hermitian negative semi-definite matrix with eigenvalues in the interval [\Gamma4ae; 0]. Then the error in the Arnoldi approximation of e -A v, i.e., is bounded in the following ways: Remark. It is instructive to compare the above error bounds with that of the conjugate gradient method applied to the linear system which is given by This bound becomes small for m AE p ae- but only with a linear decay. Proof. We use Lemma 1 with Then the conformal mapping is We start by applying the linear transformation to the interval [\Gamma1; 1]. As contour \Gamma we choose the parabola with right-most point fl that is mapped to the parabola \Pi given by the parametrization This parabola osculates to the ellipse E with foci \Sigma1 and major us the error bound Z Z where \Phi(-+ 1. The absolute value of \Phi(-) is constant along every ellipse with foci \Sigma1. Since the parabola \Pi is located outside the ellipse E , we have along \Pi Hence we obtain from (3.3) Z 1e \Gammaae- ' 2 d' s Fig. 3.1. Errors and error bounds for the symmetric example Moreover, we have r - e ff 2ffl with ff ? 0:96 for ffl - 1=2 (and ff ? 0:98 for ffl - 1=4). Minimizing e 2ae- ffl\Gammam 2ffl with respect to ffl yields Inserting this ffl in (3.4) results in the bound (with which together with " m - 2 is a sharper version of (3.1). The condition ffl - 1=2 is equivalent to m - 2ae- . To obtain the bound (3.2), we note that 1 insert in (3.4) which is close to the minimum for m AE ae- . This yields for m - 2ae- which is a sharper version of (3.2). Finally we remark, in view of the proof of Theorem 3 below, that the bounds (3.1) and (3.2) are also obtained when \Gamma is chosen as a composition of the part of the above parabola contained in the right half-plane and two rays on the imaginary axis. To give an illustration of our error bounds we consider the diagonal matrix A with equidistantly spaced eigenvalues in the interval [\Gamma40; 0] and a random unit vector v of dimension 1001. Fig. 3.1 shows the errors of the approximation to exp(A)v and those of the cg approximation to nearly a straight line. Moreover, the dashed line shows the error bounds (3.5) and (3.6), while the dotted line corresponds to 2k 1 which is the error bound of [24, Corollary 4.6] for symmetric, negative semi-definite matrices A. It is well known that Krylov subspace methods for the solution of linear systems of equations benefit from a clustering of the eigenvalues. The same is true also for the Krylov subspace approximation of e -A v. This is actually not surprising in view of the Cauchy integral representations (2.5), (2.6). As an example of such a result, we state the following theorem. This might be generalized in various directions for different types of clusterings and different types of matrices, but we will not pursue this further. Theorem 3. Let A be a Hermitian negative semi-definite matrix with eigenvalues contained in f- 1 \Gamma4ae. Then the (m 1)st error " m+1 in the Arnoldi approximation of e -A v is bounded by the right-hand sides of (3:1) and (3:2). Proof. The result is proved by using the polynomial instead of (2.12). The absolute value of the first factor is bounded by unity for z 2 [\Gamma4ae; 0] and Re- 0. Hence we obtain the same error bounds as in Theorem 2 with replaced by m \Gamma 1. For skew-Hermitian matrices A (with uniformly distributed eigenvalues) we cannot show superlinear error decay for m ! ae- . The reason is, basically, that here the conformal mapping OE maps the vertical line contour with jOE(-)j - in the symmetric negative definite case we have jOE(-)j ffl. This behavior affects equally the convergence of Krylov subspace methods for the solution of linear systems v. For skew-Hermitian A, there is, in general, no substantial error reduction for m ! ae- , and convergence is linear with a rate like Theorem 4. Let A be a skew-Hermitian matrix with eigenvalues in an interval on the imaginary axis of length 4ae. Then the error in the Arnoldi approximation of e -A v is bounded by Proof. We use Lemma 1 with the conformal mapping is After applying the linear transformation we choose the integration contour as an ellipse with foci \Sigma1 and minor semiaxis semiaxis is then and the length of the contour is bounded by 2-a. In addition, we have 1). The absolute value constant along the ellipse. With Lemma 1, we get for the error Fig. 3.2. Errors and error bounds for the skew-Hermitian example with Inserting gives the stated error bound. A sharper bound for ae- ? 1 2 , which is obtained by integrating over the parabola that osculates to the above ellipse at the right-most point, reads For a numerical illustration we choose the diagonal matrix A with 1001 equidistant eigenvalues in [\Gamma20i; 20i], and a random vector v of unit length. Fig. 3.2 shows the errors of the approximation to exp(A)v and those of the BiCG approximation to nearly a straight line. The dashed line shows the error bound (3.7), the dotted line corresponds to which is the bound given in [10, Corollary 2.2]. Theorem 5. Let A be a matrix with numerical range contained in the disk jz Then the error in the Arnoldi approximation of e -A v is bounded by Proof. We use Lemma 1 with a circle with radius rae centered at \Gammaae. Lemma 1 gives the bound Setting gives the stated result. The following is a worst-case example which shows nearly no error reduction for m - ae- . Example. Let A be the bidiagonal matrix of dimension N that has \Gamma1 on the diagonal and +1 on the subdiagonal. The numerical range of A is then contained in the disk The Arnoldi process gives as the m-dimensional version of A, so that The error vector thus contains the entries e \Gamma- k =k! for k ? m. The largest of these is close to (2-) \Gamma1=2 by Stirling's Similar to Theorem 2, the onset of superlinear convergence begins already for m- ae- when F(A) is contained in a wedge-shaped set. In particular, consider the conformal mapping which maps the exterior of the unit disk onto the exterior of the bounded sectorial set in the left half-plane S ' has a corner at 0 with opening angle '- and is symmetric with respect to the real axis. Theorem 6. For some ae ? 0 and let the numerical range of A be contained in ae \Delta S ' . Then the error in the Arnoldi approximation of e -A v is bounded by ae- with 1\Gamma' . The constants C and c ? 0 depend only on '. Proof. In the course of this proof, C denotes a generic constant which takes on different values on different occurrences. After the transformation -=ae, we use choose ffi such that r=(1 note that then ae . The right-most point of the integration contour ae Hence we have from For m AE (ae- ) ff , the right-hand side is minimized near Inserting this ffl gives (3.8) with ff ff For any r - 2 and ffi ? 0, Lemma 1 gives e /(r)ae- which becomes (3.9) upon choosing 4. Lanczos-based approximation of functions of matrices. The Arnoldi method unfortunately requires long recurrences for the construction of the Krylov basis. The Lanczos method overcomes this difficulty by computing an auxiliary basis which spans the Krylov subspace with respect to A and w 1 . The Lanczos vectors v j and w j are constructed such that they satisfy a biorthogonality condition, or block biorthogonality in case of the look-ahead version [7, 28], i.e., Dm := m Vm is block diagonal. The look-ahead process ensures that Dm is well conditioned when the index m terminates a block, which will be assumed of m in the sequel. The Lanczos vectors can be constructed by short (mostly three-term) recurrences. This results again in a matrix representation (2.1), but now with a block tridiagonal m AVm . However, unlike the Arnoldi case, neither Vm nor Wm are orthogonal matrices. It is usual to scale the Lanczos vectors to have unit norm, in which case the norms of Vm and Wm are bounded by p m. Since Hm is now an oblique projection of A, the numerical range of Hm is in general not contained in F(A). Variants of Lemma 1, which apply in this situation, are given in the following two lemmas. For the exponential function, Lemmas 7 and 8 lead to essentially the same error bounds as given for the Arnoldi method in Theorems 5 and 6, except for different constants. In Theorems 2, 3, and 4, Arnoldi and Lanczos approximations coincide. The first lemma works with the ffl-pseudospectrum of A [32], defined by Otherwise, the setting is again the one described before Lemma 1. Lemma 7. If then the error of the Lanczos approximation of f(A)v is bounded by (2:8) with Proof. The proof modifies the proof of Lemma 1. For the Lanczos process we have I , and therefore Noting m v, we thus obtain for every polynomial p m of degree - m with 1. By assumption we have that the norms of both (-I are bounded by fl \Gamma1 for - 2 \Gamma. Using further kp m (A)k - '(@E)=(2-ffl) \Delta max z2E jp m (z)j leads to (4. which in turn yields the estimate stated in the lemma. For a diagonalizable matrix A we let where X is the matrix that contains the eigenvectors of A in its columns. The following lemma involves only the spectrum (A) of A and uses once more the setting of Lemma 1. Lemma 8. Let A be diagonalizable and assume that (A) ae E, (H m \Gamma. Then the Lanczos approximation of f(A)v satisfies (2:8) with , where ffi is the minimal distance between (A) and \Gamma. Proof. The result follows from (4.1) along the lines of parts (c) and (d) of the proof of Lemma 1. Remarks. (a) It is known that in generic situations, extreme eigenvalues of A are well approximated by those of Hm for sufficiently large m [34]. For a contour \Gamma that is bounded away from (A), one can thus expect that usually k(-I uniformly bounded along \Gamma. (b) Lemmas 7 and 8 apply also to the Arnoldi method, where and kVm (c) The convexity assumption about E can be removed at the price of a larger factor M . For E a continuum containing more than one point, one can use instead of inequality (2.13) the estimate in the lemma on pp. 107f. in Volume III of [19]. The proofs of Lemmas 1, 7, and 8 provide error bounds for iterative methods for the solution of linear systems of equations whose iterates are defined by a Galerkin condition (2.3). This gives new error bounds for the biconjugate gradient method, where the Krylov basis is constructed via the Lanczos process, and for the full orthogonalization method, which is based on the Arnoldi process. The proofs can be extended to give similar error bounds also for the (quasi-) minimization methods QMR and GMRES, see [13]. 5. A class of integration methods for large systems of ODEs. In the numerical integration of very large stiff systems of ordinary differential equations y f(y), Krylov subspace methods have been used successfully for the solution of the linear systems of equations arising in fully or linearly implicit integration schemes [11, 2, 25]. These linear systems are of the form A is the Jacobian of f evaluated near the current integration point, h is the step size, and fl is a method parameter. The attraction with Krylov subspace methods lies in the fact that they require only the computation of matrix-vector products Aw. When it is convenient, these can be approximated as directional derivatives Aw : that the Jacobian A need never be formed explicitly. Our theoretical results as well as computational experiments indicate that Krylov subspace approximations of e flhA v or '(flhA)v, with converge faster than the corresponding iterations for v, at least unless a good preconditioner is at hand. This suggests the use of the following class of integration schemes, in which the linear systems arising in a linearly implicit method of Rosenbrock-Wanner type are replaced by multiplication with '(flhA). Starting from y 0 - y(t 0 ), the scheme computes an approximation y 1 of s Here are the coefficients that determine the method. The internal stages are computed one after the other, with one multiplication by '(flhA) and a function evaluation at each stage. The simplest method of this type is the well-known exponentially fitted Euler method which is of order 2 and exact for linear differential equations y A and b. It appears well suited as a basis for Richardson extrapolation. Here is another example of such a method: Theorem 9. The two-stage methods with coefficients eter), are of order 3. For arbitrary step sizes, they provide the exact solution for every linear system of differential equations with constant matrix A and constant inhomogeneity b. Proof. Taylor expansion in h of the exact and the numerical solutions shows that the order conditions up to order 3, which correspond to the elementary differentials are given by Here all sums extend from 1 to s, and we have set ff Cf. with the order conditions for Rosenbrock methods in [12], p.116, which differ from the present order conditions only in the right-hand side polynomials in fl. For the right-hand side of the last order condition vanishes, and hence this condition is automatically satisfied for every two-stage method with With ff remaining three equations yield the stated method coefficients. Direct calculation shows that the method applied to y which is the claimed property. Remarks. (a) With 3=4, the method satisfies the order condition fi 2 ff 3 which corresponds to the fourth-order elementary differential f 000 (f; f; f ). The order conditions corresponding to f are satisfied independently of ff, so that the order condition corresponding to f 00 (f; f 0 f) is then the only fourth-order condition that remains violated. (b) For non-autonomous problems y it is useful to rewrite the equation in autonomous form by adding the trivial equation t taking the Jacobian e In particular, the method is then exact for every linear equation of the form y tc, since this is rewritten as y c A y which is again a linear system with constant inhomogeneity. An efficient implementation and higher-order methods are currently under investigation Note added in the revised version. After finishing this paper we learned that Druskin and Knizhnerman [3, 4] previously obtained an estimate similar to (3.5) for the symmetric case, using a different proof. They give the asymptotic estimate a '- a a 3 with which they prove using the Chebyshev series expansion of the exponential function. In an extension of this technique to the non-Hermitian case, Knizhner- man [14] derived error bounds in terms of Faber series for the Arnoldi method (2.7). He showed k=m are the Faber series coefficients of f and the exponent ff depends on the numerical range of A. As one referee emphasizes, the Faber series approach could be put to similar use as our Lemma 1. In fact, Leonid Knizhnerman showed to us in a personal communication how it would become possible to derive a result of the type of our Theorem 6 using (5.9). Our approach via Lemma 1 makes it more obvious to see how the geometry of the numerical range comes into play. An example similar to that after Theorem 5 is given in [15, x3]. We thank Anne Greenbaum and two referees for pointing out these references and Leonid Knizhnerman for providing a commented version of the Russian paper [14]. Error bounds via Chebyshev and Faber series, for the related problem of approximating matrix functions by methods that generalize semi-iterative methods for linear systems, were given by Tal-Ezer [29, 30, 31]. Acknowledgement . We are grateful to Peter Leinen and Harry Yserentant for providing the initial motivation for this work. --R Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras Two polynomial methods of calculating functions of symmetric matrices Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic Uniform Numerical Methods for Problems with Initial and Boundary Layers On semiiterative methods generated by Faber polynomials An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices Verallgemeinerte Runge-Kutta Verfahren zur L-osung steifer Differentialgleichungen A method of exponential propagation of large systems of stiff nonlinear differential equations Efficient solution of parabolic equations by Krylov approximation methods Iterative solution of linear equations in ODE codes Solving Ordinary Differential Equations II analysis of Krylov methods in a nutshell Computation of functions of unsymmetric matrices by means of Arnoldi's method bounds in Arnoldi's method: The case of a normal matrix Propagation methods for quantum molecular dynamics Generalized Runge-Kutta processes for stable systems with large Lipschitz con- stants Theory of Functions of a Complex Variable New approach to many-state quantum dynamics: The recursive- residue-generation method Convergence of Iterations for Linear Equations Unitary quantum time evolution by iterative Lanczos reduction Krylov subspace methods for solving large unsymmetric linear systems Analysis of some Krylov subspace approximations to the matrix exponential operator Analysis of the look-ahead Lanczos algorithm Spectral methods in time for hyperbolic problems Spectral methods in time for parabolic problems Polynomial approximation of functions of matrices and applications Pseudospectra of matrices An iterative solution method for solving f(A)x A convergence analysis for nonsymmetric Lanczos algorithms --TR --CTR P. Novati, An explicit one-step method for stiff problems, Computing, v.71 n.2, p.133-151, October Vladimir Druskin, Krylov Subspaces and Electromagnetic Oil Exploration, IEEE Computational Science & Engineering, v.5 n.1, p.10-12, January 1998 Ya Yan Lu, Computing a matrix function for exponential integrators, Journal of Computational and Applied Mathematics, v.161 n.1, p.203-216, 1 December C. Gonzlez , A. Ostermann , M. Thalhammer, A second-order Magnus-type integrator for nonautonomous parabolic problems, Journal of Computational and Applied Mathematics, v.189 n.1, p.142-156, 1 May 2006 Christian Lubich, A variational splitting integrator for quantum molecular dynamics, Applied Numerical Mathematics, v.48 n.3-4, p.355-368, March 2004 Marlis Hochbruck , Alexander Ostermann, Exponential Runge-Kutta methods for parabolic problems, Applied Numerical Mathematics, v.53 n.2, p.323-339, May 2005 S. Krogstad, Generalized integrating factor methods for stiff PDEs, Journal of Computational Physics, v.203 n.1, p.72-88, 10 February 2005 F. Carbonell , J. C. Jimenez , R. Biscay, A numerical method for the computation of the Lyapunov exponents of nonlinear ordinary differential equations, Applied Mathematics and Computation, v.131 n.1, p.21-37, 10 September 2002 Serhiy Kosinov , Stephane Marchand-Maillet , Igor Kozintsev , Carole Dulong , Thierry Pun, Dual diffusion model of spreading activation for content-based image retrieval, Proceedings of the 8th ACM international workshop on Multimedia information retrieval, October 26-27, 2006, Santa Barbara, California, USA Philip W. Livermore, An implementation of the exponential time differencing scheme to the magnetohydrodynamic equations in a spherical shell, Journal of Computational Physics, v.220 n.2, p.824-838, January, 2007 M. Caliari , M. Vianello , L. Bergamaschi, Interpolating discrete advection-diffusion propagators at Leja sequences, Journal of Computational and Applied Mathematics, v.172 n.1, p.79-99, 1 November 2004 Paolo Novati, A polynomial method based on Fejr points for the computation of functions of unsymmetric matrices, Applied Numerical Mathematics, v.44 n.1-2, p.201-224, January S. Koikari, An error analysis of the modified scaling and squaring method, Computers & Mathematics with Applications, v.53 n.8, p.1293-1305, April, 2007 Roger B. Sidje, Expokit: a software package for computing matrix exponentials, ACM Transactions on Mathematical Software (TOMS), v.24 n.1, p.130-156, March 1998 James V. Lambers, Practical Implementation of Krylov Subspace Spectral Methods, Journal of Scientific Computing, v.32 n.3, p.449-476, September 2007 Hvard Berland , Brd Skaflestad , Will M. Wright, EXPINT---A MATLAB package for exponential integrators, ACM Transactions on Mathematical Software (TOMS), v.33 n.1, p.4-es, March 2007 M. A. Botchev , D. Harutyunyan , J. J. W. van der Vegt, The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations, Journal of Computational Physics, v.216 August 2006 Elena Celledoni , Arieh Iserles , Syvert P. Nrsett , Bojan Orel, Complexity theory for lie-group solvers, Journal of Complexity, v.18 n.1, p.242-286, March 2002 M. Tokman, Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods, Journal of Computational Physics, v.213

matrix exponential function;matrix-free time integration methods;arnoldi method;superlinear convergence;lanczos method;conjugate gradient-type methods;krylov subspace methods

270932

Star Unfolding of a Polytope with Applications.

We introduce the notion of a star unfolding of the surface ${\cal P}$ of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on ${\cal P}$.The first algorithm computes the edge sequences traversed by shortest paths on ${\cal P}$ in time $O(n^6 \beta (n) \log n)$, where $\beta (n)$ is an extremely slowly growing function. A much simpler $O(n^6)$ time algorithm that finds a small superset of all such edge sequences is also sketched.The second algorithm is an $O(n^{8}\log n)$ time procedure for computing the geodesic diameter of ${\cal P}$: the maximum possible separation of two points on ${\cal P}$ with the distance measured along ${\cal P}$. Finally, we describe an algorithm that preprocesses ${\cal P}$ into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter $1 \le m \le n^2$, it can preprocess ${\cal P}$ in time $O(n^6 m^{1+\delta})$, for any $\delta > 0$, into a data structure of size $O(n^6m^{1+\delta})$, so that a query can be answered in time $O((\sqrt{n}/m^{1/4}) \log n)$. If one query point always lies on an edge of ${\cal P}$, the algorithm can be improved to use $O(n^5 m^{1+\delta})$ preprocessing time and storage and guarantee $O((n/m)^{1/3} \log n)$ query time for any choice of $m$ between 1 and $n$.

Introduction The problem of computing shortest paths in Euclidean space amidst polyhedral obstacles arises in planning optimal collision-free paths for a given robot, and has been widely studied. In two dimensions, the problem is easy to solve and a number of efficient algorithms have been developed, see e.g. [SS86, Wel85, Mit93]. However, the problem becomes significantly harder in three dimen- sions. Canny and Reif [CR87] have shown it to be NP-hard, and the fastest available algorithm runs in singly-exponential time [RS89, Sha87]. This has motivated researchers to develop efficient approximation algorithms [Pap85, Cla87] and to study interesting special cases [MMP87, Sha87]. An earlier version of this paper was presented at the Second Scandinavian Workshop on Algorithm Theory [AAOS90]. Part of the work was carried out when the first two authors were at Courant Institute of Mathematical Sciences, New York University and later at Dimacs (Center for Discrete Mathematics and Theoretical Computer Sci- ence), a National Science Foundation Science and Technology Center - NSF-STC88-09648, and the fourth author was at the Department of Computer Science, The Johns Hopkins University. The work of the first author is supported by National Science Foundation Grant CCR-91-06514. The work of the second author was also partially supported by an AT&T Bell Laboratories Ph.D. Scholarship and NSF Grant CCR-92-11541. The third author's work is supported by NSF grants CCR-88-2194 and CCR-91-22169. y Computer Science Department, Duke University, Durham, NC 27706 USA z Computer Science Department, Polytechnic University, Brooklyn, NY 11201 USA x Department of Computer Science, Smith College, Northampton, MA 01063 USA - Rm. 3B-412, AT&T Bell Laboratories, 600 Mountain Ave., P.O. Box 636, Murray Hill, NJ 07974 USA One of the most widely studied special case is computing shortest paths along the surface of a convex polytope [SS86, MMP87, Mou90]; this problem was originally formulated by H. Dudeney in 1903; see [Gar61, p. 36]. Sharir and Schorr presented an O(n 3 log n) algorithm for this problem, which was subsequently improved by Mitchell et al. [MMP87] to O(n 2 log n), and then by Chen and Han to O(n 2 ) [CH90]. In this paper we consider three problems involving shortest paths on the surface P of a convex polytope in IR 3 . A shortest path on P is identified uniquely by its endpoints and the sequence of edges that it encounters. Sharir [Sha87] proved that no more than O(n 7 ) distinct sequences of edges are actually traversed by the shortest paths on P . This bound was subsequently improved to \Theta(n 4 ) [Mou85, SO88]. Sharir also gave an O(n 8 log n) time algorithm to compute an O(n 7 superset of shortest-path edge sequences. However computing the exact set of shortest-path edge sequences seems to be very difficult. Schevon and O'Rourke [SO89] presented an algorithm that computes the exact set of all shortest-path edge sequences and also identifies, in logarithmic time, the edge sequences traversed by all shortest paths connecting a given pair of query points lying on edges of P . The sequences can be explicitly generated, if necessary, in time proportional to their length. Their algorithm, however, requires O(n 9 log n) time and O(n 8 In this paper we propose two edge-sequence algorithms. The first is a very simple O(n 6 ) algorithm to compute a superset of shortest-path edge sequences, thus improving the result of [Sha87]; it is described in Section 5. The second computes the exact set of shortest-path edge sequences in O(n 6 fi(n) log n) time, where fi(:) is an extremely slowly-growing function, a significant improvement over the previously mentioned O(n 9 log n) algorithm. The computation of the collection of all shortest-path edge sequences on a polytope is an intermediate step of several algorithms [Sha87, OS89], and is of interest in its own right. The second problem studied in this paper is that of computing the geodesic diameter of P , i.e., the maximum distance along P between any two points on P . O'Rourke and Schevon [OS89] gave an O(n 14 log n) time procedure for determining the geodesic diameter of P . In [AAOS90], we presented a simpler and faster algorithm whose running time is O(n 10 ). An even faster O(n 8 log n) algorithm is presented in the current version of the paper. The third problem involves answering queries of the form: "Given x; y 2 P , determine the distance between x and y along P ." Given a parameter n 2 - s - n 4 , we present a method for preprocessing P , in O(n 4 s 1+ffi ) time, into a data structure of size O(n 4 s 1+ffi ) for any ffi ? 0, so that a query can be answered in time O( n s 1=4 log 2 n). If x is known to lie on an edge of the polytope, the preprocessing and storage requirements are reduced to O(n 3 s 1+ffi ) and query time becomes O( n s 1=3 log 2 n) for Our algorithms are based on a common geometric concept, the star unfolding. Intuitively, the star unfolding of P with respect to a point x 2 P can be viewed as follows. Suppose there exists a unique shortest path from x to every vertex of P . The object obtained after removing these n paths from P is the star unfolding of P . Remarkably, the resulting set is isometric to a simple planar polygon and the structure of shortest paths emanating from x on P corresponds to a certain Voronoi diagram in the plane [AO92]. Together with relative stability of the combinatorial structure of the unfolding as x moves within a small neighborhood on P , these properties facilitate the construction of efficient algorithms for the above three problems. preliminary version of this algorithm [SO88] erroneously claimed a time complexity of O(n 7 2 ff(n) log n); this claim was corrected in [SO89]. Hwang et al. [HCT89] claimed to have a more efficient procedure for solving the same problem, but their argument as stated is flawed, and no corrected version has appeared in the literature. Chen and Han [CH90] have independently discovered the star unfolding and used it for computing the shortest-path information from a single fixed point on the surface of a polytope. They however take the unfoldability proven by Aronov and O'Rourke [AO92] for granted. This paper is organized as follows. In Section 2, we formalize our terminology and list some basic properties of shortest paths. Section 3 defines the star unfolding and establishes some of its properties. Section 4 sketches an efficient algorithm to compute a superset of all possible shortest-path edge sequences, and in Section 5 we present an algorithm for computing the exact set of sequences; both algorithms are based on the star unfolding. In Section 6 we again use the notion of star unfolding to obtain a faster algorithm for determining the geodesic diameter of a convex polytope. Section 7 deals with shortest-path queries. Section 8 contains some concluding remarks and open problems. We begin by reviewing the geometry of shortest paths on convex polytopes. Let P be the surface of a polytope with n vertices. We refer to vertices of P as corners; the unqualified terms face and edge are reserved for faces and edges of P . We assume that P is triangulated. This does not change the number of faces and edges of P by more than a multiplicative constant, but simplifies the description of our algorithms. 2.1 Geodesics and Shortest Paths A path - on P that cannot be shortened by a local change at any point in its relative interior is referred to as a geodesic. Equivalently, a geodesic on the surface of a convex polytope is either a subsegment of an edge or a path that (1) does not pass through corners, though may possibly terminate at them, (2) is straight near any point in the interior of a face and (3) is transverse to every edge it meets in such a fashion that it would appear straight if one were to "unfold" the two faces incident on this edge until they lie in a common plane; see, for example, Sharir and Schorr [SS86]. The behavior of a geodesic is thus fully determined by its starting point and initial direction. In the following discussion we disregard the geodesics lying completely within a single edge of P . Given the sequence of edges a geodesic traverses (i.e., meets) and its starting and ending points, the geodesic itself can be obtained by laying the faces that it visits out in the plane, so that adjacent faces share an edge and lie on opposite sides of it, and then connecting the (images of) the two endpoints by a straight-line segment. In particular, the sequence of traversed edges together with the endpoints completely determine the geodesic. Trivially every shortest path along P is a geodesic and no shortest path meets a face or an edge more than once. We call the length of a shortest path between two points the geodesic distance between p and q, and denote it by d(p; q). The following additional properties of shortest paths are crucial for our analysis. Lemma 2.1 (Sharir and Schorr [SS86]) Let - 1 and - 2 be distinct shortest paths emanating from x. Let y be a point distinct from x. Then either one of the paths is a subpath of the other, or neither - 1 nor - 2 can be extended past y while remaining a shortest path. 2 Corollary 2.2 Two shortest paths cross at most once. 2 Lemma 2.3 If - are two distinct shortest paths connecting x; y 2 P, each of the two connected components of P contains a corner. Proof: First, Lemma 2.1 implies that removal of splits P into exactly two components. If one of the two components of contained no corners, - 1 and - 2 would have to traverse the same sequence of edges and faces. However, there exists at most one geodesic connecting a given pair of points and traversing a given sequence of edges and faces. 2 2.2 Edge Sequences and Sequence Trees A shortest-path edge sequence is the sequence of edges intersected by some shortest path - connecting two points on P , in the order met by -. Such a sequence is maximal if it cannot be extended in either direction while remaining a shortest-path edge sequence; it is half-maximal if no extension is possible at one of the two ends. It has been shown by Schevon and O'Rourke [SO88] that the maximum total number of half-maximal sequences is \Theta(n 3 ). Observe that every shortest-path edge sequence oe is a prefix of some half-maximal sequence, namely the one obtained by extending oe maximally at one end. Thus an exhaustive list of O(n 3 ) half-maximal sequences contains, in a sense, all the shortest-path edge-sequence information of P . More formally, given an arbitrary collection of edge sequences emanating from a fixed edge e, let the sequence tree \Sigma of this set be the tree with all distinct non-empty prefixes of the given sequences as nodes, the trivial sequence consisting solely of e as the root, and such that oe is an ancestor of oe 0 in the tree if and only if oe is prefix of oe 0 [HCT89]. The \Theta(n 3 ) bound on the number of half-maximal sequences implies that the collection of O(n) sequence trees obtained by considering all shortest-path edge sequences from each edge of P in turn, has a total of \Theta(n 3 ) leaves and \Theta(n 4 ) nodes in the worst case. 2.3 Ridge Trees and the Source Unfolding The shortest paths emanating from a fixed source x 2 P cover the surface of P in a way that can be naturally represented by "unfolding" the paths to a planar layout around x. This unfolding, the "source unfolding," has been studied since the turn of the century. We will define it precisely in a moment. A second way to organize the paths in the plane is the "star unfolding," to be defined in Section 3. This is not quite as natural, and of more recent lineage. Our algorithms will be built around the star unfolding, but some of the arguments do refer to the source unfolding as well. Given two points x; y on P , y 2 P is a ridge point with respect to x if there is more than one shortest path between x and y. Ridge points with respect to x form a ridge tree T x embedded on P , 2 whose leaves are corners of P , and whose internal vertices have degree at least three and correspond to points of P with three or more distinct shortest paths to x. In a degenerate situation where x happens to lie on the ridge tree for some corner p, then p will not be a leaf of T x , but rather lie internal to T x ; so in general not all corners will appear as leaves of T x . We define a ridge as a maximal subset of T x consisting of points with exactly two distinct shortest paths to x, and containing no corners of P . These are the "edges" of T x . Ridges are open geodesics [SS86]; a stronger characterization of ridges is given in Lemma 2.4. Figs. 1 and 2 show two views of a ridge tree on a pyramid. 2 For smooth surfaces (Riemannian manifolds), the ridge tree is known as the "cut locus" [Kob67]. A x Figure 1: Pyramid, front view: source x, shortest paths to five vertices (solid), ridge tree (dashed). Coordinates of vertices are (\Sigma1; \Sigma1; A Figure 2: Pyramid, side view of Fig. 1. The ridges incident to p 2 and p 4 lie nearly on the edge We will refer to a point y 2 P as a generic point if it is not a ridge point with respect to any corner of P . The maximal connected portion of a face (resp. an edge) of P consisting entirely of generic points will be called a ridge-free region (resp. an edgelet). x Figure 3: Ridge-free regions for Fig. 1. T p 1 are shown dashed (e.g., T p 3 is the 'X' on the bottom face). The ridge-free region containing x is shaded darker. If we cut P along the ridge tree T x and isometrically embed the resulting set in IR 2 , we obtain the source unfolding of [OS89]. 3 In the source unfolding, the ridges lie on the boundary of the unfolding, while x lies at its "center," which results in a star-shaped polygon [SS86]; see Fig. 4. Let a peel be the closure of a connected component of the set obtained by removing from P both the ridge tree T x and the shortest paths from x to all corners. A peel is isometric to a convex polygon [SS86]. Each peel's boundary consists of x, the shortest paths to two corners of P , p and p 0 , and the unique path in T x connecting p to p 0 . A peel can be thought of as the collection of all the shortest paths emanating from x between xp and xp 0 . (The peel between xp 1 and xp 5 is shaded in Fig. 4.) We need to strengthen the characterization of ridges from geodesics to shortest paths, in order to exploit Corollary 2.2. This characterization seems to be new. Lemma 2.4 Every ridge of the ridge tree T x , for any point x 2 P, is a shortest path. Proof: An edge - of the ridge tree is a geodesic consisting of points that have two different shortest paths to x [SS86]. Suppose - is not a shortest path. Then there must be two points a; b 2 - so that the portion - 0 of - between a and b is a shortest path, but there is another shortest path, say connecting them. Refer to Figure 5. By Lemma 2.1, - bg. Let ff 1 and ff 2 be the two shortest paths from x to a, and fi 1 and fi 2 be the two shortest paths from x to b. Notice that - 0 , do not meet except at the endpoints, by Lemma 2.1. In particular, we can relabel these paths so that the region bounded by (ff as illustrated in the figure, so that ff 1 and fi 1 approach - 0 "from the same side." There are two cases to consider: 3 The same object is called U(P) in [SS86], "planar layout" in [Mou85], and "outward layout" in [CH90]. For Riemannian manifolds, it is the "exponential map" [Kob67]. x Figure 4: Source unfolding for the example in Figs. 1 and 2. Shortest paths to vertices (solid), polytope edges (dashed), ridges (dotted). One peel is shaded. x a 1 a 2 a Figure 5: Illustration of the proof of Lemma 2.4. Here x is the source, and - a geodesic ridge, with shortest path. The region \Delta cannot contain any vertices of P . Case 1: x 62 - 00 . Thus - 00 does not meet ff 1 or ff 2 except at a, by Lemma 2.1. Similarly, - 00 does not meet fi 1 or fi 2 except at b. Thus, without loss of generality, we can assume that - 00 lies in the portion \Delta of P bounded by - 0 , ff 1 , and fi 1 , and not containing ff 2 or fi 2 . Now, considering the source unfolding from x, we observe that \Delta is (isometric to) a triangle contained within a single peel. \Delta is the area swept by one class of shortest paths from x to y as y ranges over In particular, \Delta contains no corners. On the other hand, paths are distinct shortest paths connecting a to b, so by Lemma 2.3 each of the two sets obtained from P by removing - 0 [- 00 has to contain a corner of P . However, one of these sets is entirely contained in \Delta-a contradiction. Case 2: x 2 - 00 . As - 00 and ff 1 can be viewed as emanating from a and having x in common, and extends past x, Lemma 2.1 implies that ff 1 is a prefix of - 00 . Similarly, ff 2 is a prefix of contradicting distinctness of ff 1 and ff 2 .Remark. Case 2 in the above proof is vacuous if x is a corner, which is the case in our applications of this lemma. As defined, T x is a tree with n vertices of degree less than 3 and thus has \Theta(n) vertices and edges. However, the worst-case combinatorial size of T x jumps from \Theta(n) to \Theta(n 2 ) if one takes into account the fact that a ridge is a shortest path comprised of as many as \Theta(n) line segments on P in the worst case-and it is possible to exhibit a ridge tree for which the number of ridge-edge incidences is For simplicity we assume that ridges intersect each edge of P transversely. 3 Star Unfolding In this section we introduce the notion of the star unfolding of P and describe its geometric and combinatorial properties. Working independently, both Chen and Han [CH90] [CH91] and Rasch [Ras90] have used the same notion, and in fact the idea can be found in Aleksandrov's work [Ale58, p.226] [AZ67, p.171]. 3.1 Geometry of the Star Unfolding be a generic point, so that there is a unique shortest path connecting x to each corner of P . These paths are called cuts and are comprised of cut points (see Fig. 1). If P is cut open along these cuts and embedded isometrically in the plane, then just as with the source unfolding, the result is a non-self-overlapping simple polygonal region, a two-dimensional complex that we call the star unfolding S x . That the star unfolding avoids overlap is by no means a straightforward claim; it was first established in [AO92]: Lemma 3.1 (Aronov and O'Rourke [AO92]) If viewed as a metric space with the natural definition of interior metric, S x is isometric to a simple polygon in the plane (with the internal geodesic metric). The polygonal boundary @S x consists entirely of edges originating from cuts. The vertices of S x derive from the corners of P and from the source x. An example is shown in Fig. 6. More complex A Figure Construction of the star unfolding corresponding to Figs. 1 and 2. S x is shaded. The superimposed dashed edges show the "natural" unfolding obtained by cutting along the four edges incident to p 3 . The A, B, C, D, and E labels indicate portions of S x derived from those faces; the relative neighborhood of each x i derives from A. examples will be shown in Fig. 10. The cuts partition the faces of P into subfaces, which map to what we call the plates of S x , each a compact convex polygon with constant number of edges. See Fig. 7. We consider these plates Figure 7: Plates corresponding to Fig. 6. The square base E is partitioned into two triangles. to be the faces of the two-dimensional complex S x . We assume that the complex carries with it labeling information consistent with P . Somewhat abusing the notation, we will freely switch between viewing S x as a complex and as a simple polygon embedded in the plane. In particular, a path - ae S x will be referred to as a segment if it corresponds to a straight-line segment in the planar embedding of S x . Note that every segment in S x is a shortest path in the intrinsic metric of the complex, but not every shortest path in S x is a segment, as some shortest paths in S x might bend at corners. For U(p) be the set of points in S x to which p maps; U is the "unfolding" map (with respect to x). U(p) is a single point whenever p is not a cut point. is a set of n distinct points in S x . A non-corner point y 2 P distinct from x and lying on a cut has exactly two unfolded images in S x . The corners of P map to single points. In particular, we have: Lemma 3.2 (Sharir and Schorr [SS86]) For a point y 2 P, any shortest path - from x to y maps to a segment - ae S x connecting an element of U(y) to an element of U(x). 2 There is a view of S x that relates it to the source unfolding: the star unfolding is just an "inside- out" version of the source unfolding, in the following sense. The star unfolding can be obtained by stitching peels together along ridges; see Fig. 8. The source unfolding is obtained by gluing them along the cuts. Compare with Fig. 4. peel was defined in Section 2.3 as a subset of P , but slightly abusing the terminology we also use this term to refer to the corresponding set of points in the source or star unfolding.) We next define the pasting tree \Pi x as the graph whose nodes are the plates of S x , with two nodes connected by an arc if the corresponding plates share an edge in S x . See Fig. 9. For a generic Figure 8: Ridge tree T x ae S x . point x, \Pi x is a tree with O(n 2 ) nodes, as it is the dual of a convex partition of a simple polygon without Steiner points. (If x were a ridge point of some corner, S x would not be connected and \Pi x would be a forest.) \Pi x has only n leaves, corresponding to the triangular plates incident to the n images of x in S x or, equivalently, to x in P x . By Lemma 3.2, any shortest path from x to corresponds to a simple path in \Pi x , originating at one of the leaves. Thus, the O(n 3 ) edge sequences corresponding to the simple paths that originate from leaves of \Pi x include all shortest path edge sequences emanating from x. In fact, there are O(n 2 ) maximal edge sequences in this set, one for each pair of leaves. In the following sections we will need the concept of the "kernel" of a star unfolding. Number the corners in the order in which cuts emanate from x. Number the n source images (elements of so that @S is the cycle x comprised of 2n segments (see Fig. 6). The kernel is a subset of S x , but to define it it would be most convenient to view S x as embedded in the plane as a simple polygon. Consider the polygonal cycle We claim that it is the boundary of a simple polygon fully contained in (the embedding of) S x . Indeed, each line segment 4 is fully contained in the peel sandwiched between x Thus the line segments p i p i+1 are segments in S x in the sense defined above and indeed form a simple cycle. The simple n-gon bounded by this cycle is referred to as the kernel K x of the star unfolding S x . An equivalent way of defining K x is by removing from S x all triangles 4p Fig. 10 illustrates the star unfolding, and its kernel for several randomly generated polytopes. 5 Note that neither set is necessarily star-shaped. We extend the definition of the map U to sets in the natural way by putting q2Q U(q). The main property of the kernel that we will need later is: 4 Here are thereafter 5 The unfoldings were produced with code written by Julie DiBiase and Stacia Wyman of Smith College. Figure 9: Pasting tree \Pi x for Fig. 7: one node per plate. Lemma 3.3 The image of the ridge tree is completely contained within the kernel, which is itself a subset of the star unfolding: U(T x Proof: Since K x can be defined by subtraction from S x , K x ae S x is immediate. The ridge tree T x can be thought of as the union of the peel boundaries that do not come from cuts. These boundaries remained when we removed triangles 4p x to form K x (see Fig. 8). 2 Recently Aronov and O'Rourke [AO92] proved that Theorem 3.4 (Aronov and O'Rourke [AO92]) U(T x ) is exactly the restriction of the planar Voronoi diagram of the set source images to within K x or, equivalently, to within S x .3.2 Structure of the Star Unfolding We now describe the combinatorial structure of S x . A vertex of S x is an image of x, of a corner of P , or of an intersection of an edge of P with a cut. An edge of S x is a maximal portion of an image of a cut or an edge of P delimited by vertices of S x . It is easy to see that S x consists of \Theta(n 2 ) plates in the worst case, even though its boundary is formed by only 2n segments, the images of the cuts. We define the combinatorial structure of S x as the 1-skeleton of S x , i.e., the graph whose nodes and arcs are the vertices and edges of S x , labeled in correspondence with the labels of S x , which are in turn derived from labels on P . The combinatorial structure of a star unfolding has the following crucial property: Lemma 3.5 Let x and y be two non-corner points lying in the same ridge-free region or on the same edgelet. Then S x and S y have the same combinatorial structure. Figure 10: Four star unfoldings: vertices, left to right, top to bottom. The kernel is shaded darker in each figure. Proof: Let f be the face containing xy in its interior. The case when xy is part of an edge is similar. As the shortest paths from any point z 2 xy to the corners are pairwise disjoint except at z (cf. Lemma 2.1) and z is confined to f , the combinatorial structure of S z is uniquely determined by (1) the circular order of the cuts around z, and (2) the sequence of edges and faces of P met by each of the cuts. We will show that (1) and (2) are invariants of S z as long as z does not cross a ridge or an edge of P . First, the set of points of f , for which some shortest path to a fixed corner traverses a fixed edge sequence, is convex-it is simply the intersection of f with the appropriate peel of the source unfolding with respect to p-implying invariance of (2). Now suppose the circular order of the cuts around z is not the same for all z 2 xy. The initial portions of the cuts, as they emanate from any z, cannot coincide, as distinct cuts are disjoint except at z. Hence there can be a change in this circular order only if one of the vectors pointing along the initial portion of the cuts changes discontinuously at some intermediate point z However, this can only happen if z 0 is a ridge point, a contradiction. 2 This lemma holds under more general conditions. Namely, instead of requiring that xy be free of ridge points, it is sufficient to assume that the number of distinct shortest paths connecting z to any corner does not change as z varies along xy. Lemma 3.6 Under the assumptions of Lemma 3.5, K x is isometric to K y , i.e., they are congruent simple polygons. Proof: K x is determined by the order of corners on @K for each i, by the choice of the shortest path p i p i+1 , if there are two or more such paths. The ordering is fixed, once combinatorial structure of S x is determined. The choice of the shortest path connecting p i to p i+1 is determined by the constraint that 4p Let R be a ridge-free region. By the above lemma, S x can be embedded in the plane in such a way that the images of the corners of P are fixed for all x 2 R, and the images of x in S x move as x varies in R ' P . This guarantees that K This is illustrated in Fig. 11. In what follows, we are going to assume such an embedding of S x , and use KR to denote K x for all points x 2 R. Similarly, define K " for an edgelet ". 3.3 How Many Different Unfoldings Are There? For the algorithms described in this paper, it will be important to bound the number of different possible combinatorial structures of star unfoldings, as we vary the position of source point x, and to efficiently compute these unfoldings (more precisely, compute their combinatorial structure plus some metric description, parametrized by the exact position of the source), as the source moves on the surface of the polytope. Two variants of this problem will be needed. In the first, we assume that the source is placed on an edge of P , and in the second the source is placed anywhere on P . In view of Lemma 3.5, it suffices to bound the number of edgelets and ridge-free regions. Lemma 3.7 In the worst case, there are \Theta(n 3 ) edgelets and they can be computed in O(n 3 log n) time. Proof: Each edge can meet a ridge of the ridge tree of a corner at most once, since ridges are shortest paths (recall that we assume that no ridge overlaps an edge-removal of this assumption does not invalidate our argument, but only adds a number of technical complications). This gives Figure 11: The star unfolding when the source x moves to y inside a ridge-free region. The unfolding shown lightly shaded; S y is shown dotted. Their common kernel K y is the central dark region. an upper bound of n \Theta O(n) \Theta O(n) on the number of edge-ridge intersections and therefore on the number of edgelets. An example are present is relatively easy to construct by modifying the lower bound construction of Mount [Mou90]. To compute the edgelets, we construct ridge trees from every corner in n \Theta O(n 2 applications of the algorithm of Chen and Han [CH90]. The edgelets are now computed by sorting the intersections of ridges along each edge. 2 Lemma 3.8 In the worst case, there are \Theta(n 4 ) ridge-free regions on P. They can be computed in Proof: The overlay of n ridge trees, one from each corner of P , produces a subdivision of P in which every region is bounded by at least three edges. Thus, by Euler's formula, the number of regions in this subdivision is proportional to the number of its vertices, which we proceed to estimate. By Lemma 2.4 ridges are shortest paths and therefore two of them intersect in at most two points (cf. Corollary 2.2) or overlap. In the latter case no new vertex of the subdivision is created, so we restrict our attention to the former. In particular, as there are n \Theta ridges, the total number of their intersection points is O(n 4 ). Refining this partition further by adding the edges of P does not affect the asymptotic complexity of the partition, as ridges intersect edges in O(n 3 ) points altogether. This establishes the upper bound. It is easily checked that in Mount's example of a polytope there regions. Hence there are \Theta(n 4 ) ridge-free regions on P in the worst case. The ridge-free regions can be computed by calculating the ridge tree for every corner, and overlaying the trees in each face of P . The first step takes O(n 3 ) time, while the second step can be accomplished in time O((r r is the number of ridge-free regions in P , using the line-sweep algorithm of Bentley and Ottmann [BO79]. If computing the ridge-free regions is a bottleneck, the last step can be improved to O(n 4 ) by using a significantly more complicated algorithm of Chazelle and Edelsbrunner [CE92]. 2 3.4 How Many Different Ridge Trees Are There? In Section 3.1, we proved that the combinatorial structure of S x is the same for all points x in a ridge-free region. As x moves in a ridge-free region, the ridge tree T x changes continuously, as a subset of P . In this subsection, we prove an upper bound on the number of different combinatorial structures of T x as the source point x varies over a ridge-free region or an edgelet. Apart from being interesting in their own right, we need these results in the algorithms described in Sections 5-7. Let R be a ridge-free region, and let x be a point in R. By Theorem 3.4, T x is the Voronoi diagram V x of clipped to lie within KR . Since ridge vertices do not lie on @S x , all changes in T x , as x varies in R, can be attributed to changes in V x . Thus it suffices to count the number of different combinatorial structures of Voronoi diagrams V x , x 2 R. g. For each x are the coordinates of a generic point y in the plane. Let the lower envelope of the f i 's. Then V x is the same as (the 1-skeleton of) the orthogonal projection of the graph of f(y) onto the y 1 y 2 -plane. We introduce an orthogonal coordinate system in R and let x have coordinates (s; t) in this system. Then positions of x i are linear functions of s; t of the form x i2 ' a i where (a are coordinates of x i when x is at the origin of the (s; t) coordinate system in R, and defines the orientation of the ith image of R in the plane. We now regard f and f i 's as 4-variate functions of s; t; y by MR the projection of the graph of f onto the (s; t; y It can be shown that total number of different combinatorial structures of V x is bounded by the number of faces in MR . Let Using (1) we obtain where, for each i, the C i 's are constants that depend solely on a denote the lower envelope of the g i 's. Since g(s; t; y the projection of the graph of g is the same as MR . Let and set Then every face of the graph of g is the intersection of the lower envelope of - 's with the surface defined by the equations (3). Since each - g i is a 8-variate linear function, by the Upper Bound Theorem for convex polyhedra, the graph of their lower envelope has O(n 4 ) faces. Hence, the number of faces in MR is also O(n 4 ). Using the algorithm of Chazelle [C91], all the faces of this lower envelope and thus also those of MR can be computed in O(n 4 ) time. Putting everything together, we conclude Lemma 3.9 The total number of different combinatorial structures of ridge trees for source points lying in a ridge-free region R is O(n 4 ). Moreover, the faces of MR can be computed in time O(n 4 ). Remark. The only reason for assuming in the above analysis that x stays away from the boundary of R was to ensure that the vertices of the Voronoi diagram avoid the boundary of KR . However, it is easy to verify that when x is allowed to vary over the closure of R, Voronoi vertices never cross the boundary of KR , but may touch it in limiting configurations. Thus the same analysis applies in that case as well. An immediate corollary of the above lemma and Lemma 3.8 is Corollary 3.10 The total number of different combinatorial structures of ridge trees for a convex polytope in IR 3 with n vertices is O(n 8 ). If the source point moves along an edgelet " rather in a ridge-free region, we can obtain a bound on the number of different combinatorial structures of ridge trees by setting Proceeding in the same way as above, each g i now becomes We now define the subdivision M " as the projection of the graph of the lower envelope of g i 's. Let i is now a 5-variate linear function, by the Upper Bound Theorem, the number of faces in " is O(n 3 ), which implies that there are O(n 3 ) distinct combinatorial structures of ridge trees as x varies along ". Moreover, M " can be computed in time O(n 3 ) [C91]. Hence, we obtain Lemma 3.11 The total number of distinct ridge trees as the source point moves in an edgelet is O(n 3 ), and the subdivision M " can be computed in O(n 3 ) time. Remarks. (i) None of Lemma 3.9, Corollary 3.10, or Lemma 3.11 are known to be tight. (ii) In the above analysis for MR (resp. M " ) the only portion of the structure that is relevant for our algorithms is that which corresponds to points with (s; We will have to "filter out" irrelevant features at a later stage in the computation. 4 Edge Sequences Superset In this section we describe an O(n 6 ) algorithm for constructing a superset of the shortest-path edge sequences, which is both more efficient and conceptually simpler than previously suggested procedures, and which produces a smaller set of sequences. Observe that all shortest-path edge sequences are realized by pairs of points lying on edges of P-any other shortest path can be contracted without affecting its edge sequence so that its endpoints lie on edges of P . Let x be a generic point lying on an edgelet ". As mentioned in Section 3.1, the pasting tree \Pi x contains all shortest path edge sequences that emanate from x. Moreover by Lemma 3.5, \Pi x is independent of choice of x in ". Therefore, the set of O(n 3 ) pasting trees is an edgeletg, each of size O(n 2 ), contains an implicit representation of a set of O(n 6 ) sequences (O(n 5 ) of which are maximal in this set), which includes all shortest-path edge sequences that emanate from generic points. Algorithm 1: Sequence Trees for each edge e of P do for each edgelet endpoint v 2 e do Compute shortest-path edge sequences \Sigma v emanating from v. for each edgelet " ae e do Choose a point x 2 ". Compute for each maximal sequence oe 2 \Pi x do for each sequence oe 2 \Sigma e do Traverse oe, augmenting T e . Stop if oe visits the same edge twice. T e is the sequence tree containing shortest path edge sequences emanating from e. Hence, we can compute a superset of shortest path edge sequences in three steps: First, partition each edge of P into O(n 3 ) edgelets in time O(n 3 ) as described in Lemma 3.7. Second, compute shortest path edge sequences from the endpoints of each edgelet, using Chen and Han's shortest path algorithm. Next, compute the star unfolding from a point in each edgelet, again using the shortest path algorithm. The total time spent in the last two steps is O(n 5 ). Finally, this representation of edge sequences is transformed into O(n) sequence trees, one for each edge (cf. Section 2.2) in time O(n 6 ) in a straightforward manner; see Algorithm 1 for the pseudocode. We thus obtain Theorem 4.1 Given a convex polytope in IR 3 with n vertices, one can construct, in time O(n 6 ), O(n) sequence trees that store a set of O(n 6 ) edge sequences, which include all shortest path edge sequences of P. 2 Remark. (i) Note that our algorithm uses nothing more complex than the algorithm of Chen and Han for computing shortest paths from a fixed point, plus some sorting and tree traversals. It achieves an improvement over previous algorithms mainly by reorganizing the computation around the star unfolding. (ii) The sequence-tree representation for just the shortest-path edge sequences is smaller by a factor of n 2 than our estimate on the size of the set produced by Algorithm 1 (cf. Section 2.2), but computing it efficiently seems difficult. In addition, it is not clear how far the actual output of our algorithm is from the set of all shortest-path edge sequences. We have a sketch of a construction for a class of polytopes that force our algorithm to sequences. 5 Exact Set of Shortest-Path Edge Sequences In this section, we present an O(n 3 fi(n) log n) algorithm, for computing the exact set of maximal shortest-path edge sequences emanating from an edgelet. Here fi(\Delta) is an extremely slowly growing asymptotically smaller than log n. Running this algorithm for all edgelets of P , the exact set of maximal shortest-path edge sequences can be computed in time O(n 6 fi(n) log n), which is a significant improvement over Schevon and O'Rourke's O(n 9 log n) algorithm [SO89]. " be an edgelet. We are interested in computing maximal shortest-path edge sequences (corresponding to paths) emanating from x, for all x 2 ". For each fixed x, the shortest paths originating from x can be subdivided according to their initial direction. If the path leaves x between xp i\Gamma1 and xp i , it corresponds to a segment in S x emanating from x i , the ith image of x. The area swept by all such segments, for a fixed x and i, is exactly the ith peel. Let us concentrate on the portion P x;i of the ith peel that lies in K One measure of how far the 'influence' of extends into K " , as x moves along ", is the union C x2" P x;i . C i is the union of a family of convex sets P x;i sharing p as a boundary edge, therefore C i is star-shaped with respect to any point z 2 p It is easily checked that the restriction of a plate of S x to not vary with x 2 ". It therefore makes sense to talk about the nodes of \Pi x visited by fl i . We say that visits a node v of \Pi x if fl i intersects the plate corresponding to v. Observe that every maximal (over all x in ") edge sequence corresponding to a shortest path emanating from x between xp i\Gamma1 and xp i is realized by some shortest path that connects x i to some point y on in fact to a ridge vertex of P x;i . This sequence is determined solely by the plate of \Pi x that contains y. Hence, it is sufficient to determine the furthest that fl i gets from x i in \Pi x . More formally, consider the minimal subtree \Pi x;i of \Pi x rooted at (the plate incident to) x i and containing all nodes of \Pi x visited by fl i . The paths from the root of \Pi x;i to its leaves correspond to desired maximal sequences. Repeating this procedure for all images x i , we can collect all maximal sequences corresponding to shortest paths from points on ". The above idea can be made algorithmic, but transforming it directly to an algorithm requires computing the sets C i (i.e., taking the union of a continuous family of convex sets P x;i , each obtained by deleting 4p from the ith peel), which is rather intricate. We therefore use a shortcut, replacing fl i by an easier-to-compute curve fl 0 . Since the desired maximal sequences are necessarily realized by shortest paths from x i to one of the ridge vertices lying on @P x;i , we only need to consider the portion of fl i that contains a vertex of T x , for some x 2 ". We now show how to compute a curve fl 0 i that contains this portion of fl i . A generic ridge vertex v is incident to three open peels c has degree more than four and exists at more than just a discrete set of positions of x 2 ", we arbitrarily pick a triple of open peels to define it. As x moves along ", the vertex traces an algebraic curve in K x . Let a lifetime of a ridge vertex v, defined by c , be a maximal connected interval implies that v is a vertex of T x . Let \Gamma i be the set of curves traced out by ridge vertices that appear on the boundary of P x;i during their lifetimes; set n j. It can be verified that the arcs in \Gamma i are the projections of those edges of the subdivision M " , defined in Section 3.4, at which g i appears on the lower envelope of g i 's. (As we mentioned in Section 3.4, may contain "irrelevant" features. In particular, we must truncate each aforementioned arc so that it corresponds to positions of the source on ". Secondly, we must verify that the Voronoi diagram vertex corresponding to the arc indeed yields a ridge vertex. It is sufficient to check, for a single point of the curve traced out by the vertex as x ranges over ", that it lies inside K " , as a ridge vertex cannot leave K " . This is easily accomplished by one point-location query per arc.) Therefore, by Lemma 3.11, summation is taken over all peels of S x , and all can be computed in time O(n 3 ). Let z be a point on p . If we introduce polar coordinates with z as the origin, each arc can be regarded as a univariate function constant number of '- monotone arcs if is not '-monotone; this is possible since g i is a portion of an algebraic curve of small degree). Let fl 00 i be the graph of the upper envelope of arcs in \Gamma i . Since fl i is a portion of the boundary of a set star-shaped with respect to z, the portion of fl i that contains a vertex of the ridge tree T x , for some x 2 ", lies on fl 00 i . However, fl 00 i is not necessarily a connected arc. Suppose are the endpoints of the connected components of fl 0 counterclockwise direction around z; we connect u 2i to u 2i+1 by a segment for each m. The resulting curve is the desired curve fl 0 i is a piecewise algebraic '-monotone arc. Since each arc i , has O(n i fi(n i is a constant depending on the maximum degree of arcs in \Gamma i , and ff(\Delta) is the inverse Ackermann function. Using a divide-and-conquer approach fl 00 i can be computed in time O(n [HS86]. We now trace fl 0 lies in the portion K " [z] of K " visible from z. Compute this portion, in linear time [EA81], and subdivide K " [z] into a linear number of triangles all incident to z. Now partition fl 0 into connected portions each fully contained in one of these triangles. This can be done in time proportional to the number of arcs constituting fl 0 and the number of triangles involved and produces at most O(n) extra arcs, as i is '-monotone and thus crosses each segment separating consecutive triangles in at most one point. Since each triangle 4 is fully contained in K " and thus encloses no images of a vertex of P , the set of plates of S x met by 4 corresponds to a subtree \Pi 4 of \Pi x of linear size, with at most one vertex of degree 3 and all remaining vertices of degree at most 2. Hence, \Pi 4 can be covered by two simple paths and they can be computed in linear time. For each - j ae 4, we determine the furthest node that - i reaches in \Pi 1 4 by binary search. An intersection between - j and a plate of K " can be detected in O(1) time, so the binary search requires only O(log n) time. The total time spent is thus O(n i fi(n) log n triangles of K " [z], and O(n 3 fi(n) log n) over all The above processing is repeated for each of the O(n 3 ) edgelets ". This completes the description of the algorithm. It is summarized in Algorithm 2. Algorithm 2: Exact Edge Sequences edgelets. for each edgelet " do of projections of edges in M " . Eliminate irrelevant features from \Gamma. for each image x i do of \Gamma at which g i appears on the lower envelope. Compute upper envelope fl 00 Convert into a connected arc Compute for each 4 in the triangulation do Compute Compute Compute 4 . Find subtrees of \Pi 1 4 visited by fl 4 . Theorem 5.1 The exact set of all shortest-path edge sequences on the surface of a 3-polytope on n vertices can be computed in O(n 6 fi(n) log n) time, where n) is an extremely slowly growing function. 6 Geodesic Diameter In this section we present an O(n 8 log n) time algorithm to determine the geodesic diameter of P . As mentioned in the introduction, this question was first investigated by O'Rourke and Schevon [OS89] who presented an O(n 14 log n) time algorithm for computing it. Their algorithm relies on the following observation: Lemma 6.1 (O'Rourke and Schevon [OS89]) If a pair of points x; y 2 P realizes the diameter of P, then either x or y is a corner of P, or there are at least five distinct shortest paths between x and y. 2 Lemma 6.1 suggests the following strategy for locating all diametral pairs. We first dispose of the possibility that either x or y is a corner in n \Theta O(n 2 just as in [OS89]. Next, we fix a ridge-free region R and let MR be the subdivision defined in Section 3.4. We need to compute all pairs of points x 2 cl (R) and y 2 K x such that there are at least five distinct shortest paths between x and y, with . By a result of Schevon [Sch89], such a pair x; y can be a diametral pair only if it is the only pair, in a sufficiently small neighborhood of x and y, with at least five distinct shortest paths between them. Such a pair of points corresponds to a vertex of MR . Hence, we use the following approach. We first compute, in O(n 4 ) time, all ridge-free regions of P (cf. Lemma 3.8). Next, for each ridge-free region R, we compute KR , vertices of MR , and f(v) for all vertices of MR (recall that f(v) is the shortest distance from v to any source image; cf. Section 3.3). Next, for each vertex lies in the closure of R and (y 1 the answer of both of these questions is 'yes', we add v to the list of candidates for diametral pairs. (This step is exactly the elimination of "irrelevant features" mentioned at the end of Section 3.4. Once the two conditions are verified, we know that f(v) is exactly d(x; y).) Finally, among all diametral candidate pairs, we choose a pair that has the largest geodesic distance. See Algorithm 3 for the pseudocode. For each ridge-free region R, KR can be computed in time O(n 2 ) and preprocessed for planar point location in additional O(n log n) time using the algorithm of Sarnak and Tarjan [ST86]. By Lemma 3.9, vertices of MR and f(v), for all vertices of MR , can be computed in time O(n 4 ). We spend O(log n) time for point location at each vertex of MR , so the total time spent is O(n 8 log n). Algorithm 3: Geodesic Diameter for each corner c of P do Construct the ridge tree T c with respect to c. for each vertex v of T c do Add d(c; v) to the list of diameter candidates. Compute the ridge-free regions. for each ridge-free region R do Compute MR and f(v) for all vertices v 2 MR . Compute Construct Preprocess KR for point location queries. Preprocess cl (R) for point location queries. for each vertex of MR do cl (R) and (y 1 to the list of diameter candidates. Find a diametral candidate pair with the maximum geodesic distance. Theorem 6.2 The geodesic diameter of a convex polytope in IR 3 with a total of n vertices can be computed in time O(n 8 log n). 7 Shortest-Path Queries In this section we discuss the preprocessing needed to support queries of the form: "Given x; y 2 P , determine d(x; y)". We assume that each point is given together with the face (or the edge) of P containing it. Two variants of the problem are considered: (1) no assumption is made about x and y and (2) x is assumed to lie on an edge of P . Our data structure is based on the following observations. Let x; y be two query points. Suppose is a generic point lying in a ridge-free region R and y is an image of y in S x . If y 2 KR , then On the other hand, if y 62 K x , then it lies in one of the triangles 4p denotes the Euclidean length of a line segment in (the planar embedding) of 4p We thus need a data structure that, given a point y, can determine whether Let -R denote the preimage of @KR on P , so U(-R partitions each face f of P into convex regions. By the nature of the way f is partitioned by -R , the regions in f can be linearly ordered (i.e., their adjacency graph is a chain), so that determining the region of f containing a given point y 2 f can be done in logarithmic time by binary search. Let \Delta ' f be such a region, then either U (\Delta) ' KR or U (\Delta) " In fact, one can prove a more interesting property of \Delta. Lemma 7.1 Let R be a ridge-free region or an edgelet, let f be a face of P, and let \Delta be a connected component of f n -R whose image is not contained in KR . Then the sequence of edges traversed by the shortest path -(x; y) is independent of the choice of x 2 R and y 2 \Delta. Proof: For the sake of a contradiction, suppose there are two points such that the sequences of edges traversed by -(x; y 0 ) and -(x; y 00 ) are distinct. Then there must exist a point with two shortest paths to x-to obtain such a point, move y from one end of y 0 y 00 to the other and observe that the shortest path from x to y changes continuously and maintains the set of edges of P that it meets, except at points y with more than one shortest path to x. Thus . However, the segment y 0 y 00 ae \Delta as \Delta is convex, implying Similarly, if x are such that the paths connecting these two points to y 2 \Delta traverse different edge sequences, there must exist x 2 x 0 x 00 which is connected to y by two shortest paths, again forcing y onto T x and yielding a contradiction. The lemma follows easily. 2 This lemma suggests the following approach to computing d(x; y) for U(y) not contained in KR . For each connected component \Delta in f n -R , whose image is not contained in KR , one can precompute the edge sequence for shortest paths from a point in R to a point in \Delta. This in turn determines the transformation of coordinates from the f-based coordinates to the coordinates usable in the planar unfolding of S x -thus we may now compute an image y of y and the peel of S x that contains y under this unfolding map, which as noted above immediately yields d(x; y). Hence, we can preprocess P in time O(n 2 log n) into a data structure of size O(n 2 ), so that one can determine in O(log n) time whether U(y) 2 K x and, if U(y) 62 K x , then it also returns d(x; y) in additional constant time. Next assume that U(y) ae KR . Note that the data structure just described can be used to compute the coordinates of U(y) even if U(y) ae KR . Now one has to compute 1-i-n are the same as in (3). Let H be the set of hyperplanes in IR 9 corresponding to the graphs of - g i 's, i.e., Then computing the value of g(s; t; y is the same as determining the first hyperplane of H intersected by the vertical ray emanating from the point (s; t; y in the positive v 5 -direction. Agarwal and Matou-sek [AM92] have described a data structure that, given a set G of n hyperplanes in IR d and a parameter n - s - n bd=2c , can preprocess G, in time O(s 1+ffi ), into a data structure of size O(s 1+ffi ), so that the first hyperplane of G intersected by a vertical ray emanating from a point with x can be computed in time O( n s 1=bd=2c log 2 n). Since in our case, we obtain a data structure of size O(s 1+ffi ) so that a query can be answered in time O( n s 1=4 log 2 n). As described in Lemma 3.8, one can partition all faces of P into ridge-free regions in time O(n 4 ). For each ridge-free region, we construct the above data structures. Finally, if x is not a generic point then, as mentioned in the remark following Lemma 3.9, we can use the data structures of any of the ridge-free regions whose boundaries contain x. It is easy to see by a continuity argument that all shortest paths from such a point are encoded equally well in the data structures of all the ridge-tree regions touching x. In summary, for a pair of points x; y 2 P , d(x; y) is computed in the following three steps. We assume that we are given the faces f x ; f y containing x; y, respectively. First find in O(log n) time the ridge-free region R of f x whose closure contains x. Next find in O(log n) time the region \Delta of f y n-R that contains y. If \Delta does not lie in KR , using the information stored at \Delta, compute d(x; y). (The data structure required for handling this case has size \Theta(n 2 ) per ridge-free region in the worst case. Hence choosing does not produce asymptotic space savings while reducing query time.) Otherwise, compute d(x; y) in time O( n s 1=4 log 2 n) using the vertical ray shooting structure. Hence, we can conclude Theorem 7.2 Given a polytope P in IR 3 with n vertices and a parameter n 2 - s - n 4 , one can construct, in time O(n 4 s 1+ffi ) for any ffi ? 0, a data structure of size O(n 4 s 1+ffi ), so that d(x; y) for any two points x; y 2 P can be computed in time O( n s 1=4 log 2 n). If x always lies on an edge, then H is a set of hyperplanes in IR 6 , so the query time of the vertical ray shooting data structure is now O( n s 1=3 log 2 n) for Moreover, we have to construct only O(n 3 ) different data structures, one for each edgelet, so we can conclude Theorem 7.3 Given a polytope P in IR 3 with n vertices and a parameter n 2 - s - n 3 , one can construct in time, O(n 3 s 1+ffi ) for any ffi ? 0, a data structure of size O(n 3 s 1+ffi ), so that for any two points x; y 2 P such that x lies on an edge of P one can compute d(x; y) in time O( n s 1=3 log 2 n). Remark. The performance can be slightly improved by employing the algorithm of Matou-sek and Schwarzkopf [MS93]. 8 Discussion and Open Problems We have shown that use of the star unfolding of a polytope leads to substantial improvements in the time complexity of three problems related to shortest paths on the surface of a convex polytope: finding edge sequences, computing the geodesic diameter, and distance queries. Moreover, the algorithms are not only theoretical improvements, but also, we believe, significant conceptual simplifications. This demonstrates the utility of the star unfolding. We conclude by mentioning some open problems 1. Can one obtain a better upper bound on the number of different combinatorial structures of ridge trees by using a more global argument? Such an improvement will yield a similar improvement in the time complexities of diameter and exact shortest path edge sequences algorithms. 2. Can one answer answer a shortest path query faster if both x and y lie on some edge of P? This special case is important for planning paths among convex polyhedra (see Sharir [Sha87]). --R Star unfolding of a polytope with applications. Konvexe Polyeder. Ray shooting and parametric search. Nonoverlap of the star unfolding. Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences Intrinsic Geometry of Surfaces. Algorithms for reporting and counting geometric intersections. New lower bound techniques for robot motion planning problems. An optimal convex hull algorithm for point sets in any fixed dimension. An optimal algorithm for intersecting line segments in the plane. Shortest paths on a polyhedron. Storing shortest paths for a polyhedron. Approximate algorithms for shortest path motion planning. A linear algorithm for computing the visibility polygon from a point. The 2nd Scientific AMerican Book of Mathematical Puzzles and Diversions Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes Finding all shortest path edge sequences on a convex polyhedron. On conjugate and cut loci. On finding shortest paths on convex polyhedra. Shortest paths among obstacles in the plane. The discrete geodesic problem. The number of shortest paths on the surface of a polyhedron. Computing the geodesic diameter of a 3-polytope An algorithm for shortest paths motion in three dimensions Shortest paths along a convex polyhedron. Shortest paths in Euclidean space with polyhedral obstacles Algorithms for geodesics on polytopes. A convex hull algorithm optimal for point sets in even dimensions. On shortest paths amidst convex polyhedra. The number of maximal edge sequences on a convex poly- tope An algorithm for finding edge sequences on a polytope. On shortest paths in polyhedral spaces. Planar point location using persistent search trees. Constructing the visibility graph for n line segments in O(n 2 --TR --CTR Sariel Har-Peled, Approximate shortest paths and geodesic diameters on convex polytopes in three dimensions, Proceedings of the thirteenth annual symposium on Computational geometry, p.359-365, June 04-06, 1997, Nice, France Yi-Jen Chiang , Joseph S. B. Mitchell, Two-point Euclidean shortest path queries in the plane, Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, p.215-224, January 17-19, 1999, Baltimore, Maryland, United States Marshall Bern , Erik D. Demaine , David Eppstein , Eric Kuo , Andrea Mantler , Jack Snoeyink, Ununfoldable polyhedra with convex faces, Computational Geometry: Theory and Applications, v.24 n.2, p.51-62, February Mark Lanthier , Anil Maheshwari , Jrg-Rdiger Sack, Approximating weighted shortest paths on polyhedral surfaces, Proceedings of the thirteenth annual symposium on Computational geometry, p.485-486, June 04-06, 1997, Nice, France Mark Lanthier , Anil Maheshwari , Jrg-Rdiger Sack, Approximating weighted shortest paths on polyhedral surfaces, Proceedings of the thirteenth annual symposium on Computational geometry, p.274-283, June 04-06, 1997, Nice, France Demaine , Martin Demaine , Anna Lubiw , Joseph O'Rourke , Irena Pashchenko, Metamorphosis of the cube, Proceedings of the fifteenth annual symposium on Computational geometry, p.409-410, June 13-16, 1999, Miami Beach, Florida, United States

geodesics;shortest paths;star unfolding;convex polytopes

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Scalable Parallel Implementations of List Ranking on Fine-Grained Machines.

AbstractWe present analytical and experimental results for fine-grained list ranking algorithms. We compare the scalability of two representative algorithms on random lists, then address the question of how the locality properties of image edge lists can be used to improve the performance of this highly data-dependent operation. Starting with Wyllie's algorithm and Anderson and Miller's randomized algorithm as bases, we use the spatial locality of edge links to derive scalable algorithms designed to exploit the characteristics of image edges. Tested on actual and synthetic edge data, this approach achieves significant speedup on the MasPar MP-1 and MP-2, compared to the standard list ranking algorithms. The modified algorithms exhibit good scalability and are robust across a wide variety of image types. We also show that load balancing on fine grained machines performs well only for large problem to machine size ratios.

Introduction List ranking is a fundamental operation in many algorithms for graph theory and computer vision problems. Moreover, it is representative of a large class of fine grained data dependent algorithms. Given a linked list of n cells, list ranking determines the distance of each cell from the head of the list. On a sequential machine, this problem can be solved in O(n) time by simply traversing the list once. However, it is much more difficult to perform list ranking on parallel machines due to its irregular and data dependent communication patterns. The problem of list ranking of random lists has been studied extensively on PRAM models and several clever techniques have been developed to implement these algorithms on existing parallel machines [2, 3, 10, 11]. In this paper, we study the scalability of these techniques on fine-grained machines. We then present efficient algorithms to perform list ranking on edge pixels in images. Performance results based on implementations on MasPar machines are discussed. Most of the algorithms proposed in the literature are either simple but not work-efficient [2, 10, 11] or are work-efficient but employ complex data structures and have large constant factors associated with them [1, 3, 11, 12]. In order to study the performance and scalability of these algorithms in an actual application scenario and on existing parallel machines, we have chosen two representative algorithms, Wyllie's algorithm and Anderson & Miller's randomized algorithm [11]. We assume that list ranking is an intermediate step in a parallel task. Therefore, in the general case, a linked list is likely to spread over the entire machine in a random fashion. We then study the performance of these algorithms in applications such as computer vision and image processing where locality of linked lists is present due to the neighborhood connectivity of edge pixels. For this purpose, we present a modified approach that takes advantage of the locality and connectivity properties. A similar technique has been described in [10] for arbitrary linked lists. In Reid-Miller's work, the assignment of list cells to processors is determined by the algorithm, whereas, for the arbitrary case, we assume pre-assigned random cell distribution. Our approach was derived independently and was motivated by the contrast between the random list case and the characteristics of edges in images. We show that in the case of random lists where cells are randomly pre-assigned to processors, the Randomized Algorithm runs faster than Wyllie's Algorithm on MasPar machines when lists are relatively long. However, Wyllie's Algorithm performs better for short lists. This agrees with the results reported on other machines [2, 10]. This also implies that in order to achieve scalability, a poly-algorithmic approach is needed That is, for long lists use the Randomized Algorithm and when the sizes of lists are reduced to a certain length, use Wyllie's Algorithm. This approach has also been used to design theoretically processor-time optimal solutions. We show, however, that for list ranking of edge pixels of images on fine-grained SIMD machines, the standard Wyllie's Algorithm and Randomized Algorithm do not take good advantage of the image edge characteristics. A modified technique described in Section 4 runs about two to ten times faster (depending on the image size) than the standard Wyllie's and Randomized algorithms on a 16K processor MasPar MP-1. Moreover, whereas the standard algorithms do not scale well on image edge lists, the modified algorithms exhibit good scalability with respect to increases in both image size and number of processors. We study the performance of the proposed algorithms on images with varying edge characteristics. In the remainder of Section 1, we briefly describe the architecture of the MasPar machines, and outline the notion of scalability used in our work. Section 2 presents an overview of parallel algorithms for list ranking. The performance results of standard Wyllie's and Randomized algorithms are discussed in Section 3 and their scalability behavior is analyzed. Efficient parallel algorithms for performing list ranking on image edge lists are presented in Section 4. Section 5 contains implementation results and discusses scalability of the modified parallel list ranking algorithms on image edge lists. 1.1 Fine-grained SIMD Machines The parallel algorithms described in this paper are implemented on MasPar MP-1 and MP-2 fine-grained SIMD machines. Fine-grained machines, in general, are characterized by a large number of processors with a fairly simple Arithmetic Logic Unit (ALU) in each processor. The MasPar machines are massively parallel SIMD machines. A MasPar MP-series system consists of a high performance Unix Workstation as a Front End and a Data Parallel Unit (DPU) consisting of 1K to 16K processing elements (PEs), each with 16 Kbytes to 1-Mbyte of memory. All PEs execute the instructions broadcast by an Array Control Unit (ACU) in lock step. PEs have indirect addressing capability and can be selectively disabled. The clock rate for both MP-1 and MP-2 machines is 12.5 MHz. Processors in the MP-2 employ a 32-bit ALU compared with a 4-bit ALU in MP-1 processors. Each PE is connected to its eight neighbors via the Xnet for local communication. Besides the Xnet, these machines have a global router network that provides direct point-to-point global communication. The router is implemented by a three-stage circuit switched network. It provides point-to-point communication between any two PEs with constant latency, regardless of the distance between them. A third network, called the global or-tree, is used to move data from individual processors to the ACU. This network can be used to perform global operations such as global maximum, prefix sum, global OR, etc. across the data in the entire array. For our experiments, we have used the MasPar MP-1 and MP-2 as being representative of fine-grained machines. The use of the router network, which provides point-to-point communication between any two processors with constant latency, ensures that our performance results are independent of any specific network topology. Also, we have run our experiments on both the MP-1 and MP-2 to show the effects of processing power of individual processors on the performance of the algorithms. We have used an extended version of sequential ANSI C, the MasPar Programming Language (MPL), to keep our implementations free of machine-dependent software features. 1.2 Scalability of Parallel Algorithms We analyze the scalability of our algorithms and implementations using several architecture and algorithm parameters. We study the performance by varying machine and problem size, by varying the characteristics of the input image, and by varying the processor speed. Several notions of scalability exist [5]. In our analyses we define scalability as follows: Consider an algorithm that runs in T (n; p) time on a p processor architecture and the input problem size is n. The algorithm is considered scalable on the architecture if T (n; p) increases linearly with an increase in the problem size, or decreases linearly with an increasing number of processors (machine size) [7, 8]. In our experiments, we use this definition intutively to study the scalability of the algorithms and implememntations. It is likely that no single algorithm is scalable over the entire range of machine and problem sizes. One of the important factors limiting the range of scalability is the sequential component of the parallel algorithm. We identify the regions of scalability for different algorithms presented in this paper and compare the analytical results with the experimental data. Parallel Algorithms for List Ranking In our implementations, we use Wyllie's parallel algorithm and Anderson & Miller's randomized algorithm for list ranking [11]. In this section, we outline these algorithms for the general case of random lists. Compared with other parallel algorithms for list ranking in the literature [11, 3, 12], we have chosen these algorithms because of their simplicity, ease of implementation, and small constant factors. In particular, the deterministic algorithms based on ruling sets and graph coloring [3] are not easily amenable to implementation on existing parallel machines. 2.1 Wyllie's Pointer Jumping Algorithm be a linked list of n cells such that then c i is the head of the list. The first element in the list which has no predecessor is referred to as the tail of the list. The list ranking problem deals with finding the rank of each cell i with respect to the head of the list. Wyllie's Algorithm uses pointer jumping or dereferencing to find the rank of a cell. In pointer jumping, the successor of a cell c i is modified to be its successor's successor. That is, given , one iteration of pointer jumping reassigns the successor of c i to be c i+2 . Each cell c i maintains a value rank[i] which is the distance of the cell c i from its current successor succ[i]. Intuitively, after each round a linked list is divided into two linked lists of half the length (see Fig. 1 a). After log(n) iterations, all the cells point to nil and rank[i] contains the exact rank of cell c i . On p processors, each processor is assigned n=p cells at random. Although this algorithm is simple and has a running time of O( n log n on a p-processor machine, it is not work-efficient compared with the serial algorithm, which takes O(n) time. In the next section we describe a work-efficient randomized algorithm. 2.2 Anderson and Miller's Randomized Algorithm Anderson & Miller's Randomized algorithm is a modified version of the work-efficient algorithm devised by Miller & Reif [9]. Assume that each processor holds n=p cells as a queue. The algorithm consists of two phases. In the first phase, called the pointer jumping phase, each processor "splices" out the cell at the top of its queue with a condition that two consecutive cells of the list assigned to two different processors are not spliced out simultaneously. In order to decide which cells to splice out, each processor tosses a coin and assigns H or T to the cell at the top of its queue. Furthermore, all the cells not at the top of the queue are assigned T. A processor splices out the cell at the top of its queue only if that cell is marked H and its predecessor is assigned T (see Fig. 1 b). The first phase ends when all of the cells in each processor are spliced out. The splicing out of cell c i consists of two atomic assignments followed by updating of rank[i] as follows: In the second phase, referred to as the reconstruction phase, the cells are put back into the a b c d a b c d a b c d (a) (b) Figure 1: (a) Pointer jumping in Wyllie's algorithms. Edge labels show the rank of the edge's predecessor cell in the current iteration; (b) Splicing condition in Anderson & Miller's algorithm. 's top cell will be spliced out. queue in reverse of the order in which they were spliced out. In reconstructing the queue, the rank of each cell c i with respect to the head of the list is updated: The expected running time of this algorithm is O(n=p), assuming n=p - log n. Additional details of the algorithm and the analysis can be found in [11]. In the rest of this paper, we refer to this algorithm as the Randomized Algorithm. 3 Implementation Results using Random Lists The algorithms presented in the previous section have been implemented on the MasPar ma- chines. In this section, we study the impact of various machine and problem parameters on the performance of these algorithms on random lists. To generate a random linked list of size n, we traverse an array of n cells in serial order, and for each cell we assign a pointer to a random location in the array. It is ensured that a cell in the array is pointed to by no more than one other cell. For the input to the Randomized Algorithm, we also assign reverse pointers in the target location to generate a doubly linked list, as required by the algorithm. The resulting random linked list is then read into the parallel processor array so that every processor holds n=p cells of the array. Since the n=p cells in the processor may point to any of the n array locations, the sublist in each processor does not generally contain successive cells in the linked list. 3.1 Scalability Analysis Fig. 2 shows the performance of the Wyllie's and Randomized algorithms on a MasPar MP-1 using various machine and problem sizes. The experimental results presented in Fig. 2 (a) and (b) are consistent with the theoretical analysis. The asymptotic complexity of Wyllie's Algorithm for a random list of size n on p processors is O((n=p) log n), while the complexity of the Randomized Algorithm is O(n=p). However, the Randomized Algorithm has larger constant factors due to the increased overhead of coin-tossing and the record keeping involved in the reconstruction stage of the algorithm. Wyllie's Algorithm is more suited for smaller linked lists due to its small constant factors. The Randomized Algorithm outperforms Wyllie's Algorithm as the problem size increases. This is because of the probabilistic splicing of the elements which generates lower congestion in the communication network. For the example shown in Fig. 2 (a), the crossover point occurs for a random list of size 94K elements on a 16K processor MP-1. Random List Size (n) Time MP-1: 16,384 Processors ___ Wyllie _._ Randomized Number of Processors (p) Time Random List ___ Wyllie _._ Randomized (a) (b) Figure 2: (a) Performance of Wyllie's and Randomized algorithms on the MP-1 for a random list of varying size; and (b) performance of the algorithms on a random list of size 16K, varying the number of processors of the MP-1 An interesting feature of Fig. 2 (b) is that the execution time of the Randomized Algorithm starts increasing when the number of processors exceeds 4K for a random list of size 16K. This is because, for a fixed problem size, as the number of processors increases beyond a certain point, the size of the queue (n=p) in each processor becomes very small. Hence it is likely that the cell at the top of the queue in a processor is pointed to by a cell at the top of the queue in another processor, thus reducing the chance that the cell is spliced out in a particular iteration. This increases the total number of iterations and thus increases the execution time of the algorithm. By varying the problem size and machine size, we are able to determine the regions of problem and machine size for which each algorithm is fastest. These experiments also demonstrate that scalability is best achieved by changing the algorithm approach as the problem size increases. This, in fact, agrees with the approaches used by algorithm designers to develop processor-time optimal solutions for the list ranking problem [11]. We performed the experiments on the MasPar MP-2 as well. The MP-1 and MP-2 use the same router communication network. The only difference between the architectures is the faster processors on the MP-2. Thus, to study the impact of processor speed, we compare the performance of the algorithms on 4096 processors of the MP-1 and MP-2. In both cases, for problem sizes greater than 16K, the Randomized Algorithm outperforms Wyllie's Algorithm. However, Wyllie's Algorithm is 35% faster and the Randomized Algorithm is 40% faster on the MP-2. In computer vision, list ranking is an intermediate step in various edge-based matching and represent them in a compact data structure for efficient processing in subsequent steps [2]. We assume input in the form of binary images with each pixel marked as an edge or non-edge pixel. Furthermore, as a result of an operation such as edge linking, each edge pixel points to the successor edge pixel in its 8-connected neighborhood, where the successor function defines the direction of the edge. An n \Theta n image is divided into p subimages of size n \Theta n , and each processor is assigned one subimage. This straightforward division and distribution of edge pixels to processors might cause load imbalance. Section 5.5 describes the effect of load imbalance, and presents an alternative data-distribution scheme that improves the load balance across the processors. 4.1 Image Edge Lists Fig. 5 shows edge maps derived from various real and synthetic images. In particular, Fig. 5 (a) is an image of a picnic scene, and Fig. 5 (b) is the edge map obtained by performing edge detection on the picnic image. This is followed by an edge linking operation that creates linked lists of contiguous edge pixels. The image edge lists used in our experiments were derived by performing sequential edge linking [4] operations on the edge maps of various images. The edge lists resulting from images typically have several properties that may affect the efficiency of the standard list ranking algorithms. For example, on the average, the number of lists is fairly large, the lists are short in length, and these edge lists exhibit spatial continuity. This implies that list cells assigned to a single processor are contiguous and they form sublists. Furthermore, one processor may contain pieces of several edge lists. On one hand, these properties can adversely affect the performance of the standard list ranking algorithms. For example, if we run Wyllie's Algorithm disregarding the connectivity property, several edge points belonging to the same sublist in a processor may compete for the communication links to access the successor information from other processors. Similarly, in the Randomized Algorithm, a processor may be tossing the coin for an edge point while its successor is already stored in the same processor. These overheads make the standard Wyllie's or Randomized algorithms unattractive for computing ranks for edge pixels in images. On the other hand, some of the image edge properties can be used to modify the standard algorithms to achieve better performance on edge lists in images, as described in the next section. 4.2 Modified Algorithms Fig. 3 compares the performance of the Wyllie's and Randomized algorithms on random lists versus their performance on edge maps of equivalent sizes obtained from the picnic image. This is achieved by counting the number of edge pixels in the picnic image, and creating a random list with the same number of elements. From Fig. 3, it is clear that the locality properties of the image edge lists cause performance degradation, especially for large image sizes. - Wyllie's on Picnic Image ___ Wyllie's on Random List Image Size Execution Time Wyllie's Algorithm - Randomized on Picnic Image ___ Randomized on Random List Image size Execution Time Randomized Algorithm (a) (b) Figure 3: Comparison of the performance of the algorithms on random lists versus equivalent sizes of the picnic image on a 16K processor MP-1. In the following, we outline a modified approach that takes advantage of the connectivity property inside a processor by reducing each sublist to a single edge, called a by-pass edge. The approach then uses the Wyllie's or Randomized algorithm over the by-pass edges in the image. In addition to eliminating redundant work within the processors, this approach also reduces the amount of communication across processors. The approach consists of three steps: Step 1. Convert each sublist of contiguous edge pixels in a subimage into a by-pass edge by performing a serial list ranking operation on all sublists within a processor (for example, see Fig. 4). Associate with each such edge the length of the sublist it is representing. For lists that span processors, their corresponding by-pass edges are connected. By-pass edge Image contour Processor i Figure 4: By-pass edges. Step 2. Run Wyllie's Algorithm or the Randomized Algorithm on the lists of by-pass edges. Step 3. Serially update the rank of each edge pixel within a subimage using the final rank of the by-pass edge that represents the pixel. The modified algorithms can thus be thought of as a combination of serial and parallel list ranking algorithms. To analyze the scalability of the list ranking algorithms on image edge lists, we assume that the image has n edge pixels uniformly distributed across p processors such that each processor holds O(n=p) edge pixels. This simplifying assumption is unlikely to be strictly true for actual images. However, since the number of edge pixels per processor is relatively low in fine-grained implementations, the extent to which the number of edge pixels per processor deviates from O(n=p) will be limited. In the modified approach, the first step takes O(n=p) computation time to form the by-pass edges. The last step takes O(n=p) time to update the rank of each edge pixel. Thus the total time taken by the serial component of the modified algorithms is O(n=p). To analyze the parallel execution time of the second step, we assume that the image has multiple edge lists of varying lengths, and the length of the longest edge is l. Note that l is a property of the underlying image from which the edge map was derived. Since the edges are assumed to be uniformly distributed across the processors, the length of the longest list consisting of by-pass edges is O(l=p). We further assume that the n=p pixels in each processor can be divided into k sets such that each set contains successive pixels of an edge list assigned to the processor (i.e. each processor contains k by-pass edges after the first step of the algorithm). Again, this is a simplifying assumption. Under these assumptions, the execution time of the second step is log l) for Wyllie's Algorithm, and O(k) expected time for the Randomized Algorithm (assuming k - log(kp)). This is the time taken by the parallel component of the modified algorithms. Therefore, the total execution time is O(n=p+k log l) for the Modified Wyllie's Algorithm, and O(n=p+ for the Modified Randomized Algorithm. Since the constant factors for the Modified Randomized Algorithm are higher, it will outperform the Modified Wyllie's Algorithm only if l is large (i.e. the image has very long edges). The algorithms presented in this section are inherently fine-grained due to the high commu- nication/computation ratio and irregular patterns of interprocessor communication. Efficient algorithms for coarse-grained machines and their implementation are described in [6]. 5 Implementation Results Using Image Edge Lists We have used the edge maps from a number of real and synthetic images to study the performance of list ranking algorithms. Fig. 5 shows the edgemaps used in our experiments. Fig. 5 (b), (c), and (d) are derived by performing edge detection and edge linking operations on gray-scale images. The edge characteristics of these images differ significantly. For example, the edges of the written text image are more local compared to the other images. Typically, the edge density (percent of edge pixels compared to the total number of pixels) of these real images is in the range of 3 to 8 percent. As a contrast to these edge maps, we have also generated synthetic edge maps of varying edge density and length. Fig. 5 (e) and (f) show synthetically generated edge maps of straight lines and a spiral, respectively. We have generated these images with edge densities ranging from 5 to 50 percent. The edge characteristics of the real and synthetic images help in gaining insight into the performance of the algorithms on images of varying edge density and edge length. 5.1 Comparison with Standard Algorithms The performance of the modified algorithms is compared to the performance of the standard algorithms in Fig. 6. Fig. 6 (a) shows the execution times for varying sizes of the picnic image on a 16K processor MP-1. The modified algorithms are significantly faster than the straightforward Wyllie's or Randomized algorithms. This is because these algorithms efficiently exploit the locality of edges in images. We have verified the results on different machine sizes of the MP-1 and MP-2. Fig. 6 (b) shows execution times of the algorithms for the dense synthetic spiral edge maps. The spiral edge map is very different from the picnic edge map because it has much higher edge density, and much longer image edges. Despite these vastly different image characteristics, the modified algorithms are significantly faster than the standard algorithms. We have also tested the algorithms on the different edge maps shown in Fig. 5 with similar results. Thus we conclude that our modifications result in significant performance improvement over the standard Wyllie's and Randomized algorithms for image edge lists. We point out that in Fig. 6, the relative performance of the modified algorithms varies depending on the image characteristics. For the spiral image, the Modified Randomized Algorithm is faster than Modified Wyllie's for images of size greater than 350 \Theta 350 pixels. On the other hand, for the picnic image, the Modified Randomized Algorithm is always slower than Modified Wyllie's. We explain this behavior in detail in Section 5.3. 5.2 Scalability Analysis The scalability of the modified algorithms has been studied using different images, and varying the image size and number of processors. Results obtained using different edge maps shown in Fig. 5 are similar. Hence we restrict our discussion to the performance of the modified algorithms (a) (b) (c) (d) Figure 5: Picnic image and detected edge contours of different images used in our experiments: (a) real picnic scene, (b) edgemap of the picnic scene (c) edgemap of written text, (d) edgemap of street scene, (e) synthetic edgemap with straight lines, and (f) synthetic edgemap of a spiral. . Randomized _. Modified Randomized ___ Modified Wyllie Image Size Execution Time Picnic Image 6 . Randomized _. Modified Randomized ___ Modified Wyllie Image Size Execution Time Dense Spiral Image (a) (b) Figure Performance of the algorithms for various sizes of (a) the picnic image, and (b) the spiral image; on 16,384 processors of the MP-1. on the edge map derived from the picnic scene. Fig. 7 examines the scalability of the Modified Wyllie and Modified Randomized algorithms with respect to increasing problem size. The plot displays the overall execution time, as well as the serial and parallel components of the modified algorithms. For small image sizes, the parallel component dominates the overall execution time. This is because of the relatively large constant factors of the parallel component compared to the sequential component. For large image sizes, as the subimage size per processor grows, the execution time of the sequential component grows much faster than the corresponding parallel component. This is consistent with the analytical results since the execution time for the sequential list ranking component grows linearly with the size of the subimage (O(n=p)), while the parallel component grows in proportion to the number of by-pass edges in a subimage. In the case of the Modified Wyllie's Algorithm, the crossover point between the execution times of sequential and parallel components occurs when the subimage size assigned to each processor is 64 pixels. In the case of the Modified Randomized Algorithm, the crossover point occurs when the subimage size is 128 pixels (see Fig. 8 (b)). Fig. 8 shows the scalability behavior of the modified algorithms with respect to changes in the machine size. We notice that the sequential component dominates the execution time for a high problem/machine size ratio, and the parallel component dominates for a low problem/machine size ratio. The results shown are for the MP-1. For the MP-2, the performance curves have . Sequential Parallel ___ Total Image Size Execution Time Modified Wyllie's Parallel ___ Total Image Size Execution Time Modified Randomized (a) (b) Figure 7: Performance of the sequential and parallel components of (a) the Modified Wyllie, and (b) Modified Randomized algorithms. (Execution time for the picnic image on the 16K processor MP-1.) similar shape. However, for the MP-2 the crossover point at which the parallel component begins to dominate the execution time occurs for a larger problem/machine size ratio due to the faster processors (higher computation/communication ratio) than the MP-1. . Sequential Parallel ___ Total Number of Processors Execution Time Modified Wyllie's . Sequential Parallel ___ Total Number of Processors Execution Time Modified Randomized (a) (b) Figure 8: Scalability with respect to number of processors for (a) the Modified Wyllie's Algo- rithm, and (b) the Modified Randomized Algorithm. (Execution time for the picnic image of size 512 \Theta 512 on the MP-1.) Fig. 9 plots the performance of the modified algorithms when the number of processors increases linearly with the image size. Thus the problem-size/machine-size ratio is constant, and the size of the subimage in a processor is constant. We observe that the sequential time, which is proportional to the size of the subimage in a processor, remains approximately the same, while there is a small increase in the parallel component. This is due to the fact that in MasPar ma-0.10.2 _. Sequential Parallel ___ Total Image Size Execution Time Modified Wyllie's Parallel ___ Total Image Size Execution Time Modified Randomized (a) (b) Figure 9: Scalability for constant problem-size/machine-size ratio (256 elements per processor) on the MP-1: (a) the Modified Wyllie's Algorithm, and (b) the Modified Randomized Algorithm. chines, with the increase in number of processors, the number of links in the router network does not change proportionally. However, since the overall increase in the execution time is small, we conclude that the modified algorithms exhibit speedup proportional to the image size, if the ratio of image-size to number of processors is constant. 5.3 Effect of Image Characteristics We have studied the performance of our algorithms on images with varying characteristics in terms of edge density and edge lengths. As discussed in Section 4.2, the total execution time is O(n=p+k log l) for the Modified Wyllie's Algorithm, and O(n=p+k) for the Modified Randomized Algorithm. Fig. 10 indicates that increase in execution time is proportional to the increase in image edge density (O(n=p)), and this is true for both Modified Wyllie's and Modified Randomized algorithms. Fig. 11 shows the behavior of modified algorithms on edge maps with varying edge lengths. In Fig. 11 (a), the Modified Randomized Algorithm is always slower than the Modified Wyllie's Algorithm. This is also the case for the picnic scene image (see Fig. 6 (a)). This is due to the large constant factors for the Modified Randomized Algorithm. However, in Fig. 11 (b), the Modified Randomized Algorithm is significantly faster than the Modified Wyllie's Algorithm beyond a certain image-size to machine-size ratio. This is because . image ___ image Image size Execution time Modified Wyllie's . image ___ image Image size Execution time Modified Randomized (a) (b) Figure 10: Performance on the synthetic line image of varying density on a 16K processor MP-1: (a) the Modified Wyllie's Algorithm, and (b) the Modified Randomized Algorithm. Modified Wyllie's ___ Modified Randomized Image size Execution time Line Image Modified Wyllie's ___ Modified Randomized Image size Execution time Spiral Image (a) (b) Figure 11: Performance of the modified algorithms on synthetic images with the same density but different edge lengths on a 16K processor MP-1: (a) line image, and (b) spiral image. the parallel execution time of the Modified Randomized Algorithm is O(k), compared to O(k log l) for the Modified Wyllie's Algorithm. Since the serial component for an equal-density line image and spiral image is the same (O(n=p)), we expect the Modified Randomized Algorithm to run faster when the length of the longest edge (l) is large as in the case of the spiral image. 5.4 MP-1 vs MP-2 The effect of the processor speed on the performance of the algorithms is studied by executing the algorithms on 4K processor MasPar MP-1 and MP-2 machines. As shown in Fig. 12, Wyllie's algorithm is faster on the MP-2 compared with MP-1. It is primarily due to increased processor speed. Similar behavior is exhibited by the Randomized algorithm. It is worth noting that although the MP-2 has lower execution times, the scalability behavior of the algorithms on the two machines is very similar. The crossover point when the sequential component dominates the overall execution time occurs for a larger image-size/machine-size ratio on the MP-2. . Sequential Parallel ___ Total Image Size Execution Time Modified Wyllie's on MP-1 . Sequential Parallel ___ Total Image Size Execution Time Modified Wyllie's on MP-2 (a) (b) Figure 12: Performance of the Modified Wyllie's Algorithm on varying sizes of the picnic image on (a) 4K processors of the MP-1, and (b) 4K processors of the MP-2 5.5 Load Balancing In an input derived from a real (as opposed to synthetic) image, it is very likely that edge contours are concentrated in a particular portion of the image. In this case, simple partitioning of the image into p subimages and assigning each subimage to a processor may yield an unbalanced load across processors. In order to study the effect of load imbalance on performance, we have experimented with various load-balancing techniques. In general, techniques based on first computing the load variance across processors then redistributing the load to the processors with light loads have failed to yield high performance. This is because the computation and data redistribution for the load-balancing step become a significant part of the total execution time. Further, regardless of the load redistribution, communication overhead remains the same. In the following, we outline a simple heuristic to address the load balancing problem and present performance results. The heuristic is based on dividing the input image into more than subimages and assigning more than one subimage to each processor. Partition the input image into kp identical sized subimages. Number the subimages in a row- shuffled order as follows. Arrange the subimages into k sets such that each set contains a grid of contiguous subimages. Number the subimages in each set in row-major order. From each set to processor j, where example row-shuffled ordering of an image on a 16 processor machine assuming shown in Figure 13. Figure 13: A heuristic partitioning scheme of an input image on a 16 processor machine. Figure 14 compares the distribution of edge pixels of the 1K \Theta 1K picnic image on a 16K processor MP-1 using simple partitioning and using the partitioning based on the above described heuristic. In the load-balanced partitioning scheme the number of processors having zero edge pixels is reduced by half. At the same time, the variance of load (edge pixels per processor) across the entire machine is reduced from 21.4 to 8.9. Figures 15 and 16 compare the performance of the Modified Wyllie's Algorithm with and without load-balancing, while varying the image and machine sizes. We observe that load-balancing pays off only for very large image sizes. In the case of the simple partitioning scheme used in the earlier sections, the sequential execution time dominates the parallel execution time for large image sizes. In the load-balanced partitioning scheme, the sequential execution time has been reduced at the expense of an increase in the parallel component. The increase in the parallel component is due to the fact that more edge pixels in a processor now have successors residing in other processors. This increases contention over the communication links during the pointer jumping phase and thus increases the parallel time. The sequential time decreases only for large ratios of N and p because the extent of load imbalance possible for small ratios will always be low since the size of the subimage assigned to a processor is very small. This claim is well supported by Figures 15 (b) and 16 (b). Number of Processors Number of Processors (a) (b) Figure 14: Histogram of edge-pixel distribution of the 1K \Theta 1K picnic image on 16K processors of the MP-1: (a) the original distribution, and (b) the distribution after load-balancing. In conclusion, a load-balancing scheme performs well for large image-size to machine-size ratios. In terms of scalability with respect to machine size as well as problem size, the behavior using the load balancing scheme is not much different than with the simple partitioning scheme. Unbalanced ___ Load-Balanced Image Size Execution Time Execution Time _. Parallel (unbalanced) ___ Parallel (load-balanced) Image Size Execution Time Sequential and Parallel Components (a) (b) Figure 15: Comparison of the execution time of the unbalanced and load-balanced Modified Wyllie's Algorithm: (a) overall execution time, and (b) sequential and parallel components. (Execution time for the picnic image on the 16K processor MP-1.) 1,024 4,096 16,3840.020.06_. Unbalanced ___ Load-Balanced Number of Processors Execution Time Execution Time 1,024 4,096 16,3840.010.030.05. Sequential (unbalanced) _. Parallel (unbalanced) ___ Parallel (load-balanced) Number of Processors Execution Time Sequential and Parallel Components (a) (b) Figure Comparison of the scalability with respect to number of processors for the unbalanced and load-balanced Modified Wyllie's Algorithm: (a) overall execution time, and (b) sequential and parallel components. (Execution time for the picnic image of size 512 \Theta 512 on the MP-1.) 6 Conclusions In this paper, we have studied the scalability of list ranking algorithms on fine grained machines and have presented efficient algorithms for list ranking of image edge lists. The Wyllie's and Anderson & Miller's algorithms for list ranking are chosen as representative of deterministic and randomized algorithms, respectively. The performance of these algorithms is studied on random lists and image edge lists. It is shown that these algorithms perform poorly for image edge lists. Also, no single algorithm covers the entire range of scalability. We show that a poly- algorithmic approach, in which the algorithmic approach changes as the data size is reduced, is required for scalability across all machine and problem sizes. For image edge lists, we have presented modified algorithms that exploit the locality property of the edge lists. Performance of our modified algorithms on actual images demonstrates the gains that can be achieved by using applications characteristics in the algorithm design. On a 16K processor MasPar MP-1, the modified algorithms run about two times faster on small images and about ten times faster on large images than the standard Wyllie's and Randomized algorithms. The modified algorithms are robust across a wide variety of images. Finally, the results of our extensive experimentation have shown that while the standard algorithms were not scalable for list ranking on image edge lists, the tailored algorithms exhibited good scalability with respect to increases both in image size and number of processors, and also with respect to changes in image characteristics. We have also shown that load balancing on fine grained machines does not pay off unless the problem to machine size ratio is very large. In summary, this study provides insight into the performance of fine-grained machines for applications that employ light computations but have intense data dependent communications. In contrast to our results for list ranking of edge lists on coarse-grained machines, the results presented here demonstrate that implementations of communication intensive problems on fine-grained machines are very sensitive to the characteristics of the input data and machine parameters --R "A simple parallel tree contraction algorithm," "Efficient parallel processing of image contours," "Faster optimal parallel prefix sums and list ranking," "Sequential edge linking," Measuring the scalability of parallel algorithms and architectures. "Contour ranking on coarse grained machines: A case study for low-level vision computations," An Introduction to Parallel Algorithms "Scalable data parallel algorithms and implementations for object recog- nition," "Parallel tree contraction and its applications," "List ranking and list scan on the Cray C-90," "List Ranking and Parallel Tree Contraction," "Efficient algorithms for list ranking and for solving graph problems on the hypercube," --TR --CTR Isabelle Gurin Lassous , Jens Gustedt, Portable list ranking: an experimental study, Journal of Experimental Algorithmics (JEA), 7, p.7, 2002

parallel algorithms;fine-grained parallel processing;image processing;scalable algorithms;list ranking;computer vision

270935

A Spectral Technique for Coloring Random 3-Colorable Graphs.

Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p $\geq$ c/n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p $\geq$ polylog(n)/n [J. Algorithms, 19 (1995), pp. 204--234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c.

Introduction A vertex coloring of a graph G is proper if no adjacent vertices receive the same color. The chromatic number -(G) of G is the minimum number of colors in a proper vertex coloring of it. The problem of determining or estimating this parameter has received a considerable amount of attention in Combinatorics and in Theoretical Computer Science, as several scheduling problems are naturally formulated as graph coloring problems. It is well known (see [13, 12]) that the problem of properly coloring a graph of chromatic number k with k colors is NP-hard, even for any fixed k - 3, and it is therefore unlikely that there are efficient algorithms for optimally coloring an arbitrary 3-chromatic input graph. On the other hand, various researchers noticed that random k-colorable graphs are usually easy to color optimally. Polynomial time algorithms that optimally color random k-colorable graphs for every fixed k with high probability, have been developed by Kucera [15], by Turner [18] and by Dyer and Frieze [8], where the last paper provides an algorithm whose average running time over all k-colorable graphs on n vertices is polynomial. Note, however, that most k-colorable graphs are quite dense, and hence easy to color. In fact, in a typical k-colorable graph, the number of common neighbors of any pair of vertices with the same color exceeds considerably that of any pair of vertices of distinct colors, and hence a simple coloring algorithm based on this fact already works with high probability. It is more difficult to color sparser random k-colorable graphs. A A preliminary version of this paper appeared in the Proc. of the 26 th ACM STOC, ACM Press (1994), 346-355. Institute for Advanced Study, Princeton, NJ 08540, USA and Department of Mathematics, Tel Aviv University, Tel Aviv, Israel. Email: [email protected]. Research supported in part by the Sloan Foundation, Grant No. 93-6-6 and by a USA-Israel BSF grant. y ATT Bell Laboratories, Murray Hill, NJ 07974. Email: [email protected]. This work was done while the author was at DIMACS. precise model for generating sparse random k-colorable graphs is described in the next subsection, where the sparsity is governed by a parameter p that specifies the edge probability. Petford and Welsh [16] suggested a randomized heuristic for 3-coloring random 3-colorable graphs and supplied experimental evidence that it works for most edge probabilities. Blum and Spencer [6] (see also [3] for some related results) designed a polynomial algorithm and proved that it colors optimally, with high probability, random 3-colorable graphs on n vertices with edge probability p provided arbitrarily small but fixed ffl ? 0. Their algorithm is based on a path counting technique, and can be viewed as a natural generalization of the simple algorithm based on counting common neighbors (that counts paths of length 2), mentioned above. Our main result here is a polynomial time algorithm that works for sparser random 3-colorable graphs. If the edge probability p satisfies p - c=n, where c is a sufficiently large absolute constant, the algorithm colors optimally the corresponding random 3-colorable graph with high probability. This settles a problem of Blum and Spencer [6], who asked if one can design an algorithm that works almost surely for p - polylog(n)=n. (Here, and in what follows, almost surely always means: with probability that approaches 1 as n tends to infinity). The algorithm uses the spectral properties of the graph and is based on the fact that almost surely a rather accurate approximation of the color classes can be read from the eigenvectors corresponding to the smallest two eigenvalues of the adjacency matrix of a large subgraph. This approximation can then be improved to yield a proper coloring. The algorithm can be easily extended to the case of k-colorable graphs, for any fixed k, and to various models of random regular 3-colorable graphs. We implemented our algorithm and tested it for hundreds of graphs drawn at random from the distribution of G 3n;p;3 . Experiments show that our algorithm performs very well in practice. The running time is a few minutes on graphs with up to 100000 nodes, and the range of edge probabilities on which the algorithm is successful is in fact even larger than what our analysis predicts. 1.1 The model There are several possible models for random k-colorable graphs. See [8] for some of these models and the relation between them. Our results hold for most of these models, but it is convenient to focus on one, which will simplify the presentation. Let V be a fixed set of kn labelled vertices. For a real kn;p;k be the random graph on the set of vertices V obtained as follows; first, split the vertices of V arbitrarily into k color classes W each of cardinality n. Next, for each u and v that lie in distinct color classes, choose uv to be an edge, randomly and independently, with probability p. The input to our algorithm is a graph G kn;p;k obtained as above, and the algorithm succeeds to color it if it finds a proper k coloring. Here we are interested in fixed k - 3 and large n. We say that an algorithm colors G kn;p;k almost surely if the probability that a randomly chosen graph as above is properly colored by the algorithm tends to one as n tends to infinity. Note that we consider here deterministic algorithms, and the above statement means that the algorithm succeeds to color almost all random graphs generated as above. A closely related model to the one given above is the model in which we do not insist that the color classes have equal sizes. In this model one first splits the set of vertices into k disjoint color classes by letting each vertex choose its color randomly, independently and uniformly among the possibilities. Next, one chooses every pair of vertices of distinct color classes to be an edge with probability p. All our results hold for both models, and we focus on the first one as it is more convenient. To simplify the presentation, we restrict our attention to the case graphs, since the results for this case easily extend to every fixed k. In addition, we make no attempt to optimize the constants and assume, whenever this is needed, that c is a sufficiently large contant, and the number of vertices 3n is sufficiently large. 1.2 The algorithm Here is a description of the algorithm, which consists of three phases. Given a graph E), define be the graph obtained from G by deleting all edges incident to a vertex of degree greater than 5d. Denote by A the adjacency matrix of G 0 , i.e., the 3n by 3n matrix (a uv ) u;v2V defined by a It is well known that since A is symmetric it has real eigenvalues - 1 and an orthonormal basis of eigenvectors e 1 . The crucial point is that almost surely one can deduce a good approximation of the coloring of G from e 3n\Gamma1 and e 3n . Note that there are several efficient algorithms to compute the eigenvalues and the eigenvectors of symmetric matrices (cf., e.g., [17]) and hence e 3n\Gamma1 and e 3n can certainly be calculated in polynomial time. For the rest of the algorithm, we will deal with G rather than G 0 . Let t be a non-zero linear combination of e 3n\Gamma1 and e 3n whose median is zero, that is, the number of positive components of t as well as the number of its negative components are both at most 3n=2. (It is easy to see that such a combination always exists and can be found efficiently.) Suppose also that t is normalized so that it's l 2 -norm is 2n. 1=2g. This is an approximation for the coloring, which will be improved in the second phase by iterations, and then in the third phase to obtain a proper 3-coloring. In iteration i of the second phase, ne, construct the color classes V i 3 as follows. For every vertex v of G, let N(v) denote the set of all its neighbors in G. In the i-th iteration, color v by the least popular color of its neighbors in the previous iteration. That is, put v in V i is the minimum among the three quantities l where equalities are broken arbitrarily. We will show that the three sets V q correctly color all but vertices. The third phase consists of two stages. First, repeatedly uncolor every vertex colored j that has less than d=2 neighbors (in G) colored l, for some l 2 f1; 2; 3g \Gamma fjg. Then, if the graph induced on the set of uncolored vertices has a connected component of size larger than log 3 n, the algorithm fails. Otherwise, find a coloring of every component consistent with the rest of the graph using brute force exhaustive search. If the algorithm cannot find such a coloring, it fails. Our main result is the following. Theorem 1.1 If p ? c=n, where c is a sufficiently large constant, the algorithm produces a proper 3-coloring of G with probability The intuition behind the algorithm is as follows. Suppose every vertex in G had exactly d neighbors in every color class other than its own. Then G G. Let F be the 2-dimensional subspace of all vectors are constant on every color class, and whose sum is 0. A simple calculation (as observed in [1]) shows that any non-zero element of F is an eigenvector of A with eigenvalue \Gammad. Moreover, if E is the union of random matchings, one can show that \Gammad is almost surely the smallest eigenvalue of A and that F is precisely the eigenspace corresponding to \Gammad. Thus, any linear combination t of e 3n\Gamma1 and e 3n is constant on every color class. If the median of t is 0 and its l 2 -norm is 2n, then t takes the values 0, 1 or \Gamma1 depending on the color class, and the coloring obtained after phase 1 of the algorithm is a proper coloring. In the model specified in Subsection 1.1 these regularity assumptions do not hold, but every vertex has the same expected number of neighbors in every color class other than its own. This is why phase 1 gives only an approximation of the coloring and phases 2 and 3 are needed to get a proper coloring. We prove Theorem 1.1 in the next two sections. We use the fact that almost surely the largest eigenvalue of G 0 is at least (1 \Gamma 2 \Gamma\Omega\Gamma d) )2d, and that its two smallest eigenvalues are at most \Gamma(1 \Gamma d) )d and all other eigenvalues are in absolute value O( d). The proof of this result is based on a proper modification of techniques developed by Friedman, Kahn and Szemer'edi in [11], and is deferred to Section 3. We show in Section 2 that it implies that each of the two eigenvectors corresponding to the two smallest eigenvalues is close to a vector which is a constant on every color class, where the sum of these three constants is zero. This suffices to show that the sets V 0 reasonably good approximation to the coloring of G, with high probability. Theorem 1.1 can then be proved by applying the expansion properties of the graph G (that hold almost surely) to show that the iteration process above converges quickly to a proper coloring of a predefined large subgraph H of G. The uncoloring procedure will uncolor all vertices which are wrongly colored, but will not affect the subgraph H . We then conclude by showing that the largest connected component of the induced subgraph of G on V \Gamma H is of logarithmic size almost surely, thereby showing that the brute-force search on the set of uncolored vertices terminates in polynomial time. We present our implementation results in Section 4. Section 5 contains some concluding remarks together with possible extensions and results for related models of random graphs. 2 The proof of the main result E) be a random 3-colorable graph generated according to the model described above. Denote by W 1 3 the three color classes of vertices of G. Let G 0 be the graph obtained from G by deleting all edges adjacent to vertices of degree greater than 5d, and let A be the adjacency matrix of G 0 . Denote by - the eigenvalues of A, and by 3n the corresponding eigenvectors, chosen so that they form an orthonormal basis of R 3n . In this section we first show that the approximate coloring produced by the algorithm using the eigenvectors e 3n\Gamma1 and e 3n is rather accurate almost surely. Then we exhibit a large subgraph H and show that, almost surely, the iterative procedure for improving the coloring colors H correctly. We then show that the third phase finds a proper coloring of G in polynomial time, almost surely. We use the following statement, whose proof is relegated to Section 3. Proposition 2.1 In the above notation, almost surely, d) )d and d) for all 2. Remark. One can show that, when 2.1 would not hold if we were dealing with the spectrum of G rather than that of G 0 , since the graph G is likely to contain many vertices of degree ?? d, and in this case the assertion of (iii) does not hold for the eigenvalues of G. 2.1 The properties of the last two eigenvectors We show in this subsection that the eigenvectors e 3n\Gamma1 and e 3n are almost constant on every color class. For this, we exhibit two orthogonal vectors constant on every color class which, roughly speaking, are close to being eigenvectors corresponding to \Gammad. Let be the vector defined by x be the vector defined by y We denote by jjf jj the l 2 -norm of a vector f . Lemma 2.2 Almost surely there are two vectors are both linear combinations of e and e 3n . Proof. We use the following lemma, whose proof is given below. Lemma 2.3 Almost surely: We prove the existence of ffi as above. The proof of the existence of ffl is analogous. Let We show that the coefficients c are small compared to jjyjj. Indeed, 3n where the last inequality follows from parts (i) and (iii) of Proposition 2.1. Define By (1) and Lemma 2.3 it follows that jjffijj O(n=d). on the other hand, y \Gamma ffi is a linear combination of e 3n\Gamma1 and e 3n . 2 Note that it was crucial to the proof of Lemma 2.2 that, almost surely, rather as is the case for some vectors in f\Gamma1; 0; 1g 3n . Proof of Lemma 2.3 To prove the first bound, observe that it suffices to show that the sum of squares of the coordinates of dI)y on W 1 is O(nd) almost surely, as the sums on W 2 and W 3 can be bounded similarly. The expectation of the vector dI)y is the null vector, and the expectation of the square of each coordinate of dI)y is O(d), by a standard calculation. This is because each coordinate of dI)y is the sum of n independent random variables, each with mean 0 and variance O(d=n). This implies that the expected value of the sum of squares of the coordinates of dI)y on W 1 is O(nd). Similarly, the expectation of the fourth power of each coordinate of dI)y is O(d 2 ). Hence, the variance of the square of each coordinate is O(d 2 ). However, the coordinates of dI)y on W 1 are independent random variables, and hence the variance of the sum of the squares of the W 1 coordinates is equal to the sum of the variances, which is O(nd 2 ). The first bound can now be deduced from Chebyshev's Inequality. The second bound can be shown in a similar manner. We omit the details. 2 The vectors are independent since they are nearly orthogonal. Indeed, if O n=d Thus Therefore, by the above lemma, the two vectors 3ne 3ne 3n can be written as linear combinations of x \Gamma ffl and y \Gamma ffi. Moreover, the coefficients in these linear combinations are all O(1) in absolute value. This is because are nearly orthogonal, and the l 2 -norm of each of the four vectors 3ne 3ne 3n is \Theta( p n). More precisely, if one of the vectors 3ne 3ne 3n is written as by the triangle inequality, n) which, by a calculation similar to the one above, implies that thus ff and fi are O(1). on the other hand, the coefficients of the vector t defined in subsection 1.2 along the vectors e 3n\Gamma1 and e 3n are at most 2n. It follows that the vector t defined in the algorithm is also a linear combination of the vectors coefficients whose absolute values are both O(1). Since both x and y belong to the vector space F defined in the proof of Proposition 2.1, this implies that be the value of f on W i , for 1 - i - 3. Assume without loss of generality that ff 1 - ff 2 - ff 3 . Since jjjjj O(n=d), at most O(n=d) of the coordinates of are greater than 0:01 in absolute value. This implies that jff 2 j - 1=4, because otherwise at least O(n=d) coordinates of t would have the same sign, contradicting the fact that 0 is a median of t. As ff 1 implies that ff 1 ? 3=4 and ff 3 ! \Gamma3=4. Therefore, the coloring defined by the sets V 0 agrees with the original coloring of G on all but at most O(n=d) ! 0:001n coordinates. 2.2 The iterative procedure Denote by H the subset of V obtained as follows. First, set H to be the set of vertices having at most 1:01d neighbors in G in each color class. Then, repeatedly, delete any vertex in H having less than 0:99d neighbors in H in some color class (other than its own.) Thus, each vertex in H has roughly d neighbors in H in each color class other than its own. Proposition 2.4 Almost surely, by the end of the second phase of the algorithm, all vertices in H are properly colored. To prove Proposition 2.4, we need the following lemma. Lemma 2.5 Almost surely, there are no two subsets of vertices U and W of V such that jU j - 0:001n, and every vertex v of W has at least d=4 neighbors in U . Proof. Note that if there are such two (not necessarily disjoint) subsets U and W , then the number of edges joining vertices of U and W is at least djW j=8. Therefore, by a standard calculation, the probability that there exist such two subsets is at most 3n !/ 3n !/ di=8 d) ):If a vertex in H is colored incorrectly at the end of iteration i of the algorithm in phase 2 (i.e. if it is colored j and does not belong to W j ), it must have more than d=4 neighbors in H colored incorrectly at the end of iteration To see this, observe that any vertex of H has at most neighbors outside H , and hence if it has at most d=4 wrongly colored neighbors in H , it must have at least 0:99d \Gamma d=4 ? d=2 neighbors of each color other than its correct color and at most d=4 0:04d neighbors of its correct color. By repeatedly applying the property asserted by the above lemma with U being the set of vertices of H whose colors in the end of the iteration are incorrect, we deduce that the number of incorrectly colored vertices decreases by a factor of two (at least) in each iteration, implying that all vertices of H will be correctly colored after dlog 2 ne iterations. This completes the proof of Proposition 2.4. We note that by being more careful one can show that O(log d n) iterations suffice here, but since this only slightly decreases the running time we do not prove the stronger statement here. 2 A standard probabilistic argument based on the Chernoff bound (see, for example, [2, Appendix A]) shows that almost surely if p - fi log n=n, where fi is a suitably large constant. Thus, it follows from Proposition 2.4 that the algorithm almost surely properly colors the graph by the end of Phase 2 if p - fi log n=n. For two sets of vertices X and Z, let e(X; Z) denote the number of edges (u; v) 2 E, with and v 2 Z. Lemma 2.6 There exists a constant fl ? 0 such that almost surely the following holds. (i) For any two distinct color classes V 1 and V 2 , and any subset X of V 1 and any subset Y of V 2 , (ii) If J is the set of vertices having more than 1:01d neighbors in G in some color class, then Proof For any subset X of is the sum of independent Bernoulli variables. By standard Chernoff bounds, the probability that there exist two color classes V 1 and V 2 , a subset X of V 1 and a subset Y of V 2 such that jX is at most/ ffln Therefore, (i) holds almost surely if fl is a sufficiently small constant. A similar reasoning applies to (ii). Therefore, both (i) and (ii) hold if fl is a sufficiently small constant. 2 Lemma 2.7 Almost surely, H has at least (1 \Gamma 2 \Gamma\Omega\Gamma d) )n vertices in every color class. Proof. It suffices to show that there are at most 7 \Delta 2 \Gammafl d n vertices outside H . Assume for contradiction that this is not true. Recall that H is obtained by first deleting all the vertices in J , and then by a deletion process in which vertices with less than 0:99d neighbors in the other color classes of H are deleted repeatedly. By Lemma 2.6 jJ almost surely, and so at least 6 \Delta 2 \Gammafl d n vertices have been deleted because they had less than 0:99d neighbors in H in some color class (other than their own.) Consider the first time during the deletion process where there exists a subset X of a color class V i of cardinality 2 \Gammafl d n, and a j 2 f1; 2; 3g \Gamma fig such that every vertex of X has been deleted because it had less than 0:99d neighbors in the remaining subset of Y be the set of vertices of V j deleted so far. Then jY j. Note that every vertex in X has less than 0:99d neighbors in We therefore get a contradiction by applying Lemma 2.6 to (X; Y ). 2 2.3 The third phase We need the following lemma, which is an immediate consequence of Lemma 2.5. Lemma 2.8 Almost surely, there exists no subset U of V of size at most 0:001n such that the graph induced on U has minimum degree at least d=2. Lemma 2.9 Almost surely, by the end of the uncoloring procedure in Phase 3 of the algorithm, all vertices of H remain colored, and all colored vertices are properly colored, i.e. any vertex colored i belongs to W i . (We assume, of course, that the numbering of the colors is chosen appropriately). Proof. By Proposition 2.4 almost surely all vertices of H are properly colored by the end of Phase 2. Since every vertex of H has at least 0:99d neighbors (in H) in each color class other than its own, all vertices of H remain colored. Moreover, if a vertex is wrongly colored at the end of the uncoloring procedure, then it has at least d=2 wrongly colored neighbors. Assume for contradiction that there exists a wrongly colored vertex at the end of the uncoloring procedure. Then the subgraph induced on the set of wrongly colored vertices has minimum degree at least d=2, and hence it must have at least 0:001n vertices by Lemma 2.8. But, since it does not intersect H , it has at most 2 \Gamma\Omega\Gamma d) n vertices by Lemma 2.7, leading to a contradiction. 2 In order to complete the proof of correctness of the algorithm, it remains to show that almost surely every connected component of the graph induced on the set of uncolored vertices is of size at most log 3 n. We prove this fact in the rest of this section. We note that it is easy to replace the term log 3 n by O( log 3 n d ), but for our purposes the above estimate suffices. Note also that if some of these components are actually components of the original graph G, as for such value of p the graph G is almost surely disconnected (and has many isolated vertices). Lemma 2.10 Let K be a graph, partition of the vertices of K into three disjoint subsets, i an integer, and L the set of vertices of K that remain after repeatedly deleting the vertices having less than i neighbors in V 1 , V 2 or V 3 . Then the set L does not depend on the order in which vertices are deleted. Proof Let L be the set of vertices that remain after a deletion process according to a given order. Consider a deletion process according to a different order. Since every vertex in L has at least neighbors in no vertex in L will be deleted in the second deletion process (otherwise, we get a contradiction by considering the first vertex in L deleted.) Therefore, the set of vertices that remain after the second deletion process contains L, and thus equals L by symmetry. 2 Lemma 2.10 implies that H does not depend on the order in which vertices are deleted. Proposition 2.11 Almost surely the largest connected component of the graph induced on has at most log 3 n vertices. Proof. Let T be a fixed tree on log 3 n vertices of V all of whose edges have their two endpoints in distinct color classes W i , Our objective is to estimate the probability that G contains T as a subgraph that does not intersect H , and show that this probability is sufficiently small to ensure that almost surely the above will not occur for any T . This property would certainly hold were a random subset of V of cardinality 2 \Gamma\Omega\Gamma d) n. Indeed, if this were the case, the probability that G contains T as a subgraph that does not intersect H would be upper bounded by the probability 2 \Gamma\Omega\Gamma djT that T is a subset of times the probability (d=n) jT j\Gamma1 that T is a subgraph of G. This bound is sufficiently small for our needs. Although H is not a random subset of V , we will be able to show a similar bound on the probability that G contains T as a subgraph that does not intersect H . To simplify the notation, we let T denote the set of edges of the tree. Let V (T ) be the set of vertices of T , and let I be the subset of all vertices degree in T is at most 4. Since T contains jV be the subset of V obtained by the following procedure, which resembles that of producing H (but depends on to be the set of vertices having at most 1:01d \Gamma 4 neighbors in G in each color class V i . Then delete from H 0 all vertices of V (T repeatedly, delete any vertex in having less than 0:99d neighbors in H 0 in some color class (other than its own.) Lemma 2.12 Let F be a set of edges, each having endpoints in distinct color classes W i , W j . Let be the set obtained by replacing E by F [T in our definition of H, and H 0 be the set obtained by replacing E by F in our definition of H 0 . Then H Proof. First, we show that the initial value of H 0 obtained after deleting the vertices with more than 1:01d \Gamma 4 neighbors in a color class of G and after deleting the vertices in V (T is a subset of the initial value of H(F [T ). Indeed, let v be any vertex that does not belong to the initial value of H(F [ T ), i.e. v has more than 1:01d neighbors in some color class of (V; F [ T ). We distinguish two cases: 1. I . In this case, v does not belong to the initial value of H 0 2. I . Then v is incident with at most 4 edges of T , and so it has more than 1:01d \Gamma 4 neighbors in some color class in (V; F ). In both cases, v does not belong to the initial value of H 0 This implies the assertion of the lemma, since the initial value of H 0 subgraph of the initial value of H(F [ T ) and hence, by Lemma 2.10, any vertex which will be deleted in the deletion process for constructing H will be deleted in the corresponding deletion process for producing H 0 as well. 2 Lemma 2.13 Pr [T is a subgraph of G and V (T is a subgraph of G] Pr Proof. It suffices to show that is a subgraph of G] - Pr But, by Lemma 2.12, F :I"H(F[T)=; Pr Pr is a subgraph of G is a subgraph of G]; where F ranges over the sets of edges with endpoints in different color classes, and F 0 ranges over those sets that do not intersect T . The third equation follows by regrouping the edge-sets F according to F noting (the obvious fact) that, for a given set F 0 that does not intersect T , the probability that F such that is equal to The fourth equation follows from the independence of the events F 0 and T is a subgraph of G. 2 Returning to the proof of Proposition 2.11 we first note that we can assume without loss of generality that d - fi log n, for some constant fi ? 0 (otherwise .) If this inequality holds by modifying the arguments in the proof of Lemma 2.7, one can show that each of the graphs (corresponding to the various choices of V (T at most 2 d) n vertices in each color class, with probability at least 1 \Gamma 2 \Gamman \Theta(1) . Since the distribution of H 0 depends only on V (T (assuming the W i 's are fixed), it is not difficult to show that this implies that Pr [I " H at most 2 \Gamma\Omega\Gamma djIj) . Since jI j - jV (T )j=2 and since the probability that T is a subgraph of G is precisely (d=n) jV (T )j\Gamma1 we conclude, by Lemma 2.13, that the probability that there exists some T of size log 3 n which is a connected component of the induced subgraph of G on H is at most\Gamma\Omega\Gamma d log 3 n) (d=n) log 3 multiplied by the number of possible trees of this size, which is 3n log 3 n (log 3 n) log 3 Therefore, the required probability is bounded by 3n log 3 n (log 3 n) log 3 n\Gamma2 2 \Gamma\Omega\Gamma d log 3 n) ( d d) ); completing the proof. 2 3 Bounding the eigenvalues In this section, we prove Proposition 2.1. Let 3 be as in Section 2. We start with the following lemma. Lemma 3.1 There exists a constant fi ? 0 such that, almost surely, for any subset X of 2 \Gammafi d n vertices, Proof As in the proof of Lemma 2.6, the probability that there exists a subset X of cardinality ffln such that e(X; is at most 3n ffln if log(1=ffl) ! d=b, where b is a sufficiently large constant. Therefore, if fi is a sufficiently small constant, this probability goes to 0 as n goes to infinity. 2 Proof of Proposition 2.1 Parts (i) and (ii) are simple. By the variational definition of eigenvalues (see [19, p. 99]), - 1 is simply the maximum of x t Ax=(x t x) where the maximum is taken over all nonzero vectors x. Therefore, by taking x to be the all 1 vector we obtain the well known result that - 1 is at least the average degree of G 0 . By the known estimates for Binomial distributions, the average degree of G is (1 + o(1))2d. on the other hand, Lemma 3.1 can be used to show that as it easily implies that the number of vertices of degree greater than 5d in each color class of G is almost surely less than 2 \Gammafi d n. Hence, the average degree of G 0 is at least (1 \Gamma 2 \Gamma\Omega\Gamma d) )2d. This proves (i). The proof of (ii) is similar. It is known [19, p. 101] that where the minimum is taken over all two dimensional subspaces F of R 3n . Let F denote the 2- dimensional subspace of all vectors denotes the number of edges of G 0 between W i and W j . Almost surely e 0 d) )nd for all 1 follows that x t Ax=(x t x) - \Gamma(1 \Gamma 2 \Gamma\Omega\Gamma d) )d almost surely for all x 2 F , implying that - 3n - d) )d, and establishing (ii). The proof of (iii) is more complicated. Its assertion for somewhat bigger p (for example, for can be deduced from the arguments of [10]. To prove it for the graph G 0 and p - c=n we use the basic approach of Kahn and Szemer'edi in [11], where the authors show that the second largest eigenvalue in absolute value of a random d-regular graph is almost surely O( d). (See also [9] for a different proof.) Since in our case the graph is not regular a few modifications are needed. Our starting point is again the variational definition of the eigenvalues, from which we will deduce that it suffices to show that almost surely the following holds. Lemma 3.2 Let S be the set of all unit vectors d) for all x 2 S. The matrix A consists of nine blocks arising from the partition of its rows and columns according to the classes W j . It is clearly sufficient to show that the contribution of each block to the sum x t Ax is bounded, in absolute value, by O( d). This, together with a simple argument based on ffl-nets (see [11], Proposition 2.1) can be used to show that Lemma 3.2 follows from the following statement. denote the set of all vectors x of length n every coordinate of which is an integral multiple of ffl= n, where the sum of coordinates is zero and the l 2 -norm is at most 1. Let B be a random n by n matrix with 0; 1 entries, where each entry of B, randomly and independently, is 1 with probability d=n. Lemma 3.3 If d exceeds a sufficiently large absolute constant then almost surely, jx t Byj - O( d) for every x; y 2 T for which x if the corresponding row of B has more than 5d nonzero entries and y if the corresponding column of B has more than 5d nonzero entries. The last lemma is proved, as in [11], by separately bounding the contribution of terms x u y v with small absolute values and the contribution of similar terms with large absolute values. Here is a description of the details that differ from those that appear in [11]. Let C denote the set of all pairs (u; v) with jx u y d=n and let . As in [11] one can show that the absolute value of the expectation of X is at most d. Next one has to show that with high probability X does not deviate from its expectation by more than c d. This is different (and in fact, somewhat easier) than the corresponding result in [11], since here we are dealing with independent random choices. It is convenient to use the following variant of the Chernoff bound. Lemma 3.4 Let a am be (not necessarily positive) reals, and let Z be the random variable randomly and independently, to be 1 with probability p and 0 with probability ce c pD for some positive constants c; S. Then Pr For the proof, one first proves the following. Lemma 3.5 Let c be a positive real. Then for every x - c, Proof. Define Therefore, f 00 (x) - 0 for all x - c and as f 0 shows that f 0 implying that f(x) is nonincreasing for x - 0 and nondecreasing for Proof of Lemma 3.4. e c pD , then, by assumption, -a i - c for all i. Therefore, by the above lemma, E(e -Z Y [pe Y Y e c e c a 2 Therefore, Applying the same argument to the random variable defined with respect to the reals \Gammaa i , the assertion of the lemma follows. 2 Using Lemma 3.4 it is not difficult to deduce that almost surely the contribution of the pairs in C to jx t Byj is O( d). This is because we can simply apply the lemma with with the a i 's being all the terms x u y v where (u; ce c d for some c ? 0. Since here jSa ce c p d ce c pD, we conclude that for every fixed vectors x and y in T , the probability that X deviates from its expectation (which is O( d)) by more than ce c d is smaller than 2e \Gammac 2 e c n=2 , and since the cardinality of T is only b n for some absolute constant can choose c so that X would almost surely not deviate from its expectation by more than ce c p d. The contribution of the terms x u y v whose absolute values exceed d=n can be bounded by following the arguments of [11], with a minor modification arising from the fact that the maximum number of ones in a row (or column) of B can exceed d (but can never exceed 5d in a row or a column in which the corresponding coordinates x u or y v are nonzero). We sketch the argument below. We start with the following lemma. Lemma 3.6 There exists a constant C such that, with high probability, for any distinct color classes any subset U of V 1 and any subset W of V 2 such that jU j - jW j, at least one of the following two conditions hold: 2. e 0 is the number of edges in G 0 between U and W , and -(U; W jd=n is the expected number of edges in G between U and W . Proof. Condition 1 is clearly satisfied if jW j - n=2, since the maximum degree in G 0 is at most 5d. So we can assume without loss of generality that Give two subsets U and W satisfying the requirements of the lemma, define to be the unique positive real number such that fi-(U; W ) log will be determined later.) Condition 2 is equivalent to e 0 (U; W ) - fi-(U; W ). Thus U; W violate Condition 1 as well as Condition 2 only if e 0 Hence, by standard Chernoff bounds, the probability of this event is at most e \Gammafl absolute constant Denoting jW j=n by b, the probability that there exist two subsets U and W that do not satisfy either condition is at mostX b: bn integer -n=2 bn b: bn integer -n=2 if C is a sufficiently large constant. 2 Kahn and Szemer'edi [11] show that for any d-regular graph satisfying the conditions of Lemma 3.6 (without restriction on the ranges of U and W ), the contribution of the terms x u y v whose absolute values exceed d=n is O( d). Up to replacing some occurences of d by 5d, the same proof shows that, for any 3-colorable graph of maximum degree 5d satisfying the conditions of Lemma 3.6, the contribution of the terms x u y v whose absolute values exceed d=n is O( d). This implies the assertion of Lemma 3.3, which implies Lemma 3.2. To deduce (iii), we need the following lemma. Lemma 3.7 Let F denote, as before, the 2-dimensional subspace of all vectors satisfying x almost surely, for all f 2 F we have Proof. Let x; y be as in the proof of Lemma 2.3. Note that x t and that both jjxjj 2 and jjyjj 2 are \Theta(n). Thus every vector f 2 F can be expressed as the sum of two orthogonal vectors x 0 and proportional to x and y respectively. Lemma 2.3 shows that implies that Similarly, it can be shown that We conclude the proof of the lemma using the triangle inequality We now show that - 2 - O( d) by using the formula - H ranges over the linear subspaces of R 3n of codimension 1. Indeed, let H be the set of vectors whose sum of coordinates is 0. Any x 2 H is of the form f and s is a multiple of a vector in S, and so As As Number of vertices d 1000 12 100000 8 Figure 1: Implementation results. O( djjsjjjjf O( O( This implies the desired upper bound on - 2 . The bound j- 3n\Gamma2 j - O( d) can be deduced from similar arguments, namely by showing that d), for any x 2 F ? . This completes the proof of Proposition 2.1. 2 4 Implementation and Experimental Results. We have implemented the following tuned version of our algorithm. The first two phases are as described in Section 1. In the third phase, we find the minimum i such that, after repeatedly un- coloring every vertex colored j that has less than i neighbors colored l, for some l 2 f1; 2; 3g \Gamma fjg, the algorithm can find a proper coloring using brute force exhaustive search on every component of uncolored vertices. If the brute force search takes more steps than the first phase (up to a multiplicative constant), the algorithm fails. Otherwise, it outputs a legal coloring. The eigenvectors e 3n and e 3n\Gamma1 are calculated approximately using an iterative procedure. The coordinates of the initial vectors are independent random variables uniformly chosen in [0; 1]. The range of values of p where the algorithm succeeded was in fact considerably larger than what our analysis predicts. Figure 1 shows some values of the parameters for which we tested our algorithm. For each of these parameters, the algorithm was run on more than a hundred graphs drawn from the corresponding distribution, and found successfully a proper coloring for all these tests. The running time was a few minutes on a Sun SPARCstation 2 for the largest graphs. The algorithm failed for some graphs drawn from distributions with smaller integral values of d than the one in the corresponding row. Note that the number of vertices is not a multiple of 3; the size of one color class exceeds the others by one. Concluding remarks 1. There are many heuristic graph algorithms based on spectral techniques, but very few rigorous proofs of correctness for any of those in a reasonable model of random graphs. Our main result here provides such an example. Another example is the algorithm of Boppana [7], who designed an algorithm for graph bisection based on eigenvalues, and showed that it finds the best bisection almost surely in an appropriately defined model of random graphs with a relatively small bisection width. Aspvall and Gilbert [1] gave a heuristic for graph coloring based on eigenvectors of the adjacency matrix, and showed that their heuristic optimally colors complete 3-partite graphs as well as certain other classes of graphs with regular structure. 2. By modifying some of the arguments of Section 2 we can show that if p is somewhat bigger (p - log 3 n=n suffices) then almost surely the initial coloring V 0 i that is computed from the eigenvectors e 3n\Gamma1 and e 3n in the first phase of our algorithm is completely correct. In this case the last two phases of the algorithm are not needed. By refining the argument in Subsection 2.2, it can also be shown that if log n=n the third phase of the algorithm is not needed, and the coloring obtained by the end of the second phase will almost surely be the correct one. 3. We can show that a variant of our algorithm finds, almost surely, a proper coloring in the model of random regular 3-colorable graphs in which one chooses randomly d perfect matchings between each pair of distinct color classes, when d is a sufficiently large absolute constant. Here, in fact, the proof is simpler, as the smallest two eigenvalues (and their corresponding are known precisely, as noted in Subsection 1.2. 4. The results easily extend to the model in which each vertex first picks a color randomly, independently and uniformly, among the three possibilities, and next every pair of vertices of distinct colors becomes an edge with probability p (? c=n). 5. If positive constant c, it is not difficult to show that almost surely G does not have any subgraph with minimum degree at least 3, and hence it is easy to 3-color it by a greedy-type (linear time) algorithm. For values of p which are bigger than this c=n but satisfy n=n), the graph G is almost surely disconnected, and has a unique component of \Omega\Gamma n) vertices, which is called the giant component in the study of random graphs (see, e.g., [2], [4]). All other components are almost surely sparse, i.e., contain no subgraph with minimum degree at least 3, and can thus be easily colored in total linear time. Our approach here suffices to find, almost surely, a proper 3-coloring of the giant component (and hence of the whole graph) for all p - c=n, where c is a sufficiently large absolute constant, and there are possible modifications of it that may even work for all values of p. At the moment, however, we are unable to obtain an algorithm that provably works for all values of p almost surely. Note that, for any constant c, if p ! c=n then the greedy algorithm will almost surely color G 3n;p;3 with a constant number of colors. Thus, our result implies that G 3n;p;3 can be almost surely colored in polynomial time with a constant number of colors for all values of p. 6. Our basic approach easily extends to k-colorable graphs, for every fixed k, as follows. Phase 2 and Phase 3 of the algorithm are essentially the same as in the case needs to be modified to extract an approximation of the coloring. Let e i , i - 1, be an eigenvector of corresponding to its ith largest eigenvalue (replace 5d by 5kd in the definition of G 0 .) Find vectors kn in and z, let W ffl be the set of vertices whose coordinates in x i are in (z \Gamma ffl; z ffl). If, for some i and z, both jW ffl k j and deviate from n by at most fi k n=d, where ffl k and fi k are constants depending on k, color the elements in W ffl k with a new color and delete them from the graph. Repeat this process until the number of vertices left is O(n=d), and color the remaining vertices in an arbitrary manner. 7. The existence of an approximation algorithm based on the spectral method for coloring arbitrary graphs is a question that deserves further investigation (which we do not address here.) Recently, improved approximation algorithms for graph coloring have been obtained using semidefinite programming [14], [5]. Acknowledgement We thank two anonymous referees for several suggestions that improved the presentation of the paper. --R Graph coloring using eigenvalue decomposition The Probabilistic Method Some tools for approximate 3-coloring Journal of Algorithms 19 Eigenvalues and graph bisection: An average case analysis The solution of some random NP-Hard problems in polynomial expected time on the second eigenvalue and random walks in random d-regular graphs on the second eigenvalue in random regular graphs Computers and intractability: a guide to the theory of NP-completeness Reducibility among combinatorial problems Approximate graph coloring by semidefinite program- ming Expected behavior of graph colouring algorithms A randomised 3-colouring algorithm A First Course in Numerical Analysis Almost all k-colorable graphs are easy to color The Algebraic Eigenvalue Problem --TR --CTR David Eppstein, Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, p.329-337, January 07-09, 2001, Washington, D.C., United States Amin Coja-Oghlan, A spectral heuristic for bisecting random graphs, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia Amin Coja-oghlan, The Lovsz Number of Random Graphs, Combinatorics, Probability and Computing, v.14 n.4, p.439-465, July 2005 Michael Krivelevich , Dan Vilenchik, Solving random satisfiable 3CNF formulas in expected polynomial time, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.454-463, January 22-26, 2006, Miami, Florida Richard Beigel , David Eppstein, 3-coloring in time O(1.3289n), Journal of Algorithms, v.54 n.2, p.168-204, February 2005 Abraham D. Flaxman , Alan M. Frieze, The diameter of randomly perturbed digraphs and some applications, Random Structures & Algorithms, v.30 n.4, p.484-504, July 2007 Amin Coja-oghlan , Andreas Goerdt , Andr Lanka, Strong Refutation Heuristics for Random k-SAT, Combinatorics, Probability and Computing, v.16 n.1, p.5-28, January 2007 Amin Coja-Oghlan , Andreas Goerdt , Andr Lanka , Frank Schdlich, Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2k-SAT, Theoretical Computer Science, v.329 n.1-3, p.1-45, 13 December 2004 Paul Beame , Joseph Culberson , David Mitchell , Cristopher Moore, The resolution complexity of random graphk-colorability, Discrete Applied Mathematics, v.153 n.1, p.25-47, 1 December 2005

graph eigenvalues;graph coloring;algorithms;random graphs

270948

Vienna-Fortran/HPF Extensions for Sparse and Irregular Problems and Their Compilation.

AbstractVienna Fortran, High Performance Fortran (HPF), and other data parallel languages have been introduced to allow the programming of massively parallel distributed-memory machines (DMMP) at a relatively high level of abstraction, based on the SPMD paradigm. Their main features include directives to express the distribution of data and computations across the processors of a machine. In this paper, we use Vienna-Fortran as a general framework for dealing with sparse data structures. We describe new methods for the representation and distribution of such data on DMMPs, and propose simple language features that permit the user to characterize a matrix as "sparse" and specify the associated representation. Together with the data distribution for the matrix, this enables the compiler and runtime system to translate sequential sparse code into explicitly parallel message-passing code. We develop new compilation and runtime techniques, which focus on achieving storage economy and reducing communication overhead in the target program. The overall result is a powerful mechanism for dealing efficiently with sparse matrices in data parallel languages and their compilers for DMMPs.

Introduction During the past few years, significant efforts have been undertaken by academia, government laboratories and industry to define high-level extensions of standard programming languages, in particular Fortran, to facilitate data parallel programming on a wide range of parallel architectures without sacrificing performance. Important results of this work are Vienna Fortran [10, 28], Fortran D [15] and High Performance Fortran (HPF) [18], which is intended to become a de-facto standard. These languages extend Fortran 77 and Fortran 90 with directives for specifying alignment and distribution of a program's data among the processors, thus enabling the programmer to influence the locality of computation whilst retaining a single thread of control and a global name space. The low-level task of mapping the computation to the target processors in the framework of the Single-Program-Multiple-Data (SPMD) model, and of inserting communication for non-local accesses is left to the compiler. HPF-1, the original version of High Performance Fortran, focussed its attention on regular com- putations, and on providing a set of basic distributions (block, cyclic and replication). Although the approved extensions of HPF-2 include facilities for expressing irregular distributions using INDIRECT , no special support for sparse data structures has been proposed. In this paper, we consider the specific requirements for sparse computations as they arise in a variety of problem areas such as molecular dynamics, matrix decompositions, solution of linear systems, image reconstruction and many others. In order to parallelize sequential sparse codes effectively, three fundamental issues must be addressed: 1. We must distribute the data structures typically used in such codes. 2. It is necessary to generalize the representation of sparse matrices on a single processor to distributed-memory machines in such a way that the savings in memory and computation are also achieved in the parallel code. 3. The compiler must be able to adapt the global computation to the local computation on each processor, resolving the additional complexity that sparse methods introduce. This paper presents an approach to solve these three problems. First, a new data type has been introduced in the Vienna Fortran language for representing sparse matrices. Then, data distributions have been explicitly designed to map this data type onto the processors in such a way that we can exploit the locality of sparse computations and preserve a compact representation of matrices and vectors, thereby obtaining an efficient workload balance and minimizing communication. Some experiments in parallelizing sparse codes by hand [2], not only confirmed the suitability of these distributions, but also the excessive amount of time spent during the development and debugging stages of manual parallelization. This encouraged us to build a compiler to specify these algorithms in a high-level data-parallel language. In this way, new elements were introduced to Vienna Fortran to extend its functionality and expressivity for irregular problems. Subsequently, compiler and runtime techniques were developed to enable specific optimizations to handle typical features of sparse code, including indirect array accesses and the appearance of array elements in loop bounds. The result is a powerful mechanism for storing and manipulating sparse matrices, which can be used in a data-parallel compiler to generate efficient SPMD programs for irregular codes of this kind. In this paper, we assume the representation and distribution of sparse data to be invariant. However, the fact the representation for sparse data is computed at runtime simplifies the additional support for handling more complex features such as dynamic redistribution or the matrix fill-in (i.e., the runtime insertion of additional non-zero elements into the sparse matrix). The rest of the paper is organized as follows. Section 2 introduces some basic formalism and background for handling sparse matrices. Section 3 presents several data distributions for sparse problems; section 4 describes new directives for the specification of these distributions in the Vienna Fortran language. Section 5 and 6 respectively outline the runtime support and compilation technology required for the implementation of these features. Sections 7 and 8 present experimental results; we finish in Sections 9 and 10 with a discussion of related work and conclusions. Representing Sparse Matrices on Distributed-Memory Machines A matrix is called sparse if only a small number of its elements are non-zero. A range of methods have been developed which enable sparse computations to be performed with considerable savings in terms of both memory and computation [16]. Solution schemes are often optimized to take advantage of the structure within the matrix. This has consequences for parallelization. Firstly, we want to retain as much of these savings as possible in the parallel code. Secondly, in order to achieve a good load balance at runtime, it is necessary to understand how this can be achieved in terms of the data structures which occur in sparse problem formulations. In this section, we discuss methods for representing sparse matrices on distributed-memory machines. We assume here that the reader is familiar with the basic distribution functions of Vienna Fortran and HPF [10, 28, 18], namely BLOCK and CYCLIC(K). Throughout this paper, we denote the set of target processors by PROCS and assume that the data is being distributed to a two-dimensional mesh PROCS of X Y processors, numbered from 0 in each dimension. Specifically, we assume that the Vienna Fortran or HPF code will include the following declaration: Note that this abstract processor declaration does not imply any specific topology of the actual processor interconnection network. 2.1 Basic Notation and Terminology Each array A is associated with an index domain which we denote by I A . A (replication-free) distribution of A is a total function that maps each array element to a processor which becomes the owner of the element and, in this capacity, stores the element in its local memory. Further, for any processor p 2 PROCS we denote by - A (p) the set of all elements of A which are local to p; this is called the local segment of A in p. In the following, we will assume that A is a two-dimensional real array representing a sparse matrix, and declared with index domain I Most of the notation introduced below will be related to A, without explicitly reflecting this dependence. We begin by defining a set of auxiliary functions. Definition 1 1. The symbolic matrix associated with A is a total predicate, ftrue; falseg, such that for all i 2 I, 2. ff :=j fiji 2 I -(i)g j specifies the number of matrix elements with a non-zero value. is a bijective enumeration of A, which numbers all elements of A consecutively in some order, starting with 1. 4. Assume an enumeration - to be selected. I is a total function such that is the t-th index under - which is associated with a non-zero element of A. By default, we will use an enumeration in the following which numbers the elements of A row- wise, i.e., we will assume -(i; When a sparse matrix is mapped to a distributed-memory machine, our approach will require two kinds of information to be specified by the user. These are: 1. The representation of the sparse matrix on a single processor. This is called a sparse format. 2. The distribution of the matrix across the processors of the machine. In this context, the concept of a distribution is used as if the matrix was dense. The combination of a sparse format with a distribution will be called a distributed sparse representation of the matrix. 2.2 Sparse Formats Before we discuss data distribution strategies for sparse data, we must understand how such data is usually represented on a single processor. Numerous storage formats have been proposed in sparse-matrix literature; for our work we have used the very commonly used CRS (Compressed Row Storage) format; the same approach can be extended to CCS (Compress Column Storage) just swapping rows and columns in the text. In the following, we will for simplicity only consider sparse matrices with real elements; this can be immediately generalized to include other element types such as logical, integer, and complex. Definition 2 The Compressed Row Storage (CRS) sparse format is determined by a triple of functions, (DA,CO,RO): total, the data function, is defined by DA(t) := A(-(t)) for all t . 1 denotes the set of real numbers. 2. CO total, the column function, is defined by CO(t) := -(t):2 for all total, the row function, is defined as follows: (a) Let i denote an arbitrary row number. Then RO(i) := at least one t with the specified property exists; otherwise RO(i) := RO(i 1). These three functions can be represented in an obvious way as vectors of ff real numbers (the data vector), ff column numbers (the column vector), and numbers in the range row vector) respectively (see Figure 1.b). The data vector stores the non-zero values of the matrix, as they are traversed in a row-wise fashion. The column vector stores the column indices of the elements in the data vector. Finally, the row vector stores the indices in the data vector that correspond to the first non-zero element of each row (if such an element exists). The storage savings achieved by this approach is usually significant. Instead of storing n m elements, we need only locations. Sparse matrix algorithms designed for the CRS format typically use a nested loop, with the outer loop iterating over the rows of the matrix and an inner loop iterating over the non-zeros in that row (see examples in Section 4). Matrix elements are identified using a two-dimensional index set, say (i,jj) , where i denotes the i-th row of the matrix and jj denotes the jj-th non-zero in that row. The matrix element referred to by (i,jj) is the one at row number R, column number CO(RO(i)+jj) and has the non-zero value stored in DA(RO(i)+jj) . The heavy use of indirect accesses that sparse representations require introduces a major source of complexity and inefficiency when parallelizing these codes on distributed-memory machines. A number of optimizations will be presented later on to overcome this. 3 Distributed Sparse Representations Let A denote a sparse matrix as discussed above, and ffi be an associated distribution. A distributed sparse representation for A results from combining ffi with a sparse format. This is to be understood as follows: The distribution ffi is interpreted in the conventional sense, i.e., as 2 For a pair z = (x; y) of numbers, y. if A were a dense Fortran array: ffi determines a locality function, -, which, for each p 2 PROCS, specifies the local segment -(p). Each -(p) is again a sparse matrix. The distributed sparse representation of A is then obtained by constructing a representation of the elements in -(p), based on the given sparse format. That is, DA, CO, and RO are automatically converted to the sets of vectors DA p , CO p , and RO p , p 2 PROCS. Hence the parallel code will save computation and storage using the very same mechanisms that were applied in the original program. For the sparse format, we use CRS to illustrate our ideas. For the data distributions, we introduce two different schemes in subsequent sections, both decomposing the sparse global domain into as many sparse local domains as required. 3.1 Multiple Recursive Decomposition (MRD) Common approaches for partitioning unstructured meshes while keeping neighborhood properties are based upon coordinate bisection, graph bisection and spectral bisection [8, 19]. Spectral bisection minimizes communication, but requires huge tables to store the boundaries of each local region and an expensive algorithm to compute it. Graph bisection is algorithmically less expen- sive, but also requires large data structures. Coordinate bisection significantly tends to reduce the time to compute the partition at the expense of a slight increase in communication time. Binary Recursive Decomposition (BRD), as proposed by Berger and Bokhari [4], belongs to the last of these categories. BRD specifies a distribution algorithm where the sparse matrix A is recursively bisected, alternating vertical and horizontal partitioning steps until there are as many submatrices as processors. Each submatrix is mapped to a unique processor. A more flexible variant of this algorithm produces partitions in which the shapes of the individual rectangles are optimized with respect to a user-determined function [7]. In this section, we define Multiple Recursive Decomposition (MRD), a generalization of the BRD method, which also improves the communication structure of the code. We again assume the processor array to be declared as be the prime factor decomposition for X Y , ordered in such a way that the prime factors of X, sorted in descending order, come first and are followed by the factors of Y, sorted in the same fashion. The MRD distribution method produces an X Y partition of matrix A in k steps, recursively performing horizontal divisions of the matrix for the prime factors of X, and vertical ones for the prime factors of Y : Matrix A is partitioned into P 1 submatrices in such a way that the non-zero elements are spread across the submatrices as evenly as possible. When a submatrix is partitioned horizontally, any rows with no non-zero entries which are not uniquely assigned to either partition are included in the lower one; in a vertical step, such columns are assigned to the right partition. Each submatrix resulting from step i-1 is partitioned into P i submatrices using the same criteria as before. When this process terminates, we have created Q k submatrices. We enumerate these consecutively from 0 to X using a horizontal ordering scheme. Now the submatrix with number r mapped to processor PROCS(r,s). 2 This distribution defines the local segment of each processor as a rectangular matrix which preserves neighborhood properties and achieves a good load balance (see Figure 2). The fact that we perform all horizontal partitioning steps before the vertical ones reduces the number of possible neighbors that a submatrix may have, and hence simplifies further analysis to be performed by the compiler and runtime system. When combined with the CRS representation for the local segments, the MRD distribution produces the MRD-CRS distributed sparse representation. This can be inmediately generalized to other storage formats; however, since we only use CRS here to illustrate our ideas, we refer to MRD-CRS as MRD itself. 3.2 BRS Distributed Sparse Representation The second strategy is based on a cyclic distribution (see Figure 4.a). This does not retain locality of access; as in the regular case, it is suitable where the workload is not spread evenly across the matrix nor presents periodicity, or when the density of the matrix varies over time. Many common algorithms are of this nature, including sparse matrix decompositions (LU, Cholesky, QR, WZ) and some image reconstruction algorithms. In this section, we assume both dimensions of A 0 to be distributed cyclically with block length 1 (see Figure 4.b). Several variants for the representation of the distribution segment in this context are described in the literature, including the MM, ESS and BBS methods [1]. Here we consider a CRS sparse format, which results in the BRS (Block Row Scatter) distributed sparse representation. A very similar distributed representation is that of BCS (Block Column Scatter) [26], where the sparse format is compressed by columns. just changing rows by columns and vice versa. The mapping which is established by the BRS choice requires complex auxiliary structures and translation schemes within the compiler. However, if such data are used together with cyclically-distributed dense arrays, then the structures are properly aligned, leading to savings in communication. Extensions for the Support of Sparse Matrix Computation 4.1 Language Considerations This section proposes new language features for the specification of sparse data in a data parallel language. Clearly, block and cyclic distributions as offered in HPF-1 are not adequate for this purpose; on the other hand, INDIRECT distributions [15, 28], which have been included in the approved extensions of HPF-2, do not allow the specification of the structure inherent in distributed sparse representations, and thus introduce unnecessary complexity in memory consumption and execution time. Our proposal makes this structure explicit by appropriate new language elements, which can be seen as providing a special syntax for an important special case of a user-defined distribution function as defined in Vienna Fortran or HPF+ [11, 12]. The new language features provide the following information to the compiler and the runtime system: ffl The name, index domain, and element type of the sparse matrix are declared. This is done using regular Fortran declaration syntax. This array will not actually appear in the original code, since it is represented by a set of arrays, but the name introduced here is referred to when specifying the distribution. ffl An annotation is specified which declares the array as being SPARSE and provides information on the representation of the array. This includes the names of the auxiliary vectors in the order data, column and row, which are not declared explicitly in the program. Their sizes are determined implicitly from the matrix index domain. ffl The DYNAMIC attribute is used in a manner analogous to its meaning in Vienna Fortran and HPF: if it is specified, then the distributed sparse representation will be determined dynamically, as a result of executing a DISTRIBUTE statement. Otherwise, all components of the distributed sparse representation can be constructed at the time the declaration is processed. Often, this information will be contained in a file whose name will be indicated in this annotation. In addition, when the input sparse matrix is not available at compile-time, it must be read from a file in some standard format and distributed at runtime. The name of this file may be provided to the compiler in an additional directive. Concrete examples for typical sparse codes illustrating details of the syntax (as well as its HPF are given in Figures 5 and 6. 4.2 Solution of a Sparse Linear System There is a wide range of techniques to solve linear systems. Among them, iterative methods use successive approximations to obtain more accurated solutions at each step. The Conjugate Gradient (CG) [3] is the oldest, best known, and most effective of the nonstationary iterative methods for symmetric positive definite systems. The convergence process can be speeded up by using a preconditionator before computing the CG itself. We include in Figure 5 the data-parallel code for the unpreconditioned CG algorithm, which involves one matrix-vector product, three vector updates, and two inner products per iteration. The input is the coefficient matrix, A, and the vector of scalars B; also, an initial estimation must be computed for Xvec, the solution vector. With all these elements, the initial residuals, R, are defined. Then, in every iteration, two inner products are performed in order to update scalars that are defined to make the sequences fulfill certain orthogonality conditions; at the end of each iteration, both solution and residual vectors are updated. 4.3 Lanczos Algorithm Figure 6 illustrates an algorithm in extended HPF for the tridiagonalization of a matrix with the Lanczos method [24]. We use a new directive, indicated by !NSD$, to specify the required declarative information. The execution of the DISTRIBUTE directive results in the computation of the distributed sparse representation. After that point, the matrix can be legally accessed in the program, where several matrix-vector and vector-vector operations are performed to compute the diagonals of the output matrix. 5 Runtime analysis Based on the language extensions introduced above, this section shows how access to sparse data can be efficiently translated from Vienna Fortran or HPF to explicitly parallel message passing code in the context of the data parallel SPMD paradigm. In the rest of the paper, we assume that the input matrix is not available at compile-time. Under such an assumption, the matrix distribution has to be postponed until runtime and this obviously enforces the global to local index translation to be also performed at runtime. To parallelize codes that use indirect addressing, compilers typically use an inspector-executor strategy [22], where each loop accessing to distributed variables is tranformed by inserting an additional preprocessing loop, called an inspector. The inspector translates the global addresses accessed by the indirection into a (processor, offset) tuple describing the location of the element, and computes a communication schedule. The executor stage then uses the preprocessed information to fetch the non-local elements and to access distributed data using the translated addresses. An obvious penalty of using the inspector-executor paradigm is the runtime overhead introduced by each inspector stage, which can become significant when multiple levels of indirection are used to access distributed arrays. As we have seen, this is frequently the case for sparse-matrix algorithms using compact storage formats such as CRS. For example, the Xvec(DA(RO(i)+jj) reference encountered in Figure 5 requires three preprocessing steps - one to access the distributed array RO , a second to access DA , and yet a third to access Xvec . We pay special attention to this issue in this section and outline an efficient solution for its parallelization. 5.1 The SAR approach Though they are based on the inspector-executor paradigm, our solution for translating CRS- like sparse indices at runtime within data-parallel compilers significantly reduces both time and memory overhead compared to the standard and general-purpose CHAOS library [23]. This technique, that we have called "Sparse Array Rolling" (SAR), encapsulates into a small descriptor information of how the input matrix is distributed across the processors. This allows us to determine the (processor, offset) location of a sparse matrix element without having to plod through the distributed auxiliary array data-structures, thus saving the preprocessing time required by all the intermediate arrays. Figure 7 provides an overview of the SAR solution approach. The distribution of the matrix represented in CRS format is carried out by a partitioner, the routine responsible for computing the domain decomposition giving as output the distributed representation as well as its associated descriptor. This descriptor can be indexed through the translation process using the row number the non-zero index ( X ) to locate the processor and offset at which the matrix element is stored. When the element is found to be non-local, the dereference process assigns an address in local memory where the element is placed once fetched. The executor stage uses the preprocessed information inside a couple of gather/scatter routines which fetch the marked non-local elements and place them in their assigned locations. Finally, the loop computation accesses the distributed data using the translated addresses. The efficiency of the translation function and the memory overheads of the descriptor are largely dependent on how the matrix is distributed. The following sections provide these details for each of the distributions studied in this paper. 5.2 MRD descriptor and translation The MRD distribution maps a rectangular portion of the dense index space (n \Theta m) onto a virtual processor space (X \Theta Y ). Its corresponding descriptor is replicated on each of the processors and consists of two parts: A vector partH stores the row numbers at which the X horizontal partitions are made and a two dimensional array partV , of size n \Theta Y , which keeps track of the number of non-zero elements in each vertical partition for each row. Example 1 For the MRD distributed matrix in Figure 3, the corresponding descriptor replicated among the processors is the following: partH(1)=8 denotes that the horizontal partition is made at row 8. Each row has two vertical partitions. The values of partV(9,1:2)= 2,3 say that the first section of row 9 has two non-zero elements while the second section has one (3 We assume partH(0)=1, partH(X)=N+1, partV(k,0)=0 and for all 1 - k - N . Given any non-zero element identified by (i,jj) we can perform a translation by means of its descriptor to determine the processor that owns the non-zero element. Assuming that processors are identified by their position (myR; myC) in the X \Theta Y virtual processor mesh, the values myR and myC of the processor that owns the element satisfies the following inequalities. Searching for the right myR and myC that satisfies these inequalities can require a search space of size X \Theta Y . The search is optimized by first checking to see if the element is local by plugging in the local processor's values for myR and myC. Assuming a high degree of locality, this check frequently succeeds immediately. When it fails, a binary search mechanism is employed. The offset at which the element is located is (Xvec-partV(i,myC) . Thus the column number of the element (i,jj) can be found at CO((Xvec-partV(i,myC)) on processor (myR; myC), and the non-zero value can be accessed from DA((Xvec-partV(i,myC)) on the same processor, without requiring any communication or additional preprocessing steps. 5.3 BRS descriptor and translation Unlike MRD, the BRS descriptor is different on each processor. Each processor (myR,myC) has elements from n=X rows mapped onto it. The BRS descriptor stores for each local row of the matrix, an entry for every non-zero element on that row, regardless of the whether that element is mapped locally or not. For those elements that are local, the entry stores the local index into DA . For non-local elements, the entry stores the global column number of that element in the original matrix. To distinguish between the local entries and non-local entries, we swap the sign of local indices so that they become negative. The actual data-structure used is a CRS-like two-vector representation - a vector called CS stores the entries of all the elements that are mapped to local rows, while another vector, RA , stores the indices at which each row starts in CS . Example 2 For the sparse matrix A and its partitioning showed in Figure 4, the values of CS and RA on processor (0,0) are the following: CS(1)=2 says that the element (53) is stored on global column 2, and is non-local. CS(2)=-1 signifies that the element (19) is mapped locally and is stored at local index 1. The remaining entries have similar interpretations. The processor owning the element R,X is identified as follows. First, the local row is identified using the simple formula X). The entry for the element is obtained using CS(RA(r)+jj) . If M is negative, then it implies that the element is local and can be accessed at DA(-M) . If it is positive, then we have the global row i and column number M of the element. This implies that the processor owning the element is We save the [i,jj] indices in a list of indices that are marked for later retrieval from processor Q. During the executor, a Gather routine will send these [i,jj] indices to Q, where a similar translation process is repeated; this time, however, the element will be locally found and sent to the requesting processor. 6 Compilation This section describes the compiler implementation within the Vienna Fortran Compilation System (VFCS). The input to the compiler is a Vienna-Fortran code extended with the sparse annotations described in Section 4. The compilation process results in a Fortran 77 code enhanced with message-passing routines as well as the runtime support already discussed in the previous section. The tool was structured in a set of modules such as shown in Figure 8. We now describe each module separately. 6.1 Front-End The first module is the only part of the tool which interacts with the declaration part of the program. It is responsible for: 1. The scanning and parsing of the new language elements presented in Section 4. These operations generate the abstract syntax tree for such annotations and a table summarizing all the compile-time information extracted from them. Once this table is built, the sparse directives are not needed any more and the compiler proceeds to remove them from the code. 2. The insertion of declarations for the local vectors and the auxiliary variables that the target code and runtime support utilize. 6.2 Parallelizer At this stage, the compiler first scans the code searching for the sparse references and extracting all the information available at compile-time (i.e., indirections, syntax of the indices, loops and conditionals inside of which the reference is, etcetera). All this information is then organized in a database for its later lookup through the parallelization process. Once this is done, the loop decomposition starts. The goal here consists of distributing the workload of the source code as evenly as possible among the processors. This task turns out to be particularly complex for a compiler when handling sparse codes, mainly because of the frequent use of indirections when accessing the sparse data and the frequent use of sparse references in loop bounds. In such cases, multiple queries to distributed sparse data are required by all processors in order to determine their own iteration space, leading to a large number of communications. To overcome this problem, we address the problem in a different way: Rather than trying to access the actual sparse values requested from the loop headers, we apply loop transformations that not only determine the local iteration space but also map such values into semantically equivalent information in the local distribution descriptor. This approach has the double advantage of reusing the compiler auxiliary structures while ensuring the locality of all the accesses performed in the loop boundaries. The result is a much faster mechanism for accessing data at no extra memory overhead. For the MRD case, for example, arrays partH and partV determine the local region for the data in a sparse matrix based on global coordinates. In this way, the loop partitioning can be driven with very similar strategies to those of BLOCK, with the only difference of the regions having a different size (but similar workload) which is determined at runtime when the descriptor is generated from the runtime support. For the BRS case the solution is not that straightforward. Let us take as example the Conjugate Gradient (CG) algorithm in Figure 5, where the dense vectors are distributed by dense CYCLIC and the sparse matrix follows a BRS scheme. Note that most of the CG loops only refer to dense structures. Its decomposition can be performed just enforcing the stride of the loops to be the number of processors on which the data dimension traversed by the loop is distributed. This is because consecutive local data in CYCLIC are always separated by a constant distance in terms of the global coordinates. However, when references to sparse vectors are included in the loops, this fact is only true for the first matrix dimension; for second one, the actual sparsity degree of the matrix determines the distance of consecutive data in terms of their global columns. Since this becomes unpredictable at compile-time (recall our assumption of not having the sparse matrix pattern available until runtime), a runtime checking defined as a function of the BRS distribution descriptor needs to be inserted for loops traversing the second matrix dimension to be successfully parallelized. This checking can be eventually moved to the inspector phase when the executor is computed through a number of iterations, thus decreasing the overall runtime overhead (see transformation in the final code generation, Figure 10). Figure 9 provides a code excerpt that outlines the loop decomposition performed within the VFCS for the two sparse loops in Figure 5. RA and CS are the vectors for the BRS descriptor on processor with coordinates (myR, myC). RA stores indices in the very same way than the local RO does, but considering all the elements placed in global rows i \Theta X + myR for any given local row i. A CYCLIC-like approach is followed to extract the local iterations from the first loop and then RA traverses all the elements in the second loop and CS delimits its local iterations in a subsequent IF. Note the different criteria followed for parallelizing both loops. In the first loop, the well-known owner's compute rule is applied. In the second loop, though, the underlying idea is to avoid the replication of the computation by first calculating a local partial sum given by the local elements and then accumulate all the values in a single reduction phase. In this way, computations are distributed based on the owner of every single DA and P value for a given index K , which makes them match always on the same processor. This achieves a complete locality 6.3 Back-End Once the workload has been assigned to each processor, the compiler enters in its last stage, whose output is the target SPMD code. To reach this goal, the code has to be transformed into inspector and executor phases for each of its loops. Figure shows the final SPMD code for the sparse loops parallelized in Figure 9. Overall, the next sequence of steps are carried out in this compiler module: 1. An inspector loop is inserted prior to each loop computation. The header for this loop is obtained through the syntax tree after the parallelization and statements inside the loop are generated to collect the indices to distributed arrays into auxiliary vectors. These vectors are then taken as input to the translation process. 2. Calls to the translate , dereference and scatter/gather routines are placed between the inspector and executor loops to complete the runtime job. 3. References to distributed variables in the executor loop are sintactically changed to be indexed by the translation functions produced as output in the inspector (see functions f and g in Figure 10). 4. Some additional I/O routines must be inserted at the beginning of the execution part to merge on each processor the local data and descriptors. In our SAR scheme, this is done by the partitioner routine. 7 Evaluation of Distribution Methods The choice of distribution strategy for the matrix is crucial in determining performance. It controls the data locality and load balance of the executor, the preprocessing costs of the inspector, and the memory overhead of the runtime support. In this section we discuss how BRS and MRD distributions affect each of these aspects for the particular case of the sparse loops in the Conjugate Gradient algorithm. To account for the effects of different sparsity structures we chose two very different matrices coming from the Harwell-Boeing collection [14], where they are identified as PSMIGR1 and BCSSTK29. The former contains population migration data and is relatively dense, whereas the latter is a very sparse matrix used in large eigenvalue problems. Matrix characteristics are summarized in Table 1. 7.1 Communication Volume in Executor Table 2 shows the communication volume in executor for 16 processors in a 4 \Theta 4 processors mesh when computing the sparse loops of the CG algorithm. This communication is necessary for accumulating the local partial products in the array Q . Such an operation has been implemented like a typical reduction operation for all the local matrix rows over each of the processor rows We note two things: first, the relation between communication volume and the processor mesh configuration and second, the balance in the communication pattern (note that comparisons of communication volumes across the two matrices should be relative to their number of rows). In general, for a X \Theta Y processor mesh and a n \Theta m sparse matrix , the communication volume is roughly proportional to (n=X) \Theta log(Y ). Thus a 8 \Theta 2 processor mesh will have 4 times less total communication volume than a 4 \Theta 4 mesh. For BRS, each processor accumulates exactly the same amount of data, while for MRD, there are minor imbalances stemming from the slightly different sizes of the horizontal partitions (see Figure 11). Communication time in executor is showed in black in Figure 13. 7.2 Loop Partitioning and Workload Balance As explained in section 6.2, each iteration of the sparse loops in the Conjugate Gradient algorithm is mapped to the owner of the DA element accessed in that iteration. This results in perfect workload balance for the MRD case, since each processor owns an equal number of non-zeros. BRS workload balance relies on the random positioning of the elements, and except for pathological cases, it too results in very good load balance. Table 3 shows the Load Balance Index for BRS (maximum variation from average divided by its average). 7.3 Memory Overhead Vectors for storing the local submatrix on each processor require similar amounts of memory in both distributions. However, the distribution descriptor used by the runtime support can require substantially different amounts of memory. Table 4 summarizes these requirements. The first row indicates the expected memory overhead and the next two rows show the actual overhead in terms of the number of integers required. The "overhead" column represents the memory overhead as a percentage of the amount of memory required to store the local submatrix. Vectors partV and CS are responsible of most overhead of its distribution, since they keep track of the positions of the non-zero elements in the MRD and BRS respectively. This overhead is much higher for BRS because the CS vector stores the column numbers even for some of the off-processor non-zeros. The length of this vector can be reduced by using processor meshes with 8 Runtime evaluation This section describes our performance evaluation of the sparse loops of the Conjugate Gradient algorithm when parallelized using the VFCS compiler under the BRS and MRD especifications. Our intent was to study the effect of the distribution choice on inspector and executor performance within a data-parallel compiler. Finally, a manual version of the application was used as a baseline to determine the overhead of a semi-automatic parallelization. Our platform was an Intel Paragon using the NXLIB communication library. In our experi- ments, we do not account for the I/O time to read in the matrix and perform its distribution. 8.1 Inspector Cost Figure 12 shows the preprocessing costs for the sparse loops of the MRD and BRS versions of the CG algorithm on the two matrices. The preprocessing overheads do reduce with increasing parallelism, though the efficiencies drop at the high end. We also note that while BRS incurs higher preprocessing overheads than MRD, it also scales better. To understand the relative costs of BRS relative to MRD, recall that the BRS translation mechanism involves preprocessing all non-zeros in local rows, while the MRD dereferencing requires a binary search through the distribution descriptor only for the local non-zeros. Though it processes fewer elements the size of the MRD search space is proportional to the size of the processor mesh, so as processors are added, each translation requires a search over a larger space. Though it is not shown in the table, our measurements indicate that the BRS inspector is actually faster than MRD for more than 64 processors. 8.2 Executor Time Since both schemes distribute the nonzeros equally across processors we found that the computational section of the executor scaled very well for both distributions until 32 processors, after which the communication overheads start to reduce efficiency. Figure 13 which shows the executor time for the sparse loops of the two CG versions indicates good load balance. In fact, we find some cases of super-linear speedup, attributable to cache effects. The executor communication time is shown in dark in Figure 13. The BRS communication overhead remains essentially invariant across all processor sizes. This suggests that the overhead of the extra communication startups is offset by the reduced communication volume, maintaining the same total overhead. For MRD, the communication is much more unbalanced and this leads to much poorer scaling of the communication costs. Indeed, this effect is particularly apparent for BCSSTK29, where the redistribution is extremely unbalanced and becomes a severe bottleneck as the processor size is increased. 8.3 Comparison to Manual Parallelization The efficiency of a sparse code parallelized within the VFCS compiler depends largely on primary factors: ffl The distribution scheme selected for the parallelization, either MRD or BRS. ffl The sparsity rate of the input matrix. ffl The cost of the inspector phase to figure out the access pattern. On the other hand, we have seen that the parallelization of the sparse loops of the CG algorithm within the VFCS leads to a target code in which the executor does not perform any communication in the gather/scatter routines as a consequence of the full locality achieved by the data distribution, its local representation and the loop partitioning strategy. Apart from the actual computation, the executor only contains the communication for accumulating the local partial products, which is implemented in a reduction routine exactly as a programmer would do. Thus, the executor time becomes an accurated estimation of the efficiency that a smart programmer can attain and the additional cost of using an automatic compilation lies intirely in the preprocessing time (inspector loops plus subsequents runtime calls in Figure 10). Figure 14 tries to explain the impact of the major factors that influence the parallel efficiency while providing a comparison between the manual and the compiler-driven parallelization. Execution times for the compiler include the cost for a single inspector plus an executor per iteration, whereas for the manual version no inspector is required. As far as the distribution itself is concerned, Figure 14 shows that the BRS introduces a bigger overhead. This is a direct consequence of its more expensive inspector because of the slower global to local translation process. However, even in the BRS case, our overall results are quite efficient through a number of iterations: In practice, the convergence in the CG algorithm starts to exhibit a stationary behaviour after no less than one hundred iterations. By that time, the inspector cost has already been widely amortized and the total compiler overhead is always kept under 10% regardless of the input matrix, the distribution chosen and the number of processors in the parallel machine. With respect to the matrix sparsity, we can conclude that the higher the degree of sparsity in a matrix is, the better is the result produced by a compiler if compared to the manual version. The overall comparison against a manual parallelization also reflects the good scalability of the manual gain for a small number of iterations. Summarizing, we can say that the cost to be paid for an automatic parallelization is worthwhile as long as the algorithm can amortize the inspector costs through a minimum number of iterations. The remaining cost of the Conjugate Gradient algorithm lies in the multiple loops dealing with dense arrays distributed by CYCLIC. However, the computational weight of this part never goes over 10% of the total execution time. Even though the compiler efficiency is expected to be improved for such cases, its influence is minimum and does not lead to a significant variation in the full algorithm. Additional experiments to demonstrate the efficiency of our schemes have been developed by Trenas [24], who implemented a manual version of the Lanczos algorithm (see Figure 6) using PVM routines and the BRS scheme. 9 Related Work Programs designed to carry out a range of sparse algorithms in matrix algebra are outlined in [3]. All these codes require the optimizations described in this paper if efficient target code is to be generated for a parallel system. There are a variety of languages and compilers targeted at distributed memory multiprocessors ([28, 9, 15, 18]). Some of them do not attempt to deal with loops that arise in sparse or irregular computation. One approach, originating from Fortran D and Vienna Fortran, is based on INDIRECT data distributions and cannot express the structure of sparse data, resulting in memory and runtime overhead. The scheme proposed in this paper provides special syntax for a special class of user-defined data distributions, as proposed in Vienna Fortran and HPF+ [12]. On the other hand, in the area of the automatic parallelization, the most outstanding tools we know (Parafrase [20], Polaris [6]) are not intended to be a framework for the parallelization of sparse algorithms such as those addressed in our present work. The methods proposed by Saltz et al. for handling irregular problems consists in endowing the compiler with a runtime library [23] to facilitate the search and capture of data located in the distributed memory. The major drawback of this approach is the large number of messages that are generated as a consequence of accessing a distributed data addressing table, and its associated overhead of memory [17]. In order to enable the compiler to apply more optimizations and simplify the task of the programmer, Bik and Wijshoff [5] have implemented a restructuring compiler which automatically converts programs operating on dense matrices into sparse code. This method postpones the selection of a data structure until the compilation phase. Though more friendly to the end user, this approach has the risk of inefficiencies that can result from not allowing the programmer to choose the most appropriate sparse structures. Our way of dealing with this problem is very different: We define heuristics that perform an efficient mapping of the data and a language extension to describe the mapping in data parallel languages [18, 28]. We have produced and benchmarked a prototype compiler, integrated into the VFCS, that is able to generate efficient code for irregular kernels. Compiler transformations insert procedures to perform the runtime optimizations. The implementation is qualitatively different from the efforts cited above in a number of important respects, in particular with respect to its use of a new data type (sparse format) and data distributions (our distributed sparse representations) for irregular computation. The basic ideas in these distributions take into account the way in which sparse data are accessed and map the data in a pseudoregular way so that the compiler may perform a number of optimizations for sparse codes. More specifically, the pseudoregularity of our distributions allows us to describe the domain decomposition using a small descriptor which can, in addition, be accessed locally. This saves most of the memory overhead of distributed tables as well as the communication cost needed for its lookup. In general, application codes in irregular problems normally have code segments and loops with more complex access functions. The most advanced analysis technique, known as slicing analysis [13], deal with multiple levels of indirection by transforming code that contains such references to code that contains only a single level of indirection. However, the multiple communication phases still remain. The SAR technique implemented inside the sparse compiler is novel because it is able to handle multiple levels of indirection at the cost of a single translation. The key for attaining this goal consists of taking advantage of the compile-time information about the the semantic relations between the elements involved in the indirect accesses. Conclusions In this paper, sparse data distributions and specific language extensions have been proposed for data-parallel languages such as Vienna Fortran or HPF to improve their handling of sparse irregular computation. These features enable the translation of codes which use typical sparse coding techniques, without any necessity for rewriting. We show in some detail how such code may be translated so that the resulting code retains significant features of sequential sparse applications. In particular, the savings in memory and computation which are typical for these techniques are retained and can lead to high efficiency at run time. The data distributions have been designed to retain data locality when appropriate, support a good load balance, and avoid memory wastage. The compile time and run time support translates these into structures which permit a sparse representation of data on the processors of a parallel system. The language extensions required are minimal, yet sufficient to provide the compiler with the additional information needed for translation and optimization. A number of typical code kernels have been shown in this paper and in [26] to demonstrate the limited amount of effort required to port a sequential code of this kind into an extended HPF or Vienna Fortran. Our results demonstrate that the data distributions and language features proposed here supply enough information to store and access the data in distributed memory, as well as to perform the compiler optimizations which bring great savings in terms of memory and communication overhead. Low-level support for sparse problems has been described, proposing the implementation of an optimizing compiler that performs these translations. This compiler improves the functionality of data-parallel languages in irregular computations, overcoming a major weakness in this field. Runtime techniques are used in the context of inspector-executor paradigm. However, our set of low-level primitives differ from those used in several existing implementations in order to take advantage of the additional semantic information available in our approach. In particular, our runtime analysis is able to translate multiple indirect array accesses in a single phase and does not make use of expensive translation tables. The final result is an optimizing compiler able to generate efficient parallel code for these computations, very close to what can be expected from a manual parallelization and much faster in comparison to existing tools in this area. --R The Scheduling of Sparse Matrix-Vector Multiplication on a Massively Parallel DAP Computer A Partitioning Strategy for Nonuniform Problems on Multiprocessors Automatic Data Structure Selection and Transformation for Sparse Matrix Computations Massively Parallel Methods for Engineering and Science Problems Vienna Fortran Compilation System. Programming in Vienna Fortran. User Defined Mappings in Vienna For- tran Extending HPF For Advanced Data Parallel Applications. Index Array Flattening Through Program Transformations. Users' Guide for the Harwell-Boeing Sparse Matrix Collection Fortran D language specification Computer Solution of Large Sparse Positive Definite Sys- tems High Performance Language Specification. Numerical experiences with partitioning of unstructured meshes The Structure of Parafrase-2: an Advanced Parallelizing Compiler for C and Fortran Data distributions for sparse matrix vector multiplication solvers Parallel Algorithms for Eigenvalues Computation with Sparse Matrices Efficient Resolution of Sparse Indirections in Data-Parallel Compilers Evaluation of parallelization techniques for sparse applications Vienna Fortran - A language Specification Version 1.1 --TR --CTR Chun-Yuan Lin , Yeh-Ching Chung , Jen-Shiuh Liu, Efficient Data Compression Methods for Multidimensional Sparse Array Operations Based on the EKMR Scheme, IEEE Transactions on Computers, v.52 n.12, p.1640-1646, December Rong-Guey Chang , Tyng-Ruey Chuang , Jenq Kuen Lee, Efficient support of parallel sparse computation for array intrinsic functions of Fortran 90, Proceedings of the 12th international conference on Supercomputing, p.45-52, July 1998, Melbourne, Australia Gerardo Bandera , Manuel Ujaldn , Emilio L. Zapata, Compile and Run-Time Support for the Parallelization of Sparse Matrix Updating Algorithms, The Journal of Supercomputing, v.17 n.3, p.263-276, Nov. 2000 Manuel Ujaldon , Emilio L. Zapata, Efficient resolution of sparse indirections in data-parallel compilers, Proceedings of the 9th international conference on Supercomputing, p.117-126, July 03-07, 1995, Barcelona, Spain Roxane Adle , Marc Aiguier , Franck Delaplace, Toward an automatic parallelization of sparse matrix computations, Journal of Parallel and Distributed Computing, v.65 n.3, p.313-330, March 2005 V. Blanco , P. Gonzlez , J. C. Cabaleiro , D. B. Heras , T. F. Pena , J. J. Pombo , F. F. Rivera, Performance Prediction for Parallel Iterative Solvers, The Journal of Supercomputing, v.28 n.2, p.177-191, May 2004 Chun-Yuan Lin , Yeh-Ching Chung, Data distribution schemes of sparse arrays on distributed memory multicomputers, The Journal of Supercomputing, v.41 n.1, p.63-87, July 2007 Bradford L. Chamberlain , Lawrence Snyder, Array language support for parallel sparse computation, Proceedings of the 15th international conference on Supercomputing, p.133-145, June 2001, Sorrento, Italy Chun-Yuan Lin , Yeh-Ching Chung , Jen-Shiuh Liu, Efficient Data Distribution Schemes for EKMR-Based Sparse Arrays on Distributed Memory Multicomputers, The Journal of Supercomputing, v.34 n.3, p.291-313, December 2005 M. Garz , I. Garca, Approaches Based on Permutations for Partitioning Sparse Matrices on Multiprocessors, The Journal of Supercomputing, v.34 n.1, p.41-61, October 2005 Thomas L. Sterling , Hans P. Zima, Gilgamesh: a multithreaded processor-in-memory architecture for petaflops computing, Proceedings of the 2002 ACM/IEEE conference on Supercomputing, p.1-23, November 16, 2002, Baltimore, Maryland Ali Pinar , Cevdet Aykanat, Fast optimal load balancing algorithms for 1D partitioning, Journal of Parallel and Distributed Computing, v.64 n.8, p.974-996, August 2004 Bradford L. Chamberlain , Steven J. Deitz , Lawrence Snyder, A comparative study of the NAS MG benchmark across parallel languages and architectures, Proceedings of the 2000 ACM/IEEE conference on Supercomputing (CDROM), p.46-es, November 04-10, 2000, Dallas, Texas, United States Ken Kennedy , Charles Koelbel , Hans Zima, The rise and fall of High Performance Fortran: an historical object lesson, Proceedings of the third ACM SIGPLAN conference on History of programming languages, p.7-1-7-22, June 09-10, 2007, San Diego, California

sparse computation;data-parallel language and compiler;distributed-memory machines;runtime support

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Structuring Communication Software for Quality-of-Service Guarantees.

AbstractA growing number of real-time applications require quality-of-service (QoS) guarantees from the underlying communication subsystem. The communication subsystem (host and network) must support real-time communication services to provide the required QoS of these applications. In this paper, we propose architectural mechanisms for structuring host communication software to provide QoS guarantees. In particular, we present and evaluate a QoS-sensitive communication subsystem architecture for end hosts that provides real-time communication support for generic network hardware. This architecture provides services for managing communication resources for guaranteed-QoS (real-time) connections, such as admission control, traffic enforcement, buffer management, and CPU and link scheduling. The design of the architecture is based on three key goals: maintenance of QoS-guarantees on a per-connection basis, overload protection between established connections, and fairness in delivered performance to best-effort traffic.Using this architecture we implement real-time channels, a paradigm for real-time communication services in packet-switched networks. The proposed architecture features a process-per-channel model that associates a channel handler with each established channel. The model employed for handler execution is one of "cooperative" preemption, where an executing handler yields the CPU to a waiting higher-priority handler at well-defined preemption points. The architecture provides several configurable policies for protocol processing and overload protection. We present extensions to the admission control procedure for real-time channels to account for cooperative preemption and overlap between protocol processing and link transmission at a sending host. We evaluate the implementation to demonstrate the efficacy with which the architecture maintains QoS guarantees on outgoing traffic while adhering to the stated design goals. The evaluation also demonstrates the need for specific features and policies provided in the architecture. In subsequent work, we have refined this architecture and used it to realize a full-fledged guaranteed-QoS communication service that performs QoS-sensitive resource management for outgoing as well as incoming traffic.

Introduction Distributed multimedia applications (e.g., video conferenc- ing, video-on-demand, digital libraries) and distributed real-time command/control systems require certain quality-of- service (QoS) guarantees from the underlyingnetwork. QoS guarantees may be specified in terms of parameters such as the end-to-end delay, delay jitter, and bandwidth delivered on each connection; additional requirements regarding packet loss and in-order delivery can also be specified. To support these applications, the communication subsys- The work reported in this paper was supported in part by the National Science Foundation under grant MIP-9203895 and the Office of Naval Re-search under grants N00014-94-1-0229. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of NSF or ONR. tem in end hosts and the network must be designed to provide per-connection QoS guarantees. Assuming that the net-work provides appropriate support to establish and maintain guaranteed-QoS connections, we focus on the design of the host communication subsystem to maintain QoS guarantees. Protocol processing for large data transfers, common in multimedia applications, can be quite expensive. Resource management policies geared towards statistical fairness and/or time-sharing can introduce excessive interference between different connections, thus degrading the delivered QoS on individual connections. Since the local delay bound at a node may be fairly tight, the unpredictability and excessive delays due to interference between different connections may even result in QoS violations. This performance degradation can be eliminated by designing the communication subsystem to provide: (i) maintenance of QoS guarantees, (ii) overload protection via per-connection traffic enforcement, and (iii) fairness to best-effort traf- fic. These requirements together ensure that per-connection QoS guarantees are maintained as the number of connections or per-connection traffic load increases. In this paper, we propose and evaluate a QoS-sensitive communication subsystem architecture for guaranteed-QoS connections. Our focus is on the architectural mechanisms used within the communication subsystem to satisfy the QoS requirements of all connections, without undue degradation in performance of best-effort traffic (with no QoS guarantees). While the proposed architecture is applicable to other proposals for guaranteed-QoS connections [3], we focus on real-time channels, a paradigm for guaranteed-QoS communication services in packet-switched networks [16]. The architecture features a process-per-channel model for protocol processing, coordinated by a unique channel handler created on successful channel establishment. While the service within a channel is FIFO, QoS guarantees on multiple channels are provided via appropriate CPU scheduling of channel handlers and link scheduling of packet transmissions. Traffic isolation between channels is facilitated via per-channel traffic enforcement and interaction between the CPU and link schedulers. APPLICATION PROGRAMMING INTERFACE ADMISSION CONTROL CONNECTION API ENTRY/EXIT PACKET TRANSMISSION ADAPTER CPU SCHEDULER signal packet transmission completion (interrupt) (device driver operations) notification packet transmission initiation of (packet transmissions) packets (fragmentation, encapsulation) data (messages) feedback (buffering, queueing) suspend/yield resume (protocol processing) (a) Overall architecture (b) Protocol processing Figure 1. Desired software architecture. We have implemented this architecture using a modified x-kernel 3.1 [14] communication executive exercising complete control over a Motorola 68040 CPU. This configuration avoids any interference from computation or other operating system activities on the host, allowing us to focus on the communication subsystem. We evaluate the implementation under different traffic loads, and demonstrate the efficacy with which it maintains QoS guarantees on real-time channels and provides fair performance for best-effort traf- fic, even in the presence of ill-behaved real-time channels. For end-to-end guarantees, resource management within the communication subsystem must be integrated with that for applications. The proposed architecture is directly applicable if a portion of the host processing capacity can be reserved for communication-related activities [21, 17]. The proposed architectural extensions can be realized as a server with appropriate capacity reserves and/or execution priority. Our implementation is indeed such a server executing in a standalone configuration. More importantly, our approach decouples protocol processing priority from that of the ap- plication. We believe that the protocol processing priority of a connection must be derived from the QoS requirements, traffic characteristics, and run-time communication behavior of the application on that connection. Integration of the proposed architecture with resource management for applications will be addressed in a forthcoming paper. Section 2 discusses architectural requirements for guaranteed-QoS communication and provides a brief description of real-time channels. Section 3 presents a QoS-sensitive communication subsystem architecture realizing these requirements, and Section 4 describes its implementation. Section 5 experimentally evaluates the efficacy of the proposed architecture. Section 6 discusses related work and Section 7 concludes the paper. 2. Architectural requirements for guaranteed- QoS communication For guaranteed-QoS communication [3], we consider unidirectional data transfer, from source to sink via intermediate nodes, with data being delivered at the sink in the order in which it is generated at the source. Corrupted, delayed, or lost data is of little value; with a continuous flow of time-sensitive data, there is insufficient time for error recovery. Thus, we consider data transfer with unreliable-datagram semantics with no acknowledgements and retransmissions. To provide per-connection QoS guarantees, host communication resources must be managed in a QoS-sensitive fashion, i.e., according to the relative importance of the connections requesting service. Host communication resources include CPU bandwidth for protocol processing, link bandwidth for packet transmissions, and buffer space. Figure 1(a) illustrates a generic software architecture for guaranteed-QoS communication services at the host. The components constituting this architecture are as follows. Application programming interface (API): The API must export routines to set up and teardown guaranteed-QoS con- nections, and perform data transfer on these connections. Signalling and admission control: A signalling protocol is required to establish/tear down guaranteed-QoS connections across the communicating hosts, possibly via multiple network nodes. The communication subsystem must keep track of communication resources, perform admission control on new connection requests, and establish connection state to store connection specific information. Network transport: Protocols are needed for unidirectional (reliable and unreliable) data transfers. Traffic enforcement: This provides overload protection between established connections by forcing an application to conform to its traffic specification. This is required at the session level, and may also be required at the link level. Link access scheduling and link abstraction: Link band-width must be managed such that all active connections receive their promised QoS. This necessitates abstracting the link in terms of transmission delay and bandwidth, and scheduling all outgoing packets for network access. The minimum requirement for provision of QoS guarantees is that packet transmission time be bounded and predictable. Assuming support for signalling, we focus on the components involved in data transfer, namely, traffic enforcement, protocol processing and link transmission. In particular, we study architectural mechanisms for structuring host communication software to provide QoS guarantees. 2.1. QoS-sensitive data transport In Figure 1(b), an application presents the API with data (messages) to be transported on a guaranteed-QoS connec- tion. The API must allocate buffers for this data and queue it appropriately. Conformant data (as per the traffic specifica- tion) is forwarded for protocol processing and transmission. Maintenance of per-connection QoS guarantees: Protocol processing involves, at the very least, fragmentation of application messages, including transport and network layer encapsulation, into packets with length smaller than a certain maximum (typically the MTU of the attached network). Additional computationally intensive services (e.g., coding, compression, or checksums) may also be performed during protocol processing. QoS-sensitive allocation of processing bandwidth necessitates multiplexing the CPU amongst active connections under control of the CPU scheduler, which must provide deadline-based or priority-based policies for scheduling protocol processing on individual connections. Non-preemptive protocol processing on a connection implies that the CPU can be reallocated to another connection only after processing an entire message, resulting in a coarser temporal grain of multiplexing and making admission control less effective. More importantly, admission control must consider the largest possible message size (maximum number of bytes presented by the application in one request) across all connections, including best-effort traffic. While maximum message size for guaranteed-QoS connections can be derived from attributes such as frame size for multimedia applications, the same for best-effort traffic may not be known a priori. Thus, mechanisms to suspend and resume protocol processing on a connection are needed. Protocol processing on a connection may also need to be suspended if it has no available packet buffers. The packets generated via protocol processing cannot be directly transmitted on the link as that would result in FIFO (i.e., QoS-insensitive) consumption of link bandwidth. In- stead, they are forwarded to the link scheduler, which must provide QoS-sensitive policies for scheduling packet trans- missions. The link scheduler selects a packet and initiates packet transmission on the network adapter. Notification of packet transmission completion is relayed to the link scheduler so that another packet can be transmitted. The link scheduler must signal the CPU scheduler to resume protocol processing on a connection that was suspended earlier due to shortage of packet buffers. Overload protection via per-connection traffic enforce- ment: As mentioned earlier, only conformant data is forwarded for protocol processing and transmission. This is necessary since QoS guarantees are based on a connection's traffic specification; a connection violating its traffic specification should not be allowed to consume communication resources over and above those reserved for it. Traffic specification violations on one connection should not affect QoS guarantees on other connections and the performance delivered to best-effort traffic. Accordingly, the communication subsystem must police per-connection traffic; in general, each parameter constituting the traffic specification (e.g., rate, burst length) must be policed individually. An important issue is the handling of non-conformant traffic, which could be buffered (shaped) until it is conformant, provided with degraded QoS, treated as best-effort traffic, or dropped altogether. Under certain situations, such as buffer overflows, it may be necessary to block the application until buffer space becomes available, although this may interfere with the timing behavior of the application. The most appropriate policy, therefore, is application-dependent. Buffering non-conformant traffic till it becomes conformant makes protocol processing non-work-conserving since the CPU idles even when there is work available; the above discussion corresponds to this option. Alternately, protocol processing can be work-conserving, with CPU scheduling mechanisms ensuring QoS-sensitive allocation of CPU bandwidth to connections. Work-conserving protocol processing can potentially improve CPU utilization, since the CPU does not idle when there is work available. While the unused capacity can be utilized to execute other best-effort activities (such as background computations), one can also utilize this CPU bandwidth by processing non-conformant traffic, if any, assuming there is no pending best-effort traf- fic. This can free up CPU processing capacity for subsequent messages. In the absence of best-effort traffic, work-conserving protocol processing can also improve the average QoS delivered to individual connections, especially if link scheduling is work-conserving. Fairness to best-effort traffic: Best-effort traffic includes data transported by conventional protocols such as TCP and UDP, and signalling for guaranteed-QoS connections. It should not be unduly penalized by non-conformant real-time traffic, especially under work-conserving processing. 2.2. Real-time channels Several models have been proposed for guaranteed-QoS communication in packet-switched networks [3]. While the architectural mechanisms proposed in this paper are applicable to most of the proposed models, we focus on real-time channels [9, 16]. A real-time channel is a simplex, fixed- route, virtual connection between a source and destination host, with sequenced messages and associated performance guarantees on message delivery. It therefore conforms to the connection semantics mentioned earlier. Traffic and QoS Specification: Traffic generation on real-time channels is based on a linear bounded arrival process [8, 2] characterized by three parameters: maximum message size (Mmax bytes), maximum message rate (Rmax messages/second), and maximum burst size (Bmax mes- sages). The notion of logical arrival time is used to enforce a minimum separation I between messages on a real-time channel. This ensures that a channel does not use more resources than it reserved at the expense of other chan- nels. The QoS on a real-time channel is specified as the desired deterministic, worst-case bound on the end-to-end delay experienced by a message. See [16] for more details. Resource Management: Admission control for real-time channels is provided by Algorithm D order [16], which uses fixed-priority scheduling for computing the worst-case delay experienced by a channel at a link. Run-time link scheduling, on the other hand, is governed by a multi-class variation of the earliest-deadline-first (EDF) policy. 2.3. Performance related considerations To provide deterministic QoS guarantees on communica- tion, all processing costs and overheads involved in managing and using resources must be accounted for. Processing costs include the time required to process and transmit a message, while the overheads include preemption costs such as context switches and cache misses, costs of accessing ordered data structures, and handling of network inter- rupts. It is important to keep the overheads low and predictable (low variability) so that reasonable worst-case estimates can be obtained. Further, resource management policies must maximize the number of connections accepted for API ENTRY/EXIT HANDLER RUN QUEUE ASSIGNMENT CPU PACKET TRANSMISSION link scheduler message queue handler API ENTRY/EXIT HANDLER RUN QUEUE ASSIGNMENT CPU PACKET RECEPTION CHANNEL MESSAGE QUEUE ASSIGNMENT handler packet queue (a) Source host (b) Destination host Figure 2. Proposed architecture. service. In addition to processing costs and implementation overheads, factors that affect admissibility include the relative bandwidths of the CPU and link and any coupling between CPU and link bandwidth allocation. In a recent paper [19], we have studied the extent to which these factors affect admissibility in the context of real-time channels. 3. A QoS-sensitive communication architecture In the process-per-message model [23], a process or thread shepherds a message through the protocol stack. Besides eliminating extraneous context switches encountered in the process-per-protocol model [23], it also facilitates protocol processing to be scheduled according to a variety of policies, as opposed to the software-interrupt level processing in BSD Unix. However, the process-per-message model introduces additional complexity for supporting QoS guarantees. Creating a distinct thread to handle each message makes the number of active threads a function of the number of messages awaiting protocol processing on each channel. Not only does this consume kernel resources (such as process control blocks and kernel stacks), but it also increases scheduling overheads which are typically a function of the number of runnable threads in dynamic scheduling envi- ronments. More importantly, with a process-per-message model, it is relatively harder to maintain channel seman- tics, provide QoS guarantees, and perform per-channel traffic policing. For example, bursts on a channel get translated into "bursts" of processes in the scheduling queues, making it harder to police ill-behaved channels and ensure fairness to best-effort traffic. Further, scheduling overhead becomes unpredictable, making worst-case estimates either overly conservative or impossible to provide. Since QoS guarantees are specified on a per-channel ba- sis, it suffices to have a single thread coordinate access to resources for all messages on a given channel. We employ a process-per-channel model, which is a QoS-sensitive extension of the process-per-connection model [23]. In the process-per-channel model, protocol processing on each channel is coordinated by a unique channel handler, a lightweight thread created on successful establishment of the channel. With unique per-channel handlers, CPU scheduling overhead is only a function of the number of active channels, those with messages waiting to be trans- ported. Since the number of established channels, and hence the number of active channels, varies much more slowly compared to the number of messages outstanding on all active channels, CPU scheduling overhead is significantly more predictable. As we discuss later, a process-per-channel model also facilitates per-channel traffic enforcement. Fur- ther, since it reduces context switches and scheduling over- heads, this model is likely to provide good performance to connection-oriented best-effort traffic. Figure 2 depicts the key components of the proposed architecture at the source (transmitting) and destination (re- ceiving) hosts; only the components involved in data transfer are shown. Associated with each channel is a message queue, a FIFO queue of messages to be processed by the channel handler (at the source) or to be received by the application (at the destination). Each channel also has associated with it a packet queue, a FIFO queue of packets waiting to be transmitted by the link scheduler (at the source) or to be reassembled by the channel handler (at the destination). Transmission-side processing: In Figure 2(a), invocation of message transmission transfers control to the API. After traffic enforcement (traffic shaping and deadline assign- ment), the message is enqueued onto the corresponding channel's message queue for subsequent processing by the channel handler. Based on channel type, the channel handler is assigned to one of three CPU run queues for execution (described in Section 3.1). It executes in an infinite loop, dequeueing messages from the message queue and performing protocol processing (including fragmentation). The packets thus generated are inserted into the channel packet queue and into one of three (outbound) link packet queues for the corresponding link, based on channel type and traffic gener- ation, to be transmitted by the link scheduler. Reception-side processing: In Figure 2(b), a received packet is demultiplexed to the corresponding channel's packet queue, for subsequent processing and reassembly. As in transmission-side processing, channel handlers are assigned to one of three CPU run queues for execution, and execute in an infinite loop, waiting for packets to arrive in the channel packet queue. Packets in the packet queue are processed and transferred to the channel's reassembly queue. Once the last packet of a message arrives, the channel handler completes message reassembly and inserts the message into the corresponding message queue, from where the application retrieves the message via the API's receive routine. At intermediate nodes, the link scheduler relays arriving packets to the next node along the route. While we focus on transmission-side processing at the sending host, the following discussion also applies to reception-side processing. Message queue Message queue semaphore Packet queue Packet queue semaphore deadline/priority status type Proxy: process id deadline/priority Traffic specification QoS specification Relative channel priority Local delay bound Handler: process id Application requirements Admission control Buffer management Protocol processing messages from application channel message queue message processed completely channel packet queue inherit message deadline if message early else initiate message processing initialize block count if packet buffers available enqueue packet else suspend until buffer available decrement block count if block count zero if yield condition true, yield CPU else reset block count continue else continue message not processed completely dequeue message process packet enqueue packet link packet queues suspend until message current (a) Channel state (b) Handler execution profile Figure 3. Channel state and handler profile. 3.1. Salient features Figure 3(a) illustrates a portion of the state associated with a channel at the host upon successful establishment. Each channel is assigned a priority relative to other channels, as determined by the admission control procedure. The local delay bound computed during admission control at the host is used to compute deadlines of individual messages. Each handler is associated with a type, and execution deadline or priority, and execution status (runnable, blocked, etc. In addition, two semaphores are allocated to each channel han- dler, one to synchronize with message insertions into the channel's message queue (the message queue semaphore), and the other to synchronize with availability of buffer space in the channel's packet queue (the packet queue semaphore). Channel handlers are broadly classified into two types, best-effort and real-time. A best-effort handler is one that processes messages on a best-effort channel. Real-time handlers are further classified as current real-time and early real-time. A current real-time handler is one that processes on-time messages (obeying the channel's rate specification), while an early real-time handler is one that processes early messages (violating the channel's rate specification). Figure 3(b) shows the execution profile of a channel handler at the source host. The handler executes in an infinite loop processing messages one at a time. When initialized, it simply waits for messages to process from the message queue. Once a message becomes available, the handler dequeues the message and inherits its deadline. If the message is early, the handler computes the time until the message will become current and suspends execution for that dura- tion. If the message is current, the handler initiates protocol processing of the message. After creating each packet, the handler checks for space in the packet queue (via the packet queue semaphore); it is automatically blocked if space is not available. The packets created are enqueued onto the chan- nel's packet queue, and if the queue was previously empty, the link packet queues are also updated to reflect that this channel has packets to transmit. Handler execution employs cooperative preemption, where the currently-executing handler relinquishes the CPU to a waiting higher-priority handler after processing a block of packets, as explained below. While the above suffices for non-work-conserving protocol processing, a mechanism is needed to continue handler execution in the case of work-conserving protocol process- ing. Accordingly, in addition to blocking the handler as be- fore, a channel proxy is created on behalf of the handler. A channel proxy is a thread that simply signals the (blocked) channel handler to resume execution. It competes for CPU access with other channel proxies in the order of logical arrival time, and exits immediately if the handler has already woken up. This ensures that the handler is made runnable if the proxy obtains access to the CPU before the handler becomes current. Note that an early handler must still relinquish the CPU to a waiting handler that is already current. Maintenance of QoS guarantees: Per-channel QoS guarantees are provided via appropriate preemptive scheduling of channel handlers and non-preemptive scheduling of packet transmissions. While CPU scheduling can be priority-based (using relative channel priorities), we consider deadline-based scheduling for channel handlers and proxies. Execution deadline of a channel handler is inherited dynamically from the deadline of the message to be pro- cessed. Execution deadline of a channel proxy is derived from the logical arrival time of the message to be processed. Channel handlers are assigned to one of two run queues based on their type (best-effort or real-time), while channel proxies (representing early real-time traffic) are assigned to a separate run queue. The relative priority assignment for handler run queues is such that on-time real-time traffic gets the highest protocol processing priority, followed by best-effort traffic and early real-time traffic in that order. Provision of QoS guarantees necessitates bounded delays in obtaining the CPU for protocol processing. As shown in [19], immediate preemption of an executing lower-priority handler results in expensive context switches and cache misses; channel admissibility is significantly improved if preemption overheads are amortized over the processing of several packets. The maximum number of packets processed in a block is a system parameter determined via experimentation on a given host architecture. Cooperative preemption provides a reasonable mechanism to bound CPU access delays while improving utilization, especially if all handlers execute within a single (kernel) address space. Link bandwidth is managed via multi-class non-preemptive EDF scheduling with link packet queues organized similar to CPU run queues. Link scheduling is non-work-conserving to avoid stressing resources at downstream hosts; in general, the link is allowed to "work ahead" in a limited fashion, as per the link horizon [16]. Overload protection: Per-channel traffic enforcement is performed when new messages are inserted into the message queue, and again when packets are inserted into the link packet queues. The message queue absorbs message bursts on a channel, preventing violations of Bmax and Rmax Csw context switch time Ccm cache miss penalty 90 -s 1st processing cost 420 -s processing cost 170 -s C l per-packet link scheduling cost 160 -s packets between preemption points 4 pkts S maximum packet size 4K bytes Table 1. Important system parameters. on this channel from interfering with other, well-behaved channels. During deadline assignment, new messages are checked for violations in Mmax and Rmax . Before inserting each message into the message queue, the inter-message spacing is enforced according to I min . For violations in Mmax , the (logical) inter-arrival time between messages is increased in proportion to the extra packets in the message. The number of packet buffers available to a channel is determined by the product of the maximum number of packets constituting a message (derived from Mmax ) and the maximum allowable burst length Bmax . Under work-conserving processing, it is possible that the packets generated by a handler cannot be accommodated in the channel packet queue because all the packet buffers available to the channel are exhausted. A similar situation could arise in non-work-conserving processing with violations of Mmax . Handlers of such violating channels are prevented from consuming excess processing and link capacity, either by blocking their execution or lowering their priority relative to well-behaved channels. Blocked handlers are subsequently woken up when the link scheduler indicates availability of packet buffers. Blocking handlers in this fashion is also useful in that a slowdown in the service provided to a channel propagates up to the application via the message queue. Once the message queue fills up, the application can be blocked until additional space becomes available. Al- ternately, messages overflowing the queue can be dropped and the application informed appropriately. Note that while scheduling of handlers and packets provides isolation between traffic on different channels, interaction between the CPU and link schedulers helps police per-channel traffic. Fairness: Under non-work-conserving processing, early real-time traffic does not consume any resources at the expense of best-effort traffic. With work-conserving process- ing, best-effort traffic is given processing and transmission priority over early real-time traffic. 3.2. CPU preemption delays and overheads The admission control procedure (D order) must account for CPU preemption overheads, access delays due to cooperative preemption, and other overheads involved in managing resources. In addition, it must account for the overlap between CPU processing and link transmission, and hence the relative bandwidths of the CPU and link. In a companion paper [19], we presented extensions to D order to account for the above-mentioned factors. Table 1 lists the important system parameters used in the extensions. 3.3. Determination of P, S, and L x P and S determine the granularity at which the CPU and link, respectively, are multiplexed between channels, and thus determine channel admissibility at the host [19]. Selection of P is governed by the architectural characteristics of the host CPU (Table 1). For a given host architecture, P is selected such that channel admissibility is maximized while delivering reasonable data transfer throughput. S is selected either using end-to-end transport protocol performance or host/adapter design characteristics. In general, the latency and throughput characteristics of the adapter as a function of packet size can be used to pick a packet size that minimizes delivering reasonable data transfer throughput. For a typical network adapter, the transmission time for a packet of size s, L x (s), depends primarily on the overhead of initiating transmission and the time to transfer the packet to the adapter and on the link. The latter is a function of packet size and the data transfer bandwidth available between host and adapter memories. Data transfer band-width itself is determined by host/adapter design features (pipelining, queueing on the adapter) and the raw link band- width. If C x is the overhead to initiate transmission on an adapter feeding a link of bandwidth B l bytes/second, L x can be approximated as L x is the data transfer bandwidth available to/from host mem- ory. B x is determined by factors such as the mode (di- rect memory access (DMA) or programmed IO) and efficiency of data transfer, and the degree to which the adapter pipelines packet transmissions. C x includes the cost of setting up DMA transfer operations, if any. 4. Implementation We have implemented the proposed architecture using a modified x-kernel 3.1 communication executive [14] that exercises complete control over a 25 MHz Motorola 68040 CPU. CPU bandwidth is consumed only by communication-related activities, facilitating admission control and resource management for real-time channels. 1 x-kernel (v3.1) employs a process-per-message protocol-processingmodel and a priority-based non-preemptive scheduler with levels; the CPU is allocated to the highest-priority runnable thread, while scheduling within a priority level is FIFO. 4.1. Architectural configuration Real-time communication is accomplished via a connection-oriented protocol stack in the communication executive (see Figure 4(a)). The API exports routines for channel establishment, channel teardown, and data it also supports routines for best-effort data transfer. Network transport for signalling is provided by a 1 Implementation of the reception-side architecture is a slight variation of the transmission-side architecture. APPLICATIONS RPC Network Layer Link Access Layer Application Programming Interface Name Service RTC Signalling & Traffic Enforcement Synchronization Clockcurrent best-effort early context switch between handlers designated priority level x-kernel context switch x-kernel run queues xkernel process EDF handler run queues for active handlers x-kernel scheduler (selection of process to run) (selection of handler to run) EDF scheduler (a) x-kernel protocol stack (b) Layered EDF scheduler Figure 4. Implementation environment. (resource reservation) protocol layered on top of a remote procedure call (RPC) protocol derived from x-kernel's CHAN protocol. Network data transport is provided by a fragmentation which packetizes large messages so that communication resources can be multiplexed between channels on a packet-by-packet basis. The FRAG transport protocol is a modified, unreliable version of x-kernel's BLAST protocol with timeout and data retransmission operations disabled. The protocol stack also provides protocols for clock synchronization and network layer encapsulation. The network layer protocol is connection-oriented and provides network-level encapsulation for data transport across a point-to-point communication network. The link access layer provides link scheduling and includes the network device driver. More details on the protocol stack are provided in [15]. 4.2. Realizing a QoS-sensitive architecture Process-per-channel model: On successful establishment, a channel is allocated a channel handler, space for its message and packet queues, and the message and packet queue semaphores. If work-conserving protocol processing is de- sired, a channel proxy is also allocated to the channel. A channel handler is an x-kernel process (which provides its thread of control) with additional attributes such as the type of channel (best-effort or real-time), flags encoding the state of the handler, its execution priority or deadline, and an event identifier corresponding to the most recent x-kernel timer event registered by the handler. In order to suspend execution until a message is current, a handler utilizes x- timer event facility and an event semaphore which is signaled when the timer expires. A channel proxy is also an x-kernel process with an execution priority or deadline. The states of all established channels are maintained in a linked list that is updated during channel signalling. We extended x-kernel's process management and semaphore routines to support handler creation, termi- nation, and synchronization with message insertions and availability of packet buffers after packet transmissions. Each packet of a message must inherit the transmission Category Available Policies Protocol process-per-channel processing work-conserving, non-work-conserving CPU fixed-priority with scheduling multi-class earliest-deadline-first Handler cooperative preemption (configurable execution number of packets between preemptions) Link multi-class earliest-deadline-first scheduling (options 1, 2 and Overload block handler, decay handler deadline, protection enforce I min , drop overflow messages Table 2. Implementation policies. deadline assigned to the message. We modified the BLAST protocol and message manipulation routines in x-kernel to associate the message deadline with each packet. Each outgoing packet carries a global channel identifier, allowing efficient packet demultiplexing at a receiving node. CPU scheduling: For multi-class EDF scheduling, three distinct run queues are maintained for channel handlers, one for each of the three classes mentioned in Section 3.1, similar to the link packet queues. Q1 is a priority queue implemented as a heap ordered by handler deadline while Q2 is implemented as a FIFO queue. Q3, utilized only when the protocol processing is work-conserving, is a priority queue implemented as a heap ordered by the logical arrival time of the message being processed by the handler. Channel proxies are also realized as x-kernel threads and are assigned to Q3. Since Q3 has the lowest priority, proxies do not interfere with the execution of channel handlers. The multi-class EDF scheduler is layered above the x-kernel scheduler (Figure 4(b)). When a channel handler or proxy is selected from the EDF run queues, the associated x-kernel process is inserted into a designated x-kernel priority level for CPU allocation by the x-kernel sched- uler. To realize this design, we modified x-kernel's context switch, semaphore, and process management routines ap- propriately. For example, a context switch between channel handlers involves enqueuing the currently-active handler in the EDF run queues, picking another runnable handler, and invoking the normal x-kernel code to switch process con- texts. To support cooperative preemption, we added new routines to check the EDF and x-kernel run queues for waiting higher-priority handlers or native x-kernel processes, re- spectively, and yield the CPU accordingly. Link scheduling: The implementation can be configured such that link scheduling is performed via a function call in the currently executing handler's context or in interrupt context (option 1), or by a dedicated process/thread (option 2), or by a new thread after each packet transmission (option 3). As demonstrated in [19], option 1 gives the best performance in terms of throughput and sensitivity of channel admissibility to P and S; we focus on option 1 below. The organization of link packet queues is similar to that of handler run queues, except that Q3 is used for early packets when protocol processing is work-conserving. After inserting a packet into the appropriate link packet queue, channel handlers invoke the scheduler directly as a function call. If the link is busy, i.e., a packet transmission is in progress, the function returns immediately and the handler continues execution. If the link is idle, current packets (if any) are transferred from Q3 to Q1, and the highest priority packet is selected for transmission from Q1 or Q2. If Q1 and Q2 are empty, a wakeup event is registered for the time when the packet at the head of Q3 becomes current. Scheduler processing is repeated when the network adapter indicates completion of packet transmission or the wakeup event for early packets expires. Traffic enforcement: A channel's message queue semaphore is initialized to Bmax ; messages overflowing the message queue are dropped. The packet queue semaphore is initialized to Bmax \Delta N pkts , the maximum number of outstanding packets permitted on a channel. On completion of a packet's transmission, its channel's packet queue semaphore is signalled to indicate availability of packet buffers and enable execution of a blocked handler. If the overflow is due to a violation in Mmax , the handler's priority/deadline is degraded in proportion to the extra packets in its payload, so that further consumption of CPU bandwidth does not affect other well-behaved channels. Table 2 summarizes the available policies and options. 4.3. System parameterization Table 1 lists the system parameters for our implementation. Selection of P and S is based on the tradeoff between available resources and channel admissibility [19]. The packet time model presented in Section 3.3 requires that C x and B x be determined for a given network adapter and host architecture. An evaluation of the available networking hardware revealed significant performance-related deficiencies (poor data transfer throughput; high and unpredictable packet transmission time) [15]. These deficiences in the adapter design severely limited our ability to demonstrate the capabilities of our architecture. Given our focus on unidirectional data transfer, it suffices to ensure that transmission of a packet of size s takes L x units. This can be achieved by emulating a network adapter by consuming units for each packet being transmitted. We have implemented such a device emulator, the null device [19], that can be configured to emulate a desired packet transmission time. We have used it to study a variety of tradeoffs,such as the effects of the relationship between CPU and link processing bandwidth, in the context of QoS-sensitive protocol processing [19]. We experimentally determined C x to be - 40-s. For the experiments we select to correspond to a link (and data transfer) speed of 50 ns per byte, for an effective packet transmission band-width (for 4KB packets) of 16 MB/s. 5. Evaluation We evaluate the efficacy of the proposed architecture in isolating real-time channels from each other and from best- Traffic Specification Ch Mmax Bmax Rmax I min Deadline 2, RT Table 3. Workload used for the evaluation. effort traffic. The evaluation is conducted for a subset of the policies listed in Table 2, under varying degrees of traffic load and traffic specification violations. In particular, we evaluate the process-per-channel model with non-work- conserving multi-class EDF CPU scheduling and non-work- conserving multi-class EDF link scheduling using option 1 (Section 4.2). Overload protection for packet queue overflows is provided via blocking of channel handlers; messages overflowing the message queues are dropped. The parameter settings of Table 1 are used for the evaluation. 5.1. Methodology and metrics We chose a workload that stresses the resources on our plat- form, and is shown in Table 3. Similar results were obtained for other workloads, including a large number of channels with a wide variety of deadlines and traffic specifications. Three real-time channels are established (channel establishment here is strictly local) with different traffic specifica- tions. Channels 0 and 1 are bursty while channel 2 is periodic in nature. Best-effort traffic is realized as channel 3, with a variable load depending on the experiment, and has similar semantics as the real-time traffic, i.e., it is unreliable with no retransmissions under packet loss. Messages on each real-time channel are generated by an x-kernel process, running at the highest priority, as specified by a linear bounded arrival process with bursts of up to Bmax messages. Rate violations are realized by generating messages at rates that are multiples of Rmax . The best-effort traffic generating process is similar, but runs at a priority lower than that of the real-time generating processes and higher than the x-kernel priority assigned to channel handlers. Each experiment's duration corresponds to 32K packet transmissions; only steady-state behavior is evaluated by ignoring the first 2K and last 2K packets. All experiments reported here have traffic enforcement and CPU and link scheduling enabled. The following metrics measure per-channel performance. Throughput refers to the service received by each channel and best-effort traffic. It is calculated by counting the number of packets successfully transmitted within the experiment duration. Message laxity is the difference between the transmission deadline of a real-time message and the actual time that it completes transmission. Deadline misses measures the number of real-time packets missing deadlines. Packet drops measures the number of packets dropped for both real-time and best-effort traffic. Deadline misses and packet drops account for QoS violations on individual channels. | 4096 | | | | | | | | | | load (KB/s) Throughput (KB/s) (mean)# RT Channel 2 (min) | 50 | | | | | | | | | | load (KB/s) Message laxity (ms) (a) Throughput (b) Message laxity Figure 5. Maintenance of QoS guarantees when traffic specifications are honored. 5.2. Efficacy of the proposed architecture Figure 5 depicts the efficacy of the proposed architecture in maintaining QoS guarantees when all channels honor their traffic specifications. Figure 5(a) plots the throughput of each real-time channel and best-effort traffic as a function of offered best-effort load. Several conclusions can be drawn from the observed trends. First, all real-time channels receive their desired bandwidth; since no packets were dropped (not shown here) or late (Figure 5(b)), the QoS requirements of all real-time channels are met. Increase in offered best-effort load has no effect on the service received by real-time channels. Second, best-effort traffic throughput increases linearly until system capacity is exceeded; real-time traffic (early and current) does not deny service to best-effort traffic. Third, even under extreme overload condi- tions, best-effort throughput saturates and declines slightly due to packet drops, without affecting real-time traffic. Figure 5(b) plots the message laxity for real-time traffic, also as a function of offered best-effort load. No messages miss their deadlines, since minimum laxity is non-negative for all channels. In addition, the mean laxity for real-time messages is largely unaffected by an increase in best-effort load, regardless of whether the channel is bursty or not. Figure 6 demonstrates the same behavior even in the presence of traffic specification violations by real-time chan- nels. Channel 0 generates messages at a rate faster than specified while best-effort traffic is fixed at - 1900 KB/s. In Figure 6(a), not only do well-behaved real-time channels and best-effort traffic receive their expected service, channel also receives only its expected service. The laxity behavior is similar to that shown in Figure 5(b). No real-time packets miss deadlines, including those of channel 0. How- ever, channel 0 overflows its message queue and drops excess messages (Figure 6(b)). None of the other real-time channels or best-effort traffic incur any packet drops. 5.3. Need for cooperative preemption The preceding results demonstrate that the architectural features provided are sufficient to maintain QoS guarantees. The following results demonstrate that these features are also necessary. In Figure 7(a), protocol processing for best-effort traffic is non-preemptive. Even though best-effort traffic is processed at a lower priority than real-time traf- fic, once the best-effort handler obtains the CPU, it continues to process messages from the message queue regardless of any waiting real-time handlers, making CPU scheduling QoS-insensitive. As can be seen, this introduces a significant number of deadline misses and packet drops, even at low best-effort loads. The deadline misses and packet drops increase with best-effort load until the system capacity is reached. Subsequently, all excess best-effort traffic is dropped, while the drops and misses for real-time channels decline. The behavior is largely unpredictable, in that different channels are affected differently, and depends on the mix of channels. This behavior is exacerbated by an increase in the buffer space allocated to best-effort traffic; the best-effort handler now runs longer before blocking due to buffer overflow, thus increasing the window of non-preemptibility. Figure 7(b) shows the effect of processing real-time messages with preemption only at message boundaries. Early handlers are allowed to execute in a work-conserving fashion but at a priority higher than best-effort traffic. Note that all real-time traffic is still being shaped since logical arrival time is enforced. Again, we observe significant deadline misses and packet drops for all real-time channels. Best-effort throughput also declines due to early real-time traffic having higher processing priority. This behavior worsens when the window of non-preemptibility is increased by draining the message queue each time a handler executes. Discussion: The above results demonstrate the need for cooperative preemption, in addition to traffic enforcement and CPU scheduling. While CPU and link scheduling were always enabled, real-time traffic was also shaped via traffic enforcement. If traffic was not shaped, one would observe significantly worse real-time and best-effort performance due to non-conformant traffic. We also note that a fully-preemptive kernel is likely to have larger, unpredictable costs for context switches and cache misses. Cooperative preemption provides greater control over preemption 3000 4000 | 5000 6000 4096 | | | | | | | | | | | Offered load (Ch Throughput (KB/s) 2000 3000 4000 30000 | | | | | | | | | | | | | Offered load (Ch Number of packets dropped (a) Throughput (b) Number of packets dropped Figure 6. Maintenance of QoS guarantees under violation of Rmax . points, which in turn improves utilization of resources that may be used concurrently. For example, a handler can initiate transmission on the link before yielding to any higher priority activity; arbitrary preemption may occur before the handler initiates transmission, thus idling the link. 6. Related work While we have focused on host communication subsystem design to implement real-time channels, our implementation methodology is applicable to other proposals for providing QoS guarantees in packet-switched networks. A detailed survey of the proposed techniques can be found in [3]. Similar issues are being examined for provision of integrated services on the Internet [7, 6]. The expected QoS requirements of applications and issues involved in sharing link bandwidthacross multiple classes of traffic are explored in [24, 10]. The issues involved in providing QoS support in IP-over-ATM networks are also being explored [5, 22]. The Tenet protocol suite [4] provides real-time communication on wide-area networks (WANs), but does not incorporate protocol processing overheads into their network-level resource management policies. In particular, it does not provide QoS-sensitive protocol processing inside end hosts. The need for scheduling protocol processing at priority levels consistent with the communicating application was highlighted in [1] and some implementation strategies demonstrated in [12]. Processor capacity reserves in Real-Time Mach [21] have been combined with user-level protocol processing [18] for predictable protocol processing inside hosts [17]. Operating system support for multimedia communication is explored in [25, 13]. However, no explicit support is provided for traffic enforcement or decoupling of protocol processing priority from application priority. The Path abstraction [11] provides a rich framework for development of real-time communication services. 7. Conclusions and future work We have proposed and evaluated a QoS-sensitive communication subsystem architecture for end hosts that supports guaranteed-QoS connections. Using our implementation of real-time channels, we demonstrated the efficacy with which the architecture maintains QoS guarantees and delivers reasonable performance to best-effort traffic. While we evaluated the architecture for a relatively lightweight stack, such supportwould be necessary if computationallyintensive services such as coding, compression, or checksums are added to the protocol stack. The usefulness of the features also depends on the relative bandwidths of the CPU and the link. The proposed architectural features are independent of our platform, and are generally applicable. Our work assumes that the network adapter (i.e., the underlying network) does not provide any explicit support for QoS guarantees, other than providing a bounded and predictable packet transmission time. This assumption is valid for a large class of networks prevalent today, such as FDDI and switch-based networks. Thus, link scheduling is realized in software, requiring lower layers of the protocol stack to be cognizant of the delay-bandwidth characteristics of the network. A software-based implementation also enables experimentation with a variety of link sharing policies, especially if multiple service classes are supported. The architecture can also be extended to networks providing explicit support for QoS guarantees, such as ATM. We are now extending the null device into a sophisticated network device emulator providing link bandwidth management, to explore issues involved when interfacing to adapters with support for QoS guarantees. For true end-to- guarantees, scheduling of channel handlers must be integrated with application scheduling. We are currently implementing the proposed architecture in OSF Mach-RT, a microkernel-based uniprocessor real-time operating sys- tem. Finally, we have extended this architecture to shared-memory multiprocessor multimedia servers [20]. --R Structure and scheduling in real-time protocol implementations Support for continuousmedia in the DASH sys- tem The Tenet real-time protocol suite: Design Integration of real-time services in an IP-ATM network architecture Integrated services in the Internet architecture: An overview. Supporting real-time applications in an integrated services packet network: Architecture and mechanism A Calculus for Network Delay and a Note on Topologies of Interconnection Networks. A scheme for real-time channel establishment in wide-area networks Programming with system resources in support of real-time distributed applications Scheduling and IPC mechanisms for continuous media. Workstation support for real-time multimedia communication Design tradeoffs in implementing real-time channels on bus-based multiprocessor hosts Predictable communication protocol processing in Real-Time Mach Resource management for real-time communication: Making theory meet practice Processor capacity reserves for multimedia operating systems. ATM signaling support for IP over ATM. Transport system architecture services for high-performance communications systems A scheduling service model anda schedulingarchitecture for an integrated services packet network. The Heidelberg resource administration technique design philosophy and goals. --TR --CTR Binoy Ravindran , Lonnie Welch , Behrooz Shirazi, Resource Management Middleware for Dynamic, DependableReal-Time Systems, Real-Time Systems, v.20 n.2, p.183-196, March 2001 Christopher D. Gill , Jeanna M. Gossett , David Corman , Joseph P. Loyall , Richard E. Schantz , Michael Atighetchi , Douglas C. Schmidt, Integrated Adaptive QoS Management in Middleware: A Case Study, Real-Time Systems, v.29 n.2-3, p.101-130, March 2005 Christopher D. Gill , David L. Levine , Douglas C. Schmidt, The Design and Performance of a Real-Time CORBA SchedulingService, Real-Time Systems, v.20 n.2, p.117-154, March 2001

traffic enforcement;CPU;QoS-sensitive resource management;and link scheduling;real-time communication

271016

A Multiframe Model for Real-Time Tasks.

AbstractThe well-known periodic task model of Liu and Layland [10] assumes a worst-case execution time bound for every task and may be too pessimistic if the worst-case execution time of a task is much longer than the average. In this paper, we give a multiframe real-time task model which allows the execution time of a task to vary from one instance to another by specifying the execution time of a task in terms of a sequence of numbers. We investigate the schedulability problem for this model for the preemptive fixed priority scheduling policy. We show that a significant improvement in the utilization bound can be established in our model.

Introduction The well-known periodic task model by Liu and Layland(L&L) [1] assumes a worst-case execution time bound for every task. While this is a reasonable assumption for process-control-type real-time applications, it may be overly conservative [4] for situations where the average-case execution time of a task is significantly smaller than that of the worst-case. In the case where it is critical to ensure the completion of a task before its deadline, the worst-case execution time is used at the price of excess capacity. Other approaches have been considered to make better use of system resources when there is substantial excess capacity. For example, many algorithms have been developed to schedule best-effort tasks for resources unused by hard-real-time periodic tasks; aperiodic task scheduling has been studied extensively and different aperiodic server algorithms have been developed to schedule them together with periodic tasks [6, 7, 8, 9]. In [10], etc., the imprecise computation model is used when a system cannot schedule all the desired computation. We have also investigated an adaptive scheduling model where the timing parameters of a real-time task may be parameterized [3]. However, none of the work mentioned above addresses the scheduleability of real-time tasks when the execution time of a task may vary greatly but follows a known pattern. In this paper, we propose a multiframe task model which takes into account such execution time patterns; we shall show that better schedulability bounds can be obtained. In the multiframe model, the execution time of a task is specified by a finite list of numbers. By repeating this list, a periodic sequence of numbers is generated such that the execution time of each instance (frame or job) of the task is bounded above by the corresponding number in the periodic sequence. Consider the following example. Suppose a computer system is used to track vehicles by registering the status of every vehicle every 3 time units. To get the complete picture, the computer takes 3 time units to perform the tracking execution, i.e., the computer is 100% utilized. Suppose in addition, the computer is required to execute some routine task which takes 1 time unit and the task is to be executed every 5 time units. Obviously, the computer cannot handle both tasks. Routine: Tracking: Figure 1: Schedule of the Vehicle Tracking System However, if the tracking task can be relaxed so that it requires only 1 time unit to execute every other period, then the computer should be able to perform both the tracking and routine tasks (see the timing diagram in figure 1). This solution cannot be obtained by the L&L model since the worst-case execution time of the tracking task is 3, so that the periodic task set in the L&L model is given by f(3; 3); (1; 5)g (the first component in a pair is the execution time and the second the period). This task set has utilization factor of 1:2 and is thus unscheduleable. Also notice that we cannot replace the tracking task by a pair of periodic tasks f(3; 6); (1; 6)g since a scheduler may defer the execution of the (3; that its first execution extends past the interval [0,3], while in fact it must be finished by time=3. In this paper, we shall investigate the schedulability of tasks for our multiframe task model under the preemptive fixed priority scheduling policy. For generality, we allow tasks to be sporadic instead of periodic. A sporadic task is one whose requests may not occur periodically but there is a minimum separation between any two successive requests from the same task. A periodic task is the limiting case of a sporadic task whose requests arrive at the maximum allowable rate. We shall establish the utilization bounds for our model which will be shown to subsume the L&L result [1]. To obtain these results, however, we require the execution times of the multiframe tasks to satisfy a fairly liberal constraint. It will be seen that the schedulability bounds can be improved substantially if there is a large variance between the peak and non-peak execution time of a task. Using the multiframe model, we can safely admit more real-time tasks than the L&L model. The paper is organized as follows. Section 2 presents our multiframe real-time task model, defines some terminology, and prove some basic results about scheduling multiframe tasks. Section 3 investigates the schedulability bound of the fixed priority scheduler for the multiframe model. Section 4 is the conclusion. 2 The Multiframe Task Model For the rest of the paper, we shall assume that time values have the domain the set of non-negative real numbers. All timing parameters in the following definitions are non-negative real numbers. We remark that all our results will still hold if the domain of time is the non-negative integers. real-time task is a tuple (\Gamma; P ), where \Gamma is an array of N execution times is the minimum separation time, i.e., the ready times of two consecutive frames (requests) must be at least P time units apart. The execution time of the ith frame of the task is C ((i\Gamma1) mod N) , where 1 - i. The deadline of each frame is P after its ready time. For example, is a multiframe task with a minimum separation time of 2. Its execution time alternates between 2 and 1. When the separation between two consecutive ready times is always P and the ready time of the first frame of a task is at time=0, the task reduces to a periodic task. In the proofs to follow, we shall often associate a multiframe task whose \Gamma has only one element (i.e., N=1) with a periodic task in the L&L model which has the same execution time and whose period is the same as the minimum separation of the multiframe task. For example, task ((1); 5) has only one execution time and its corresponding L&L task is (1; 5). We shall call a periodic task in the L&L model an L&L task, and whenever there is no confusion, we shall call a multiframe task simply a task. Consider a task which has more than one execution time. Let We shall call C m the peak execution time of task T . We shall call the pair corresponding L&L task of the multiframe task T . For a set S of n tasks fT We call U the peak utilization factor of S. We call U the maximum average utilization factor of S: Given a scheduling policy A, we call U m A the utilization bound of A if for any task set S, S is scheduleable by A whenever U m - U m A , We note that U m is also the utilization factor of S's corresponding L&L task set. Example 1 Consider the task set 5)g. Its corresponding L&L task set S 0 isfT 0 5)g. The peak utilization factor of S is U 1:2. The maximum average utilization factor of S is U A pessimistic way to analyze the scheduleability of a multiframe task set is to consider the schedulability of its corresponding L&L task set. This, however, may result in rejecting many task sets which actually are scheduleable. For example, the task set in Example 1 will be rejected if we use the L&L model, whereas it is actually scheduleable by a fixed priority scheduler under RMA (Rate Monotonic Assignment), as we shall show later. respect to a scheduling policy A, a task set is said to be fully utilizing the processor if it is scheduleable by A, but increasing the execution time of any frame of any task will result in the modified task set being unscheduleable by A. We note that U m A is the greatest lower bound of all fully utilizing task sets with respect to the scheduling policy A. Lemma 1 For any scheduling policy A, U m A - 1. Proof. We shall prove this by contradiction. Suppose there is an U m A larger than 1, we arbitrarily form a task set A where C m i is the peak execution time of T i . So we have \Sigma n When the peak frames of all the tasks start at the same time, they cannot be all finished within P by any scheduler, which violates the definition of A . So U m A cannot exceed 1. QED. Suppose A is a scheduling policy which can be used to schedule both multiframe and L&L task sets. Let the utilization bound of A be U m A for multiframe task sets. Let the utilization bound of A for the corresponding L&L task sets be U c A . Then U m A . Proof. Proof is by contradiction. Consider a task set S of size n. Suppose U m - U c A and the set is unscheduleable. Its corresponding L&L task set S 0 has the same utilization factor as U m . S 0 is scheduleable. Suppose the ith frame of task T j miss its deadline at time t j . For every task T locate the time point t k which is the ready time of the latest frame of T k such that t k - t j . We transform the ready time pattern as follows. In the interval from 0 to t k , we push the ready times of all frames toward t k so that the separation times of all consecutive frames are all equal to P k . We now set all execution times to be the peak execution time. If t some k, we add more peak frames of T k at its maximum rate in the interval between t k and t j . The transformed ready time pattern is at least as stringent as the original case. So the ith frame of T j still misses its deadline. However, the transformed case is actually a ready time pattern of S 0 which should be scheduleable, hence a contradiction QED. Is the inequality in Lemma 2 strict? Intuitively, if U m of a task set is larger than U c A and there is not much frame-by-frame variance in the execution times of the tasks in the set, then the task set is unlikely to be scheduleable. However, if the variance is sufficiently big, then the same scheduling policy will admit more tasks. This can be quantified by determining the utilization bound for our task model. We shall show how to establish an exact bound if the execution times of the tasks satisfy a rather liberal restriction. be the maximum in an array of execution times (C array is said to be AM (Accumulatively Monotonic) if \Sigma m+j is said to be AM if its array of execution times is AM. Intuitively, an AM task is a task whose total execution time for any sequence of L - 1 frames is the largest among all size-L frame sequences when the first frame in the sequence is the frame with the peak execution time. For instance, all tasks in Example 1 are AM. We note that tasks in multimedia applications usually satisfy this restriction. In the following section, we assume that all tasks satisfy the AM property. It will be seen that without loss of generality, we can assume that the first component of the array of execution time of every task is its peak execution time, i.e., C m =C 0 Fixed Priority Scheduling In this section, we shall show that, for the preemptive fixed-priority scheduling policy, the multiframe task model does have a higher utilization bound than the L&L model if we consider the execution time variance explicitly. The utilization bound for the L&L model is given by the following theorem in the much cited paper [1]. Theorem 1 (Theorem 5 from [1]) For L&L task sets of size n, the utilization bound of the preemptive fixed priority schuduling policy is n(2 1=n \Gamma 1). Definition 5 The critical instance of a multiframe task is the period when its peak execution time is requested simultaneously with the peak execution times of all higher priority tasks, and all higher priority tasks request execution at the maximum rate. Theorem 2 For the preemptive fixed priority scheduling policy, a multiframe task is scheduleable if it is scheduleable in its critical instance. Proof. Suppose a task T scheduleable in its critical instance. We shall prove that all its frames are scheduleable regardless of their ready times. First, we prove that the first frame of T k is scheduleable. Let T k be ready at time t and its first frame finishes at t end . We trace backward in time from time=t to locate a point t 0 when none of the higher priority tasks was being executed. t 0 always exists, since at time 0 no task is scheduled. Let us pretend that T k 's first frame becomes ready at time t 0 . It will still finish at time t end . Now let us shift the ready time pattern of each higher priority task such that its frame which becomes ready after t 0 now becomes ready at t 0 . This will only postpone the finish time of T k 's first frame to a point no earlier than t end , say t end . In other words, T end - t 0 end . Then for each higher priority task, we shift the ready time of every frame after t 0 toward time=0, so that the separation between two consecutive frames is always the minimum separation time. This will further postpone the finish time of T k 's first frame to no earlier than T 0 end . In other words, t 0 end . Now, we shift all higher priority tasks by by frames until the peak frame starts at t 0 . Since \Gamma k is AM, this shifting has the effect of postponing the finish time of T k to t 000 end . By construction, the resulting request pattern is the critical instance for T k . Since T is scheduleable in its critical instance, we have t 000 first frame is scheduleable. Next, we prove that all other frames of T k are also scheduleable. This is done by induction. Induction base case: The first frame of T k is scheduleable. Induction step: Suppose first i frames of T k are scheduleable. Let us consider the (i+1)th frame and apply the same argument as before. Suppose that this frame starts at time t and finishes at t end . Again, we trace backward from t along the time line until we hit a point t 0 when no higher priority tasks is being executed. t 0 always exists, since no higher priority task is being executed at the finish time of the ith frame. Let the (i + 1)th frame start at time t 0 . It will still finish at time t end . Now shift the requests of each higher priority task such that its frame which starts after t 0 now starts at t 0 . This will only postpone the finish time of T k 's (i 1)th frame to a point in time no earlier than t end , say t end . Then for each higher priority task, we shift the ready time of every frame after t 0 toward time=0 so that the separation time between any two consecutive frames is always the minimum separation time of the task. This will further postpone the finish time of T k 's (i 1)th frame to no earlier than T 0 end . In other words, end . Now for all higher priority tasks, we shift them by frames until the peak frames start at t 0 . Again, since \Gamma is AM, this further postpones the finish time of T k 's (i 1)th frame to t 000 end . This last case is actually the critical instance for T k . Since T k is scheduleable in its critical instance, we have t 000 1)th frame is also scheduleable. We have thus proved the theorem. QED. We shall say that a task passes its critical instance test if it is scheduleable in its critical instance. Corollary 1 A task set is scheduleable by a fixed priority scheduler if all its tasks pass the critical instance test. From now on, we can assume, without loss of generality that C 0 is the peak execution time of a task without affecting the schedulability of the task set. This is because we can always replace a task T whose peek execution time is not in the first frame by one whose execution time array is obtained by rotating T 's array so that the peek execution time is C 0 . From the argument in the proof of theorem 2, it is clear that such a task replacement does not affect the result of the critical instance test. Example 2 Task set f((2; 1); 3); ((3); 7)g is scheduleable under rate-monotonic assignment. U tasks pass their critical instance test. However, its corresponding L&L task set f(2; 3); (3; 7)g is unscheduleable under any fixed priority assignment. Example 3 The L&L task set f(3; 3); (1; 5)g with utilization factor 1:2 is obviously unscheduleable by any scheduling policy. However, if the requirement of the first task is relaxed such that every other frame needs only 1 time unit, the task set becomes scheduleable by RMA. This is because the relaxed case is given by the multiframe task set f((3; 1); 3); ((1); 5)g which passes the critical instance test. We remark that the Example 3 above specifies the vehicle tracking system mentioned at the beginning of this paper. From the above argument, we can now establish its schedulability. These examples also show that even if the total peak utilization exceeds 1, a task set may still be schedulable. Of course, the average utilization must not be larger than 1 for scheduleability. The complexity of the scheduleability test based on Corollary 1 is O(P ), where P is the biggest period. Theorem 3 If a feasible priority assignment exists for some multiframe task set, the rate-monotonic priority assignment is feasible for that task set. Proof. Suppose a feasible priority assignment exists for a task set. Let T i and T j be two tasks of adjacent priority in such an assignment with T i being the higher priority one. Suppose that us interchange the priorities of T i and T j . It is not difficult to see that the resultant priority assignment is still feasible by checking the critical instances. The rate-monotonic priority assignment can be obtained from any priority ordering by a finite sequence of pairwise priority reordering as above. QED. To compute the utilization bound, we need the following lemma. Definition 6 Let \Psi(n; ff) denote the minimum of the expression \Sigma subject to the constraint: 2. Proof. With the substitution x Pn , we can compute minimize \Sigma n subject to x This is a strictly convex problem. There is a unique critical point which is the absolute minimum. The symmetry of the minimization problem in its variables means that all x i 's are equal in the solution. So we have x Definition 7 A task set is said to be extremely utilizing the processor if it is scheduleable but increasing peak execution time of the lowest priority task by any amount will result in a task set which is unscheduleable. We shall use U e to denote the greatest lower bound of all extremely utilizing task sets. It is important to note the distinction between fully utilizing and extremely utilizing task sets. It is crucial to the proof of Lemma 4 and Lemma 5. Lemma 4 Consider all task sets of size n satisfying the restriction r Proof. From Theorem 2 and Theorem 3, we only need to consider the case where all tasks start at time 0 and request at their maximum rates thereafter. We can use rate-monotonic priority assignment and check for scheduleability in the interval from time 0 to P n . Since we know that only C 0 and C 1 are involved in all the critical instance tests. First, the utilization bound corresponds to the case where every C 0 =C 1 equals r, since we can increase C 1 without changing U m . And increasing C 1 will only take more CPU time. So in the following proof we assume that all the ratios are equal to r. For any scheduleable and extremely utilizing task set S with U shall prove four claims. 1: The second request of T must be finished before P n . Suppose ffi of C 1 i is scheduled after P n , we can derive a new task set S 0 by only changing the following execution times of T i and T n , and arbitrarily reducing other execution times of T i to maintain the AM property of the execution time arrays. It is easy to show that S 0 is schedulable and also extremely utilizes the processor. This contradicts the assumption that U e is the minimum of all extremely utilizing task set. So the second request of any T i should be completed before P n . 2: If 0, we can derive a new task set S 0 by only changing the following execution times of T i and T n , and arbitrarily reducing other execution times of T i to maintain the AM property of the execution time arrays. It is easy to check that S 0 is scheduleable and also extremely utilizes the processor. This contradicts the assumption that U e is the minimum. So C 0 3: If n should be finished before P i . Instead of proving claim 3, we prove the following equivalent claim: Consider an extreme utilizing task set S satisfying claim 1 and claim 2. If the last part of C 0 finishes between P i and P i+1 , and does not correspond to the minimal case. As in claim 2, we can derive a new task set S 0 by only changing the following execution times of T i and T n , and arbitrarily reducing other execution times of T n to maintain the AM property of the execution time arrays. Suppose P j is the smallest value satisfying According to claim 1 and claim 2, the second requests of all tasks other than T n are scheduled between P j and P n . Since we know the first requests of all tasks other than T n are all scheduled before P j . extremely utilizes the CPU, we know that the part of C n scheduled before P j is larger than that scheduled after P j . This guarantees that the new task set S 0 is still scheduleable and extremely utilizes the CPU. r Hence, the task set S cannot be the minimal case. This establishes claim 3. then the second request of T should be completed exactly at time P i+1 . If the second request of T completes ahead of P i+1 , the processor will idle between its completion time and P i+1 , which shows S does not extremely utilize processor. So this cannot be true. If ffi of the second request of T derive a new task set S 0 by only changing the following execution times of T i and T i+1 , and arbitrarily reducing other execution times of T i to maintain the AM property of the execution time arrays. Again it is easy to check that S 0 is schedulable and also extremely utilizes the process. This contradicts the assumption that U e is the minimum. So the second request of T should be completed exactly at time T i+1 . From these four claims and Lemma 3, we can conclude: r sets of size n, U r Proof. Again, we assume all C 0 =C 1 equals r, and all tasks request at the maximum rate. For any task T i in an extremely utilizing task set with P i i such that P 0 increase C 0 n by the amount needed to again extremely utilize the processor. This increase is smaller than C 1 1). Let the old and new utilization factors be U m and U 0m respectively. Therefore we can conclude that the minimum utilization occurs among task sets in which the longest period is no larger than twice of the shortest period. This establishes Lemma 5. QED Theorem 4 Let sets of size n, the utilization bound is given by r r Proof. By definition, the least upper bound is the minimum of the U e for task sets of size ranging from 1 to n, and we have min n r r We observe that Liu and Layland's Theorem 1 is a special case of Theorem 4 with the frame separation time equals the period. The following tables summarize the relative advantage of using the multiframe model over the L&L model in determining whether a set of task is scheduleable. The column under UL&L gives the utilization bound in the L&L model. 5 0.743 13.6 19.5 22.8 24.9 26.3 27.4 28.2 28.9 29.4 34.5 50 0.698 16.7 24.0 28.2 30.8 32.7 34.1 35.2 36.0 36.7 43.3 100 0.696 16.8 24.3 28.5 31.2 33.1 34.5 35.5 36.4 37.1 43.8 Table 1: Utilization Bound Percentage Improvement CONCLUSION AND FUTURE RESEARCH 17 Table 1 shows the percentage improvement of our bound over the Liu and Layland bound. Specifically, the table entries denote 100 (U m =UL&L \Gamma 1), for different combination of r (the ratio of the peak execution time to the execution time of the second frame) and n (the number of tasks in the task set). For example, suppose we have a system capable of processing one Gigabyte of data per second, and a set of tasks each of which needs to process one Megabyte of data per second. Using a utilization bound of ln 2, we can only allow 693 tasks. By Theorem 4, we can allow at least 863 tasks (over 24% improvement) when r - 3. As r increases, the bound improvement increases. Actually, as r ! 1, a simple calculation shows that the bound ! 1. This says that our model excels when the execution time of the task varies sharply. It is also interesting to compare the maximum average utilization with L&L bound. However, the maximum average utilization factor may be arbitrarily low even if the maximum utilization factor is very high. One simple example is f((10; 5; 10)g. So, we take instead the average of the first two frames of the task. In Table 2 we calculate 100 ( 1 r ))=U L&L , which is the ratio of the biggest possible maximum average utilization factor to Liu and Layland bound. Table 3 shows we can still maintain good overall system utilization when task execution time varies. 4 Conclusion and Future Research In this paper, we give a multiframe model for real-time tasks which is more amenable to specifying tasks whose execution time varies from one instance to another. In our model, the execution times of successive instances of a task is specified by a finite array of numbers rather than a single number which is the worst-case execution time of the classical Liu and Layland model. Using the new model, we derive the utilization bound for the preemptive fixed priority scheduler, under the assumption that the execution time array of the tasks satisfies the AM (Accumulative CONCLUSION AND FUTURE RESEARCH 3 0.780 83.5 77.4 74.3 72.3 71.0 70.0 69.3 68.8 68.3 64.1 5 0.743 85.2 79.7 76.7 74.9 73.7 72.8 72.1 71.6 71.2 67.3 100 0.696 87.6 82.8 80.3 78.7 77.6 76.8 76.2 75.8 75.4 71.9 Table 2: Ratio of maximum average to L&L bound Monotonic) property. This property is rather liberal and is consistent with common encoding schemes in multimedia applications, where one of the execution times in an array "dominates" the others. We show that significant improvement in the utilization bound over the Liu and Layland model results from using our model. This is useful in dynamic applications where the number of tasks can vary and the figure of merit for resource allocation is the number of tasks that the system can admit without causing timing failures. Work is under way to apply this model to real-life applications such as video stream scheduling and will be reported in the future. --R Scheduling Algorithms for Multiprogramming in a Hard- Real-Time Environment Fundamental Design Problems of Distributed systems for the Hard-Real-Time Envi- ronment Load Adjustment in Adaptive Real-Time Systems The Rate Monotonic Scheduling Algorithm - Exact Characterization and Average Case Behavior Assigning Real-Time Tasks to Homogeneous Multiprocessor Systems The Deferrable Server Algorithm for Enhanced Aperiodic Responsiveness in Hard Real-Time Environments A Practical Method for Increasing Processor Utilization Aperiodic Servers in a Deadline Scheduling Environment Aperiodic Task Scheduling for Hard Real-Time Systems Scheduling Periodic Jobs That Allow Imprecise Results --TR --CTR Tei-Wei Kuo , Li-Pin Chang , Yu-Hua Liu , Kwei-Jay Lin, Efficient Online Schedulability Tests for Real-Time Systems, IEEE Transactions on Software Engineering, v.29 n.8, p.734-751, August Marek Jersak , Rafik Henia , Rolf Ernst, Context-Aware Performance Analysis for Efficient Embedded System Design, Proceedings of the conference on Design, automation and test in Europe, p.21046, February 16-20, 2004 Jrn Migge , Alain Jean-Marie , Nicolas Navet, Timing analysis of compound scheduling policies: application to posix1003.1b, Journal of Scheduling, v.6 n.5, p.457-482, September/October Samarjit Chakraborty , Thomas Erlebach , Simon Knzli , Lothar Thiele, Schedulability of event-driven code blocks in real-time embedded systems, Proceedings of the 39th conference on Design automation, June 10-14, 2002, New Orleans, Louisiana, USA Sanjoy K. Baruah, Dynamic- and Static-priority Scheduling of Recurring Real-time Tasks, Real-Time Systems, v.24 n.1, p.93-128, January Chang-Gun Lee , Lui Sha , Avinash Peddi, Enhanced Utilization Bounds for QoS Management, IEEE Transactions on Computers, v.53 n.2, p.187-200, February 2004 Michael A. Palis, The Granularity Metric for Fine-Grain Real-Time Scheduling, IEEE Transactions on Computers, v.54 n.12, p.1572-1583, December 2005 Tarek F. Abdelzaher , Vivek Sharma , Chenyang Lu, A Utilization Bound for Aperiodic Tasks and Priority Driven Scheduling, IEEE Transactions on Computers, v.53 n.3, p.334-350, March 2004 Christopher D. Gill , David L. Levine , Douglas C. Schmidt, The Design and Performance of a Real-Time CORBA SchedulingService, Real-Time Systems, v.20 n.2, p.117-154, March 2001 Chin-Fu Kuo , Tei-Wei Kuo , Cheng Chang, Real-Time Digital Signal Processing of Phased Array Radars, IEEE Transactions on Parallel and Distributed Systems, v.14 n.5, p.433-446, May Lui Sha , Tarek Abdelzaher , Karl-Erik rzn , Anton Cervin , Theodore Baker , Alan Burns , Giorgio Buttazzo , Marco Caccamo , John Lehoczky , Aloysius K. Mok, Real Time Scheduling Theory: A Historical Perspective, Real-Time Systems, v.28 n.2-3, p.101-155, November-December 2004

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An interaction of coherence protocols and memory consistency models in DSM systems.

Coherence protocols and memory consistency models are two improtant issues in hardware coherent shared memory multiprocessors and softare distributed shared memory(DSM) systems. Over the years, many researchers have made extensive study on these two issues repectively. However, the interaction between them has not been studied in the literature. In this paper, we study the coherence protocols and memory consistency models used by hardware and software DSM systems in detail. Based on our analysis, we draw a general definition for memory consistency model, i.e., memory consistency model is the logical sum of the ordering of events in each processor and coherence protocol. We also point that in hardware DSM system the emphasis of memory consistency model is relaxing the restriction of event ordering, while in software DSM system, memory consistency model focuses mainly on relaxing coherence protocol. Taking Lazy Release Consistency(LRC) as an example, we analyze the relationship between coherence protocols and memory consistency models in software DSM systems, and find that whether the advantages of LRC can be exploited or not depends greatly on it's corresponding protocol. We draw the conclusion that the more relaxed consistency model is, the more relaxed coherence protocol needed to support it. This conclusion is very useful when we design a new consistency model. Furthermore, we make some improvements on traditional multiple writer protocol, and as far as we aware, we describe the complex state transition for multiple writer protocol for the first time. In the end, we list the main research directions for memory consistency models in hardware and software DSM systems.

Introduction Distributed Shared Memory(DSM) systems have gained popular acceptance by combining the scalability and low cost of distributed system with the ease of use of the single address space. Generally, there are two methods to implement DSM systems: hardware vs software. Cache-Coherent Non- Uniform-Memory-Access Multiprocessors(CC-NUMA) are the dominant direction of hardware DSM The work of this paper is supported by the CLIMBING Program and the President Foundation of the Chinese Academy of Sciences. systems. To date, there are many commercial and research systems, such as Stanford DASH[34], Stanford FLASH[31], MIT Alewife[5], StartT-Voyager[10], SGI Origin series. While in software DSM alternatives, the general method is supplying a high-level shared memory abstraction on the top of underlying messaging passing based system, such as multicomputers and LAN-connected NOW. For example, Rice Munin[13], Rice TreadMarks[19], Princeton IVY[33], CMU Midwary[12], Utah Quarks[28], Marland CVM[26], DIKU CarlOS[29] are current commercial or research software DSM systems in the world. Cache coherences and memory consistency models are two important issues in CC-NUMA ar- chitecture. Cache coherence is mainly used to keep multiple copies of one cache block consistent to all processors, while the role of memory consistency models is specifying how memory behaviors with respect to read and write operations from multiple processors. Both of these two issues have gained extensive study in the past 20 years. There are several cache coherence protocols and memory consistency models have been proposed in the literature. Coherence protocols include write-invalidate, write-update, delay-update. Memory consistency models include sequential consistency[32], processor consistency[22] [21], weak ordering[3], release consistency[21] et al. Coherent problem remains in software DSM systems. The difference between hardware and software implementation is the granularity of coherence unit. In software DSM systems the coherence unit is page, while in hardware DSM systems the coherence unit is cache block line. Since the role of page in software DSM systems is similar to that of cache in hardware DSM systems, for simplicity, we will use cache coherence in both cases in the rest of this paper. The coherence protocols widely adopted in software DSM systems include multiple writer protocol (i.e., write-shared protocol)[13], single writer protocol[26][8]. The memory consistency models are mainly used to reduce the frequency of communication and message traffic[27]. Examples of relaxed consistency models adopted in software DSM systems include eager release consistency[13], lazy release consistency[27], entry consistency[12], automatic update release consistency[23], scope consistency[24], home-based lazy release consistency[43], singer-writer lazy release consistency [26], message-driven release consistency[29], and affinity entry consistency[6]. In fact, both cache coherence protocols and memory consistency models describe the behaviours of DSM systems. They are interdependent on each other and the relationship between them is very complex. Up to now as we know, there is no research on this problem. In this paper, we first describe a clear understanding about coherence and memory consistency model and propose a general definition for memory consistency model. We also point that in hardware DSM system the emphasis of memory consistency model is relaxing the restriction of event ordering, while in software DSM system, memory consistency model focuses mainly on relaxing coherence protocol. Taking Lazy Release Consistency(LRC) as an example, we analyze the relationship between coherence protocols and memory consistency models in software DSM systems. We draw the conclusion that the more relaxed consistency model is, the more relaxed coherence protocol needed to support it. This conclusion is very useful when we design a new consistency model. Furthermore, we make some improvements on traditional multiple writer protocol by adding a new state, and as far as we aware, we describe the complex state transition for multiple writer protocol for the first time. In the end, we list the main research directions for memory consistency models in hardware and software DSM systems. The remainder of the paper is organized as follows. In section 2 we briefly overview the development of coherence protocols and memory consistency models, and propose a general definition for memory consistency model. Taking LRC as an example, the relationship between coherence protocols and memory consistency models is deduced in section 3. Furthermore, the state transition graph for new improved multiple writer protocol is shown in section 3. Related works and conclusions remarks are listed in section 4 and section 5 respectively. General Definition for Memory Consistency Model Censier and Feautrier defined a coherent memory scheme as follows[14]: Definition 2.1: A memory scheme is coherent if the value returned on a LOAD instruction is always the value given by the latest STORE instruction with the same address. This definition, while intuitively appealing, is vague and simplistic. The reality is much more complex. In a computer system where STORE can be buffered in a store buffer, it is not clear whether "last STORE" refer to the execution of the STORE by a processor, or to the update of memory. In fact, the above definition contains two different aspects of memory system behaviours, both of which are critical to writing correct shared memory programs. The first aspect, called coherence, defines what values can be returned by a read operation. The second aspect, called event ordering in each processor, determines when a written value will be returned by a read operation. Coherence ensures that multiple processors see a coherent view of the same location, while event ordering in a processor describes that when other processor sees a value that has been updated by this processor. In [38] Hennessy and Patterson presented the sufficient conditions to coherence as follows. 1. A read by a processor, P, to a location X that follows a write by P to X, with no writes of X by another processor occurring between the write and the read by P, always returns the value written by P. 2. A read by a processor to location X that follows a write by another processor to X returns the written value if the read and write are sufficiently separated and no other writes to X occur between the two accesses. 3. Writes to the same location are serialized: that is, two writes to the same location by any two processors are seen in the same order by all processors. For example, if the values 1 and then are written to a location, processors can never read the value of the location as 2 and then later read it as 1. The above three conditions only guarantee that all the processors have a coherent view about position X. However, Since they don't tell us the event ordering in each processor, we can't determine when a written value will be seen by other processors. As such, in order to capture the behaviors of memory system accurately, we need to restrict both the coherence and the event ordering, this is the role of memory consistency model. We will use the strictest memory consistency model -sequential consistency as an example to explain this in the following. Scheurish and Dubois[39] described a sufficient condition to sequential consistency as follows: 1. All processors issue memory accesses in program order. 2. If a processor issues a STORE, then the processor may not issue another memory access until the value written has become accessible by all other processors. 3. If a processor issues a LOAD, then the processor may not issue another memory access until the value which is to be read has both been bound to the LOAD operation and become accessible to all other processors. From above two sufficient conditions for coherence and consistency, we see that coherence and consistency are related tightly. The conditions of coherence is a subset of the conditions of memory consistency model. The former considers the different events from different processors to the same location only, while consistency model not only considers the different events to the same location but also imposes constraints on the ordering of event within each processor, that is, the execution order in each processor. As such, the combination of cache coherence and event ordering will determine the behaviour of whole memory system. Based on our understanding, we propose a general definition for memory consistency model to reveal the relationship between them. memory consistency model = coherence protocol event ordering in each processor Some researchers use event ordering and memory consistency model interchangely. However, we believe that our new understanding can describe the memory system behaviours more accurately than them. In our new definition, the role of coherence is to ensure that the coherent view of the given memory location by multiple processors. Ordering of events describes the happen sequence of memory events issued by each processor. Here, we use memory events to represent the read and store operation to the memory[17]. Memory consistency model is the logical sum of coherence protocol and event ordering in each processor. For example, sequential consistency, defined by Lamport in 1979[32], can be viewed as two conditions[2]: 1. all memory access appear to execute automatically in some total order. 2. all memory accesses of each processor appear to execute in an order specified by its programmer The first condition, atomicity, is ensured by coherence protocol. There are two base kinds of cache coherence protocols: write invalidate and write update[41]. Write update protocol ensures that all processors which keep copies will see the new value simultaneously with modified processor, while invalidate protocol reaches the coherent view by invalidate all other copies. The second condition, program order, is determined by the executing order of events. If we restrict all the events in one processor to be issued, performed in program order, this condition will be satisfied. However, the above constraint is too strict to improve the performance. Especially, many hardware and compiler optimizations, such as write buffer, look-up free, non-binding reading, register allocation, dynamic scheduling, multiple issues, can't be utilized in this memory consistency model. These two conditions for sequential consistency model show that there are two directions for us to relax this strictest consistency model. In hardware DSM systems, over the years, there are several consistency models proposed to exploit hardware and complier optimization techniques. Such as, processor consistency, weakly order, weak ordering(new definition), release consistency, DRF1, PLpc model[20]. Almost all the hardware and compiler optimization techniques have been exploited in PLpc memory consistency model. In these relaxed consistency models, however, only the event ordering in each processor is relaxed, such as W!R, W!W, R!R, R!W operations in successive synchronization operation, while the atomicity demand changes little. For example, in RC, all the operations between two synchronization operations can be completed out of order if no data dependence is violated. However, the atomicity is maintained by write invalidate protocol which is similar to that in sequential consistency model. In software DSM systems, the communication overhead is so expensive that the cost for maintaining atomicity is more expensive than that in hardware DSM systems. Therefore, reducing the frequency of communication and message traffic is more important in software DSM systems than in hardware DSM systems. Furthermore, since the coherence unit in software DSM system is page or larger than page[9], the false sharing problem is more serious than that in hardware DSM sys- tems. As such, how to eliminate the false sharing problem effectively is also an important issue in software DSM systems. Before discussing the solution for false sharing, we first give a clear description about false sharing. False sharing occurs when two processors access logically unrelated data that happen to fall on the same page and at least one of them writes the data, causing the coherence mechanisms to ping-pong the page between these processors. It includes two categories: write-write false sharing and write-read false sharing. False sharing will entail many useless communication among processors[25]. In order to reduce the communication overhead in software DSM systems, several new memory consistency models are proposed in the past. The represented consistency models for software DSM systems include Eager Release Consistency(ERC) in Munin[13], Lazy Release Consis- in TreadMarks [27], Entry Consistency(EC) in Midway[12], Automatic Update Release Consistency(AURC) in SHRIMP[23], Scope Consistency(ScC)[24], Home-based Lazy Release Consistency(HLRC)[43], Singer Writer Lazy Release Consistency(SW-LRC)[26], Message-Driven Release Consistency(MDRC) in CarlOS [29], and Affinity Entry Consistency[6] in NCP2. Among these consistency models, the ordering of events in each processor is similar to one another, in other words, the main difference of these models is the method to maintain coherence. This includes two aspects: (1) Which coherence protocol is used;(2) How this protocol is imple- mented. All the coherence protocols adopted in software DSM systems are more relaxed than the strict single writer protocol used in hardware DSM systems. Relaxed coherence protocols include write-shared protocol(multiple-writer)[13], relaxed single-writer protocol(or, delay invalidate)[26] 1 . For example, both TreadMarks and Munin systems use multiple writer protocol, while CVM adopts relaxed single writer protocol. Among above described memory consistency models, the main difference lies in when the up-dated value made by one processor can be available by other processors. For example, in LRC, when other processors execute an acquire operation , they will see all the updated value modified before it in happen-before-1 partial order. However, in EC and ScC, only the shared data associate with the same synchronization object are available when one processor acquires a synchronization object. Based on the analysis of hardware and software DSM systems in this section, we see the coherence protocol and event ordering in each processor are two important parts of memory consistency model. In particular, in software DSM system the difference among different consistency models depends greatly on corresponding coherence protocols. we will analyze their relationship in detail in the following section.In order to differentiate this improved single writer protocol from strict single writer protocol, we use relaxed single-writer to represent it here. In next section, the operation procedure of relaxed single writer protocol will be shown. 3 Interaction between Coherence and Consistency in LRC In this section, we take lazy release consistency as an example to analyze the relationship between coherence protocol and consistency model in detail. 3.1 Lazy Release Consistency LRC[27] is an all-software, page-based, write-invalidated based multiple writer memory consistency model. It has been implemented in TreadMarks system[19]. Since the objective of this section is analyzing the interaction between coherence protocol and consistency model, we describe the key idea of lazy release consistency in this subsection , and multiple writer protocol will be described in the following subsection. LRC is a lazy implementation of RC[21] or ERC[13]. It delays the propagation of modifications to a processor until that processor executes an acquire operation. The main idea of the consistency model is to use timestamps, or intervals to establish the happen- before-1 ordering between causal-related events. Local intervals are delimited by synchronization events, such as LOCKs, BARRIERs. LRC uses the happen- before-1 partial order to maintain ordering of events. The happen-before-1 partial order is the union of the total processor order of the memory accesses on each individual processor and the partial order of release-acquire pairs. Vector timestamps are used to represent the partial order. When a processor executes an acquire, it sends its current vector timestamp in the acquire message. The last releaser then piggybacks on its response a set of write notices. The write notices include the processor id, the vector timestamp for the interval during which the page was modified and the modified page number. A faulting processor uses write notices to locate and apply the modifications required to update its copy of the page. Both write-invalidate and write-update protocols can be used to implement above algorithms. In the following discussion, we assume that write invalidate protocol is used. 3.2 Multiple Writer Protocol Multiple Writer(MW) protocol has been developed to address the write-write false sharing problem. With a MW protocol, two or more processors can modify their local copies of a shared page simultaneously. Their modifications are merged at the next synchronization operation. Therefore the effect of false sharing is reduced. There are two key issues to implement MW protocol: write trapping and write collection. Where write trapping is the method to detect what shared memory location has been changed, twinning and software dirty bit[12] are two general methods used in software DSM systems. Write collection refers to the mechanism used for determining what modified data needs to be propagated to the acquirer. Timestamp and diffing are two methods used in Midway and TreadMarks respectively. For more details, please refer to [8]. In TreadMarks twining and diffing mechanisms are adopted. Although multiple writer protocol was first introduced by Carter in Munin, this protocol is not a new idea. It is derived from the idea of delay update in [11]. In [11], Bennett et al. pointed out In order to avoid unnecessary synchronization that is not required by the pro- gram's semantics. When a thread modifies a shared object, we can delay sending the update to remote copies of the object until remote threads could otherwise indirectly detect that the object has been modified. This is original description about delay update protocol. It is very vague for understanding, for example, what "indirectly" means? Whether two or more processors can modify the same unit si- multaneously? If we assign different understanding for this idea, we will obtain different coherence protocols. For example, if two or more processor 2 are allowed to modify the same page simultane- ously, the delay update protocol will evolve multiple writer protocol. If at any time there is only one writer is allowed to exist, and multiple reader can coexisted with single writer, the delay update protocol evolves relaxed single writer protocol. This protocol has been proved very useful in some applications [26]. The implementation of these protocol will be shown in the next subsection. Multiple writer protocol has several benefits as follows: multiple nodes can update the same page simultaneously, the protocol can greatly reduce the protocol overhead due to false sharing. ffl The protocol can reduce the communication traffic due to data transfer: instead of transfering the whole page each time, it transfers diffs only. ffl The protocol can relaxed the ordering of events in an interval further since without the ownership is required before a write operation. 3.3 Interaction between LRC and its corresponding Protocol According to above description of LRC and multiple-writer protocol, we find that LRC and multiple writer protocol are closely related. We will analyze the relationship between them step by step, from the strict single writer protocol to the most relaxed multiple writer protocol. If LRC uses the strict singer-writer single-reader protocol which is adopted in release consistency, i.e., before one processor writes a cache block, it must obtain the ownership first, and when other processors receive invalidate message, they must invalidate their local pages immediately. Therefore, at release operation, no write notices are needed to record because that all other processors have already known the modification executed in this processor which will release the lock. When other processor executes an acquire operation in the following, no write notices are needed since all the pages should be invalidated in lazy release consistency have been invalidated already. As such, the lazy property can not be exploited at all. In this case, the write notices and vector timestamps are only used to help faulting processors to locate the valided page, just like the role of "home" in CC-NUMA machine. Furthermore, the write-write and write-read false sharing cannot be resolved. In this case, the performance of LRC will be similar to that of RC. If the relaxed single writer protocol (i.e., singer-writer multiple-reader protocol used in [26]) is adopted, when one processor receives invalidate messages, it doesn't invalidate those pages immedi- ately, on the contrary, it keeps its own copies appear valid to itself until the next acquire operation. 2 In this paper, we assume that each processor has only one thread, therefore, we use thread, process, and processor interchangely. In this case, the write-read false sharing is eliminated partially, while write-write false sharing remains too. In order to depict this protocol accurately, we must introduce a new state stale, which is used to represent the state of a page during the interval between receiving an invalidate message about this page and next acquire operation. The messages received at an acquire operation in this coherence protocol include 3 categories :(1)necessary messages, (2) unnecessary messages entailed by false sharing, (3) unrelated messages about other pages which will be not used by the requiring processor. For example, in Fig 1, p1, p2 are two processors, variable x0,x1,x2,x3,x4 are 5 shared data, where x0 and x2 are allocated on the page 0, x1 and x3 are allocated on the page 1, x4 is allocated on page 2 solely. Lock L0, L2 are 2 locks used by users to create critical sections to protect the use of shared data. With relaxed single writer protocol, when p1 wants to write x1, it must obtain the ownership before write operation and the procedure of invalidating other copies can be overlapped with other operations following this write operation. When p2 receives the invalidate message, it doesn't invalidate page 1 immediately, and keeps it appear to valid until next lock acquiring operation, therefore, at the time when p2 read x3, it doesn't cause page fault error, this read-write false sharing case can be eliminated. However, when p2 wants to read x3 after an acquire operation, it will cause a page fault error. Therefore, the read-write false sharing problem does not be eliminated completely in relaxed single writer protocol. On the other hand, when p2 writes x2, it must cause a write fault since only one writer is allowed to write a page at a given time. As such, the write-write false sharing problem remains. In fig 1, the message 1 and 2 belong to both first category and second category, while message 3 is unrelated message, message 4 and 5 are the unnecessary messages entailed by false sharing. In above example, if the strict single-writer protocol is used , the page 0 in p2 will be invalidated immediately, and the first read of x3 will cause false sharing. So, we find that the relaxed single writer protocol is better than strict single writer protocol. On the other hand, in relaxed single writer protocol, when page fault occurs, the whole page will be transfered, which results in a great message traffic. In order to solve false sharing problem completely and reduce large communication traffic, multiple writer protocol is proposed and widely used in software DSM systems. In traditional MW protocol, however, when the acquiring processor receives the write notices, it will invalidate its corresponding pages immediately, which results in write-read false sharing as entailed by strict single writer protocol. Therefore, we improve the multiple writer protocol by combining the traditional MW protocol and relaxed single writer protocol. When improved multiple writer coherence protocol is adopted, the advantages of LRC can be exploited completely. Almost all the false sharing effects will be eliminated. Furthermore, with improved MW protocol, the write operations within one interval can forward without waiting the ownership. In this case, the messages received at an acquire operation include:(1)necessary message(such as write notices) , (2) some unrelated messages about other pages which will be not used by requiring processor, such as invalidate message for page 2 in Fig 1. These unrelated messages will be solved in Entry consistency and our new NLLS consistency model[40]. This is beyond the scope of this paper. In multiple writer protocol, since two or more writers can modify the same page simultaneously, the state of each page will be more complex than above two protocols. For example, when two writers write the same page, whch one is the owner? When a third processor want to write a page, to whom it should to inform? Diffing and twinning mechanism only tell us the implementation method for certain protocol, however, its don't tell us how to maintain the state transition in multiple writer protocol. We will describe the state transition in next subsection. From above analysis, we draw the following conclusions as shown in Table 1. For completeness, page 0 page 1 page 2 x3 Invalidate page 1 invalidate page 2 Invalidate page 0 release (L0) acquire (L2)24ask for ownership send ownership no page fault cccur page fault occur page fault occur Figure 1: An example of write-invalidate-based relaxed single writer protocol. we list the strictest sequential consistency as the base for comparison. memory consistency model corresponding coherence protocol sequential consistency (1)require all memory operations atomically, (2)send and receive invalidate messages immediately, and(3)stop until the acknowledgement is received. release consistency (1)the atomicity demand relaxed, but the ownership must obtain before write, (2)send and receive invalidate message immediately, and (3)the receiving of acknowledgement can be delayed until the following release synchronization operation LRC(single writer) (1)the atomicity demand relaxed, but the ownership must obtain immediately before write, (2)send invalidate message, the receiver delay accept the invalidate message until next acquire synchronization operation, and (3)the receiving of acknowledgement can be delayed until the following release synchronization operation LRC(multiple writer) (1)the atomicity demand was relaxed further, no ownership is needed before write(if this processor has already had a copy of this page), (2)both send and receive of invalidate messages can be delayed, and(3)the receiving of acknowledgement can be delayed until the following release synchronization operation Table 1: The relationship between memory consistency model and coherence protocol. According to the performance comparisons presented by other researchers[8],[36][26] and our analysis shown in Table 1, we find that the advantages of consistency models depend closely on their corresponding coherence protocols. The more relaxed consistency model is, the more relaxed coherence protocol needed to support it. This conclusion is very useful when we design a new consistency model. For example, with the support of MW protocol, scope consistency [24] is more relaxed than LRC, It combines the advantages of EC and LRC, distinguishes different Locks, i.e., the acquiring processor obtains the modified data from the releasers which use the same Lock. Therefore, many useless messages in above three protocols can be reduced greatly 3 . 3.4 The State Transition for Invalidate-based Multiple Writer Protocol As described above, in order to depict the coherence transition for relaxed single writer protocol and improved multiple writer protocol, we must add a new state named stale, which means that this coherence unit is modified by other processors, however, it appears to valid for this processor itself. Fig 2 shows a state transition graph for write-invalidated-based multiple writer protocol for LRC. As far as we aware, this is the first time a whole state transition graph for multiple writer protocol is shown. Exclusive shared Invalid stale Ri, Wj, Rj Ri,Rj.W, Ri: Read from local processor Wi: write from local processor Rj: Read from remote processor Acquire(l):Acquire lock Release (l): Release lock l. Wj: Write from remote processor Create DIFFi (create DIFFi and send it to exclusive node) Wi(create Twin,. keep write notices) Figure 2: The state transition graph for write-invalidated-based multiple writer protocol for LRC. 3 Although in [24], the authors didn't tell us the multiple writer protocol is adopted, from the examples shown in that paper we deduce that scope consistency uses multiple writer protocol too. 4 Related Works Dubios et.al in [17] analyzed the relationship between synchronization, coherence and event order- ing. Although they separated the concepts of coherence and event ordering, they didn't present the relationship between them, and they equalized event ordering with memory consistency model, which is different from our viewpoint. They defined strong ordering and weak ordering in that paper and presented that strong ordering is the same as sequential consistency. In fact, this viewpoint is wrong because the cache coherence protocol is not considered, Adve and Hill gave an example in [4] to demonstrate this wrong case. Per Stenstrom in 1990 presented an excellent survey about cache coherence protocols[41]. Adve and Gharachorloo discuss an extensive survey about memory consistency models[2]. However, both of them considered the case for hardware DSM systems only. In our paper, we consider both hardware and software DSM systems together, and study the relationship between coherence protocol and memory consistency model. Recently, Zhou et.al discussed the relationship between relaxed consistency model and coherence granularity in DSM systems[44]. They only consider the granularity of coherence protocol, while never consider the coherence protocol. Dubios et.al in [18] proposed delay consistency model for a release consistent system where an invalidation is buffered at the receiving processor until a subsequent acquire is executed by the processor. The delay consistency model is a coherence protocol which includes two categories:delay receive and delay-send delay receive, where delay receive is the same as relaxed single writer protocol, delay send and delay receive is similar to multiple writer protocol. They presented the state transition for hardware shared memory in detail. In this paper we describe the state transition for software DSM systems for the first time. 5 Conclusion and Future Work In this paper, starting with a classical coherent memory scheme, we point out that memory coherent includes two issues:coherency and event ordering in each processor. Based on a clear description of these two concepts, we define a general definition about memory consistency model, which is the logical sum of coherence protocol and event ordering in each processor. Second, we analyze the consistency models used in hardware DSM systems and software DSM systems under our new definition. We point out that in hardware DSM systems, the relaxed consistency model is devoted to relax the ordering of events, such as W!R, W!W, R!R, R!W operations, to utilize the hardware and complier optimization techniques. While coherence protocol does not make much progress in the past years. In software DSM systems, the main obstacle to performance is high communication overhead and the useless coherence-related messages entailed by the large coherence granularity. Therefore, the main purpose of the consistency model in software DSM system is to reduce the number of message and message traffic. while the event ordering in each processor of all new consistency models are similar to that of release consistency, Third, taking the LRC as an example, we analyze the relationship between coherence protocol and consistency model in software DSM systems, and conclude that these two issues are closed related. The more relaxed consistency model is, the more relaxed coherence protocol needed to support it. This conclusion is very useful when we design a new consistency model. Fourth, we make some improvements on traditional multiple writer protocol by adding a new state, and describe the state transition graph for invalidate-based multiple writer protocol for the first time. Finally, based on the analysis in this paper, we propose that the main directions of memory consistency model research in the future as following: For hardware DSM systems: ffl Relaxing the coherence protocol further, such as allowing multiple writers in hardware- coherent DSM system. This is possible because of the much progress of semi-conductor technology[37]. ffl Considering the hybrid hierarchical DSM MPP system, where each node uses hardware to implement DSM , while shared memory abstraction among nodes is supported by software DSM. For software DSM systems: ffl Relaxing coherence protocol further. ffl Using hybrid coherence protocols for different shared data, such as shared data protected by locks and shared data protected by barriers. ffl Intergrating more memory consistency model together to support different applications. ffl Considering the interaction with other latency tolerate techniques, such as multithreading and prefetching. --R A Comparison of Entry Consistency and Lazy Release Consistency Implementations. Shared Memory Consistency Models: A Tutorial. Weak Ordering:A new definition. Implementing Sequential Consistency In Cache Based Systems. The MIT Alewife Machine: Architecture and Performance. The Affinity Entry Consistency Protocol. TreadMarks: Shared Memory Computing on Networks of Workstations. Software DSM Protocols that Adapt between Single Writer and Multiple Writer. Tradeoffs between False Sharing and Aggregation in Software Distributed Shared Memory. Larry Rudolph and Arvind. Munin: Distributed Shared Memory Based on Type-Specific Memory Coherence The Midway Distributed Shared Memory System. Implementation and Performance of Munin. A New Solution to Coherence Problems in Multicache Systems. Parallel Computer Architecture (alpha version). Memory Access Buffering in Multiprocessors. Jin Chin Wang TreadMarks Distributed Shared Memory on standard workstations and operating systems. Programming for Different Memory Models. Memory consistency and event ordering in scalable shared memory multiprocessors. Cache consistency and sequential consistency. Improving Release-Consistent Shared Virtual Memory using Automatic Update Understanding Application performance on Shared Virtual Memory systems. The Relative Importance of Concurrent Writes and Weak Consistency Mod- els Lazy Release Consistency for software Distributed Shared Memory. Portable Distributed Shared Memory on UNIX. Lazy Release Consistency for Hardware-Coherent Multiprocessor The Stanford FLASH Multiprocessor. How to Make a Multiprocessors Computer That Correctly Executes Multiprocessor Programs. IVY:A Shared Virtual Memory System for Parallel Computing. The Standard Dash Multiprocessor. ADSM: A hybrid DSM Protocol that Efficiently Adapts to Sharing Patterns. An Evaluation of Memory Consistency Models for Shared Memory Systems with ILP Processors. Intelligent RAM (IRAM): Chips that Remember and Compute Revised Computer Architecture: A Quantitative Approach. Correct Memory Operation of Cache-Based Mul- tiprocessors Memory Consistency Models for Distributed Shared Memory Systems A Survey of Cache Coherence Schemes for Multiprocessors. The SPLASH-2 Programs: Characterization and Methodological Considerations Performance Evaluation of Two Home-based Lazy Release Consistency Protocols for Shared virtual Memory Systems --TR --CTR optimization and integration in DSM, ACM SIGOPS Operating Systems Review, v.34 n.3, p.29-39, July 2000

event ordering;software DSM systems;memory consistency models;coherence protocol;hardware DSM systems

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Analysis and Reduction for Angle Calculation Using the CORDIC Algorithm.

AbstractIn this paper, we consider the errors appearing in angle computations with the CORDIC algorithm (circular and hyperbolic coordinate systems) using fixed-point arithmetic. We include errors arising not only from the finite number of iterations and the finite width of the data path, but also from the finite number of bits of the input. We show that this last contribution is significant when both operands are small and that the error is acceptable only if an input normalization stage is included, making unsatisfactory other previous proposals to reduce the error. We propose a method based on the prescaling of the input operands and a modified CORDIC recurrence and show that it is a suitable alternative to the input normalization with a smaller hardware cost. This solution can also be used in pipelined architectures with redundant carry-save arithmetic.

INTRODUCTION The CORDIC (Coordinate Rotation DIgital Computer) algorithm is an iterative technique that permits computing several transcendental functions using only additions and shifts operations [15] [16]. Among these functions are included trigonometric functions, like sine, cosine, tan- gent, arctangent and module of a vector, hyperbolic functions, like sinh, cosh, tanh, arctanh, and several arithmetic functions. Due to the simplicity of its hardware implementation several signal processing algorithms, such as digital filters, orthogonal transforms and matrix factorization have been formulated with Cordic arithmetic and, therefore, several Cordic-based VLSI architectures have been proposed to solve these related signal processing problems [7]. In applications requiring high speed, pipelining and/or redundant arithmetic are introduced in the implementation of the Cordic algorithm, in such a way that each iteration of the algorithm is evaluated in a different stage of the pipeline. On the other hand, carry ripple adders are replaced by redundant adders, carry save (CS) or signed digit (SD), where the carry propagation within the adders is eliminated [4] [11] [13] [14]. As the control of the algorithm requires the exact determination of the sign of some variable, a Modified Cordic algorithm which facilitates the determination of the sign with redundant arithmetic has been proposed [5]. analysis of the Cordic algorithm is fundamental for the efficient design of Cordic- based architectures. To achieve a good performance, it is important to know the behaviour of the error and to take into account the effect that it will have on the hardware implementation to obtain a specified accuracy. The different sources of error of the Cordic algorithm have been analyzed in detail [8] [9] [10]. In this paper we will focus on the errors appearing in the computation of the inverse tangent function (angle calculation) with fixed-point arithmetic. This operation can be computed with the Cordic algorithm in the vectoring mode and does not require the final scaling inherent to the Cordic algorithm, since no scale factor is introduced in this mode of operation. The angle calculation is useful in algorithms for matrix factorization like eigenvalue and singular value decomposition (SVD) [6] and in the implementation of some digital filters [1] [12]. Matrix factorization requires the angle computation in the circular coordinate system, while digital filters may need angle computation in both circular and hyperbolic coordinates. The rounding error that is accumulated in the control coordinate through all the iterations of the Cordic algorithm may result in large error in the evaluation of the inverse tangent. This error can be important in applications such the Cordic based SVD algorithm, where the inverse tangent function is evaluated using fixed-point format with unnormalized data [10], or in some kind of filters where hyperbolic inverse tangent need to be computed [1] [12]. In [10] a technique, called partial normalization, is proposed to bound this error. However, this technique is hard to implement with redundant arithmetic and does not take into consideration the initial error due to the rounding of the input operands. We perform an analysis of the error considering the following three components: ffl Error due to the rounding of the input data ffl Error due to the finite number of iterations ffl Error due to the finite datapath width. We show that the first of these components, which has not been considered in previous analyses, is significant when both input operands are small. As a consequence, the solution proposed in [10] might not be appropriate. Because of the above, for small input operands it seems that the only suitable solution is to perform a normalization of the input operands which includes additional bits. We present a solution that performs a prescaling of the operands and modifies the CORDIC recurrence [5]. We show that this solution is simpler than normalization and produces a smaller total error. The Cordic algorithm consists in the rotation of a vector in a circular or hyperbolic coordinate system. The rotation is performed by decomposing the angle into a sequence of preselected elementary angles. Specifically, the basic Cordic iteration or microrotation is [15] [16] where m is an integer taking values +1 or \Gamma1 for circular or hyperbolic coordinates, respectively, are the variables x, y and z before the microrotation, tan is the microrotation angle. The shifting sequence s(m; i) in a circular coordinate system is s(1; in a hyperbolic coordinate system a sequence such as may be chosen, starting with microrotations have to be repeated. The Cordic iterations in coordinates x and y can be rewritten in matrix notation as, being the input vector at iteration i, v[i the output vector and Depending on parameter m, the Cordic algorithm may evaluate trigonometric functions selecting the y coordinate is reduced to zero. This permits evaluating the angle of a vector that is accumulated in z n+1 , in such a way that z coordinates and z coordinates. In the following we derive numerical bounds for the overall error in the computation of the inverse tangent function in the circular, coordinate systems. In these cases, the y coordinate is reduced to zero by chosing according to equation (4) and the angle computed, ', is stored in variable z. We follow the notation introduced in [8] and [10]. The accuracy of the Cordic algorithm and its influence on processor design was first analyzed by Walther [16]. He concluded that, to obtain a precision of n bits, data paths with bits are needed. Later, more precise bounds in all the modes of the Cordic algorithm have been obtained [8] [9] [10]. Hu [8] performs a thorough analysis of several errors in all the modes of the Cordic algorithm and derives numerical error bounds for each type of error. However, no results on the inverse tangent and inverse hyperbolic tangent computations are included. In [10] it has been shown that the numerical errors in inverse tangent computation using fixed point arithmetic with small operands can be too large. The overall error is split into two components, an approximation error and a rounding error. The approximation error is due to the angle quantization. That is, the decomposition of the angle into a finite number of microrotation angles produces an error in the representation of the angle. On the other hand, the rounding error is caused by the finite word length of any data path. If we denote as ideal Cordic the mathematically defined Cordic algorithm, with infinite precision in the data path and infinite number of microrotations, and as real Cordic the practical implementation of the Cordic algorithm, with finite precision in the data path and finite number of microrotations, it is possible to define an intermediate Cordic that uses infinite precision arithmetic and the same number of microrotations as the real Cordic. Figure 1 shows the relationship among the three definitions. In this way, the approximation error is the difference between the output of the ideal Cordic and the output of the intermediate Cordic. The rounding error is the difference between the output of the intermediate Cordic and that of the real Cordic. Usually, the error due to the rounding of the input operands has been neglected. That is, it has been considered that there are no rounding errors in the input data. But often, the real situation is that the input data is obtained from another hardware module or from the rounding of variables with larger precision. In such cases, there is as initial rounding error in the input data that has to be considered when deriving the bounds. This source of error can become very important in applications involving small inputs. We consider fixed-point arithmetic with n fractional bits in the input operands and b fractional bits in the representation of the x, y and z data paths inside the Cordic, to obtain n-bit results. The wordlengths used are illustrated in figure 2. This way, the initial rounding error in the input operands is and the rounding error introduced in each microrotation is For circular coordinates the value of coordinate y obtained with the intermediate Cordic after microrotations is being ff the angle calculated with the intermediate Cordic. On the other hand, the angle ' computed by the ideal Cordic is given by Then, the approximation error is [10] being jv[0]j the module of the input vector On the other hand, jy y n+1 is the value of y n+1 with finite precision and f c n+1 is the rounding error in coordinate y after microrotations. For convergence, j"y Then, considering the rounding error in the z datapath ((n conclude that the angle error is bounded by, Now, we have to find a bound for f c n+1 . Following the derivation in [8], the rounding error is bounded by, Y Y where P 1 (i) is given by equation (3) and k \Delta k is the l 2 norm, defined as the square root of the largest eigenvalue of the matrix [8]. The rounding error is composed of two parts: the rounding error produced by the initial rounding error in the input operands (first term), and the rounding error accumulated in n+ 1 microrotations considering no initial rounding error in the input data (second and third terms). Therefore, where \Gamman is due to the initial rounding error (first term of equation (11)) and 1:5 is due to the accumulated rounding error (second and third terms of equation (11)). Then, replacing equation (12) in equation (10) the bound for the overall error in the computation of the inverse tangent function is obtained, \Gamman 1:5 Equation 13 shows that the error in the computation of the inverse tangent does not have a constant bound as it depends on the norm of the input vector, jv[0]j. The error becomes larger the smaller the norm of the input vector, in such a way that when x 0 and y 0 are close to zero, a large error results. Consequently, the error is not bounded if the input operands are not bounded. A similar equation may be obtained for the overall error of the hyperbolic vectoring being m the number of Cordic iterations in hyperbolic coordinates [16] and ' max is the maximum input angle. In this case, the error becomes large when the hyperbolic norm of the input vector, 0 , is small. By means of the input normalization, the error can be bounded since a lower bound on jv[0]j is enforced. However, the implementation of the normalization requires extra hardware to determine the amount by which the components of the vector can be shifted, two leading-zero encoders and a comparator, and barrel shifters to perform the shifts in a single cycle. Therefore, the normalization is very hardware consuming. In [10] an alternative solution is proposed for the circular vectoring, the partial normal- ization, that involves the modification of the Cordic unit to include a normalization step, which is integrated with the Cordic iterations. This solution bounds the error. The main drawbacks of this solution are that the initial rounding error is not considered and it is very difficult to implement efficiently with redundant arithmetic. Figure 3 illustrates the partial normalization. As the input normalization is distributed along the Cordic iterations, the normalization is performed by introducing zeros, and not real bits, in the least significant positions of the input data. That is, when the input data is known with a precision larger than the precision used in the Cordic iteration, b bits, only b bits of the input are considered for normalization and the extra bits of the input are ignored, resulting in a large error not considered in the analysis performed in [10]. Figure 4 shows the error produced with the partial normalization when an initial rounding error is considered, the error with the conventional Cordic algorithm and the error produced using the Cordic with input normalization. In this latter case, the error considering normalization introducing zeros and normalization with real bits are considered. The figure plots the error, expressed as the precision obtained in the angle, versus the module of the input vector. Although the error is lower than the error produced with the standard Cordic algorithm, it is still very significant for small inputs, and higher than the error of the Cordic algorithm with input normalization, because the partial normalization is performed introducing zeros. On the other hand, the microrotations are modifed to include the normalization, resulting in microrotations that include comparisons to choose the maximum and minimum of two variables and variable shifts to perform the normalization. Therefore, this solution is not adequate for redundant arithmetic and/or pipelined architectures. In the next sections, new approaches are developed to bound the error of the angle calcu- lation. These approaches are suitable for word-serial or pipelined architectures with redundant or non-redundant arithmetic, and require little extra hardware cost. 4 MODIFIED CORDIC ALGORITHM The introduction of redundant arithmetic in the angle computation with the Cordic algorithm has motivated the development of modified Cordic microrotations for the circular coordinate system [5], where the recurrences are transformed making Then the microrotations (equation (1)) are transformed into and the selection of the ffi i is performed according to the following equation Figure 5 illustrates the modified Cordic algorithm. The w coordinate is not reduced to zero, because of the left shift that is performed over this coordinate at each microrotation. This transformation facilitates the implementation of the Cordic algorithm with redundant arithmetic. In redundant arithmetic, CSA or SDA, the exact determination of the sign of coordinate y is time consuming because of the redundant representation of this coordinate. With this transformation it is possible to use an estimate of the redundant representation of w i in the determination of ffi i , instead of its fully assimilated value. To make this possible it is necessary to use a redundant representation of the angle, allowing ffi i to take values in the set \Gamma1; 0:1. The corresponding selection functions for using carry-save or signed-digit representations can be found in [5]. Moreover, the hardware is reduced since one of the shifters is eliminated. In this work, we propose this same change in the Cordic equations but with the aim of reducing the errors in the computation of the inverse tangent function. In the following, we obtain the error bounds of the angle computation with the modified Cordic algorithm, based on variable w, in the circular and hyperbolic coordinates. We take into account the initial rounding error in the input operands. 4.1 Angle Error Analysis in Circular Coordinate System The modified Cordic iterations in circular coordinate system, given by equation (16), can be rewritten as being the input vector, v[i the output vector and Pw;1 (i) the transformation matrix Following a similar derivation as for the standard Cordic algorithm, it is possible to find a bound for the overall error. This way, the value of coordinate w after microrotations is being ff the angle calculated with the intermediate Cordic. Then, equation (20) can be rewritten as, where ' is the angle of vector v[0]. Therefore, and the approximation error is Then, considering the rounding error in the z datapath, the angle error with the modified algorithm in circular coordinates is bounded by, w;n+1 is the rounding error in coordinate w after Now, we have to find a bound for f c w;n+1 . We can use equation (11), but considering that the transformation matrix is given by equation 19, Y Y Moreover, the rounding error introduced in each microrotation of the modified Cordic algorithm is bounded by that is, no rounding error is introduced in coordinate w, as it is multiplied by a factor of 2 each microrotation and there are no right shifts. To evaluate the l 2 norm of the matrix product, k i=j Pw;1 (i) k, the following relation [2] has to be taken into account n Y Y where and As k the rounding error in the modified Cordic algorithm is bounded by being the first term the contribution of the initial rounding error and the second one the accumulated error in the microrotations. Replacing this result in equation (24) the error in the computation of the angle with the modified algorithm is obtained, As it can be seen by comparing equations (13) and (30), there is a reduction in the error of the computation of the angle, tan using the modified Cordic iterations. This is due, mainly, to the elimination of the rounding error in the w coordinate, that results in a lower accumulated rounding error after n iterations. Figure 6a shows the errors observed for several initial values with the standard and the modified Cordic algorithms, with and several different values of b. In both cases, the error becomes more important as jv[0]j decreases, although the error with the standard Cordic is always slighty larger than the error with the modified Cordic. However, although the diference between the errors with the standard and the modified algorithms is small, the modified Cordic algorithm results in a simpler hardware imple- mentation, since one shifter is eliminated, and is suitable for implementations with redundant arithmetic. 4.2 Angle Error Analysis in Hyperbolic Coordinate System The analysis that has been developed in the previous section can be extended to the evaluation of the error in the angle computation in hyperbolic coordinates. Now, the function we are calculating is tanh iteration i=0 is not evaluated. Similarly to circular coordinates, we can define the modified Cordic algorithm, by means of the transformation given in equation (15). Then, the modified microrotation is Similarly to the circular coordinate case, we find that the quantization error is given by Therefore, to obtain a bound in the error of the angle calculation, we have to find a bound for the accumulated rounding error, f h w;n+1 . We use equation (25), but considering that the transformation matrix, Pw;\Gamma1 (i), is Similarly to the case of circular coordinate system, to evaluate k into account equation (26) with matrix A(nj) defined as in equation (27) and matrix Q(ij) defined as where the values of q 0 and q 1 depend on the type of microrotation performed as follows if is a repetition and is a repetition and i ? j Therefore, the l 2 norm of this matrix is if it is a repetition and This way, the rounding error in hyperbolic coordinates is obtained as Then, considering the rounding error of the z data path, the overall error in the hyperbolic angle computation with the modified Cordic algorithm is The errors of the standard and modified Cordic algorithms operating in hyperbolic vectoring with are shown in figure 6b. As in the circular vectoring mode of operation, the error is reduced by means of the utilization of the modified Cordic algorithm as the rounding error in the w coordinate is eliminated. On the other hand, from the observation of figure 6, we find that the error in hyperbolic vectoring is larger than in circular vectoring. The hyperbolic coordinate scale factor, K h , is less than unity and reduces the operands and the hyperbolic module of the vector decreases with the mapping of the vector over the x-axis. Therefore, the rounding error is more important in hyperbolic coordinates than in circular coordinates. According to equation (38), the hyperbolic vectoring error would be large even with normalized inputs, when x 1 and y 1 are similar. However, the range of convergence of the algorithm imposes a limit to the value of x considering the basic sequence of microrotations of the Cordic algorithm for hyperbolic coordinates [16], which results in ' Therefore, the error will only be significant when the inputs operands are small. Although the modified Cordic algorithm reduces the overall error in the angle computa- tion, both in circular and in hyperbolic coordinate system, this error is still unbounded and it becomes very important when the module of the input vector is small. The smaller the module of the input vector, the larger the error. Therefore, it is necessary to develop solutions that efficiently reduce the error, with a low hardware and timing cost. The error in the angle calculation with the modified Cordic algorithm is still unbounded and large when the module of the input vector is small. The more natural solution to this problem is the normalization of the input operands, however this requires two leading-zeros coders, a comparator and two barrel shifters. That is, it is very hardware consuming. We propose a solution for the minimization of the angle calculation error with the modified algorithm in circular and hyperbolic coordinates, both in non-redundant and redundant arithmetic. We develop a solution based in an operand pre-scaling, where the error is bounded and close to the precision of the algorithm. The hardware implementation is simpler than the implementation of the partial normalization technique and the standard input normalization. Moreover, unlike to the solution developed in [10], this solution may applied to the Cordic algorithm with redundant arithmetic. The angle calculation error is only important when the module of the input vector is small. Therefore, if the input vector module is forced to take large values, the angle error is reduced and the output of the Cordic algorithm is within the precision required. The operand pre-scaling technique multiplies the module of the input vector by a constant factor, in such a way that the resulting module is large enough to minimize the angle calculation error. The pre-scaling is carried out before starting the Cordic iterations, thus being a preprocessing stage. In the following we consider that, to perform the operand pre-scaling, b bits of the input operands are known, where b is the internal wordlength of the Cordic and the p least significant bits are used in the pre-scaling to shift to the right the input vectors, as shown in figure 7. As it has been shown before, with the solution presented in [10] is not possible to perform the normalization considering additional bits of the input. Taking into account these considerations, the pre-scaling is as follows being (x in ; w in ) the input variables and the inputs to the Cordic processor after the pre-scaling. Then, the module of the input vector is multiplied by a scaling factor M . This way, the error in the computation of the inverse tangent function is reduced by an important factor, because we are reducing the rounding error and we are also imposing a lower bound for v in of M \Delta 2 \Gamman . The pre-scaling should only be carried out when the input vector module is small. There- fore, M is defined as, 1 if jx in j 2 \Gammas or jw in j 2 \Gammas That is, the pre-scaling is only performed if the module of the input vector is lower than 2 \Gammas . In this case, we multiply the input vector times 2 s to obtain a large module. The value of s should be chosen in such a way that the error is minimized. That is, the error in the angle computation of an input vector with module jv in j 2 \Gammas must already be bounded and close to the precision of the algorithm. This way, the maximum overall error for the angle computation in circular coordinates with pre-scaling is obtained by replacing jv[0]j in equation considering in this later case that the minimum input vector module is jv in This results in sin sin K1 \Delta2 \Gamma(16\Gammas) Figure 8 shows the error with several different pre-scalings, corresponding to pre-scaling), 9. The precision of the algorithm is and the internal precision of the modified Cordic arquitecture is Moreover, the input operands are rounded to bits after the pre-scaling; that is, the pre-scaling is carried out considering that, at least, b bits of the input are known. While the module of the input vector is less than 2 \Gammas , since the module is not modified, the error is the same as without pre-scaling when the module is larger than 2 \Gammas , that is, the pre-scaling is carried out, the error is significantly reduced, since the module has been enlarged with the pre-scaling. Similar results can be obtained for hyperbolic coordinate system. However, the error in the angle, although has been reduced, is still large in vectors with module jv in j ! 2 \Gamma2s . For example, when the pre-scaling is performed with the error is large for input vectors with a small module, less than, approximately 2 \Gamma10 . On the other hand, when the scaling factor is large, some input vectors with module less than 2 \Gammas present an important error. For example, with pre-scaling the error is large when the module is between 2 \Gamma6 and 2 \Gamma9 , since the pre-scaling has not been carried out. From expression (43) and the illustration of Figure 8, it can be seen that there is no single value of s that produces an acceptable error for the whole range of jv[0]j. This can be achieved by a double pre-scaling in which two different scaling factors are used so that 1 if jx in j 2 \Gammas1 or jw in j 2 \Gammas1 with s1 s2. Figure 9 shows the error for circular and hyperbolic coordinates, respectively, considering double pre-scaling with 9. This way, the maximum error has been reduced to, approximately, 2 \Gamma15 , that is very close to the precision of the algorithm, for every input vector module. Moreover, a common pre-scaling for hyperbolic and circular coordinates has been used, which facilitates the VLSI implementation of the modified Cordic architecture. Figure compares the error in the angle calculation using the partial normalization technique, described in [10], and the pre-scaling technique with double pre-scaling and modified equations. The error with the pre-scaling technique is always lower than the error with the partial normalization, because, as said before, the partial normalization can not use bits, corresponding to precision less than 2 \Gammab , of the input to perform the normalization. The pre-scaling can be extended to other precision, although for high precision, n 32, it is necessary to consider at least three values for the scaling factor. 5.1 Pre-Scaling Technique with Non-Redundant Arithmetic If the Cordic algorithm is implemented using non-redundant arithmetic, the hardware implementation of the pre-scaling technique consists in the comparison of the module of the vector to the scaling factors and the corresponding shifting of the input operands. The comparison of the module to the scaling factor is performed according equation (44), that is, the two input coordinates are compared to 2 \Gammas1 and 2 \Gammas2 , and the corresponding scaling factor is obtained. As an example, the implementation of the double pre-scaling technique with shown in figure 11. As it can be seen, it is necessary to include only a small number of control gates and two rows of multiplexers, that are in charge of selecting the scaled or unscaled operand. The first row of multiplexers performs the shift 2 5 , according to the result of checking bits 0 to 5 of x in and w in , in such a way that the shift is carried out if these bits are equal to 1 or 0, that is, negative positive numbers less than 2 \Gamma5 . In the second pair of multiplexers an additional shift of 2 4 is performed if bits 6 to 9 are also 1 or 0, performing in this case a total shift of 2 9 . 5.2 Pre-Scaling Technique with Redundant Arithmetic The implementation of the Cordic algorithm with redundant arithmetic requires the assimilation of a certain number of bits of the w variable, to obtain an estimation of the sign, which is used in the determination of the direction of each microrotation [5] [11] [13]. This can only be applied to normalized data because the position of the more significant bit has to be known, in such a way that if the data is not normalized it is necessary to perform a previous normalization. In [3] a Cordic architecture that avoids the assimilation and checking of a certain number of bits is proposed. This architecture is based on the More-Significant-Bit-First calculation of the absolute value of the y coordinate and in the detection of magnitude changes in this calculation. The calculation of the direction of the microrotation requires the propagation of a "carry" from the most-significant to the least-significant bit of the y coordinate. For this reason additional registers are needed for the skew of the data. An important characteristic of this architecture is that the input data do not need to be normalized. The angle calculation error is as described in previous sections. Although this architecture uses the y coordinate as control variable, it can be modified to use variable w instead of the y coordinate, resulting in an high-speed implementation of the modified Cordic algorithm. On the other hand, the pre-scaling can be performed with the scheme shown in figure 11. This way, the pre-scaling technique may be incorporated as a pre-processing stage of this architecture to obtain a pipelined redundant arithmetic Cordic which permits the computation of the inverse tangent function in circular and hyperbolic coordinates without errors over unnormalized data, without any need for performing an initial normalization. 5.3 Evaluation of the Pre-Scaling Technique The hardware complexity of the double pre-scaling is lower than the complexity of a standard normalization stage. Actually, the pre-scaling technique can be considered as an incomplete normalization, where only two different shifts are possible. On the other hand, this hardware complexity is also lower than the partial normalization [10]. The partial normalization implies the introduction of additional hardware into the Cordic architecture. If the x-y registers are b bits long, it is necessary to introduce two normalization shift registers with a maximum shift of (m=2) 1=2 . In addition two leading-zero encoders which operate over the (m=2) 1=2 most significant digits of coordinates x and y are required. An increase in the clock cycle is produced because of the inclusion of two normalization barrel shifters in the critical path of the word-serial implementation. The solution we have presented, the pre-scaling technique, does not introduce additional hardware in the Cordic processors, since the pre-scaling is a pre-processing stage. Moreover, the utilization of variable w instead of coordinate y permits reducing the global hardware cost of the Cordic architecture because one of the shifters is eliminated. Finally, the pre-scaling technique can be used in pipelined processors with redundant arithmetic, whereas the partial normalization technique is restricted to a word-serial architecture with conventional non-redundant arithmetic. A pipelined architecture for the Cordic algorithm with the partial normalization technique would need two barrel shifters in each stage of the pipeline, in addition to the hardware implementation of the Cordic microrotations. The implementation of the partial normalization with redundant arihtmetic is inefficient, since there are several comparisons involved. 6 CONCLUSIONS A thorough analysis of the errors appearing in the calculation of the inverse tangent and the hyperbolic inverse tangent functions with the Cordic algorithm shows that large numerical errors can result when the inputs are unnormnalized and the module of the input vector is small. We have shown, by means of an error analysis, that the utilization of a modified Cordic algorithm based on the w iteration siginificantly reduces the numerical errors in the angle computation with circular and hyperbolic coordinates. In this analysis the rounding error in the input operands has been taken into account, error source neglected in some previous analyse in the literature, and it has been shown that this error becomes very important in applications involving unnormalized inputs. On the other hand, we propose a solution to the problem, operand pre-scaling, that results in a low cost VLSI implementation. The operand pre-scaling technique consists of a pre-processing stage, before the cordic microrotations, where the the input vector is scaled by a constant factor if the module is small enough to result in a large error in the angle computation. This solution can be used in pipelined and word-serial cordic processors with redundant or non-redundant arithmetic. Moreover, the pre-scaling can be chosen in such a way that the same scale factor is used for circular and hyperbolic coordinate systems. --R "A VLSI Speech Analysis Chip Set Based on Square-Root Normalized Ladder Forms" "Unnormalized Fixed-Point Cordic Arithmetic for SVD Processors" "High Speed Bit-Level Pipelined Architectures for Redundant Cordic Implementations" "The Cordic Algorithm: New Results for Fast VLSI Implementation" "Redundant and On-Line Cordic: Application to Matrix Triangularization and SVD" "Redundant and On-Line Cordic for Unitary Transforma- tions" "Cordic-Based VLSI Architectures for Digital Signal Processing" "The Quantization Effects of the Cordic Algorithm" "A Neglected Error Source in the Cordic Algorithm" "Numerical Accuracy and Hardware Tradeoffs for Cordic Arithmetic for Special-Purpose Processors" "Constant-Factor Redundant Cordic for Angle Calculation and Rotation" "2-Fold Normalized Square-Root Schur RLS Adaptive Filter" "Redundant Cordic Methods with a Constant Scale Factor for Sine and Cosine Computation" "Low Latency Time Cordic Algorithms" "The Cordic Trigonometric Computing Technique" "A Unified Algorithm for Elementary Functions" --TR --CTR Toms Lang , Elisardo Antelo, CORDIC Vectoring with Arbitrary Target Value, IEEE Transactions on Computers, v.47 n.7, p.736-749, July 1998

error analysis;operand normalization;angle computation;redundant arithmetic;CORDIC algorithm

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Parallel Cluster Identification for Multidimensional Lattices.

AbstractThe cluster identification problem is a variant of connected component labeling that arises in cluster algorithms for spin models in statistical physics. We present a multidimensional version of Belkhale and Banerjee's Quad algorithm for connected component labeling on distributed memory parallel computers. Our extension abstracts away extraneous spatial connectivity information in more than two dimensions, simplifying implementation for higher dimensionality. We identify two types of locality present in cluster configurations, and present optimizations to exploit locality for better performance. Performance results from 2D, 3D, and 4D Ising model simulations with Swendson-Wang dynamics show that the optimizations improve performance by 20-80 percent.

Introduction The cluster identification problem is a variant of connected component labeling that arises in cluster algorithms for spin models in statistical mechanics. In these applications, the graph to be labeled is a d-dimensional hypercubic lattice of variables called spins, with edges (bonds) that may exist between nearest-neighbor spins. A cluster of spins is a set of spins defined by the transitive closure of the relation "is a bond between". Cluster algorithms require the lattice to be labeled such that any two spins have the same label if and only if they belong to the same cluster. Since the cluster identification step is often the bottleneck in cluster spin model applica- tions, it is a candidate for parallelization. However, implementation on a distributed memory parallel computer is problematic since clusters may span the entire spatial domain, requiring global information propagation. Furthermore, cluster configurations may be highly irregular, preventing a priori analysis of communication and computation patterns. Parallel algorithms for cluster identification must overcome these difficulties to achieve good performance. We have developed a multidimensional extension of Belkhale and Banerjee's Quad algorithm [1, 2], a 2D connected component labeling algorithm developed for VLSI circuit extraction on a hypercube multiprocessor. This paper presents performance results from applying the algorithm to Ising model simulations with Swendson-Wang dynamics [3] in 2D, 3D, and 4D. Our extension abstracts away extraneous spatial information so that distributed data structures are managed in a dimension-independent manner. This strategy considerably simplifies implementation in more than two dimensions. To our knowledge, this implementation is the first parallelization of cluster identification in 4D. To improve performance, we identify two types of locality present in Swendson-Wang cluster configurations and present optimizations to exploit each type of locality. The optimizations work with an abstract representation of the spatial connectivity information, so they are no more complicated to implement in d ? 2 dimensions than in 2D. Performance results show that the optimizations effectively exploit cluster locality, and can improve performance by 20-80% for the multidimensional Quad algorithm. The remainder of his paper proceeds as follows. Section 2 discusses previous approaches to the cluster identification problem on parallel computers. Section 3 describes the Ising model and the Swendson-Wang dynamics. Section 4 reviews Belkhale and Banerjee's Quad algorithm and presents extensions for more than two dimensions. Section 5 presents two optimizations to exploit cluster locality, and section 6 gives performance results in 2D, 3D, and 4D. Related Work Several algorithms for 2D cluster identification on distributed memory MIMD computers have been presented in recent years. Flanigan and Tamayo present a relaxation algorithm for a block domain decomposition [4]. In this method, neighboring processors compare cluster labels and iterate until a steady state is reached. Baillie and Coddington consider a similar approach in their self- labeling algorithm [5]. Both relaxation methods demonstrate reasonable scaleup for 2D problems, but for critical cluster configurations the number of relaxation iterations grows as the distance between the furthest two processors (2 P for block decompositions on P processors). Other approaches similar to relaxation have been presented with strip decompositions [6, 7]. Strip decompositions result in only two external surfaces per processor. However, the distance between two processors can be as large as P , which increases the number of stages to reach a steady state. Multigrid methods to accelerate the relaxation algorithm for large clusters have been presented for SIMD architectures [8, 9]. Host-node algorithms involve communicating global connectivity information to a single processor. This host processor labels the global graph and then communicates the results to other processors. Host-node algorithms [10, 11, 5] do not scale to more than a few processors since the serialized host process becomes a bottleneck. Hierarchical methods for connected component labeling are characterized by a spatial domain decomposition and propagation of global information in log P stages. Our approach is based on the hierarchical Quad algorithm for VLSI circuit extraction on a hypercube multiprocessor [1]. Other hierarchical methods for distributed memory computers have been used for image component labeling [12, 13]. Baillie and Coddington consider a MIMD hierarchical algorithm for the Ising model, but do not achieve good parallel efficiency [5]. Mino presents a hierarchical labeling algorithm for vector architectures [14]. There has been comparably little work evaluating MIMD cluster identification algorithms in more than two dimensions. Bauernfeind et al. consider both relaxation and a host- node approaches to the 3D problem [15]. They introduce the channel reduction and net list optimizations to reduce communication and computation requirements in 3D. They conclude that the host-node approach is inappropriate for 3D due to increased memory requirements on the host node. Fink et al. present 2D and 3D results from a preliminary implementation of the multidimensional Quad algorithm [2]. This paper includes 4D results and introduces issues pertaining to a dimension-independent implementation. Ising Model Many physical systems such as binary fluids, liquid and gas systems, and magnets exhibit phase transitions. In order to understand these "critical phenomena," simple effective models have been constructed in statistical mechanics. The simplest such model, the Ising model, gives qualitative insights into the properties of phase transitions and sometimes can even provide quantitative predictions for measurable physical quantities [16]. The Ising model can be solved exactly in 2D [17]. In more than two dimensions, exact solutions are not known and numerical simulations are often used to obtain approximate results. For example, numerical simulations of the 3D Ising model can be used to determine properties of phase transitions in systems like binary liquids [18]. The 4D Ising model is a prototype of a relativistic field theory and can be used to learn about non-perturbative aspects, in particular phase transitions, of such theories [19]. In d dimensions, the Ising model consists of a d-dimensional lattice of variables (called spins) that take discrete values of \Sigma1. Neighboring spins are coupled, with a coupling strength - which is inversely proportional to the temperature T . Monte Carlo simulations of the Ising model generate a sequence of spin configurations. In traditional local-update Monte Carlo Ising model simulations, a spin's value may or may not change depending on the values of its neighbors and a random variable [5]. Since each spin update depends solely on local information, these algorithms map naturally onto a distributed memory architecture. The interesting physics arises from spin configurations in the critical region, where phase transitions occur. In these configurations, neighboring spins form large clusters in which all spins have the same value. Unfortunately, if - is the length over which spins are correlated (the correlation length), then the number of iterations required to reach a statistically independent configuration grows as - z . For local update schemes the value z (the dynamical critical exponent) is z - 2. Thus, even for correlation lengths - as small as 10 to 100, critical slowing-down severely limits the effectiveness of local-update algorithms for the Ising model [20]. In order to avoid critical slowing-down, Swendson and Wang's cluster algorithm updates whole regions of spins simultaneously [3]. This non-local update scheme generates independent configurations in fewer iterations that the conventional algorithms. The cluster algorithm has a much smaller value of z, often approaching 0. Therefore, it eliminates critical slowing-down completely. The Swendson-Wang cluster algorithm proceeds as follows: 1. Compute bonds between spins. A bond exists with probability adjacent spins with the same value. 2. Label clusters of spins, where clusters are defined by the transitive closure of the relation "is a bond between". 3. Randomly assign all spins in each cluster a common spin value, \Sigma1. These steps are repeated in each iteration. On a distributed memory computer, a very large spin lattice must be partitioned spatially across processors. With a block decomposition, step 1 is simple to parallelize, since we only compute bonds between neighboring spins. Each processor must only communicate spins on the boundaries to neighboring processors. The work in step 3 is proportional to the number of clusters, which is typically much less than the number of lattice sites. Step 2 is the bottleneck in the computation. A single cluster may span the entire lattice, and thus the entire processor array. To label such a cluster requires global propagation of information. Thus the labeling step is not ideally matched to a distributed memory architecture, and requires an efficient parallel algorithm. 4.1 2D Quad Algorithm Our cluster identification method is based on Belkhale and Banerjee's Quad algorithm for geometric connected component labeling [1], which was developed to label connected sets of rectangles that represent VLSI circuits in a plane. It is straightforward to apply the same algorithm to label clusters in a 2D lattice of spin values. A brief description of the Quad algorithm as applied to a 2D lattice of spins is presented here. For a more complete description of the Quad algorithm, see [1]. The cluster labeling algorithm consists of a local labeling phase and a global combining phase. First, the global 2D lattice is partitioned blockwise across the processor array. Each processor labels the clusters in its local partition of the plane with some sequential labeling algorithm. The Quad algorithm merges the local labels across processors to assign the correct global label to each spin site on each processor. On P processors, the Quad algorithm takes log P stages, during which each processor determines the correct global labels for spins in its partition of the plane. Before each stage, each processor has knowledge of a rectangular information region that spans an ever-growing section of the plane. Intuitively, processor Q's information region represents the portion of the global domain from which Q has already collected the information necessary to label Q's local spins. The data associated with an information region consists of ffl A list of labels of clusters that touch at least one border of the information region. These clusters are called CCOMP sets. ffl For each of the four borders of the information region, a list representing the off- processor bonds that touch the border. Each bond in a border list connects a spin site in the current information region with a spin site that is outside the region. Each bond is associated with the CCOMP set containing the local spin site. The border list data structure is a list of offsets into the list of CCOMP set labels, where each offset represents one bond on the border. This indirect representation facilitates Union-Find cluster mergers, which are described below (see figure 1). l 1 l l l l l CCOMP Labels Border Bond Lists Figure 1: Fields of an information region data structure. The initial information region for a processor consists of the CCOMP set labels and border lists for its local partition of the plane. At each stage, each processor Q 1 exchanges messages with a processor Q 2 such that Q 1 and Q 2 's information regions are adjacent. The messages contain the CCOMP set labels and border lists of the current information region. Processor merges the CCOMP sets on the common border of the two information regions using a Union-Find data structure [21]. The other border lists of the two information regions are concatenated to form the information region for processor Q 1 in the next stage. In this manner, the size of a processor's information region doubles at each stage so after log P stages each processor's information region spans the entire plane. Figure 2 illustrates how the information region grows to span the entire global domain. For a planar topology, a processor's global combining is complete when its information region spans the entire plane. If the global domain has a toroidal topology, clusters on opposite borders of the last information region are merged in a post-processing step. current information region stage 1 stage 2 stage 4 stage 3 partner information region done Figure 2: Information regions in each stage of the Quad algorithm. There are sixteen processors. At each stage, the information region of the processor in the top left corner is the current information region. The partner processor and its information region are also shown. In each stage, the two information regions are merged, forming the information region for the subsequent stage. 4.2 Extending the Quad Algorithm to Higher Dimensions A straightforward extension of the Quad algorithm to more than two dimensions results in fairly complex multidimensional information region data structures. To simplify implemen- tation, we present a multidimensional extension using an abstract dimension-independent information region representation. The divide-and-conquer Quad algorithm strategy can be naturally extended to d ? 2 dimensions by partitioning the global domain into d-dimensional blocks, and assigning them one to a processor. Each processor performs a sequential labeling method on its local domain, and then the domains are translated into information regions for the global combining step. An information region represents a d-dimensional subset of the d-dimensional global domain. These d-dimensional information regions are merged at each stage of the algorithm, so after log P stages the information region spans the entire global domain. In two dimensions, the list of bonds on each border is just a 1D list, corresponding to the 1D border between two 2D information regions. Since bonds do not exist at every lattice site, the border lists are sparse. For a 3D lattice, the border lists must represent sparse 2D borders. In general, the border between two d-dimensional information regions is a d \Gamma 1- dimensional hyperplane. Thus a straightforward 3D or 4D implementation would be much more complex than in two dimensions, because sparse multidimensional hyperplanes must be communicated and traversed in order to merge clusters. To avoid this complication, note that if we impose an order on the bonds touching an information region border, the actual spatial location of each bond within the border is not needed to merge sets across processors. As long as each processor stores the border bonds in the same order, we can store the bonds in a 1D list and merge clusters from different processors by traversing corresponding border lists in order. Figure 3 illustrates this for 3D lattices. This concept was first applied by Fink et al. to the 3D Quad algorithm [2], and a Figure 3: The 2D borders of a 3D information region are linearized by enumerating the border bonds in the same order on each processor. similar optimization was applied to 3D lattices by Bauernfeind et al.[15]. We define an order on the border bonds by considering each (d \Gamma 1)-dimensional border as a subset of the d-dimensional global lattice. Enumerate the bonds touching a dimensional border in column-major order relative to the d-dimensional global lattice. Since each processor enumerates the sites relative to the same global indices, each processor stores the sets on a border in the same order, without regard to the orientation of the border in space. This ordering linearizes (d \Gamma 1)-dimensional borders, resulting in an abstract information region whose border representations are independent of the problem dimensionality. When two of these information regions are merged, the order of bonds in the new border lists is consistent on different processors. Therefore, the logic of merging clusters on a border of two information regions does not change for multidimensional lattices. No sparse hyperplane data structures are required, and a 2D cluster implementation can be extended to 3D and 4D with few modifications. 4.3 Performance analysis Belkhale and Banerjee show that the 2D Quad algorithm for VLSI circuit extraction runs in is the number of processors, ff() is the inverse of Ackerman's function, t s is the message startup time, t b is the communication time per byte, and B is is the number of border rectangles along a cross section of the global domain [1]. The number of border rectangles in VLSI circuit extraction applications corresponds to the number of border bonds in cluster identification applications. For cluster identification on a lattice, let N be the lattice size and p be the probability that there is a bond between two adjacent lattice points. Then giving a running time of O(log P t s N)). For a d-dimensional problem, define N and p as above. Assume the global domain is a d-dimensional hypercube with sides N 1=d , which is partitioned onto the d-dimensional logical hypercube of processors with sides P 1=d . Suppose at stage i a processor's information region is a hypercube with sides of length a. Then at stage i + d the information region is a hypercube with sides of length 2a. Thus, the surface area of information region increases by a factor of 2 d\Gamma1 every d stages. Let b(i) be the surface area of the information region at stage i. Then b(i) is at most d e (1) It is easy to see that d . The total number of bonds on the border of an information region is proportional to the surface area. Summing over log P stages, we find the total number of bytes that a processor communicates during the algorithm is O(2dpN d ). There are log P message starts, so the total time spent in communication is O(log The total number of Union-Find operations performed by a processor at each stage is equal to the number of bonds on a border of the information region. Using the path compression and union by rank optimizations of Union-Find operations [21], the total work spent merging clusters is O(pdN d ff(pdN d )). (Our implementation uses the path compression heuristic but not union-by-rank.) Adding together communication and computation, the running time for global combining is O(log d ff(pdN d )). Breadth-First Search(BFS) has been shown to be an efficient algorithm to perform the sequential local labeling step [5]. Since BFS runs in O(jV j+jEj) [21], the local labeling phase runs in O(( N )). Thus, for any dimension lattice, the time for the local phase will dominate the time for the global phase as long as N is large. However, as d increases, the global time increases relative to the local time for a fixed problem size. We must therefore scale the problem size along with the problem dimensionality in order to realize equivalent parallel efficiency for higher dimensional lattices. Optimizations One limitation of the Quad algorithm is that the surface area of the information region grows in each stage. By the last stage, each processor must handle a cross-section of the entire global domain. With many processors and large problem sizes, this can degrade the algorithm's performance [1]. To mitigate this effect, we have developed optimizations that exploit properties of the cluster configuration for better performance. In Monte Carlo Ising model simulations, the cluster configuration structure depends heavily on the coupling constant -. Recall that the probability that a bond exists between two adjacent same-valued spins is . For subcritical (low) -, bonds are relative sparse and most clusters are small. For supercritical (high) -, bonds are relatively dense and one large cluster tends to permeate the entire lattice. At criticality, the system is in transition between these two cases, and the cluster configurations are combinations of small and large clusters. How any particular spin affects the labels of other spins depends on the cluster configuration properties. We identify the following two types of locality that may exist in a cluster configuration: clusters only affect cluster labels in a limited area. ffl Type 2: Adjacent lattice points are likely to belong the same cluster. Subcritical configurations exhibit Type 1 locality, and supercritical configurations exhibit Type 2 locality. Configurations at criticality show some aspects of both types. Belkhale and Banerjee exploit Type 1 locality in two dimensions with the Overlap Quad algorithm [1]. In this algorithm, information regions overlap and only clusters that span the overlap region must be merged. Intuitively, small clusters are eliminated in early stages of the algorithm, leaving only large clusters to merge in later stages. The Overlap Quad algorithm requires that the positions of bonds within borders be maintained, precluding use of an abstract dimension-independent information region data structure. Instead, we present two simpler optimizations, Bubble Elimination and Border Compression. These optimizations work with the abstract border representations, so they are no more complicated to implement in d ? 2 dimensions than in 2D. 5.1 Bubble Elimination Bubble Elimination exploits Type 1 locality by eliminating small clusters in a preprocessing phase to the Quad algorithm. A local cluster that touches only one border of the information region is called a bubble. Immediately after initializing its information region, each processor identifies the bubbles along each border. This information is exchanged with each neighbor, and clusters marked as bubbles on both sides of a border are merged and deleted from the borders. Thus, small clusters are eliminated from the information regions before performing the basic Quad algorithm. During the course of the Quad algorithm, communication and computation is reduced since the bubble clusters are not considered. Bubble elimination incurs a communication overhead of 3 d \Gamma1 messages for a d-dimensional problem. If we communicate with Manhattan neighbors only, the communication overhead drops to 2d messages. Although bubbles on the corners and edges of an information region are not eliminated, this effect is insignificant if the granularity of the problem is sufficiently large. 5.2 Border Compression Border Compression exploits Type 2 locality by changing the representation of the border lists. We compress the representation of each list using run-length encoding [22]. That is, a border list of set labels is replaced by a sequence of pairs ((l where s(l i ) is the number of times value l i appears in succession in a border list. If Type 2 locality is prevalent, border compression aids performance in two ways: it reduces the length of messages, and we can exploit the compressed representation to reduce the number of Union-Find operations that are performed. Before two compressed borders are merged, they are decompressed to form two corresponding lists of cluster labels to combine. From the compressed representation, it is simple to determine when two clusters are merged together several times in succession. During decompression, it is simple to filter redundant mergers out of the lists, reducing the number of Union-Find mergers to be performed. Thus, border compression reduces both communication volume and computation. For some cluster configurations, bubble elimination increases the effectiveness of border compression. Suppose the global cluster configuration resembles swiss cheese, in that there are many small clusters interspersed with one large cluster. This phenomenon occurs in Ising model cluster configurations with - at or above criticality. Bubble elimination removes most small clusters during preprocessing, leaving most active clusters belonging to the one large cluster. In this case, there will be long runs of identical labels along a border of an information region. Border compression collapses these runs, leaving small effective information region borders. 6 Performance Results 6.1 Implementation We have implemented the cluster algorithm and Ising model simulation in 2D, 3D, and 4D with C++. The global lattice has a toroidal topology in all directions. When using bubble elimination, only Manhattan neighbors are considered. The local labeling method is Breadth-First Search [21]. According to the Swendson-Wang algorithm, clusters must be flipped randomly after the cluster identification step. For a spatially decomposed parallel implementation, it is necessary that all processors obtain consistent pseudorandom numbers when generating the new spins for clusters that span more than one processor. In our implementation, we generate the new random spins for each local cluster prior to global cluster merging, and store the new spin as the high-order bit in the cluster label. Thus, after cluster merging, all spins in a cluster are guaranteed to be consistent. To simplify implementation in more than 2 dimensions, we use the LPARX programming library, version 1.1 [23]. LPARX manages data distribution and communication between Cartesian lattices, and greatly simplifies the sections of the code that manages the regular spin lattice. The kernel of the cluster algorithm is written in a message-passing style using the message-passing layer of LPARX[24], a generic message-passing system resembling the Message Passing Interface[25]. Since the cluster algorithm is largely dimension-independent, the message-passing code is almost identical for each problem dimensionality. In fact, the same code generates the 2D,3D, and 4D versions; compile-time macros determine the problem dimensionality. The code was developed and debugged on a Sun workstation using LPARX mechanisms to simulate parallel processes and message-passing. All performance results were obtained from runs on an Intel Paragon under OSF/1 R1.2, with 32MB/node. The code was compiled with gcc v.2.5.7 with optimization level 2. 6.2 Performance The total cluster identification time consists of a local stage, to perform local labeling with a sequential algorithm, and a global stage, to combine labels across processors using the multidimensional Quad algorithm. All times reported are wall clock times and were obtained with the Paragon dclock() system call, which has an accuracy of 100 nanoseconds. Intuitively, we expect the benefits from bubble elimination and border compression to vary with -, the coupling constant. Figures 4, 5, and 6 show the global stage running times at varying values of -. For the problem sizes shown, the critical region occurs at - c - 0:221 in 2D, - c - 0:111 in 3D, and - c - 0:08 in 4D. Since the surface area-to-volume ratio is larger in 3D and 4D than in 2D, the optimizations are more important for these problems. As expected, figures 5 and 6 show that bubble elimination is effective in the subcritical region, and border compression is effective in the supercritical region. In the critical region, the two optimizations together are more effective than either optimization alone. Presumably this is due to the "swiss cheese" effect discussed in Section 5. Together the optimizations improve performance by 35-85%, depending on -, in both 3D and 4D. Kappa1.03.05.07.0 Global Combining Time Per 4D Global Combining Time Nodes, 68x68x34x34 Lattice Bubble Elimination Border Compression Both In 2D, the optimizations improve performance by 20-70%, but do not show the intuitive dependence on - as in 3D and 4D. We suspect this is due to cache effects. As - increases, the number of global clusters decreases. Thus, during cluster merging, more union-find data structure accesses will be cache hits at higher - since a greater portion of the union- find data structure fits in cache. In 2D, the surface area-to-volume ratio is low, so these union-find accesses become the dominant factor in the algorithm's performance. In 3D and 4D, information region borders are much larger, overflowing the cache and causing many more cache misses traversing the borders of the information region. Since these borders are larger than the union-find data structures, union-find data structure memory accesses are less critical. Figure 7 shows the relative costs of the local stage and global stages with and without optimizations. The breakdown shows that in 2D, the local labeling time dominates the global time, so the benefit from optimizations is limited by Amdahl's law [26]. However, in 3D and 4D, the global stage is the bottleneck in the computation, so the two optimizations have a significant impact on performance. Timing results are instructive, but depend on implementation details and machine ar- chitecture. To evaluate the optimizations with a more objective measure, Table 1 shows the total number of bytes transmitted in the course of the Quad algorithm. Since the amount of work to merge clusters is directly proportional to the length of messages, these numbers give a good indication of how successfully the optimizations exploit various cluster configu- rations. The communication volume reduction varies depending on the cluster configuration structure, ranging up to a factor of twenty to one. Since physicists are interested in using parallel processing capability to run larger problems than possible on a single workstation, it is appropriate to examine the algorithm's performance as the problem size and number of processors grow. For an ideal linear parallel Time per Site (ns) Normalized Labeling Perfomance Local Labeling Global Labeling Optimization Both Optimizations subcritical critical supercritical Figure 7: Breakdown of algorithm costs, normalized per spin site. All runs here are with 64 processors of an Intel paragon. The lattice sizes are 4680x4680 in 2D, 280x280x280 in 3D, and 68x68x34x34 in 4D. For subcritical runs, in 3D, and 0.04 in 4D. For critical runs, in 2D, 0.111 in 3D, and 0.08 in 4D. For supercritical runs, in 2D, 0.2 in 3D, and 0.2 in 4D. Opt. Elim. Compress 2D 4680x4680 lattice 3D 280x280x280 lattice 4D 68x68x34x34 lattice Table 1: Total number of bytes transmitted during global combining. All runs are with 64 processors. algorithm, if the problem size and number of processors are scaled together asymptotically, the running time remains constant. Due to the global nature of the cluster identification problem, the basic Quad algorithm cannot achieve ideal scaled speedup in practice. Since the Quad algorithm takes log P stages, the global work should increase by at least log P . A further difficulty is that in d dimensions, the work in the last stage of the algorithm doubles every d stages. However, the bubble elimination and border compression optimizations vastly reduce the work in later stages of the algorithm. Thus, with the optimizations, we can get closer to achieving ideal scaled speedup. Table 2 shows these scaled speedup results for a fixed number of spins sites per processor for critical cluster configurations. The results show that as the number of processors and problem size are scaled together, the performance benefit from the optimizations increases. In 2D, the scaled speedup with the optimizations is nearly ideal. The 3D and especially 4D versions do not scale as well, although figure 7 shows that better performance is achieved away from criticality. Although the optimizations were developed with the multidimensional Quad algorithm in mind, we conjecture that they would also be effective for other cluster identification algorithms, such as relaxation methods [4, 6, 7]. The multidimensional Quad algorithm and optimizations may be also be appropriate for other variants of connected component labeling. One open question is whether the border compression and bubble elimination optimizations would effectively exploit the graph structure of other applications, such as image component labeling applications. 7 Conclusion We have presented an efficient multidimensional extension to Belkhale and Banerjee's Quad algorithm for connected component labeling. Since the extension deals with abstract spatial Number of No Optimizations Both Optimizations Processors 2D 3D 4D 2D 3D 4D Table 2: Global combining time, in seconds, when the lattice size and number of processors is scaled together. Each processor's partition is 585x585 in 2D, 70x70x70 in 3D, and 17x17x17x17 in 4D. All runs are at - c . connectivity information, distributed data structures are managed in a dimension-independent manner. This technique considerably simplifies implementations in more than two dimen- sions. We introduced two optimizations to the basic algorithm that effectively exploit locality in Ising model cluster configurations. Depending on the structure of cluster configurations, the optimizations improve performance by 20-80% on the Intel Paragon. With the opti- mizations, large lattices can be labeled on many processors with good parallel efficiency. The optimizations are especially important in more than two dimensions, where the surface area-to-volume ratio is high. --R "Parallel algorithms for geometric connected component labeling on a hypercube multiprocessor," "Cluster identification on a distributed memory multiprocessor," "Nonuniversal critical dynamics in monte carlo simu- lations," "A parallel cluster labeling method for monte carlo dy- namics," "Cluster identification algorithms for spin models - sequential and parallel," "Parallelization of the Ising model and its performance evaluation," "Swendson-wang dynamics on large 2d critical Ising models," "A multi-grid cluster labeling scheme," "A parallel multigrid algorithm for percolation clusters," "Parallel simulation of the Ising model," "Paralleliza- tion of the 2d swendson-wang algorithm," "Evaluation of connected component labeling algorithms on shared and distributed memory multiprocessors," "Component labeling algorithms on an intel ipsc/2 hypercube," "A vectorized algorithm for cluster formation in the Swendson-Wang dynam- ics," "3D Ising model with swendson-wang dynamics: A parallel approach," Statistical Field Theory. "Crystal statistics. i. a two-dimensional model with an order-disorder tran- sition," "Numerical investigation of the interface tension in the three-dimensional Ising model," "Broken phase of the 4-dimensional Ising model in a finite volume," Computer simulation methods in theoretical physics. Introduction to Algorithms. "A robust parallel programming model for dynamic, non-uniform scientific computation," "The LPARX user's guide v2.0," Message Passing Interface Forum Computer Architecture A Quantitative Approach. --TR --CTR Scott B. Baden, Software infrastructure for non-uniform scientific computations on parallel processors, ACM SIGAPP Applied Computing Review, v.4 n.1, p.7-10, Spring 1996

swendson-wang dynamics;parallel algorithm;cluster identification;ising model;connected component labeling

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Prior Learning and Gibbs Reaction-Diffusion.

AbstractThis article addresses two important themes in early visual computation: First, it presents a novel theory for learning the universal statistics of natural imagesa prior model for typical cluttered scenes of the worldfrom a set of natural images, and, second, it proposes a general framework of designing reaction-diffusion equations for image processing. We start by studying the statistics of natural images including the scale invariant properties, then generic prior models were learned to duplicate the observed statistics, based on the minimax entropy theory studied in two previous papers. The resulting Gibbs distributions have potentials of the form $U\left( {{\schmi{\bf I}};\,\Lambda ,\,S} \right)=\sum\nolimits_{\alpha =1}^K {\sum\nolimits_{ \left( {x,y} \right)} {\lambda ^{\left( \alpha \right)}}}\left( {\left( {F^{\left( \alpha \right)}*{\schmi{\bf I}}} \right)\left( {x,y} \right)} \right)$ with being a set of filters and the potential functions. The learned Gibbs distributions confirm and improve the form of existing prior models such as line-process, but, in contrast to all previous models, inverted potentials (i.e., (x) decreasing as a function of |x|) were found to be necessary. We find that the partial differential equations given by gradient descent on U(I; , S) are essentially reaction-diffusion equations, where the usual energy terms produce anisotropic diffusion, while the inverted energy terms produce reaction associated with pattern formation, enhancing preferred image features. We illustrate how these models can be used for texture pattern rendering, denoising, image enhancement, and clutter removal by careful choice of both prior and data models of this type, incorporating the appropriate features.

texture pattern rendering, denoising, image enhancement and clutter removal by careful choice of both prior and data models of this type, incorporating the appropriate features. Song Chun Zhu is now with the Computer Science Department, Stanford University, Stanford, CA 94305, and David Mumford is with the Division of Applied Mathematics, Brown University, Providence, RI 02912. This work started when the authors were at Harvard University. 1 Introduction and motivation In computer vision, many generic prior models have been proposed to capture the universal low level statistics of natural images. These models presume that surfaces of objects be smooth, and adjacent pixels in images have similar intensity values unless separated by edges, and they are applied in vision algorithms ranging from image restoration, motion analysis, to 3D surface reconstruction. For example, in image restoration general smoothness models are expressed as probability distributions [9, 4, 20, 11]: where I is the image, Z is a normalization factor, and r x I(x; r y I(x; are differential operators. Three typical forms of the potential function /() are displayed in figure 1. The functions in figure 1b, 1c have flat tails to preserve edges and object boundaries, and thus they are said to have advantages over the quadratic function in figure 1a. Figure Three existing forms for /(). a, Quadratic: 2 . b, Line process: These prior models have been motivated by regularization theory [26, 18], 1 phys- 1 Where the smoothness term is explained as a stabilizer for solving "ill-posed" problems [32]. ical modeling [31, 4], 2 Bayesian theory [9, 20] and robust statistics [19, 13, 3]. Some connections between these interpretations are also observed in [12, 13] based on effective energy in statistics mechanics. Prior models of this kind are either generalized from traditional physical models [37] or chosen for mathematical convenience. There is, however, little rigorous theoretical or empirical justification for applying these prior models to generic images, and there is little theory to guide the construction and selection of prior models. One may ask the following questions. 1. Why are the differential operators good choices in capturing image features? 2. What is the best form for p(I) and /()? 3. A relevant fact is that real world scenes are observed at more or less arbitrary scales, thus a good prior model should remain the same for image features at multiple scales. However none of the existing prior models has the scale-invariance property on the 2D image lattice, i.e., is renormalizable in terms of renormalization group theory [36]. In previous work on modeling textures, we proposed a new class of Gibbs distributions of the following form [40, 41], e \GammaU (I; ;S) ; (2) In the above equation, is a set of linear filters, and is a set of potential functions on the features extracted by S. The central property of this class of models is that they can reproduce the marginal distributions of F (ff) I estimated over a set of the training images I - while having the maximum entropy - and the best set of features fF (1) ; F (2) ; :::; F (K) g is If /() is quadratic, then variational solutions minimizing the potential are splines, such as flexible membrane or thin plate models. selected by minimizing the entropy of p(I) [41]. The conclusion of our earlier papers is that for an appropriate choice of a small set of filters S, random samples from these models can duplicate very general classes of textures - as far as normal human perception is concerned. Recently we found that similar ideas of model inference using maximum entropy have also been used in natural language modeling[1]. In this paper, we want to study to what extent probability distributions of this type can be used to model generic natural images, and we try to answer the three questions raised above. We start by studying the statistics of a database of 44 real world images, and then we describe experiments in which Gibbs distributions in the form of equation (2) were constructed to duplicate the observed statistics. The learned potential functions can be classified into two categories: diffusion terms which are similar to figure 1c, and reaction terms which, in contrast to all previous models, have inverted potentials (i.e. -(x) decreasing as a function of jxj). We find that the partial differential equations given by gradient descent on are essentially reaction-diffusion equations, which we call the Gibbs Reaction And Diffusion Equations (GRADE). In GRADE, the diffusion components produce denoising effects which is similar to the anisotropic diffusion [25], while reaction components form patterns and enhance preferred image features. The learned prior models are applied to the following applications. First, we run the GRADE starting with white noise images, and demonstrate how GRADE can easily generate canonical texture patterns such as leopard blobs and zebra stripe, as the Turing reaction-diffusion equations do[34, 38]. Thus our theory provides a new method for designing PDEs for pattern synthesis. Second, we illustrate how the learned models can be used for denoising, image enhancement and clutter removal by careful choice of both prior and noise models of this type, incorporating the appropriate features extracted at various scales and orientations. The computation simulates a stochastic process - the Langevin equations - for sampling the posterior distribution. This paper is arranged as follows. Section (2) presents a general theory for prior learning. Section (3) demonstrates some experiments on the statistics of natural images and prior learning. Section (4) studies the reaction-diffusion equations. Section (5) demonstrates experiments on denoising, image enhancement and clutter removal. Finally section (6) concludes with a discussion. 2 Theory of prior learning 2.1 Goal of prior learning and two extreme cases We define an image I on an N \Theta N lattice L to be a function such that for any pixel (x; y), I(x; y) 2 L, and L is either an interval of R or L ae Z. We assume that there is an underlying probability distribution f(I) on the image space L N 2 for general natural images - arbitrary views of the world. Let NI 2::; Mg be a set of observed images which are independent samples from f(I). The objective of learning a generic prior model is to look for common features and their statistics from the observed natural images. Such features and their statistics are then incorporated into a probability distribution p(I) as an estimation of f(I), so that p(I), as a prior model, will bias vision algorithms against image features which are not typical in natural images, such as noise distortion and blurring. For this objective, it is reasonable to assume that any image features have equal chance to occur at any location, so f(I) is translation invariant with respect to (x; y). We will discuss the limits of this assumption in section (6). To study the properties of images fI obs we start from exploring a set of linear filters which are characteristic of the observed images. The statistics extracted by S are the empirical marginal distributions (or histograms) of the filter responses. Given a probability distribution f(I), the marginal distribution of f(I) with respect to F (ff) is, Z Z F (ff) \LambdaI(x;y)=z where 8z 2 R and ffi() is a Dirac function with point mass concentrated at 0. Given a linear filter F (ff) and an image I, the empirical marginal distribution (or histogram) of the filtered image F (ff) I(x; y) is, We compute the histogram averaged over all images in NI obs as the observed statistics obs (z) =M H (ff) (z; I obs If we make a good choice of our database, then we may assume that - (ff) obs (z) is an unbiased estimate for f (ff) (z), and as M ! 1, - (ff) obs (z) converges to f (ff) Now, to learn a prior model from the observed images fI obs immediately we have two simple solutions. The first one is, Y obs is the observed average histogram of the image intensities, i.e., the filter used. Taking / 1 obs (z), we rewrite equation (4) as, The second solution is: Let kI obs to the Potts model [37]. These two solutions stand for two typical mechanisms for constructing probability models in the literature. The first is often used for image coding [35], and the second is a special case of the learning scheme using radial basis functions (RBF) [27]. 3 Although the philosophies for learning these two prior models are very differ- ent, we observe that they share two common properties. 1. The potentials / 1 (), / 2 () are built on the responses of linear filters. In equation (7), I obs are used as linear filters of size N \Theta N pixels, which we denote by F n . 2. For each filter F (ff) chosen, p(I) in both cases duplicates the observed marginal distributions. It is trivial to prove that E p [H (ff) (z; obs (z), thus as M increases, This second property is in general not satisfied by existing prior models in equation (1). p(I) in both cases meets our objective for prior learning, but intuitively they are not good choices. In equation (5), the ffi() filter does not capture spatial structures of larger than one pixels, and in equation (7), filters F (obsn) are too specific to predict features in unobserved images. In fact, the filters used above lie in the two extremes of the spectrum of all linear filters. As discussed by Gabor [7], the ffi filter is localized in space but is extended uniformly in frequency. In contrast, some other filters, like the sine waves, are well 3 In RBF, the basis functions are presumed to be smooth, such as a Gaussian function. Here, using ffi () is more loyal to the observed data. localized in frequency but are extended in space. Filter F (obsn) includes a specific combination of all the components in both space and frequency. A quantitative analysis of the goodness of these filters is given in table 1 at section (3.2). 2.2 Learning prior models by minimax entropy To generalize the two extreme examples, it is desirable to compute a probability distribution which duplicates the observed marginal distributions for an arbitrary set of filters, linear or nonlinear. This goal is achieved by a minimax entropy theory studied for modeling textures in our previous papers [40, 41]. Given a set of filters fF (ff) observed statistics f- (ff) obs Kg, a maximum entropy distribution is derived which has the following Gibbs form: In the above equation, we consider linear filters only, and is a set of potential functions on the features extracted by S. In practice, image intensities are discretized into a finite gray levels, and the filter responses are divided into a finite number of bins, therefore - (ff) () is approximated by a piecewisely constant functions, i.e., a vector, which we denote by - (ff) The - (ff) 's are computed in a non-parametric way so that the learned p(I; ; S) reproduces the observed statistics: Therefore as far as the selected features and their statistics are concerned, we cannot distinguish between p(I; ; S) and the "true" distribution f(I). Unfortunately, there is no simple way to express the - (ff) 's in terms of the - (ff) obs 's as in the two extreme examples. To compute - (ff) 's, we adopted the Gibbs sampler [9], which simulates an inhomogeneous Markov chain in image space L jN 2 j . This Monte Carlo method iteratively samples from the distribution p(I; ; S), followed by computing the histogram of the filter responses for this sample and updating the - (ff) to bring the histograms closer to the observed ones. For a detailed account of the computation of - (ff) 's, the readers are referred to [40, 41]. In our previous papers, the following two propositions are observed. Proposition 1 Given a filter set S, and observed statistics f- (ff) there is an unique solution for Kg. Proposition 2 f(I) is determined by its marginal distributions, thus if it reproduces all the marginal distributions of linear filters. But for computational reasons, it is often desirable to choose a small set of filters which most efficiently capture the image structures. Given a set of filters S, and an ME distribution p(I; ; S), the goodness of p(I; ; S) is often measured by the Kullback-Leibler information distance between p(I; ; S) and the ideal distribution Z Z f(I) log f(I) Then for a fixed model complexity K, the best feature set S is selected by the following criterion, where S is chosen from a general filter bank B such as Gabor filters at multiple scales and orientations. Enumerating all possible sets of features S in the filter bank and comparing their entropies is computational too expensive. Instead, in [41] we propose a step-wise greedy procedure for minimizing the KL-distance. We start from a uniform distribution, and introduce one filter at a time. Each added filter is chosen to maximally decrease the KL-distance, and we keep doing this until the decrease is smaller than a certain value. In the experiments of this paper, we have used a simpler measure of the "infor- mation gain" achieved by adding one new filter to our feature set S. This is roughly an L 1 -distance (vs. the L 2 -measure implicit in the Kullback-Leibler distance), the readers are referred to [42] for a detailed account). and p(I; ; S) defined above, the information criterion (IC) for each filter F (fi) 2 B=S at step K kH (fi) (z; I obs kH (fi) (z; I obs obs (z)k we call the first term average information gain (AIG) by choosing F (fi) , and the second term average information fluctuation (AIF ). Intuitively, AIG measures the average error between the filter responses in the database and the marginal distribution of the current model p(I; ; S). In practice, we need to sample p(I; ; S), thus synthesize images fI syn estimate E p(I; ;S) [H (fi) (z; I)] by - (fi) n ). For a filter F (fi) , the bigger AIG is, the more information F (fi) captures as it reports the error between the current model and the observations. AIF is a measure of disagreement between the observed images. The bigger AIF is, the less their responses to F (ff) have in common. 3 Experiments on natural images This section presents experiments on learning prior models, and we start from exploring the statistical properties of natural images. Figure 6 out of the 44 collected natural images. 3.1 Statistic of natural images It is well known that natural images have statistical properties distinguishing them from random noise images [28, 6, 24]. In our experiments, we collected a set of 44 natural images, six of which are shown in figure 2. These images are from various sources, some digitized from books and postcards, and some from a Corel image database. Our database includes both indoor and outdoor pictures, country and urban scenes, and all images are normalized to have intensity between 0 and 31. As stated in proposition (2), marginal distributions of linear filters alone are capable of characterizing f(I). In the rest of this paper, we shall only study the following bank B of linear filters. 1. An intensity filter ffi(). 2. Isotropic center-surround filters, i.e., the Laplacian of Gaussian filters. const 2oe stands for the scale of the filter. We denote these filters by LG(s). A special filter is LG( ), which has a 3 \Theta 3 window [0; we denote it by \Delta. 3. Gabor filters with both sine and cosine components, which are models for the frequency and orientation sensitive simple cells. const It is a sine wave at frequency 2- s modulated by an elongated Gaussian function, and rotated at angle '. We denote the real and image parts of G(x; and Gsin(s; '). Two special Gsin(s; ') filters are the gradients r x ; r y . 4. We will approximate large scale filters by filters of small window sizes on the high level of the image pyramid, where the image in one level is a "blown-down" version (i.e., averaged in 2 \Theta 2 blocks) of the image below. We observed three important aspects of the statistics of natural images. First, for some features, the statistics of natural images vary widely from image to image. We look at the ffi() filter as in section (2.1). The average intensity histogram of the 44 images - (o) obs is plotted in figure 3a, and figure 3b is the intensity histogram of an individual image (the temple image in figure 2). It appears that obs (z) is close to a uniform distribution (figure 3c), while the difference between figure 3a and figure 3b is very big. Thus IC for filter ffi() should be small (see table 1). Second, for many other filters, the histograms of their responses are amazingly consistent across all 44 natural images, and they are very different from the histograms of noise images. For example, we look at filter r x . Figure 4a is the average histogram of 44 filtered natural images, figure 4b is the histogram of an individual filtered image (the same image as in figure 3b), and figure 4c is the histogram of a filtered uniform noise image. The average histogram in figure 4a is very different from a Gaussian distribution. Figure 3 The intensity histograms in domain [0; 31], a, averaged over 44 natural images, b, an individual natural image, c, a uniform noise image. Figure 4 The histograms of r x I plotted in domain [-30, 30], a. averaged over 44 natural images, b, an individual natural image, c, a uniform noise image. Figure 5 a. The histogram of r x I plotted against Gaussian curve (dashed) of same mean and variance in domain [\Gamma15; 15]. b, The logarithm of the two curves in a. To see this, figure 5a plots it against a Gaussian curve (dashed) of the same mean and same variance. The histogram of natural images has higher kurtosis and heavier tails. Similar results are reported in [6]. To see the difference of the tails, figure 5b plots the logarithm of the two curves. Third, the statistics of natural images are essentially scale invariant with respect to some features. As an example, we look at filters r x and r y . For each image I obs build a pyramid with I [s] n being the image at the s-th layer. We set I [0] n , and let I [s+1] I [s] The size of I [s] n is N=2 s \Theta N=2 s . -22 a b c Figure 2. b. log - x;s c. histograms of a filtered uniform noise image at scales: curve), and (dashed curve). For the filter r x , let - x;s (z) be the average histogram of r x I [s] Figure 6a plots - x;s (z), for and they are almost identi- cal. To see the tails more clearly, we display log - x;s (z); in figure 6c. The difference between them are still small. Similar results are observed for - y;s (z) the average histograms of r y I obs n . In contrast, figure 6b plots the histograms of r x I [s] with I [s] being a uniform noise image at scales 2. Combining the second and the third aspects above, we conclude that the histograms of r x I [s] are very consistent across all observed natural images and across scales 2. The scale invariant property of natural images is largely caused by the following facts: 1). natural images contains objects of all sizes, 2). natural scenes are viewed and made into images at arbitrary distances. 3.2 Empirical prior models In this section, we learn the prior models according to the theory proposed in section (2), and analyze the efficiency of the filters quantitatively. Experiment I. a b c Figure 7 The three learned potential functions for filters a. \Delta, b. r x , and c. r y . Dashed curves are the fitting functions: a. / 1 and c. / 3 We start from We compute the AIF , AIG and IC for all filters in our filter bank. We list the results for a small number of filters in table 1. The filter \Delta has the biggest IC (= 0:642), thus is chosen as F (1) . An ME distribution p 1 (I; ; S) is learned, and the information criterion for each filter is shown in the column headed p 1 (I) in table 1. We notice that the IC for the filter \Delta drops to near 0, and IC also drops for other filters because these filters are in general not independent of \Delta. Some small filters like LG(1) have smaller ICs than others, due to higher correlations between them and \Delta. Figure 8 A typical sample of p 3 (I) (256 \Theta 256 pixels). The big filters with larger IC are investigated in Experiment II. In this experi- ment, we choose both r x and r y to be F (2) ; F (3) as in other prior models. Therefore a prior model p 3 (I) is learned with potential: are plotted in figure 7. Since - (1) we only plot - (1) (z) for z 2 [\Gamma9:5; 9:5] and - (2) (z); - (3) (z) for z 2 [\Gamma22; 22]. These three curves are fitted with the functions synthesized image sampled from p 3 (I) is displayed in figure 8. So far, we have used three filters to characterize the statistics of natural images, and the synthesized image in figure 8 is still far from natural ones. Especially, even though the learned potential functions - (ff) (z); tails to 4 In fact, - (1) obs obs , with N \Theta N being the size of synthesized image. Filter Filter Size AIF AIG IC AIG IC AIG IC AIG IC I Table 1 The information criterion for filter selection. preserve intensity breaks, they only generate small speckles instead of big regions and long edges as one may expect. Based on this synthesized image, we compute the AIG and IC for all filters, and the results are list in table 1 in column p 3 (I). Experiment II. It is clear that we need large-scale filters to do better. Rather than using the large scale Gabor filters, we chose to use r x and r y on 4 different scales and impose explicitly the scale invariant property that we find in natural images. Given an image I defined on an N \Theta N lattice L, we build a pyramid in the same way as before. Let I 3 be four layers of the pyramid. Let H x;s (z; x; y) denote the histogram of r x I [s] (x; y) and H y;s (z; x; y) the histogram of r y I [s] (x; y). We ask for a probability model p(I) which satisfies, 3: 3: where L s is the image lattice at level s, and - -(z) is the average of the observed histograms of r x I [s] and r y I [s] on all 44 natural images at all scales. This results in a maximum entropy distribution p s (I) with energy of the following form, U s (I) =X Figure 3. At the beginning of the learning process, all - x;s (); are of the form displayed in figure 7 with low values around zero to encourage smoothness. As the learning proceeds, gradually - x;3 () turns "up- side down" with smaller values at the two tails. Then - x;2 () and - x;1 () turn upside down one by one. Similar results are observed for - y;s (); 3. Figure 11 is a typical sample image from p s (I). To demonstrate it has scale invariant properties, in figure 10 we show the histograms H x;s and log H x;s of this synthesized image for 3. The learning process iterates for more than 10; 000 sweeps. To verify the learned -()'s, we restarted a homogeneous Markov chain from a noise image using the learned model, and found that the Markov chain goes to a perceptually similar image after 6000 sweeps. Remark 1. In figure 9, we notice that - x;s () are inverted, i.e. decreasing functions of j z j for distinguishing this model from other prior models in computer vision. First of all, as the image intensity has finite range [0; 31], r x I [s] is defined in [\Gamma31; 31]. Therefore we may define - x;s still well-defined. Second, such inverted potentials have significant meaning in visual computation. In image restoration, when a high intensity difference r x I [s] (x; y) is present, it is very likely to be noise if However this is not true for 3. Additive noise can hardly pass to the high layers of the pyramid because at each layer the 2 \Theta 2 averaging operator reduces the variance of the noise by 4 times. When r x I [s] (x; y) is large for it is more likely to be a true a b -5 c d Figure 9 Learned potential functions - x;s (); 3. The dashed curves are fitting functions: -22 a b Figure a. The histograms of the synthesized image at 4 scales-almost indistinguishable. b. The logarithm of histograms in a. Figure 11 A typical sample of p s (I) (384 \Theta 384 pixels). edge and object boundary. So in p s (I), - x;0 () suppresses noise at the first layer, while - x;s (); encourages sharp edges to form, and thus enhances blurred boundaries. We notice that regions of various scales emerge in figure 11, and the intensity contrasts are also higher at the boundary. These appearances are missing in figure 8. Remark 2. Based on the image in figure 11, we computed IC and AIG for all filters and list them under column p s (I) in table 1. We also compare the two extreme cases discussed in section (2.1). For the ffi() filter, AIF is very big, and AIG is only slightly bigger than AIF . Since all the prior models that we learned have no preference about the image intensity domain, the image intensity has uniform distribution, but we limit it inside [0; 31], thus the first row of table 1 has the same value for IC and AIG. For filter I (obsi) , M i.e. the biggest among all filters, and AIG ! 1. In both cases, ICs are the two smallest. 4 Gibbs reaction-diffusion equations 4.1 From Gibbs distribution to reaction-diffusion equations The empirical results in the previous section suggest that the forms of the potentials learned from images of real world scenes can be divided into two classes: upright curves -(z) for which -() is an even function increasing as jzj increases and inverted curves for which the opposite happens. Similar phenomenon was observed in our learned texture models [40]. In figure 9, - x;s (z) are fit to the family of functions (see the dashed curves), are respectively the translation and scaling constants, and kak weights the contribution of the filter. In general the Gibbs distribution learned from images in a given application has potential function of the following form, OE (ff) ff=n d +1 Note that the filter set is now divided into two parts Kg. In most cases S d consists of filters such as r x ; r y ; \Delta which capture the general smoothness of images, and S r contains filters which characterize the prominent features of a class of images, e.g. Gabor filters at various orientations and scales which respond to the larger edges and bars. Instead of defining a whole distribution with U , one can use U to set up a variational problem. In particular, one can attempt to minimize U by gradient descent. This leads to a non-linear parabolic partial differential equation: I F (ff) ff=n d +1 F (ff) with F (ff) \Gammay). Thus starting from an input image I(x; I in , the first term diffuses the image by reducing the gradients while the second term forms patterns as the reaction term. We call equation (14) the Gibbs Diffusion And Reaction Equation (GRADE). Since the computation of equation (14) involves convolving twice for each of the selected filters, a conventional way for efficient computation is to build an image pyramid so that filters with large scales and low frequencies can be scaled down into small ones in the higher level of the image pyramid. This is appropriate especially when the filters are selected from a bank of multiple scales, such as the Gabor filters and the wavelet transforms. We adopt this representation in our experiments. For an image I, let I [s] be an image at level of a pyramid, and I I, the potential function becomes, s ff s s ff s s I [s] (x; y)) We can derive the diffusion equations similarly for this pyramidal representation. 4.2 Anisotropic diffusion and Gibbs reaction-diffusion This section compares GRADEs with previous diffusion equations in vision. In [25, 23] anisotropic diffusion equations for generating image scale spaces are introduced in the following form, I where div is the divergence operator, i.e., div( ~ and Malik defined the heat conductivity c(x; as functions of local gradients, for example: I I x I y ); (16) Equation (16) minimizes the energy function in a continuous form, Z Z are plotted in figure 12. Similar forms of the energy functions are widely used as prior distributions [9, 4, 20, 11], and they can also be equivalently interpreted in the sense of robust statistics [13, 3] x -0.4 a Figure In the following, we address three important properties of the Gibbs reaction-diffusion equations. First, we note that equation (14) is an extension to equation (15) on a discrete lattice by defining a vector field, ~ and a divergence operator, Thus equation (14) can be written as, I Compared to equation (15) which transfers the "heat" among adjacent pixels, equation transfers the "heat" in many directions in a graph, and the conductivities are defined as functions of the local patterns not just the local gradients. Second, in figure 13, OE(-) has round tip for fl - 1, and a cusp occurs at (0) can be any value in (\Gamma1; 1) as shown by the dotted curves in figure 13d. An interesting fact is that the potential function learned from real world images does have a cusp as shown in figure 9a where the best fit is 0:7. This cusp forms because large part of objects in real world images have flat intensity appearances, and OE(-) with produce piecewise constant regions with much stronger forces than fl - 1. By continuity, OE 0 (-) can be assigned any value in the range [\Gamma!; !] for - 2 [\Gammaffl; ffl] In numerical simulations, for - 2 [\Gamma!; !] we take \Gammaoe if oe 2 [\Gamma!; !] a -0.4 -0.4 x Figure 13 The potential function (-). a,c, where oe is the summation of the other terms in the differential equation whose values are well defined. Intuitively when (0) forms an attractive basin in its neighborhood N (ff) (x; y) specified by the filter window of F (ff) . For a pixel (u; v) 2 N (ff) (x; y), the depth of the attractive basin is k!F (ff) a pixel is involved in multiple zero filter responses, it will accumulate the depth of the attractive basin generated by each filter. Thus unless the absolute value of the driving force from other well-defined terms is larger than the total depth of the attractive basin at (u; v), I(u; v) will stay unchanged. In the image restoration experiments in later sections, performance in forming piecewise constant regions. Third, the learned potential functions confirmed and improved the existing prior models and diffusion equations, but more interestingly reaction terms are first dis- covered, and they can produce patterns and enhance preferred features. We will demonstrate this property in the experiments below. 4.3 Gibbs reaction-diffusion for pattern formation In the literature, there are many nonlinear PDEs for pattern formation, of which the following two examples are interesting. (I) The Turing reaction-diffusion equation which models the chemical mechanism of animal coats [33, 21]. Two canonical patterns that the Turing equations can synthesize are leopard blobs and zebra stripes [34, 38]. These equations are also applied to image processing such as image halftoning [29] and a theoretical analysis can be found in [15]. (II) The Swindale equation which simulates the development of the ocular dominance stripes in the visual cortex of cats and monkey [30]. The simulated patterns are very similar to the zebra stripes. In this section, we show that these patterns can be easily generated with only 2 or 3 filters using the GRADE. We run equation (14) starting with I(x; as a uniform noise image, and GRADE converges to a local minimum. Some synthesized texture patterns are displayed in figure 14. For all the six patterns in figure 14, we choose F (1) the Laplacian of Gaussian filter at level 0 of the image pyramid as the only diffusion filter, and we fix (-). For the three patterns in figure 14 a,b,c we choose isotropic center-surround filter LG( of widow size 7 \Theta 7 pixels as the reaction filter F (2) 1 at level 1 of the image pyramid, and we set (a = \Gamma6:0; (-). The differences between these three patterns are caused by - forms the patterns with symmetric appearances for both black and white part as shown in figure 14a. As - negative, black blobs begin to form as shown in figure 14b where - positive - blobs in the black background as shown in figure 14c where - 6. The general smoothness appearance of the images is attributed to the diffusion filter. Figure 14d is generated with two reaction filters - LG( 2) at level 1 and level 2 respectively, a b c Figure 14 Leopard blobs and zebra stripes synthesized by GRADEs. therefore the GRADE creates blobs of mixed sizes. Similarly we selected one cosine Gabor filter Gcos(4; pixels oriented at 1 as the reaction filter F (2) Figure 14f is generated with two reaction filters Gcos(4; It seems that the leopard blobs and zebra stripes are among the most canonical patterns which can be generated with easy choices of filters and parameters. As shown in [40], the Gibbs distribution are capable of modeling a large variety of texture patterns, but filters and different forms for /(-) have to be learned for a given texture pattern. 5 Image enhancement and clutter removal So far we have studied the use of a single energy function U(I) either as the log likelihood of a probability distribution at I or as a function of I to be minimized by gradient descent. In image processing, we often need to model both the underlying images and some distortions, and to maximize a posterior distribution. Suppose the distortions are additive, i.e., an input image is, I in = I +C: In many applications, the distortion images C are often not i.i.d. Gaussian noise, but clutter with structures such as trees in front of a building or a military target. Such clutter will be very hard to handle by edge detection and image segmentation algorithms. We propose to model clutter by an extra Gibbs distribution, which can be learned from some training images by the minimax entropy theory as we did for the underlying image I. Thus an extra pyramidal representation for I in \Gamma I is needed in a Gibbs distribution form as shown in figure 15. The resulting posterior distributions are still of the Gibbs form with potential function, U where UC () is the potential of the clutter distribution. Thus the MAP estimate of I is the minimum of U . In the experiments which we use the Langevin equation for minimization, a variant of simulated annealing where w(x; is the standard Brownian motion process, i.e., w(x; T (t) is the "temperature" which controls the magnitude of the random fluctuation. Under mild conditions on U , equation (19) approaches a global minimum of U at target features image pyramid for clutters image pyramid for targets clutter features observed image target image clutter image Figure 15 The computational scheme for removing noise and clutter. a low temperature. The analyses of convergence of the equations can be found in [14, 10, 8]. The computational load for the annealing process is notorious, but for applications like denoising, a fast decrease of temperature may not affect the final result very much. Experiment I In the first experiment, we take UC to be quadratic, i.e. C to be an i.i.d. Gaussian noise image. We first compare the performance of the three prior models potential functions are respectively: U l the 4-scale energy in equation (12) (22) l () and / t () are the line-process and T-function displayed in figure 1b and 1c respectively. Figure demonstrates the results: the original image is the lobster boat displayed in figure 2. It is normalized to have intensity in [0; 31] and Gaussian noise from N(0; 25) are added. The distorted image is displayed in figure 16a, where we keep the image boundary noise-free for the convenience of boundary condition. The restored images using p l (I), p t (I) and p s (I) are shown in figure 16b, 16c, 16d respectively. p s (I), which is the only model with a reaction term, appears to have the best effect in recovering the boat, especially the top of the boat, but it also enhances the water. Experiment II In many applications, i.i.d. Gaussian models for distortions are not sufficient. For example, in figure 17a, the tree branches in the foreground will make image segmentation and object recognition extremely difficult because they cause strong edges across the image. Modeling such clutter is a challenging problem in many applications. In this paper, we only consider clutter as two dimensional pattern despite its geometry and 3D structure. We collected a set of images of buildings and a set of images of trees all against clean background - the sky. For the tree images, we translate the image intensities to sky. In this case, since the trees are always darker than the build, thus the negative intensity will approximately take care of the occlusion effects. We learn the Gibbs distributions for each set respectively in the pyramid, then such models are respectively adopted as the prior distribution and the likelihood as in equation (18). We recovered the underlying images by maximizing a posteriori distribution using the stochastic process. For example, figure 17b is computed using 6 filters with 2 filters for I: fr and 4 filters for I C i.e., the potential for I C is, In the above equation, OE (-) and / (-) are fit to the potential functions learned from the set of tree images, a a c d Figure a. The noise distorted image, b. c. d. are respectively restored images by prior models p l (I). Figure 17 a. the observed image, b, the restored image using 6 filters. So the energy term OE (I(x; y)) forces zero intensity for the clutter image while allowing for large negative intensities for the dark tree branches. Figure 18b is computed using 8 filters with 4 filters for I and 4 filters for I C . 13 filters are used for figure 19 where the clutter is much heavier. As a comparison, we run the anisotropic diffusion process [25] on figure 19a, and images at iterations are displayed in figure 20. As we can see that as becomes a flat image. A robust anisotropic diffusion equation is recently reported in [2]. 6 Conclusion In this paper, we studied the statistics of natural images, based on which a novel theory is proposed for learning the generic prior model - the universal statistics of real world scenes. We argue that the same strategy developed in this paper can be used in other applications. For example, learning probability models for MRI Figure a. an observed image, b. the restored image using 8 filters. a b Figure 19 a. the observed image, b. the restored image using 13 filters. a Figure 20 Images by anisotropic diffusion at iteration images and 3D depth maps. The learned prior models demonstrate some important properties such as the "inverted" potentials terms for patterns formation and image enhancement. The expressive power of the learned Gibbs distributions allow us to model structured noise-clutter in natural scenes. Furthermore our prior learning method provides a novel framework for designing reaction-diffusion equations based on the observed images in a given application, without modeling the physical or chemical processes as people did before [33]. Although the synthesized images bear important features of natural images, they are still far from realistic ones. In other words, these generic prior models can do very little beyond image restoration. This is mainly due to the fact that all generic prior models are assumed to be translation invariant, and this homogeneity assumption is unrealistic. We call the generic prior models studied in this paper the first level prior. A more sophisticated prior model should incorporate concepts like object geometry, and we call such prior models second level priors. Diffusion equations derived from this second level priors are studied in image segmentation [39], and in scale space of shapes [16]. 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clutter modeling;reaction-diffusion;anisotropic diffusion;gibbs distribution;image restoration;texture synthesis;visual learning

271577

Proximal Minimization Methods with Generalized Bregman Functions.

We consider methods for minimizing a convex function f that generate a sequence {xk} by taking xk+1 to be an approximate minimizer of f(x)+Dh(x,xk)/ck, where ck > 0 and Dh is the D-function of a Bregman function h. Extensions are made to B-functions that generalize Bregman functions and cover more applications. Convergence is established under criteria amenable to implementation. Applications are made to nonquadratic multiplier methods for nonlinear programs.

Introduction We consider the convex minimization problem is a closed proper convex function and X is a nonempty closed convex set in IR n . One method for solving (1.1) is the proximal point algorithm (PPA) [Mar70, Roc76b] which generates a sequence starting from any point x is the Euclidean norm and fc k g is a sequence of positive numbers. The convergence and applications of the PPA are discussed, e.g., in [Aus86, CoL93, EcB92, GoT89, G-ul91, Lem89, Roc76a, Roc76b]. Several proposals have been made for replacing the quadratic term in (1.2) with other distance-like functions [BeT94, CeZ92, ChT93, Eck93, Egg90, Ius95, IuT93, Teb92, TsB93]. In [CeZ92], (1.2) is replaced by Research supported by the State Committee for Scientific Research under Grant 8S50502206. Systems Research Institute, Newelska 6, 01-447 Warsaw, Poland ([email protected]) where D h (x; is the D-function of a Bregman function h [Bre67, CeL81], which is continuous, strictly convex and differentiable in the interior of its domain (see x2 for a full definition); here h\Delta; \Deltai is the usual inner product and rh is the gradient of h. Accordingly, this is called Bregman proximal minimization (BPM). The convergence of the BPM method is discussed in [CeZ92, ChT93, Eck93, Ius95, TsB93], a generalization for finding zeros of monotone operators is given in [Eck93], and applications to convex programming are presented in [Cha94, Eck93, Ius95, NiZ92, NiZ93a, NiZ93b, This paper discusses convergence of the BPM method using the B-functions of [Kiw94] that generalize Bregman functions, being possibly nondifferentiable and infinite on the boundary of their domains (cf. x2). Then (1.3) involves D k , where fl k is a subgradient of h at x k . We establish for the first time convergence of versions of the BPM method that relax the requirement for exact minimization in (1.3). (The alternative approach of [Fl-a94], being restricted to Bregman functions with Lipschitz continuous gradients, cannot handle the applications of xx7-9.) We note that in several important applications, strictly convex problems of the form (1.3) may be solved by dual ascent methods; cf. references in [Kiw94, Tse90]. The application of the BPM method to the dual functional of a convex program yields nonquadratic multiplier methods [Eck93, Teb92]. By allowing h to have singularities, we extend this class of methods to include, e.g., shifted Frish and Carroll barrier function methods [FiM68]. We show that our criteria for inexact minimization can be implemented similarly as in the nonquadratic multiplier methods of [Ber82, Chap. 5]. Our convergence results extend those in [Eck93, TsB93] to quite general shifted penalty functions, including twice continuously differentiable ones. We add that the continuing interest in nonquadratic modified Lagrangians stems from the fact that, in contrast with the quadratic one, they are twice continuously differentiable, and this facilitates their minimization [Ber82, BTYZ92, BrS93, BrS94, CGT92, CGT94, GoT89, IST94, JeP94, Kiw96, NPS94, Pol92, PoT94, Teb92, TsB93]. By the way, our convergence results seem stronger than ones in [IST94, PoT94] for modified barrier func- tions, resulting from a dual application of (1.3) with D k replaced by an entropy-like OE-divergence. The paper is organized as follows. In x2 we recall the definitions of B-functions and Bregman functions and state their elementary properties. In x3 we present an inexact BPM method. Its global convergence under various conditions is established in xx4-5. In x6 we show that the exact BPM method converges finitely when (1.1) enjoys a sharp minimum property. Applications to multiplier methods are given in x7. Convergence of general multiplier methods is studied in x8, while x9 focuses on two classes of shifted penalty methods. Additional aspects of multiplier methods are discussed in x10. The Appendix contains proofs of certain technical results. Our notation and terminology mostly follow [Roc70]. IR m ? are the nonnegative and positive orthants of IR m respectively. For any set C in IR n , cl C, - ri C and bd C denote the closure, interior, relative interior and boundary of C respectively. ffi C is the indicator function of C (ffi C xi is the support function of C. For any closed proper convex function f on IR n and x in its effective domain is the ffl- subdifferential of f at x for each ffl - 0, is the ordinary subdifferential of f at x and f 0 denotes the derivative of f in any direction . By [Roc70, Thms 23.1-23.2], f 0 (x; d) - \Gammaf 0 (x; \Gammad) and The domain and range of @f are denoted by C @f and im@f respectively. By [Roc70, Thm ri C f ae C @f ae C f . f is called cofinite when its conjugate f is real-valued. A proper convex function f is called essentially smooth if - differentiable on - C f . If f is closed proper convex, its recession function f0 positively homogeneous [Roc70, Thm 8.5]. B-functions We first recall the definitions of B-functions [Kiw94] and of Bregman functions [CeL81]. For any convex function h on IR n , we define its difference functions By convexity (cf. (1.4)), h(x) - h(y) h and D ] h generalize the usual D-function of h [Bre67, CeL81], defined by since Definition 2.1. A closed proper (possibly nondifferentiable) convex function h is called a B-function (generalized Bregman function) if (a) h is strictly convex on C h . (b) h is continuous on C h . (c) For every ff 2 IR and x 2 C h , the set L 1 (d) For every ff 2 IR and x 2 C h , if fy k g ae L 1 h (x; ff) is a convergent sequence with limit Definition 2.2. Let S be a nonempty open convex set in IR n . Then cl S, is called a Bregman function with zone S, denoted by h 2 B(S), if (i) h is continuously differentiable on S. (ii) h is strictly convex on - S. (iii) h is continuous on - S. (iv) For every ff 2 IR, ~ S, the sets L 2 fy are bounded. (v) If fy k g ae S is a convergent sequence with limit y , then D h (y (vi) If fy k g ae S converges to y , fx k g ae - S is bounded and D h (Note that the extension e of h to IR n , defined by is a B-function with C ri C e (\Delta; e (\Delta; h and D ] h are used like distances, because for and D [ strict convexity. Definition 2.2 (due to [CeL81]), which requires that h be finite-valued on - S, does not cover Burg's entropy [CDPI91]. Our Definition 2.1 captures features of h essential for algorithmic purposes. As shown in [Kiw94], condition (b) implies (c) if h is cofinite. Sometimes one may verify the following stronger version of condition (d) C @h oe fy k by using the following three lemmas proven in [Kiw94]. Lemma 2.3. (a) Let h be a closed proper convex function on IR n , and let S 6= ; be a compact subset of ri C h . Then there exists ff 2 IR s.t. joe @h(y) (b) Let is the indicator function of a convex polyhedral set S 6= ; in IR n . Then h satisfies condition (2.5). (c) Let h be a proper polyhedral convex function on IR n . Then h satisfies condition (2.5). (d) Let h be a closed proper convex function on IR. Then h is continuous on C h , and fy k g ae C h . Lemma 2.4. (a) Let are closed proper convex functions s.t. h are polyhedral and " j condition (c) of Def. 2.1, then so does h. If condition (d) of Def. 2.1 or (2.5), then so does h. If h 1 is a B-function, h are continuous on C and satisfy condition (d) of Def. 2.1, then h is a B-function. In particular, h is a B-function if so are h (b) Let h be B-functions s.t. " j ri C h i 6= ;. Then function. (c) Let h 1 be a B-function and let h 2 be a closed proper convex function s.t. C h 1 ae ri C h 2 is a B-function. (d) Let h closed proper strictly convex functions on IR s.t. L 1 (t; ff) is bounded for any t; ff 2 IR, Lemma 2.5. Let h be a proper convex function on IR. Then L 1 h (x; ff) is bounded for each Lemma 2.6. (a) If / is a B-function on IR then / is essentially smooth and C / (b) If OE closed proper convex essentially smooth and C C OE then OE is a B-function with ri C OE ae imrOE ae C OE . Proof. (a): This follows from Def. 2.1, Lem. 2.5 and [Roc70, Thm 26.3]. (b): By [Roc70, Thms 23.4, 23.5 and 26.1], ri C OE ae C @OE strictly convex on C @OE , and hence on C OE by an elementary argument. Since OE is closed proper convex and OE Thm 12.2], the conclusion follows from Lems. 2.3(d) and 2.5. Examples 2.7. Let In each of the examples, it can be verified that h is an essentially smooth B-function. ff =ff. Then h 2. ff =ff if p. 106]. 3 ('x log x'-entropy) [Bre67]. Kullback-Liebler entropy). [Roc70, p. 105] and D h is the Kullback-Liebler entropy. (1\Gammay 2 6 (Burg's entropy) [CDPI91]. 3 The BPM method We make the following standing assumptions about problem (1.1) and the algorithm. Assumption 3.1. (i) f is a closed proper convex function. (ii) X is a nonempty closed convex set. (iii) h is a (possibly nonsmooth) B-function. is the essential objective of (1.1). is a sequence of positive numbers satisfying is a sequence of nonnegative numbers satisfying lim l!1 Consider the following inexact BPM method. At iteration k - 1, having find x k+1 , fl k+1 and p k+1 satisfying We note that x k+1 - arg minffX +D k with by (2.1), (2.2), (3.2) and (3.3); in fact x k+1 is an ffl k -minimizer of as shown after the following (well-known) technical result (cf. [Roc70, Thm 27.1]). Lemma 3.2. A closed proper and strictly convex function OE on IR n has a unique minimizer iff OE is inf-compact, i.e., the ff-level set L OE is bounded for any ff 2 IR, and this holds iff L OE (ff) is nonempty and bounded for one ff 2 IR. Proof. If x 2 Arg min OE then, by strict convexity of OE, L OE is inf-compact (cf. [Roc70, Cor. 8.7.1]). If for some ff 2 IR, L OE (ff) 6= ; is bounded then it is closed (cf. [Roc70, Thm 7.1]) and contains Arg min OE 6= ; because OE is closed. Lemma 3.3. Under the above assumptions, we have: (i) OE k is closed proper and strictly convex. is cofinite. In particular, OE k is inf-compact if (fl ri C f (v) If OE k is inf-compact and either ri C f X ri C h 6= ;, or C f X ri C h 6= ; and fX is polyhedral, then there exist - x if C @f X ae - C h or C essentially smooth. (vi) The assumptions of (v) hold if either ri C f X ae - ae - and Proof. (i) Since f , ffi X and h are closed proper convex, so are h (\Delta; x k )=c k (cf. [Roc70, Thm 9.3]), having nonempty domains C f " X, C h and respectively (cf. Assumption 3.1(iv)). D k are strictly convex, since so is h (cf. Def. 2.1(a)). (ii) For any x, add the inequality D k (cf. (3.3), (3.4)) divided by c k to fX (x) - fX (cf. (3.6)) and use (3.5) to get OE k (x) - OE k (iii) By part (i), closed proper strictly convex, and L / by strict convexity of h (cf. Def. 2.1(a), (2.2) and (1.4)), so / is inf-compact (cf. Lem. 3.2). (cf. (3.9)). The last set is bounded, since / is inf-compact, so OE k is inf-compact by part (i) and Lem. 3.2. ~ closed proper and strictly convex (so is D k cf. part (i)), and ~ Thm 23.8]). Hence ~ / is inf-compact (cf. Lem. 3.2), and so is OE k , since OE k - ~ / from yi. To see that strict convexity of h (cf. Def. 2.1(a)) implies C h , we note that - Thms 26.3 and 26.1], and @h by [Roc70, Thm 23.5], so that C @h is cofinite. The second assertion follows from ri C f ae C @f (v) By part (i) and Lem. 3.2, - x defined. The rest follows from (cf. (3.8)), the fact due to our assumptions on C f X and ri C h (cf. [Roc70, Thm 23.8]), and [Roc70, Thm 26.1]. (vi) If inf is inf-compact by parts (iii)-(iv). If ri ri ri C ri C f X 6= ;, since C f X Assumption 3.1(iv)). Remark 3.4. Lemma 3.3(v,vi) states conditions under which the exact BPM method (with x in (3.6)) is well defined. Our conditions are slightly weaker than those in [Eck93, Thm 5], which correspond to ri C f X ae - cl being finite, continuous and bounded below on X. Example 3.5. Let implies that - x k+1 is well defined. Example 3.6. Let ri C f ri C f " - const for c Although h is not a Bregman function, this is a counterexample to [Teb92, Thm 3.1]. 4 Convergence of the BPM method We first derive a global convergence rate estimate for the BPM method. We follow the analysis of [ChT93], which generalized that in [G-ul91]. Let s Lemma 4.1. For all x 2 C h and k - l, we have l l l Proof. The equality in (4.1) follows from (3.3), and the inequality from (cf. (3.5)) and p k+1 2 @ ffl k fX since c k ? 0. (4.2) is a consequence of (4.1). Summing (4.1) over l we obtain l l l Use fX in (4.5) to get (4.3). (4.4) follows from (4.3) and the fact D k We shall need the following two results proven in [TsB91]. Lemma 4.2 ([TsB91, Lem. 1]). Let h : be a closed proper convex function continuous on C h . Then: (a) For any y 2 C h , there exists ffl ? 0 s.t. closed. (b) For any y 2 C h and z s.t. y any sequences y k ! y and z k ! z s.t. Lemma 4.3. Let h : be a closed proper convex function continuous on C h . If fy k g ae C h is a bounded sequence s.t., for some y 2 C h , fh(y k bounded from below, then fh(y k )g is bounded and any limit point of fy k g is in C h . Proof. Use the final paragraph of the proof of [TsB91, Lem. 2]. Lemmas 4.2-4.3 could be expressed in terms of the following analogue of (2.1) Lemma 4.4. Let h : be a closed proper strictly convex function continuous on C h . If y 2 C h and fy k g is a bounded sequence in C h s.t. D 0 Proof. Let y 1 be the limit of a subsequence fy k g k2K . Since h(y k h(y by Lem. 4.3 and h(y k ) K \Gamma! h(y 1 ) by continuity of h on C h . Then by Lem. 4.2(b), yields strict convexity of h. Hence By (1.4), (3.2), (3.3), (2.2) and (4.6), for all k Lemma 4.5. If is bounded and fx k g ae L 1 (ii) Every limit point of fx k g is in C h . converges to some x Proof. (i) We have D l ae C @h (cf. (3.1)), so fx k g ae L 1 h (x; ff), a bounded set (cf. Def. 2.1(c)). (4.6), (4.7)), so the desired conclusion follows from continuity of h on C h (cf. Def. 2.1(b)), being bounded in C h (cf. (3.1) and part (i)) and Lem. 4.3. (iii) By parts (i)-(ii), a subsequence fx l j g converges to some x But fX and fX is closed (cf. Assumption 3.1(i,ii)). Hence for l ? l j , D l (cf. (4.2)) with Finally, if x does not converge, it has a limit point x 0 6= x 1 (cf. parts (i)-(ii)), and replacing x and x 1 by respectively in the preceding argument yields a contradiction. We may now prove our main result for the inexact BPM descent method (3.1)-(3.7). Theorem 4.6. Suppose Assumption 3.1(i-ii,iv-v) holds with h closed proper convex. (a) If lim l!1 Hence ae C h . If ri C h " ri C f X cl C h fX . If ri C f X ae cl C h (e.g., C @f X ae cl C h ) then cl C h oe cl C f X and Arg min X f ae cl C h . (b) If h is a B-function, fX fX is nonempty then fx k g converges to some x (c) If fX Proof. (a) For any x 2 C h , taking the limit in (4.4) yields lim l!1 fX using Assumption 3.1(v)) and Hence fX 7.3.2]). If ri C h " ri (cf. [Roc70, Thm 6.5]) and inf C h cl C h fX , so cl C h fX . If ri C f X ae cl C h then cl C f X ae cl C h (cf. [Roc70, Thm 6.5]). (b) If x 2 X then fX (c) If jx k j 6! 1, fx k g has a limit point x with fX (x) - inf C h fX / fX closed; cf. Assumption 3.1(i,ii)), so C f Remark 4.7. For the exact BPM method (with ffl k j 0), Thm 4.6(a,b) subsumes [ChT93, Thm 3.4], which assumes ri C f X ae - C h and C cl C h . Thm 4.6(b,c) strengthens [Eck93, Thm 5], which only shows that fx k g is unbounded if cl C f X ae - and Lem. 3.3 subsume [Ius95, Thm 4.1], which assumes that h is essentially smooth, f is continuous on C f , cl C h , Arg min X f For choosing fffl k g (cf. Assumption 3.1(vi)), one may use the following simple result. Lemma 4.8. (i) If ffl k ! 0 then (ii) If Proof. (i) For any ffl ? 0, pick - k and - l ? - k s.t. ffl k - ffl for all k - k and for all l - l; then (ii) We have 5 Convergence of a nondescent BPM method In certain applications (cf. x7) it may be difficult to satisfy the descent requirement (3.7). Hence we now consider a nondescent BPM method, in which (3.7) is replaced by By Lem. 3.3(ii), (5.1) holds automatically, since it means OE k Lemma 5.1. For all x 2 C h and k - l, we have l l l Proof. (4.1)-(4.2) still hold. (5.2) follows from D k . Multiplying this inequality by s and summing over l l l s Subtract (5.5) from (4.5) and rearrange, using s to get (5.3). (5.4) follows from (5.3) and the fact D k Theorem 5.2. Suppose Assumption 3.1(i-ii,iv-v) holds with h closed proper convex. (a) If 5.3 for sufficient conditions), then fX Hence the assertions of Theorem 4.6(a) hold. (b) If h is a B-function, fX fX is nonempty then fx k g converges to some x ae C h . (c) If fX ae C h and X Proof. (a) The upper limit in (5.4) for any x 2 C h yields lim sup l!1 fX using (b) If x 2 X then fX Assertions (i)-(iii) of Lem. 4.5 still hold, since the proofs of (i)-(ii) remain valid, whereas in the proof of (iii) we have x and fX (c) Use the proof of Thm 4.6(c). Lemma 5.3. (i) Let fff k g, ffi k g and f" k g be sequences in IR s.t. 0 - ff (ii) If Proof. (i) See, e.g., [Pol83, Lem. 2.2.3]. (ii) Use part (i) with ff l = (iii) Use part (ii) with c l =s l 2 [c min =lc 6 Finite termination for sharp minima We now extend to the exact BPM method the finite convergence property of the PPA in the case of sharp minima (cf. [Fer91, Roc76b] and [BuF93]). Theorem 6.1. Let f have a sharp minimum on X, i.e., X there exists x. Consider the exact BPM method applied to (1.1) with a B-function h s.t. C f X ae Crh , ffl k j 0 and inf k c k ? 0. Then there exists k s.t. Proof. By Thm 4.6, x continuity of rh on Crh [Roc70, Thm 25.5]) and @fX (cf. (3.5)-(3.6)). But if for Hence for some k, jp We note that piecewise linear programs have sharp minima, if any (cf. [Ber82, x5.4]). 7 Inexact multiplier methods Following [Eck93, Teb92], this section considers the application of the BPM method to dual formulations of convex programs of the form presented in [Roc70, x28]: minimize f(x); subject to g i (x) - 0; under the following Assumption 7.1. f , are closed proper convex functions on IR n with C f ae and ri C f ae ri C g i Letting we define the Lagrangian of (7.1) and the dual functional . Assume that -. The dual problem to (7.1) is to maximize d, or equivalently to minimize q(-) over - 0, where \Gammad is a closed proper convex function. We will apply the BPM method to this problem, using some B-function h on IR m . We assume that IR m is a B-function (cf. Lem. 2.4(a)). The monotone conjugate of h (cf. [Roc70, p. 111]) defined by h nondecreasing (i.e., h coincides with the convex conjugate h of h+ , since h (\Delta). We need the following variation on [Eck93, Lem. A3]. Its proof is given in the Appendix. Lemma 7.2. If h is a closed proper essentially strictly convex function on IR m with ri C h 6= ;, then h + is closed proper convex and essentially smooth, @h all is continuous on C @h where I is the identity operator and N IR m is the normal cone operator of IR m i.e., additionally im@h oe IR m ? then h+ is cofinite, C continuously differentiable. , to find inf 0 q(-) via the BPM method we replace in (3.1)- our inexact multiplier method requires finding - k+1 and x k+1 s.t. with for some p k+1 and fl k+1 . Note that (7.2) implies since \Delta; g(x k+1 ) \Gammag(x and (7.6), (7.4)-(7.5) hold if we take p since then Using (7.3) and (@h+ (Lem. 7.2), we have so we may take ~ fl choices will be discussed later. Further insight may be gained as follows. Rewrite (7.3) as where Let cf. Assumption 7.1), L k Lemma 7.3. Suppose inf C f e.g., the feasible set C of (7.1) is nonempty. Then L k is a proper convex function and If - (g(-x)). The preceding assertions hold when inf C f (cf. Lem. 7.2). Proof. Using ? and ~ u. Then, since P k is nondecreasing (so is ri C f ae ri C g i (cf. Assumption 7.1), Lem. A.1 in the Appendix yields im@P k ae IR m and (7.13), using @P (cf. Lem. 7.2). Hence if @L k (x) so ri C f ae ri C g i implies (cf. [Roc70, Thm 23.8]) @ x L(x; -). Finally, when C then for any ~ x 2 C f we may pick ~ with u, since C f ae (Assumption 7.1) and C The exact multiplier method of [Eck93, Thm 7] takes x assuming h is smooth, - ? and imrh oe IR m ? . Then (7.2) holds with Our inexact method only requires that x k+1 ~ in the sense that (7.2) holds for a given ffl k - 0. Thus we have derived the following Algorithm 7.4. At iteration k - 1, having ae oe s.t. (7.2) holds, choose fl k+1 satisfying (7.7) and set p To find x k+1 as in [Ber82, x5.3], suppose f is strongly convex, i.e., for some - ff ? 0 Adding subgradient inequalities of g i , using (7.14) yields for all x 7.3). Minimization in (7.16) yields so (7.2) holds if Thus, as in the multiplier methods of [Ber82, x5.3], one may use any algorithm for minimizing L k that generates a sequence fz j g such that lim inf j!1 setting ff is unknown, it may be replaced in (7.18) by any fixed ff ? 0; this only scales fffl k g.) Of course, the strong convexity assumption is not necessary if one can employ the direct criterion (7.2), i.e., L(z (cf. (7.10)), where d(-) may be computed with an error that can be absorbed in ffl k . Some examples are now in order. Example 7.5. Suppose are B-functions on IR with C h i oe 2.4(d)). For each i, let - so that (cf. [Eck93, Ex. 6]) h Using (7.9) and "maximal" fl k+1 in (7.7), Alg. 7.4 may be written as Remark 7.6. To justify (7.19c), note that if we had would not penalize constraint violations An ordinary penalty method (cf. [Ber82, p. 354]) would use (7.19a,b) with u and c k " 1. Thus (7.19) is a shifted penalty method, in which the shifts fl k should ensure convergence even for sup k c k ! 1, thus avoiding the ill-conditioning of ordinary penalty methods. Example 7.7. Suppose C @h " , so that from IR m (cf. [Roc70, Thms 23.8 and 25.1]). Then we may use since the maximal shift due to (7.9). Thus Alg. 7.4 becomes ae oe In the separable case of Ex. 7.5, the formulae specialize to Example 7.8. Let a B-function on IR with Cr/ oe IR ? . Using (7.7) and (7.9) as in Ex. 7.5, we may let fl k+1 m. Thus Alg. 7.4 becomes Example 7.9. For becomes Even if f and all g i are smooth, for the objective of (7.21a) is, in general, only once continuously differentiable. This is a well-known drawback of quadratic augmented Lagrangians (cf. [Ber82, TsB93]). However, for we obtain a cubic multiplier method [Kiw96] with a twice continuously differentiable objective. Example 7.10 ([Eck93, Ex. 7]). For reduces to i.e., to an inexact exponential multiplier method (cf. [Ber82, x5.1.2], [TsB93]). Example 7.11. For reduces to i.e., to an inexact shifted logarithm barrier method (which was also derived heuristically in [Cha94, Ex. 4.2]). This method is related, but not indentical, to ones in [CGT92, GMSW88]; cf. [CGT94]. Example 7.12. If reduces to corresponds to a shifted Carroll barier method. 8 Convergence of multiplier methods In addition to Assumption 7.1, we make the following standing assumptions. Assumption 8.1. (i) h+ is a B-function s.t. C h+ oe IR m (e.g., so is h; cf. Lem. 2.4(a)). is a sequence of positive numbers s.t. s Remark 8.2. Under Assumption 8.1, q is closed proper convex, - cl C cl q. Hence for the BPM method applied to the dual problem sup with a B-function h+ we may invoke the results of xx3-6 (replacing f , X and h by q, IR m and h+ respectively). Theorem 8.3. If Proof. This follows from Rem. 8.2 and Thm 5.2, since C h+ "Arg maxd ae Arg Theorem 8.4. Let Crh oe IR m and if inf k c k ? 0 then lim sup d(-) and lim sup and every limit point of fx k g solves (7.1). If Proof. Since C h oe Crh oe IR m , the assertions about f- k g follow from Thm 8.3. Suppose 7.7), we have (cf. Lem. 7.2) and is continuous on IR m Hence (cf. (7.2)) means f(x any x, in the limit some x 1 and K ae are closed), so by weak duality, f(x 1 Remark 8.5. Let C denote the optimal solution set for (7.1). If (7.1) is consistent (i.e., is nonempty and compact iff f and g i , direction of recession [Ber82, x5.3], in which case (8.1) implies that fx k g is bounded, and hence has limit points. In particular, if C in Thm 8.4. Remark 8.6. Theorems 8.3-8.4 subsume [Eck93, Thm 7], which additionally requires that ffl k j 0, imrh oe IR m ? and each g i is continuous on C f . Theorem 8.7. Let (7.1) be s.t. \Gammad has a sharp minimum. Let Crh oe IR m k. Then there exists k s.t. Proof. Using the proof of Thm 6.1 with the conclusion follows from the proof of Thm 8.4. Remark 8.8. Results on finite convergence of other multiplier methods are restricted to only once continuously differentiable augmented Lagrangians [Ber82, x5.4], whereas Thm 8.7 covers Ex. 7.9 also with fi ? 2. Applications include polyhedral programs. We shall need the following result, similar to ones in [Ber82, x5.3] and [TsB93]. Lemma 8.9. With u k+1 := g(x k+1 ), for each k, we have Proof. As for (8.2), use (7.12), (7.3), (2.3) and convexity of h + to develop yields (8.3), and (8.4) holds with by the convexity of h + . (8.5) follows from (8.2)-(8.4) and (7.2). Theorem 8.4 only covers methods with Crh oe IR m , such as Exs. 7.7 and 7.9. To handle other examples in x9, we shall use the following abstraction of the ergodic framework of [TsB93]. For each k, define the aggregate primal solution Since g is convex and c j g(x j+1 Lemma 8.10. Suppose sup i;k fl k Then lim sup If f-x k g has a limit point x 1 (e.g., C 6= ; is bounded; cf. Rem. 8.11), then x 1 solves and each limit point of f- k g maximizes d. Proof. By 1. By (8.6) and convexity of f , d 1 from (8.5), so are closed). Hence by weak duality, solves (7.1). Since d(- k and d is closed, each cluster of f- k g maximizes d. Remark 8.11. If C 6= ; is bounded then (8.8) implies that f-x k g is bounded (cf. Rem. 8.5). In particular, if C 9 Classes of penalty functions Examples 7.10-7.12 stem from B-functions of the form B-function on IR s.t. Since may also be derived by choosing suitable penalty functions OE on IR and letting 2.6). We now define two classes of penalty functions and study their relations with B-functions. Definition 9.1. We say OE closed proper convex essentially smooth, - t, t 0 strictly convexg and \Phi strictly convex on (t 0 \Gamma1g. Remark 9.2. If OE 2 \Phi then OE is nondecreasing (imrOE ae is increasing on (t 0 closed proper convex, . (For the "if" part, note that rOE(t k and rOE is nondecreasing.) Lemma 9.3. If OE 2 \Phi then OE is a B-function with lim . If OE 2 \Phi s then OE is essentially smooth, C @OE Proof. By Def. 9.1 and Lem. 2.6, IR ? ae imrOE ae IR + and OE is a B-function with ri C OE ae imrOE ae C OE , so and rOE is nondecreasing, lim Since OE is closed and proper, OE0 [Roc70, Thm 13.3] with oe C OE cl C OE and cl C OE and closedness of OE; otherwise lim t"t OE [Roc70, Thm 8.5]. By [Roc70, Thm 26.1], C OE . If OE 2 \Phi s then OE is essentially smooth [Roc70, Thm 26.3], so @OE and CrOE [Roc70, Thm 26.1]. If OE 2 \Phi 0 then @OE is increasing on (t 0 is single-valued on IR and hence @OE Lemma 9.4. Let / be a B-function on IR s.t. C / oe IR ? . Then Cr/ oe IR ? . If then /+ is essentially smooth there exists a B-function - Proof. is a B-function (Lem. 2.4(a)) and 2.6(a)). Also / + is nondecreasing and essentially smooth (Lem. 7.2), so imr/ By strict convexity of / (cf. Def. 2.1(a)), is increasing on IR ? , so r/ is increasing on (t strictly convex on (t essentially smooth [Roc70, Thm 26.3]. Otherwise, t be a strictly convex quadratic function s.t. - Corollary 9.5. If OE 2 \Phi 0 then the method of Ex. 7.8 with coincides with the method of Ex. 7.7 with / is the smooth extension of / described in Lem. 9.4, so that Crh oe IR m and Thms 8.4 and 8.7 apply. Proof. We have OE so OE and / 0 (t; Remark 9.6. In terms of OE 2 \Phi 0 , the method of Ex. 7.8 with OE where OE 0 In view of Cor. 9.5, we restrict attention to methods generated by OE 2 \Phi s . Example 9.7. Choosing OE 2 \Phi s and in Ex. 7.8 yields the method OE with and / (rOE) \Gamma1 by Def. 9.1 and [Roc70, Thms 26.3 and 26.5].) Note that The following results will ensure that 0, as required in Lem. 8.10. Definition 9.8. We say OE 2 \Phi is forcing on [t 0 sequences ft 0 Lemma 9.9. If OE 2 \Phi s , then OE is forcing on [\Gamma1; t 00 Proof. Replace OE by OE \Gamma inf OE, so that inf positive and increasing (cf. Rem. 9.2), so is OE. Let [OE OE . If \Gamma! 1. \Gamma! 1. Therefore, Lemma 9.10. The following functions are forcing on [\Gamma1; t 00 Proof. Let forcing. Invoke Lem. 9.9 for OE 1 and OE 3 . Example 9.11. Let OE 2 \Phi s be s.t. OE is not forcing on [\Gamma1; \Gamma1], although lim Lemma 9.12. Consider Ex. 9.7 with OE 2 \Phi s , t t and t . Then 1). In general, t is bounded. Proof. This follows from the facts - k 1 and strict monotonicity of rOE; cf. Rem. 9.2, Lem. 9.3 and Ex. 9.7. Lemma 9.13. Suppose in Ex. 9.7 OE 2 \Phi s is forcing on (\Gamma1; t fl ] with t Proof. Since rOE is nondecreasing and h we deduce from (8.4) that and [OE 0 (fl k (cf. Ex. 9.7) yields sup i;k ffl k so the preceding relation and the forcing property of OE give - k Theorem 9.14. Consider Ex. 9.7 with OE 2 \Phi s s.t. inf OE ? \Gamma1. Suppose Arg maxd 6= ;, holds. If f-x k g has a limit point x 1 (e.g., C 6= ; is bounded; cf. Rem. 8.11), then x 1 solves (7.1) and f(x 1 Proof. Let We have , so the assertions about f- k g follow from Thm 8.3. Then t by Lem. 9.12 (f- k g is bounded), so OE is forcing on [\Gamma1; t fl 9.13. The conclusion follows from Lem. 8.10. Remark 9.15. For the exponential multiplier method (Ex. 7.10 with 8.3 and 9.14 subsume [TsB93, Prop. 3.1] (in which Arg maxd 6= ;, C 6= ; is bounded, Theorem 9.16. Consider Ex. 9.7 with OE 2 \Phi s forcing on (\Gamma1; t OE ) 6= IR (e.g., holds. If f-x k g has a limit point x 1 (e.g., C 6= ; is bounded; cf. Rem. 8.11), then and each limit point of f- k g maximizes d. Proof. By Lem. 9.12, t so OE is forcing on (\Gamma1; t fl ]. Since d(- k 9.13. Since t fl - t OE ! 1, the conclusion follows from Lem. 8.10. Remark 9.17. Suppose 2.2.3]). If d duality. If is bounded iff so is Arg maxd This observation may be used in Lem. 8.10 and Thm 9.16. Theorem 9.18. Consider Ex. 9.7 with OE 2 \Phi s s.t. inf OE ? \Gamma1. Suppose some x 2 C f , holds. If f-x k g has a limit point x 1 (e.g., C 6= ; is bounded; cf. Rem. 8.11), then x 1 solves and each limit point of f- k g maximizes d. If d Proof. Since and Arg maxd 6= ; are bounded (Rem. 9.17), we get, as in the proof of Thm 9.14, C Hence the first two assertions follow from Lem. 8.10, and the third one from Thm 8.3. Theorem 9.19. Consider Ex. 9.7 with OE 2 \Phi s forcing on (\Gamma1; t 00 and holds. If f-x k g has a limit point x 1 (e.g., C 6= ; is bounded; cf. Rem. 8.11), and each limit point of f- k g maximizes d. If Proof. Use the proof of Thm 9.18, without asserting that C . Remark 9.20. It is easy to see that we may replace OE 2 \Phi s by OE 2 \Phi 0 and Ex. 9.7 by Ex. 7.8 with Lems. 9.9, 9.12, 9.13 and Thms 9.14, 9.16, 9.18, 9.19. (In the proof of Lem. 9.9, OE , since OE 0 and OE are positive and increasing on proving Lem. 9.12, recall the proof of Cor. 9.5; in the proof of Lem. 9.13, use Such results complement Thms 8.4 and 8.7; cf. Cor. 9.5. Additional aspects of multiplier methods Modified barrier functions can be extrapolated quadratically to facilitate their minimiza- tion; cf. [BTYZ92, BrS93, BrS94, NPS94, PoT94]. We now extend such techniques to our penalty functions, starting with a technical result. Lemma 10.1. Let OE 1 ; OE 2 2 \Phi be s.t. for some t s 2 (t 0 is forcing on (\Gamma1; t s ] and OE 2 is forcing on is forcing on (\Gamma1; t 00 Proof. Suppose (other cases being trivial). Since OE 0 1 and OE 0 are nondecreasing, so is OE 0 ; therefore, all terms in are nonnegative and tend to zero. Thus OE 0 Hence t 0 yield the first assertion. For the second one, use Def. 9.1 and Rem. 9.2. Examples 10.2. Using the notation of Lem. 10.1, we add the condition OE 00 to make OE twice continuously differentiable. In each example, OE 2 \Phi s [ \Phi 0 is forcing on Lems. 9.9-9.10 and Rem. 9.20. 12ts only grows as fast as OE 2 in Ex. 7.9 with but is smoother. b. This OE does not grow as fast as e t in Ex. 7.10. 3 (log-quadratic). This OE allows arbitrarily large infeasibilities, in contrast to OE 1 in Ex. 7.11. Again, this OE has C in contrast to OE 1 in Ex. 7.12. s Remark 10.3. Other smooth penalty functions (e.g., cubic-log-quadratic) are easy to derive. Such functions are covered by the various results of x9. Their properties, e.g., may also have practical significance; this should be verified experimentally. The following result (inspired by [Ber82, Prop. 5.7]) shows that minimizing L k (cf. in Alg. 7.4 is well posed under mild conditions (see the Appendix for its proof). Lemma 10.4. Let is a B-function with C / oe IR ? . Suppose is nonempty and compact iff f and have no common direction of recession, and if C 0 6= ; then this is equivalent to having a nonempty and compact set of solutions. We now consider a variant of condition (7.18), inspired by one in [Ber82, p. 328]. Lemma 10.5. Under the strong convexity assumption (7.15), consider (7.17) with replacing (7.18), where ff Next, suppose is bounded. Proof. By (7.17) and (10.1), (10.2) holds with L(x follows from (8.5). Similarly, L(x ff yields L(x ff (Lem. 4.8(i)). Remark 10.6. In view of Lem. 10.5, suppose in the strongly convex case of (7.15), (10.1) is used with (cf. (10.3)), the results of xx8-9 may invoke, instead of Thm 5.2 with The latter condition holds automatically if lim k!1 d(- k 1. Thus we may drop the conditions: Thms 8.3, 8.4, 9.14, ffl k ! 0 from Lem. 8.10 and Thm 9.16, and Thms 9.18-9.19. Instead of we may assume that fc k j k g is bounded in Thms 8.3, 8.4, 9.14 and 9.18-9.19. Condition (10.1) can be implemented as in [Ber82, Prop. 5.7(b)]. Lemma 10.7. Suppose f is strongly convex, inf C f is continuous on C f . Consider iteration k of Ex. 7.5 with is a B-function s.t. Cr/ oe IR ? . If is not a Lagrange multiplier of (7.1), fz j g is a sequence converging to - satisfying the stopping criterion (10.1). Proof. By Lemmas 9.3-9.4, Ex. 7.5 has - u). Then, as in (8.2), Suppose (-x). By (10.5), (2.3) and convexity of h m. Therefore, since OE is strictly convex on [t 0 with OE (Def. 9.1), and fl k OE , for each i, either fl k h- Combining this with -) (Lem. 7.3), we see (cf. [Roc70, Thm 28.3]) that - k is a Lagrange multiplier, a contradiction. Therefore, we must have strict inequality in (10.5). Since the stopping criterion will be satisfied for sufficiently large j. A Appendix Proof of Lemma 7.2. IR m (cf. [Roc70, Thm 23.8]), so and h+ is essentially strictly convex (cf. [Roc70, p. 253]). Hence (cf. is closed proper essentially smooth, so @h by [Roc70, Thm 26.1] and rh + is continuous on - by [Roc70, Thm 25.5]. By [Roc70, Thm 23.5], @h nondecreasing, as the union of open sets. That and ~ (-) then and and ~ Hence OE is inf-compact and We need the following slightly sharpened version of [GoT89, Thm 1.5.4]. Lemma A.1 (subdifferential chain rule). Let f be proper convex functions on ri C f i 6= ;. Let OE be a proper convex nondecreasing function on IR m s.t. y for some ~ , and for each - f Proof. For any x 1 and hence is convex. Since /(x) ? \Gamma1 for all x, / is proper. Let . For any x, f(x) - f(-x) yields To prove the opposite inclusion, let - fl 2 @/(-x). Consider the convex program By the monotonicity of OE and the definition of subdifferential, (-x; - solves (A.2), which satisfies Slater's condition (cf. f(~x) ! ~ y), so (cf. [Roc70, Cor. 28.2.1]) it has a Kuhn-Tucker point - xi 8x yields - ri C f i Thm 23.8]). Thus @/(-x) ae Q, i.e., To see that im@OE ae IR m , note that if Proof of Lemma 10.4. Let OE i m. Each OE i is closed: for any ff 2 IR, is closed nondecreasing and lim t"t / is closed (so is g i ). We have and OE i closed proper and L k 6j 1, so L k is closed and L k Thm 9.3]. Suppose Lem. 9.4 and Def. 9.1) and g i is closed, there is x 2 ri C Hence (cf. Lemmas 9.3-9.4) yield ff t. Then from . Thus OE Therefore, otherwise. The proof may be finished as in [Ber82, x5.3]. --R Numerical methods for nondifferentiable convex optimization New York Partial proximal minimization algorithms for convex program- ming The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming Computational experience with modified log-barrier methods for nonlinear programming Penalty/barrier multiplier methods for minimax and constrained smooth convex programs Weak sharp minima in mathematical programming Optimization of Burg's entropy over linear constraints An iterative row action method for interval convex programming Proximal minimization algorithm with D-functions A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds Nouvelles m'ethodes s'equentielles et parall'eles pour l'optimisation de r'eseaux 'a co-ots lin'eaires et convexes Convergence analysis of a proximal-like minimization algorithm using Bregman functions Convergence of some algorithms for convex minimization On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators Nonlinear proximal point algorithms using Bregman functions Multiplicative iterative algorithms for convex programming Finite termination of the proximal point algorithm Sequential Unconstrained Minimization Techniques Equilibrium programming using proximal-like algorithms Shifted barrier methods for linear programming Theory and Optimization Methods On the convergence of the proximal point algorithm for convex minimization On some properties of generalized proximal point methods for quadratic and linear programming On the convergence rate of entropic proximal optimization algorithms The convergence of a modified barrier method for convex programming The proximal algorithm R'egularisation d'in'equations variationelles par approximations successives Massively parallel algorithms for singly constrained convex programs A numerical comparison of barrier and modified barrier methods for large-scale bound-constrained optimization English transl. Nonlinear rescaling and proximal-like methods in convex opti- mization Convex Analysis Entropic proximal mappings with applications to nonlinear programming Relaxation methods for problems with strictly convex costs and linear constraints Dual ascent methods for problems with strictly convex costs and linear constraints: A unified approach --TR --CTR Lin He , Martin Burger , Stanley J. Osher, Iterative Total Variation Regularization with Non-Quadratic Fidelity, Journal of Mathematical Imaging and Vision, v.26 n.1-2, p.167-184, November 2006 Hatem Ben Amor , Jacques Desrosiers, A proximal trust-region algorithm for column generation stabilization, Computers and Operations Research, v.33 n.4, p.910-927, April 2006 G. Birgin , R. A. Castillo , J. M. Martnez, Numerical Comparison of Augmented Lagrangian Algorithms for Nonconvex Problems, Computational Optimization and Applications, v.31 n.1, p.31-55, May 2005 Krzysztof C. Kiwiel , P. O. Lindberg , Andreas Nu, Bregman Proximal Relaxation of Large-Scale 01 Problems, Computational Optimization and Applications, v.15 n.1, p.33-44, Jan. 2000 A. B. Juditsky , A. V. Nazin , A. B. Tsybakov , N. Vayatis, Recursive Aggregation of Estimators by the Mirror Descent Algorithm with Averaging, Problems of Information Transmission, v.41 n.4, p.368-384, October 2005 N. H. Xiu , J. Z. Zhang, Local convergence analysis of projection-type algorithms: unified approach, Journal of Optimization Theory and Applications, v.115 n.1, p.211-230, October 2002 Naihua Xiu , Jianzhong Zhang, Some recent advances in projection-type methods for variational inequalities, Journal of Computational and Applied Mathematics, v.152 n.1-2, p.559-585, 1 March

convex programming;nondifferentiable optimization;b-functions;bregman functions;proximal methods

271602

A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries.

Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating time-dependent incompressible inviscid flow which combines a projection method with a "Cartesian grid" approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volume-of-fluid representation. A redistribution procedure is used to eliminate time-step restrictions due to small cells where the boundary intersects the mesh. The projection step uses an approximate projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and for the time-dependent algorithm in two dimensions. The method is also demonstrated on flow past a half-cylinder with vortex shedding.

Introduction In this paper, we present a numerical method for solving the unsteady incompressible Euler equations in domains with irregular boundaries. The underlying discretization method is a projection method [22, 5]. Discretizations of the nonlinear convective terms and lagged pressure gradient are first used to construct an approximate update to the velocity field; the divergence constraint is subsequently imposed to define the velocity and pressure at the new time. The irregular boundary is represented using the Cartesian mesh approach [58], i.e. by intersecting the boundary with a uniform Cartesian grid, with irregular cells appearing only adjacent to the boundary. The extension of the basic projection methodology to the Cartesian grid setting exploits the separation of hyperbolic and elliptic terms of the method in [5] to allow us to use previous work on discretization of hyperbolic and elliptic PDE's on Cartesian grids. The treatment of the hyperbolic terms is based on algorithms developed for gas dynamics, and closely follows the algorithm of Pember et al. [56, 57]. The Cartesian grid projection uses the techniques developed by Young et al [72] for full potential transonic flow to discretize the elliptic equation that is used to enforce the incompressibility constraint. The overall design goals of the method are to be able to use the high-resolution finite difference methods based on higher-order Godunov methods for the advective terms in the presence of irregular boundaries that effectively add a potential flow component to the solution. Cartesian grid methods were first used by Purvis and Burkhalter [58] for solving the equations of transonic potential flow; see also [71, 40, 62, 72]. Clarke et al. [23] extended the methodology to steady compressible flow; see also [30, 29, 28, 52]. Zeeuw and Powell [74], Coirier and Powell [24], Coirier [25], Melton et al. [51], and Aftosmis et al. [1] have developed adaptive methods for the steady Euler and Navier-Stokes equations. For time-dependent hyperbolic problems, the primary difficulty in using the Cartesian grid approach lies in the treatment of the cells created by the intersection of the irregular boundary with the uniform mesh. There are no restrictions on how the boundary intersects the Cartesian grid (unlike the "stair step" approach which defines the body as aligned with cell edges), and as a result cells with arbitrarily small volumes can be created. A standard finite-volume approach using conservative differencing requires division by the volume of each cell; this is unstable unless the time step is reduced proportionally to the volume. Although in the projection method convective differencing is used for the hyperbolic terms in the momentum equation, we base our methodology for incompressible flow on the experience gained for compressible flow in the handling of small cells. (In addition, we will wish to update other quantities conservatively.) The major issues, then, in designing such a method are how to maintain accuracy, stability, and conservation in the irregular cells at the fluid-body interface while using a time step restricted by CFL considerations using the regular cell size alone. We refer to this as the "small cell problem." Noh [54] did early work in this area in which he used both cell merging techniques and redistribution. LeVeque [42, 43] and Berger and LeVeque [12] have developed explicit methods which use the large time step approach developed by LeVeque [41] to overcome the small cell problem. Berger and LeVeque [13, 14] have also studied approaches in which the small cell problem is avoided by the use of a rotated difference scheme in the cells cut by the fluid-body interface. Both the large time step and the rotated difference schemes are globally second-order and better than first-order but less than second-order at the fluid- body interface. Cell merging techniques were also used by Chiang et al. [20], Bayyuk et al. [4], and Quirk [60, 59] to overcome the small cell problem. Results for this method suggest it is globally second order accurate and first order at the boundary. Two other approaches for unsteady compressible flow are based on flux-vector splitting (Choi and Grossman [21]) and state-vector splitting ( - Oks-uzo-glu [55], Gooch and - Oks-uzo-glu [31]), but neither of these approaches avoids the small cell problem. The advection step of the method presented here uses a different approach for handling irregular cells based on ideas previously developed for shock tracking by Chern and Colella [17] and Bell, Colella, and Welcome [8], and extended by Pember et al. [56, 57]. In this approach the boundary is viewed as a "tracked front" in a regular Cartesian grid with the fluid dynamics at the boundary governed by the boundary conditions of a stationary reflecting wall. The basic integration scheme for hyperbolic problems consists of two steps. In the first step, a reference state is computed using fluxes generated by a higher-order Godunov method in which the fluid-body boundary is essentially "ignored". In the second step, a correction is computed to the state in each irregular cell. A stable, but nonconservative, portion of this correction is applied to the irregular cell. Conservation is then maintained by a variation of the algebraic redistribution algorithm in [17] which distributes the remainder of the correction to those regular and irregular cells that are immediate neighbors of the irregular cell. This redistribution procedure allows the scheme to use time steps computed from CFL considerations on the uniform grid alone. We adapt this scheme for the advection step of the projection method. The projection step requires solution of an elliptic equation on an irregular grid. The finite-element-based projection developed by Almgren et al. [3] for a regular grid is extended here to accommodate embedded boundaries using the techniques developed by Young et al. [72]. The approach of Young et al. defines the approximating space using standard bilinear elements on an enlarged domain that includes nodes of mixed cells lying outside the domain. Quadratures in the weak form are restricted to the actual domain to define the discretization. The resulting linear system can be readily solved with a straightforward extension of multigrid. A second-order Cartesian grid projection method has been developed recently by Tau [67] for the incompressible Navier-Stokes equations. In this formulation velocities are defined on a staggered grid (which is incompatible with the cell-centered form of the Godunov method), and the tangential velocity must vanish at the boundaries (i.e. the formulation is not extensible to the Euler equations). Comments and results in [67] indicate that the accuracy of the method again is in general first-order at the boundary but globally second- order. No mention is made of the small cell problem for advecting quantities other than momentum. In addition to the approach taken here, there are two other basic approaches to the treatment of irregular domains: and locally body-fitted grids obtained from mesh map- pings, and unstructured grids. For body-fitted structured or block-structured grids for compressible flow, the literature is quite extensive. Typical examples include [37, 35, 16, 11, 27, 7, 65, 68, 9, 61, 73]. The primary objection to mapped grids is the difficulty in generating them for complex geometries, particularly in three dimensions. There have been a number of approaches to making mapped grids more flexible, such as using combinations of of unstructured and structured grids [69, 70, 49, 64, 53, 36, 48], multiblock-structured grids [2, 66, 63] and overlapping composite grids [10, 19, 34]. On the topic of unstructured grids, we refer the reader to [39, 38, 45, 46, 50] for compressible flow. L-ohner et al [44] have developed an unstructured finite element algorithm for high-Reynolds number and inviscid incompressible flows. In comparison with the other approaches discussed above, the advantages of the Cartesian grid approach include not requiring special grid generation techniques to fit arbitrary boundaries, the ability to use a standard time-stepping algorithm at all cells with no additional work other than at cells at or near the boundary, and the ability to incorporate efficient solvers for the elliptic equation. The most serious disadvantage is a loss of accuracy at the boundary (numerical results show a reduction to first-order accuracy at the boundary). The loss of second-order accuracy at the boundary indicates that the Cartesian grid approach is not ideal for calculations designed to resolve viscous boundary layers, but should be satisfactory for flows where the primary effect of the boundary is to introduce a potential flow component to the velocity corresponding to the effect of the no-normal flow condition at the boundary. Problems where this is the case include flows in large containers, such as utility burners and boilers, and atmospheric flows over irregular orography. In the next section we review the basic fractional step algorithm, and introduce the notation of the Cartesian grid method. The subsequent two sections contain descriptions of the advection step and the projection step, respectively, for flows with embedded boundaries. In the final two sections we present numerical results and conclusions. All results and detailed discussion will be for two spatial dimensions; the extension to three dimensions is straightforward and will be presented in later work. 2 Basic Algorithm 2.1 Overview of Fractional Step Formulation The incompressible Euler equations written in convective form are and r Alternatively, (2.1) could be written in conservative form as The projection method is a fractional step scheme for solving these equations, composed of an advection step followed by a projection. In the advection step for cells entirely in the flow domain we solve the discretization of (2.1), \Deltat for the intermediate velocity U : For small cells adjoining the body we modify the velocity update using the conservative formulation of the nonlinear terms (see section 3). The pressure gradient at t n\Gamma 1 was computed in the previous time step and is treated as a source term in (2.4). The advection terms in (2.4), namely [(U \Delta r)U 2 , are approximated at time 2 to second-order in space and time using an explicit predictor-corrector scheme; their construction is described in section 3. The velocity field U is not, in general, divergence-free. The projection step of the algorithm decomposes the result of the first step into a discrete gradient of a scalar potential and an approximately divergence-free vector field. They correspond to the update to the pressure gradient and the update to the velocity, respectively. In particular, if P represents the projection operator then \Deltat \Deltat \Deltat Note that the pressure gradient is defined at the same time as the time derivative of velocity, and therefore at half-time levels. 2.2 Notation We first introduce the notation used to describe how the body intersects the computational domain. The volume fraction i;j for each cell is defined as the fraction of the computational cell B i;j that is inside the flow domain. The area fractions a i+ 1 2 ;j and a i;j+ 1specify the fractions of the (i respectively, that lie inside the flow domain. We label a cell entirely within the fluid ( as a full cell or fluid cell, a cell entirely outside of the flow domain ( as a body cell, and a cell partially in the fluid (0 ! as a mixed cell. A small cell is a mixed cell with a small volume fraction. We note that it is possible for the area fraction of an edge of a fluid cell to be less than one if the fluid cell abuts a body or mixed cell. In the method presented here, the state of the fluid at time t n is defined by U n ij ); the velocity field in cell B i;j at time t n ; and p n\Gamma the pressure at node (i\Gamma 2 . Gp n\Gamma 1ij is the pressure gradient in cell B i;j at time t For the construction of the nonlinear advective terms at t n+ 1 velocities are also defined on all edges of full and mixed cells at t n+ 1 process requires values of the velocity and pressure gradients in the cells on either side of an edge at t n : We must therefore define, at each time step, extended states in the body cells adjoining mixed or full cells. We do this in a volume-weighted fashion: U ext The pressure gradient is extended in the same manner. Here nbhd(B i;j ), defined as the neighborhood of B i;j ; refers to the eight cells (in two dimensions) that share an edge or corner with 3 Advection step 3.1 Momentum The algorithm is a predictor-corrector method, similar to that used in [5], but with some modifications as discussed in [6]. The details of the current version without geometry are given in [3]. For simplicity we will assume that the normal velocity on the embedded boundary is zero; the treatment of a more general Dirichlet boundary condition such as inflow is straightforward. In the predictor we extrapolate the velocity to the cell edges at t n+ 1 using a second-order Taylor series expansion in space and time. The time derivative is replaced using (2.1). For edge (i U n;lim extrapolated from B i;j , and U n;lim extrapolated from Analogous formulae are used to predict values at each of the other edges of the cell. In evaluating these terms the first-order derivatives normal to the edge (in this case U n;lim x are evaluated using a monotonicity-limited fourth-order slope approximation [26]. The limiting is done on the components of the velocity individually, with modifications as discussed below in cells near the body. The transverse derivative terms ( d vU y in this case) are evaluated as in [6], by first extrapolating from above and below to construct edge states, using normal derivatives only, and then choosing between these states using the upwinding procedure defined below. In particular, we define \Deltay U n;lim U n;lim where U n;lim y are limited slopes in the y direction, with similar formulae for the lower edge of B i;j . In this upwinding procedure we first define the normal advective velocity on the edge: (We suppress the spatial indices on bottom and top states here and in the next equation.) We now upwind b U based on b U i;j+ 1= After constructing b similar manner, we use these upwind values to form the transverse derivative in (3.1): 2\Deltay (b v adv U We use a similar upwinding procedure to choose the appropriate states U i+ 1 given the left and right states, U n+ 1;L U We follow a similar procedure to construct U i\Gamma 1 In general the normal velocities at the edges are not divergence-free. In order to make these velocities divergence-free we apply a MAC projection [6] before the construction of the convective derivatives. The equation solved for OE in all full or mixed cells, with a a i;j+ 1v n+ 1i;j+ 1\Gamma a \Deltax \Deltay Note that in regions of full cells the D MAC G MAC stencil is simply a standard five-point cell-centered stencil for the Laplacian. For edges within the body that are needed for the corrector step below, the MAC gradient is linearly extrapolated from edges that lie partially or fully in the fluid. For example, if edge (i lies completely within the body, but edges lie at least partially within the fluid, then we define While it is possible this could lead to instabilities, none have yet been observed in numerical testing. We solve this linear system using standard multigrid methods (see [15]), specifically, V-cycles with Jacobi relaxation. In the corrector step, we form an approximation to the convective derivatives in (2.1) 2\Deltay (v MAC where The intermediate velocity U at time n+ 1 is then defined on all full and mixed cells as The upwind method is an explicit difference scheme and, as such, requires a time-step restriction. When all cells are full cells, a linear, constant-coefficient analysis shows that for stability we require \Deltay where oe is the CFL number. The time-step restriction of the upwind method is used to set the time step for the overall algorithm. As mentioned earlier, for incompressible flow the two forms of the momentum equation, and (2.3), are analytically equivalent. For flows without geometry we have successfully used the convective difference form of the equations, (2.4). By contrast, we could use the conservative form of the equations, \Deltat +r \Delta F n+ 1 2 ); and evaluate the conservative update as R In the presence of embedded boundaries, the convective update form of the equations is stable even for very small cells, but this update is calculated as if the cell were a full cell, i.e. it doesn't ``see'' the body other than through the MAC-projected normal advection velocities (and the modification of the limited slopes as discussed at the end of this section). To compute the conservative update, however, one ignores entirely the portion of the cell not in the fluid, and integrates the flux only along the parts of the edges of the cell that lie within the fluid. Thus the presence of the body is correctly accounted for, but using the conservative form requires that the time step approach zero as the cell volume goes to zero, which is the small-cell time step restriction which we seek to avoid. A solution in this case is to use a weighted average of the convective and conservative updates, effectively allowing as much momentum to pass into a small cell in a time step as will keep the scheme stable. The momentum that does not pass into the small cell is redistributed to the neighboring cells, to maintain conservation in the advection step. This approach is modeled on the algorithm of [57, 56], and is based on the algebraic redistribution scheme of Chern and Colella [18]. The algorithm is as follows: (1) First, in all cells construct the reference state, e U , defined as e using the advective algorithm described for full cells. (2) On mixed cells only construct an alternative e e U using a conservative U (a (a i;j+ 1F y \Deltay where the fluxes are defined as This solution enforces no-flow across the boundary of the body, but is not necessarily stable. Define the difference ffiM U U (3) The conservative solution can be written e e U U ; however, this solution is not stable for - 1: Define instead b U U ffiM i;j on mixed cells, b U U i;j on full cells. This allows the mixed cell state to keep the fraction of ffiM which will keep the scheme stable given that the time step is set by the full-cell CFL constraint. Now redistribute the remaining fraction of ffiM from each mixed cell, the amount to the fluid and mixed cells among the eight neighbors of B i;j in a volume-weighted fashion. Since we redistribute the extensive rather than intensive quantity (e.g. momentum rather than momentum density), the resulting redistribution has the form c c U U i;j where (5) Subtract the pressure gradient term from the solution for all full and mixed cells, treating it as a source term: U A second modification to the algorithm for cells at or near the boundary is to the slope calculations; this is based on the principle that no state information from cells entirely within the body should be used in calculating slopes. Despite the use of extended states in the creation of edge states, the slope calculation uses state information only from full and mixed cells. In a mixed or full cell for which the fourth-order stencil would require using an extended state, the slope calculation reduces to the second-order monotonicity-limited formula. If the second-order formula would require an extended state, then the slope is set to zero. This has the effect of adding dissipation to the scheme; if the slopes throughout the domain were all set to zero the method would reduce to first-order. Two comments about the algorithm are in order here. First, we note that while the use of extended states seems to be a low-order approach to representing the body, numerical results show that it is adequate to maintain first-order accuracy at the boundary. An alternative to extended states is the so-called "thin wall approximation" of [57]; with this approximation extended states are not defined, rather upwinding always chooses the values coming from the fluid side (rather than the body side) of the fluid-body interface. In our numerical testing, however, we have found that the extended states version of the algorithm performs slightly better. Second, we are unable to provide a proof that the redistribution algorithm is stable, but in extensive numerical testing of the redistribution algorithm for compressible flow (see [57, 32, 47]) no stability problems have been observed, and none have been observed in our incompressible flow calculations. It is clear that the algorithm removes the original small-cell stability problem by replacing division by arbitrarily small volume fractions with division by one in step (3). The redistribution algorithm is such that the amount redistributed into a cell is proportional to the volume of that cell; this feature combined with the observation in our calculations that the values of ffiM are small relative to the values of U in the mixed cells gives a heuristic argument as to why redistribution would be stable. 3.2 Passive Scalars The algorithm for advecting momentum extends naturally to linear scalar advection. In the third numerical example shown in this paper, a passive scalar enters the domain at the inflow boundary and is advected around the body. We now briefly describe the scalar advection routine, assuming the scalar s is a conserved quantity, i.e. First, at each time step extended states are defined for the passive scalar just as for velocity. In the predictor we extrapolate the scalar to the cell edges at t n+ 1 using a second-order Taylor series expansion in space and time. For edge (i \Deltat extrapolated from B i;j , and s n+ 1;R \Deltat extrapolated from Analogous formulae are used to predict values at each of the other edges of the cell. As with velocity, derivatives normal to the edge (in this case s n;lim x are evaluated using the monotonicity-limited fourth-order slope approximation [26], with modifications due to geometry. The transverse derivative terms (vc s y in this case) are defined using c where and We use a similar procedure to choose s n+ 1i+ 1;j given s n+ 1 if u MAC s n+ 1;R if u MAC In the corrector step for full cells, we update s conservatively: s \Deltat \Deltax \Deltat \Deltay (v MAC In the corrector step for mixed cells, we follow the three steps below: (1) Construct the reference state e s using the convective formulation: \Deltat \Deltat 2\Deltay (v MAC (2) Construct e e s using the conservative formulation: (a i+ 1;j u MAC \Deltax (a i;j+ 1u MAC \Deltay Define the difference ffis s i;j Construct b s s Finally, redistribute the remaining fraction of ffis from each mixed cell, the amount to the full and mixed cells among the eight neighbors of B i;j in a volume-weighted fashion: In full cells not adjacent to any mixed cells, set s n+1 Projection Step Since U is defined on cell centers, the projection used to enforce incompressibility at time must include a divergence operator that acts on cell-centered quantities, unlike the MAC projection. The projection we use here is approximate; i.e. P 2 6= P: The operator, as well as the motivation for using an approximate rather than exact projection, is described in detail in [3]. Here we outline the algorithm for the case of no-flow physical boundaries with no geometry, and for the case of embedded boundaries. The basic approach for the embedded boundaries utilizes the same discretization as was used by Young et al [72] for full potential transonic flow. The necessary modifications for inflow-outflow conditions are described in [3] and their use is demonstrated in the numerical examples. This projection is based on a finite element formulation. In particular, we consider the scalar pressure field to be a C 0 function that is bilinear over each cell; i.e., the pressure is in s (x) is the space of polynomials of degree t in the x direction on each cell with C s continuity at x-edges. For the velocity space we define in x and a discontinuous linear function of y in each cell, with a similar form for v: For mixed cells, we think of the representations as only being defined on the portion of each cell within the flow domain,\Omega ; although the spaces implicitly define an extension of the solution over the entire cell. Note that in the integral used to define the projection (see (4.3) below) the domain of integration is limited to the actual flow domain; we do not integrate over the portion of mixed cells outside the domain. For use in the predictor and corrector, the velocity and pressure gradient are considered to be average values over each cell. The vector space, V h , contains additional functions that represent the linear variation within each cell. These additional degrees of freedom make large enough to contain rOE for OE 2 S h . We establish a correspondence between these two representations by introducing an orthogonal decomposition of V h . In particular, for each we define a piecewise constant component V and the variation V so that for each cell B i;j ; R construction these two components are orthogonal in L 2 so they can be used to define a decomposition of V h into two subspaces represent the cell averages and the orthogonal linear variation, respec- tively. The decomposition of V h induces a decomposition of rOE for all OE 2 S h ; namely, We now define a weak form of the projection on V h , based on a weak divergence on V h . In particular, we define a vector field V d in V h to be divergence-free in the domain\Omega if Z\Omega Using the definition (4.2) we can then project any vector field V into a gradient rOE and weakly divergence-free field V d (with vanishing normal velocities on boundaries) by solving Z Z for setting V Here we define the /'s to be the standard basis functions for S h , namely / i+ 1;j+ 1(x) is the piecewise bilinear function having node values / i+1=2;j+1=2 For the purposes of the fractional step scheme we wish to decompose the vector field \Deltat into its approximately divergence free-part \Deltat and the update to the pressure, Since the finite-difference advection scheme is designed to handle cell-based quantities that are considered to be average values over each cell, the quantity (GOE) ? is discarded at the end of the projection step. This makes the projection approximate, i.e. if D(V then in practice, we solve the system defined by (4.3) (again using standard multigrid techniques with V-cycles and Jacobi relaxation), and set Z as the approximation to U n+1 \GammaU n \Deltat in (2.5). (See [3] for more details.) The left-hand-side of equation (4.3) is, in discrete form, a nine-point stencil approximating the Laplacian of OE and the right hand side, for is a standard four-point divergence stencil. Without embedded boundaries, the method reduces to constant coefficient difference stencils for divergence, gradient, and the Laplacian operator. These stencils are, for (DV and, letting (DGOE) Body cells contribute nothing to the integrals in (4.3); in mixed cells the integrals are computed only over the portion of each cell that lies in the fluid. This calculation can be optimized by pre-computing the following integrals in each mixed cell; these integrands result from the products of the gradients of the basis functions in (4.3). We define I Z I Z I Z "\Omega are functions defined on each cell such that x 0 at the center of \Sigma1=2 at the left and right edges of B i;j ; and y \Sigma1=2 at the top and bottom edges of B Linear combinations of these integrals form the coefficients used in the divergence, gradient and Laplacian operators. For example, the divergence of a barred vector becomes (DV I a I a I a I a )=\Deltay If we let (GOE) full be (GOE) as it would be defined if the cell were a full cell, then in a mixed cell the average of the gradient over the fluid portion of the cell is full I b full I a where, for The full Laplacian operator requires the third integral I c because of the quadratic terms which results from the inner products of OE i+ 1;j+ 1and OE k+1=2;'+1=2 ; we leave the derivation to the reader. Note that for a full cell and so the stencils above for mixed cells reduce to those for full cells. Note that solving (4.3) defines OE i+ 1;j+ 1at each node (i which the support of intersects the fluid region. Thus, the pressure is defined even at nodes contained in the body region, as long as they are within one cell of the fluid region. 5 Numerical Results In this section we present results of calculations done using the Cartesian grid representation of bodies and/or boundaries of the domain. The first two sets of results are convergence studies; the first tests the accuracy of the projection alone, the second tests the accuracy of the full method for the Euler equations. In both cases, the results are presented in the form of tables which show the norms of the errors, as well as the calculated rates of convergence. The error for a calculation on a grid of spacing h is defined as the difference between the solution on that grid and the averaged solution from the same calculation on a grid of spacing h=2: In the first convergence study, the column 128-256 refers to the errors of the solution on the 128 2 grid as calculated by comparing the solution on the 128 2 grid to the solution on the 256 2 grid; similarly for 256-512. Once the errors are computed pointwise for each calculation, the L 1 , L 2 , and L1 norms of the errors are calculated. The convergence rates of the method can be estimated by taking the log 2 of the ratio of these norms. This provides a heuristic estimate of the convergence rate assuming that the method is operating in its asymptotic range. We present two separate measures of the error. In the first we compute the norms over the entire domain ("All cells"). In the second we examine the error in an interior subdomain in order to measure errors away from the boundary. For the subdomain we have selected the region covered by full cells in the 128 2 grid ("Full 128 cells"). The first convergence study tests the accuracy of the projection alone. The domain is the unit square, and three bodies are placed in the interior. A circle is centered at (:75; :75) and has radius .1; ellipses are centered at (:25; :625) and (:5; :25) and have axes (:15; :1) and respectively. The boundary conditions are inflow at and no-flow boundaries at In this study, the initial data inside the numerical domain is (u; this data is then projected to define a velocity field which is approximately divergence-free in the region of the domain not covered by the bodies. The pressure gradient which is used to correct the initial data represents the deviation of the potential flow solution from uniform flow. In Tables 1 and 2 we present the results of this convergence study. The almost-second- order convergence in the L 1 norm, first-order convergence in the L1 norm, and approximately h 1:5 convergence in the L 2 norm, are what we would expect of a solution which is second-order in most of the domain and first-order at boundaries. This is consistent with the observation that the maximum absolute error on the mixed cells is an order of magnitude higher than the maximum error on the full cells for the finer grids. Figure 1 shows a contour plot of the magnitude of the error of the 256 2 calculation. The second convergence study is of flow through a diverging channel. In this case we evaluate the velocity field at time in order to demonstrate the order of the complete algorithm for flow that is smooth in and near the mixed cells. The problem domain is 4 x 1, and the fluid is restricted to flow between the curves y top and y bot ; defined as 3: 3: - x - 4: and with The calculation is run at CFL number 0.9, and the flow is initialized with the potential flow field corresponding to inflow velocity 1: at the left edge, as computed by the initial projection. Tables 3 and 4 show the errors and convergence rates for the velocity components; here 128-256 refers to the errors of the solution on the 128x32 grid as calculated by comparing the solution on the 128x32 grid to the solution on the 256x64 grid; similarly for 256-512. As before, we compute the errors both on the full domain and on a subdomain defined as the region covered by full cells on the 128x32 grid. (In this case, the norms are scaled, as appropriate, by the area of the domain.) Again we see rates corresponding to global second-order accuracy but first-order accuracy near the boundaries. Figures 2a-b are contour plots of the log 10 of the error in the velocity components. The error is clearly concentrated along the fluid-body boundaries; for the sake of the figures we have defined all errors less than 1% of the maximum error in each (including the values in the body cells) to be equal to 1% of the maximum error. There are ten contour intervals in each figure spanning these two orders of magnitude. In time the error advects further along but does not contaminate the interior flow. We note that for this particular flow the velocity field near the boundary is essentially parallel to the boundary, and in a more general case we might expect to see more contamination of the interior flow. Note, however, that the maximum error is less than :3% of the magnitude of the velocity. The third example we present is that of flow past a half-cylinder. There is an extensive experimental and computational literature on the subject of flow past a cylinder in an infinite domain at low to moderate Reynolds number; see, for example [33] and [61] for recent experimental and computational results, respectively. Since the methodology presented here is for inviscid flow, we compute flow past a half-cylinder rather than a full cylinder so as to force the separation point to occur at the trailing edge. However, we present this calculation to demonstrate the type of application for which the Cartesian grid methodology would be most useful: a calculation in which one might be interested most in the flow features away from the body (i.e. the shed vortices downstream from the cylinder), but which requires the presence of the body in order to generate or modify those features. The resolution of our calculation is 256 x 64, the domain is 4 x 1, and the diameter of the half-cylinder is .25. The inflow velocity at the left edge is the boundary conditions are outflow at the right edge, no-flow boundaries at the top and bottom. The initial conditions are the potential flow with uniform inflow as calculated from the initial projection, combined with a small vortical perturbation to break the symmetry of the problem. Shown here are "snapshots" of the vorticity (Figure 3a) and a passively advected scalar (Figure 3b) at late enough time that the flow is periodic, and that the perturbation has been advected through the domain. The scalar was advected in from the center of the inflow edge. The Strouhal number is calculated to be approximately D=(U1T is the observed period of vortex shedding, is the cylinder diameter, and is the free-stream (i.e. inflow) velocity. One would expect a value between .2 and .4 (see [33] and the references cited there), so this seems a reasonable value given the limitations of the comparison. All cells Full 128 cells Table 1: Errors and convergence rates for the x-component of the pressure gradient. All cells Full 128 cells Table 2: Errors and convergence rates for the y-component of the pressure gradient. All cells Full 128 cells L1 2.36e-3 .80 1.35e-3 1.28e-3 1.08 6.04e-4 Table 3: Errors and convergence rates for the x-component of the velocity. All cells Full 128 cells Table 4: Errors and convergence rates for the y-component of the velocity. 6 Conclusion We have presented a method for calculation of time-dependent incompressible inviscid flow in a domain with embedded boundaries. This approach combines the basic projection method, using an approximate projection, with the Cartesian grid representation of ge- ometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The adaptation of the higher-order upwind method to include geometry is modeled on the Cartesian grid method for compressible flow. The discretization of the body near the flow uses a volume-of-fluid representation with a redistribution procedure. The approximate projection incorporates knowledge of the body through volume and area fractions, and certain integrals over the mixed cells. Convergence results indicate that the method is first-order at and near the body, and globally second-order away from the body. The method is demonstrated on flow past a half-cylinder with vortex shedding, and is shown to give a reasonable Strouhal number. The method here is presented in two dimensions; the extension to r \Gamma z and three dimensions and to variable density flows, and the inclusion of this representation with an adaptive mesh refinement algorithm for incompressible flow are being developed. The techniques developed here are also being modified for use in more general low Mach number models. In particular, we are using this methodology to model low Mach number combustion in realistic industrial burner geometries and to represent terrain in an anelastic model of the atmosphere. Acknowledgements We would like to thank Rick Pember for the many conversations on the subtleties of the Cartesian grid method. --R Adaptation and surface modeling for cartesian mesh methods Techniques in multiblock domain decomposition and surface grid generation A numerical method for the incompressible Navier-Stokes equations based on an approximate projection A simulation technique for 2-d unsteady inviscid flows around arbitrarily moving and deforming bodies of arbitrary geometry A second-order projection method for the incompressible Navier-Stokes equations An efficient second-order projection method for viscous incompressible flow Adaptive mesh refinement on moving quadrilateral grids Conservative front-tracking for inviscid compressible flow A projection method for viscous incompressible flow on quadrilateral grids A flexible grid embedding technique with application to the Euler equations Automatic adaptive refinement for the Euler equations An adaptive Cartesian mesh algorithm for the Euler equations in arbit rary geometries Progress in finite-volume calculations for wind- fuselage combinations A conservative front tracking method for hyperbolic conservation laws A conservative front tracking method for hyperbolic conservations laws Composite overlapping meshes for the solution of partial differential equations Simulation of unsteady inviscid flow on an adaptively refined Cartesian grid Numerical solution of the navier-stokes equations Euler calculations for multielement airfoils using Cartesian grids An accuracy assessment of Cartesian-mesh approaches for the Euler equ ations A direct Eulerian MUSCL scheme for gas dynamics Cartesian Euler methods for arbitrary aircraft configurations An adaptive multigrid applied to supersonic blunt body flow Euler calculations for wings using Cartesian grids. An adaptive multifluid interface-capturing method for compressible flow in complex geometries An experimental study of the parallel and oblique vortex shedding from circular cylinders A fourth-order accurate method for the incompressible Navier-Stokes equations on overlapping grids Inviscid and viscous solutions for airfoil/cascade flows using a locally implicit algorithm on adaptive meshes Transonic potential flow calculations using conservative form Improvements to the aircraft Euler method. Calculation of inviscid transonic flow over a complete aircraft. A large time step generalization of Godunov's method for systems of conservation laws Multidimensional numerical simulation of a pulse combustor A hybrid structured-unstructured grid method for unsteady turbomachinery flow compuations A solution-adaptive hybrid-grid method for the unsteady analysis of turbomachinery Accurate multigrid solution of the Euler equations on unstructured and adaptive meshes 3d applications of a cartesian grid euler method A finite difference solution of the Euler equations on non-body fitted grids Cel: A time-dependent Prediction of critical Mach number for store config- urations A Cartesian grid approach with hierarchical refinement for compressible flows A fractional step solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems A zonal implicit procedure for hybrid structured-unstructured grids Computations of unsteady viscous compressible flows using adaptive mesh refinement in curvilinear body-fitted grid systems A general decomposition algorithm applied to multi-element airfoil grids A second-order projection method for the incompressible Navier-Stokes equations in arbitrary domains Boundary fitted coordinate systems for numerical solution of partial differential equations - A review A three-dimensional struc- tured/unstructured hybrid Navier-Stokes method for turbine blade rows Mixed structured-unstructured meshes for aerodynamic flow sim- ulation A method for solving the transonic full-potential equations for general configurations A locally refined rectangular grid finite element method: Application to computational fluid dynamics and computational physics fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coor- dinates An adaptively refined Cartesian mesh solver for the Euler equations --TR --CTR Caroline Gatti-Bono , Phillip Colella, An anelastic allspeed projection method for gravitationally stratified flows, Journal of Computational Physics, v.216 August 2006 Yu-Heng Tseng , Joel H. Ferziger, A ghost-cell immersed boundary method for flow in complex geometry, Journal of Computational Physics, v.192 December Jiun-Der Yu , Shinri Sakai , James Sethian, A coupled quadrilateral grid level set projection method applied to ink jet simulation, Journal of Computational Physics, v.206 n.1, p.227-251, 10 June 2005 Xiaolin Zhong, A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity, Journal of Computational Physics, v.225 n.1, p.1066-1099, July, 2007 R. Ghias , R. Mittal , H. Dong, A sharp interface immersed boundary method for compressible viscous flows, Journal of Computational Physics, v.225 n.1, p.528-553, July, 2007 Stphane Popinet, Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, Journal of Computational Physics, v.190 n.2, p.572-600, 20 September Anvar Gilmanov , Fotis Sotiropoulos, A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, Journal of Computational Physics, v.207 August 2005 M. P. Kirkpatrick , S. W. Armfield , J. H. Kent, A representation of curved boundaries for the solution of the Navier-Stokes equations on a staggered three-dimensional Cartesian grid, Journal of Computational Physics, v.184 n.1, p.1-36, S. Marella , S. Krishnan , H. Liu , H. S. Udaykumar, Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations, Journal of Computational Physics, v.210 n.1, p.1-31, 20 November 2005

cartesian grid;incompressible Euler equations;projection method

271621

The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers.

The implementation and performance of a class of divide-and-conquer algorithms for computing the spectral decomposition of nonsymmetric matrices on distributed memory parallel computers are studied in this paper. After presenting a general framework, we focus on a spectral divide-and-conquer (SDC) algorithm with Newton iteration. Although the algorithm requires several times as many floating point operations as the best serial QR algorithm, it can be simply constructed from a small set of highly parallelizable matrix building blocks within Level 3 basic linear algebra subroutines (BLAS). Efficient implementations of these building blocks are available on a wide range of machines. In some ill-conditioned cases, the algorithm may lose numerical stability, but this can easily be detected and compensated for.The algorithm reached 31% efficiency with respect to the underlying PUMMA matrix multiplication and 82% efficiency with respect to the underlying ScaLAPACK matrix inversion on a 256 processor Intel Touchstone Delta system, and 41% efficiency with respect to the matrix multiplication in CMSSL on a Thinking Machines CM-5 with vector units. Our performance model predicts the performance reasonably accurately.To take advantage of the geometric nature of SDC algorithms, we have designed a graphical user interface to let the user choose the spectral decomposition according to specified regions in the complex plane.

Introduction A standard technique in parallel computing is to build new algorithms from existing high performance building blocks. For example, the LAPACK linear algebra library [1] is writ- Department of Mathematics, University of Kentucky, Lexington, KY 40506. y Computer Science Division and Mathematics Department, University of California, Berkeley, CA 94720. z Department of Computer Science, University of Tennessee, Knoxville, TN 37996 and Mathematical Sciences Section, Oak Ridge National Laboratory, Oak Ridge, TN 37831. x Department of Computer Science, University of Tennessee, Knoxville, TN 37996. - Department of Mathematics, University of California, Berkeley, CA 94720. Computer Science Division, University of California, Berkeley, CA 94720. ten in terms of the Basic Linear Algebra Subroutines (BLAS)[38, 23, 22], for which efficient implementations are available on many workstations, vector processors, and shared memory parallel machines. The recently released ScaLAPACK 1.0(beta) linear algebra library [26] is written in terms of the Parallel Block BLAS (PB-BLAS) [15], Basic Linear Algebra Communication Subroutines (BLACS) [25], BLAS and LAPACK. ScaLAPACK includes routines for LU, QR and Cholesky factorizations, and matrix inversion, and has been ported to the Intel Gamma, Delta and Paragon, Thinking Machines CM-5, and PVM clusters. The Connection Machine Scientific Software Library (CMSSL)[54] provides analogous functionality and high performance for the CM-5. In this work, we use these high performance kernels to build two new algorithms for finding eigenvalues and invariant subspaces of nonsymmetric matrices on distributed memory parallel computers. These algorithms perform spectral divide and conquer, i.e. they recursively divide the matrix into smaller submatrices, each of which has a subset of the original eigenvalues as its own. One algorithm uses the matrix sign function evaluated with Newton iteration [8, 42, 6, 4]. The other algorithm avoids the matrix inverse required by Newton iteration, and so is called the inverse free algorithm [30, 10, 44, 7]. Both algorithms are simply constructed from a small set of highly parallelizable building blocks, including matrix multiplication, QR decomposition and matrix inversion, as we describe in section 2. By using existing high performance kernels in ScaLAPACK and CMSSL, we have achieved high efficiency. On a 256 processor Intel Touchstone Delta system, the sign function algorithm reached 31% efficiency with respect to the underlying matrix multiplication (PUMMA [16]) for 4000-by-4000 matrices, and 82% efficiency with respect to the underlying ScaLAPACK 1.0 matrix inversion. On a Thinking Machines CM-5 with vector units, the hybrid Newton-Schultz sign function algorithm obtained 41% efficiency with respect to matrix multiplication from CMSSL 3.2 for 2048-by-2048 matrices. The nonsymmetric spectral decomposition problem has until recently resisted attempts at parallelization. The conventional method is to use the Hessenberg QR algorithm. One first reduces the matrix to Schur form, and then swaps the desired eigenvalues along the diagonal to group them together in order to form the desired invariant subspace [1]. The algorithm had appeared to required fine grain parallelism and be difficult to parallelize [5, 27, 57], but recently Henry and van de Geijn[32] have shown that the Hessenburg QR algorithm phase can be effectively parallelized for distributed memory parallel computers with up to 100 processors. Although parallel QR does not appear to be as scalable as the algorithms presented in this paper, it may be faster on a wide range of distributed memory parallel computers. Our algorithms perform several times as many floating point operations as QR, but they are nearly all within Level 3 BLAS, whereas implementations of QR performing the fewest floating point operations use less efficient Level 1 and 2 BLAS. A thorough comparison of these algorithms will be the subject of a future paper. Other parallel eigenproblem algorithms which have been developed include earlier par- allelizations of the QR algorithm [29, 50, 56, 55], Hessenberg divide and conquer algorithm using either Newton's method [24] or homotopies [17, 39, 40], and Jacobi's method [28, 47, 48, 49]. All these methods suffer from the use of fine-grain parallelism, instability, slow or misconvergence in the presence of clustered eigenvalues of the original problem or some constructed subproblems [20], or all three. The methods in this paper may be less stable than QR algorithm, and may fail to converge in a number of circumstances. Fortunately, it is easy to detect and compensate for this loss of stability, by choosing to divide the spectrum in a slightly different location. Compared with the other approaches mentioned above, we believe the algorithms discussed in this paper offer an effective tradeoff between parallelizability and stability. The other algorithms most closely related to the approaches used here may be found in [3, 9, 36], where symmetric matrices, or more generally matrices with real spectra, are treated. Another advantage of the algorithms described in this paper is that they can compute just those eigenvalues (and the corresponding invariant subspace) in a user-specified region of the complex plane. To help the user specify this region, we will describe a graphical user interface for the algorithms. The rest of this paper is organized as follows. In section 2, we present our two algorithms for spectral divide and conquer in a single framework, show how to divide the spectrum along arbitrary circles and lines in the complex plane, and discuss implementation details. In section 3, we discuss the performance of our algorithms on the Intel Delta and CM-5. In section 4, we present a model for the performance of our algorithms, and demonstrate that it can predict the execution time reasonably accurately. Section 5 describes the design of an X-window user interface. Section 6 draws conclusions and outlines our future work. Parallel Spectral Divide and Conquer Algorithms Both spectral divide and conquer (SDC) algorithms discussed in this paper can be presented in the following framework. Let be the Jordan canonical form of A, where the eigenvalues of the l \Theta l submatrix J+ are the eigenvalues of A inside a selected region D in the complex plane, and the eigenvalues of the are the eigenvalues of A outside D. We assume that there are no eigenvalues of A on the boundary of D, otherwise we reselect or move the region D slightly. The invariant subspace of the matrix A corresponding to the eigenvalues inside D are spanned by the first l columns of X . The matrix I 0 is the corresponding spectral projector. Let QR\Pi be the rank revealing QR decomposition of the matrix P+ , where Q is unitary, R is upper triangular, and \Pi is a permutation matrix chosen so that the leading l columns of Q span the range space of P+ . Then Q yields the desired spectral decomposition: A 11 A 12 where the eigenvalues of A 11 are the eigenvalues of A inside D, and the eigenvalues of A 22 are the eigenvalues of A outside D. By substituting the complementary projector I \Gamma P+ for P+ in (2.2), A 11 will have the eigenvalues outside D and A 22 will have the eigenvalues inside D. The crux of a parallel SDC algorithm is to efficiently compute the desired spectral projector P+ without computing the Jordan canonical form. 2.1 The SDC algorithm with Newton iteration The first SDC algorithm uses the matrix sign function, which was introduced by Roberts [46] for solving the algebraic Riccati equation. However, it was soon extended to solving the spectral decomposition problem [8]. More recent studies may be found in [11, 42, 6]. The matrix sign function, sign(A), of a matrix A with no eigenvalues on the imaginary axis can be defined via the Jordan canonical form of A (2.1), where the eigenvalues of J+ are in the open right half plane D, and the eigenvalues of J \Gamma are in the open left half plane D. Then sign(A) is I 0 It is easy to see that the matrix is the spectral projector onto the invariant subspace corresponding to the eigenvalues of A in D. l is the number of the eigenvalues of A in D. I \Gamma is the spectral projector corresponding to the eigenvalues of A in - D. Now let QR\Pi be the rank revealing QR decomposition of the projector P+ . Then Q yields the desired spectral decomposition (2.3), where the eigenvalues of A 11 are the eigenvalues of A in D, and the eigenvalues of A 22 are the eigenvalues of A in - D. Since the matrix sign function, sign(A), satisfies the matrix equation we can use Newton's method to solve this matrix equation and obtain the following simple iteration: The iteration is globally and ultimately quadratically convergent with lim j!1 A provided A has no pure imaginary eigenvalues [46, 35]. The iteration fails otherwise, and in finite precision, the iteration could converge slowly or not at all if A is "close" to having pure imaginary eigenvalues. There are many ways to improve the accuracy and convergence rate of this basic iteration [12, 33, 37]. For example, if may use the so called Newton-Schulz iteration to avoid the use of the matrix inverse. Although it requires twice as many flops, it is more efficient whenever matrix multiply is at least twice as efficient as matrix inversion. The Newton-Schulz iteration is also quadratically convergent provided that hybrid iteration might begin with Newton iteration until kA 2 then switch to Newton-Schulz iteration (we discuss the performance of one such hybrid later). Hence, we have the following algorithm which divides the spectrum along the pure imaginary axis. Algorithm 1 (The SDC Algorithm with Newton Iteration) Let A For convergence or j ? j max do if End Compute Compute A 11 A 12 Compute Here - is the stopping criterion for the Newton iteration (say, " is the machine precision), and j max limits the maximum number of iterations (say j return, the generally nonzero quantity measures the backward stability of the computed decomposition, since by setting E 21 to zero and so decoupling the problem into A 11 and A 22 , a backward error of For simplicity, we just use the QR decomposition with column pivoting to reveal rank, although more sophisticated rank-revealing schemes exist [14, 31, 34, 51]. All the variations of the Newton iteration with global convergence still need to compute the inverse of a matrix explicitly in one form or another. Dealing with ill-conditioned matrices and instability in the Newton iteration for computing the matrix sign function and the subsequent spectral decomposition is discussed in [11, 6, 4] and the references therein. 2.2 The SDC algorithm with inverse free iteration The above algorithm needs an explicit matrix inverse. This could cause numerical instability when the matrix is ill-conditioned. The following algorithm, originally due to Godunov, Bulgakov and Malyshev [30, 10, 44] and modified by Bai, Demmel and Gu [7], eliminates the need for the matrix inverse, and divides the spectrum along the unit circle instead of the imaginary axis. We first describe the algorithm, and then briefly explain why it works. Algorithm 2 (The SDC Algorithm with Inverse Free Iteration) Let A For convergence or j ? j max do !/ R j! End Compute Compute A 11 A 12 Compute As in Algorithm 1, we need to choose a stopping criterion - in the inner loop, as well as a limit j max on the maximum number of iterations. On convergence, the eigenvalues of A 11 are the eigenvalues of A inside the unit disk D, and the eigenvalues of A 22 are the eigenvalues of A outside D. It is assumed that no eigenvalues of A are on the unit circle. As with Algorithm 1, the quantity measures the backward stability. To illustrate how the algorithm works we will assume that all matrices we want to invert are invertible. From the inner loop of the algorithm, we see that 22 A j R j! so 22 A j or B j A \Gamma1 22 . Therefore A so the algorithm is simply repeatedly squaring the eigenvalues, driving the ones inside the unit circle to 0 and those outside to 1. Repeated squaring yields quadratic convergence. This is analogous to the sign function iteration where computing (A+A \Gamma1 )=2 is equivalent to taking the Cayley transform and taking the inverse Cayley transform. Further explanation of how the algorithm works can be found in [7]. An attraction of this algorithm is that it can equally well deal with the generalized nonsymmetric eigenproblem A \Gamma -B, provided the problem is regular, i.e. not identically zero. One simply has to start the algorithm with B Regarding the QR decomposition in the inner loop, there is no need to form the entire 2n \Theta 2n unitary matrix Q in order to get the submatrices Q 12 and Q 22 . Instead, we can compute the QR decomposition of the 2n \Theta n matrix (B H implicitly as Householder vectors in the lower triangular part of the matrix and another n dimensional array. We can then apply Q - without computing it - to the 2n \Theta n matrix (0; I) T to obtain the desired matrices Q 12 and Q 22 . We now show how to compute Q in the rank revealing QR decomposition of computing the explicit inverse subsequent products. This will yield the ultimate inverse free algorithm. Recall that for our purposes, we only need the unitary factor Q and the rank of It turns out that by using the generalized QR decomposition technique developed in [45, 2], we can get the desired information without computing In fact, in order to compute the QR decomposition with pivoting of first compute the QR decomposition with pivoting of the matrix A and then we compute the RQ factorization of the matrix Q H From (2.7) and (2.8), we have (R The Q is the desired unitary factor. The rank of R 1 is also the rank of the matrix 2.3 Spectral Transformation Techniques Although Algorithms 1 and 2 only divide the spectrum along the pure imaginary axis and the unit circle, respectively, we can use M-obius and other simple transformations of the input matrix A to divide along other more general curves. As a result, we can compute the eigenvalues (and corresponding invariant subspace) inside any region defined as the intersection of regions defined by these curves. This is a major attraction of this kind of algorithm. Let us show how to use M-obius transformations to divide the spectrum along arbitrary lines and circles. Transform the eigenproblem Az = -z to Then if we apply Algorithm 1 to A fiI) we can split the spectrum with respect to a region 0: If we apply Algorithm 2 to I), we can split along the curve For example, by computing the matrix sign function of then Algorithm 1 will split the spectrum of A along a circle centered at - with radius r. If A is real, and we choose - to be real, then all arithmetic will be real. If A will split the spectrum of A along a circle centered at - with radius r. If A is real, and we choose - to be real, then all arithmetic in the algorithm will be real. Y a O X Figure 1: Different Geometric Regions for the Spectral Decomposition Other more general regions can be obtained by taking A 0 as a polynomial function of A. For example, by computing the matrix sign function of , we can divide the spectrum within a "bowtie" shaped region centered at ff. Figure 1 illustrates the regions which the algorithms can deal with assuming that A is real and the algorithms use only real arithmetic. 2.4 Tradeoffs Algorithm 1 computes an explicit inverse, which could cause numerical instability if the matrix is ill-conditioned. The provides an alternative approach for achieving better numerical stability. There are some very difficult problems where Algorithm 2 gives a more accurate answer than Algorithm 1. Numerical examples can be found in [7]. However, neither algorithm avoids all accuracy and convergence difficulties associated with eigenvalues very close to the boundary of the selected region. The stability advantage of the inverse free approach is obtained at the cost of more storage and arithmetic. Algorithm 2 needs 4n 2 more storage space than Algorithm 1. This will certainly limit the problem size we will be able to solve. Furthermore, one step of the Algorithm 2 does about 6 to 7 times more arithmetic than the one step of Algorithm 1. QR decomposition, the major component of Algorithm 2, and matrix inversion, the main component of Algorithm 1, require comparable amounts of communication per flop. (See table 4 for details.) Therefore, Algorithm 2 can be expected to run significantly slower than Algorithm 1. Algorithm 1 is faster but somewhat less stable than Algorithm 2, and since testing stability is easy (compute use the following 3 step algorithm: 1. Try to use Algorithm 1 to split the spectrum. If it succeeds, stop. 2. Otherwise, try to split the spectrum using Algorithm 2. If it succeeds, stop. 3. Otherwise, use the QR algorithm. This 3-step approach works by trying the fastest but least stable method first, falling back to slower but more stable methods only if necessary. The same paradigm is also used in other parallel algorithms [19]. If a fast parallel version of the QR algorithm[32] becomes available, it would probably be faster than the inverse free algorithm and hence would obviate the need for the second step listed above. Algorithm 2 would still be of interest if only a subset of the spectrum is desired (the QR algorithm necessarily computes the entire spectrum), or for the generalized eigenproblem of a matrix pencil A \Gamma -B. 3 Implementation and Performance We started with a Fortran 77 implementation of Algorithm 1. This code is built using the BLAS and LAPACK for the basic matrix operations, such as LU decomposition, triangular inversion, QR decomposition and so on. Initially, we tested our software on SUN and IBM RS6000 workstations, and then the CRAY. Some preliminary performance data of the matrix sign function based algorithm have been reported in [6]. In this report, we will focus on the implementation and performance evaluation of the algorithms on distributed memory parallel machines, namely the Intel Delta and the CM-5. We have implemented Algorithm 1, and collected a large set of data for the performance of the primitive matrix operation subroutines on our target machines. More performance evaluation and comparison of these two algorithms and their applications are in progress. 3.1 Implementation and Performance on Intel Touchstone The Intel Touchstone Delta computer system is 16 \Theta 32 mesh of i860 processors with a wormhole routing interconnection network [41], located at the California Institute of Technology on behalf of the Concurrent Supercomputing Consortium. The Delta's communication characteristics are described in [43]. In order to implement Algorithm 1, it was natural to rely on the ScaLAPACK 1.0 library (beta version) [26]. This choice requires us to exploit two key design features of this package. First, the ScaLAPACK library relies on the Parallel Block BLAS (PB-BLAS)[15], which hides much of the interprocessor communication. This hiding of communication makes it possible to express most algorithms using only the PB-BLAS, thus avoiding explicit calls to communication routines. The PB-BLAS are implemented on top of calls to the BLAS and to the Basic Linear Algebra Communication Subroutines (BLACS)[25]. Second, ScaLAPACK assumes that the data is distributed according to the square block cyclic decomposition scheme, which allows the routines to achieve well balanced computations and to minimize communication costs. ScaLAPACK includes subroutines for LU, QR and Cholesky factorizations, which we use as building blocks for our implementation. The PUMMA routines [16] provide the required matrix multiplication. matrix order time ScaLAPACK on 256 PEs Intel Touchstone Delta (timing in second) GEMM INV QRP 1000 2000 3000 4000 5000 6000 7000 800051525matrix order mflops per node ScaLAPACK on 256 PEs Intel Touchstone Delta (Mflops per Node) GEMM INV QRP Figure 2: Performance of ScaLAPACK 1.0 (beta version) subroutines on 256 (16 \Theta 16) PEs Intel Touchstone Delta system. The matrix inversion is done in two steps. After the LU factorization has been computed, the upper triangular U matrix is inverted, and A \Gamma1 is obtained by substitution with L. Using blocked operations leads to performance comparable to that obtained for LU factorization. The implementation of the QR factorization with or without column pivoting is based on the parallel algorithm presented by Coleman and Plassmann [18]. The QR factorization with column pivoting has a much larger sequential component, processing one column at a time, and needs to update the norms of the column vectors at each step. This makes using blocked operations impossible and induces high synchronization overheads. However, as we will see, the cost of this step remains negligible in comparison with the time spent in the Newton iteration. Unlike QR factorization with pivoting, the QR factorization without pivoting and the post- and pre-multiplication by an orthogonal matrix do use blocked operations. Figure 2 plots the timing results obtained by the PUMMA package using the BLACS for the general matrix multiplication, and ScaLAPACK 1.0 (beta version) subroutines for the matrix inversion, QR decomposition with and without column pivoting. Corresponding tabular data can be found in the Appendix. To measure the efficiency of Algorithm 1, we generated random matrices of different sizes, all of whose entries are normally distributed with mean 0 and variance 1. All computations were performed in real double precision arithmetic. Table 1 lists the measured results of the backward error, the number of Newton iterations, the total CPU time and the megaflops rate. In particular, the second column of the table contains the backward errors and the number of the Newton iterations in parentheses. We note that the convergence rate is problem-data dependent. From Table 1, we see that for a 4000-by-4000 matrix, the algorithm reached 7.19/23.12=31% efficiency with respect to PUMMA matrix multiplica- tion, and 7.19/8.70=82% efficiency with respect to the underlying ScaLAPACK 1.0 (beta) matrix inversion subroutine. As our performance model shows, and tables 9, 10, 11, 12, and 14 confirm, efficiency will continue to improve as the matrix size n increases. Our Table 1: Backward accuracy, timing in seconds and megaflops of Algorithm 1 on a 256 node Intel Touchstone Delta system. Timing Mflops Mflops GEMM-Mflops INV-Mflops (iter) (seconds) (total) (per node) (per node) (per node) 1000 2000 3000 4000 5000 6000 7000 8000 9000135matrix size Gflops The Newton Iteration based Algorithm on Intel Delta System (Gflops) Figure 3: Performance of Algorithm 1 on the Intel Delta system as a function of matrix size for different numbers of processors. performance model is explained in section 4. Figure 3 shows the performance of Algorithm 1 on the Intel Delta system as a function of matrix size for different numbers of processors. Table gives details of the total CPU timing of the Newton iteration based algorithm, summarized in Table 1). It is clear that the Newton iteration (sign function) is most expensive, and takes about 90% of the total running time. To compare with the standard sequential algorithm, we also ran the LAPACK driver routine DGEES for computing the Schur decomposition (with reordering of eigenvalues) on one i860 processor. It took 592 seconds for a matrix of order 600, or 9.1 megaflops/second. Assuming that the time scales like n 3 , we can predict that for a matrix of order 4000, if the matrix were able to fit on a single node, then DGEES would take 175,000 seconds (48 hours) to compute the desired spectral decomposition. In contrast, Algorithm 1 would only take 1,436 seconds (24 minutes). This is about 120 times faster! However, we should note that DGEES actually computes a complete Schur decomposition with the necessary reordering of the spectrum. Algorithm 1 only decomposes the spectrum along the pure imaginary axis. In some applications, this may be what the users want. If the decomposition along a finer region or a complete Schur decomposition is desired, then the cost of the Newton iteration based algorithms will be increased, though it is likely that the first step just described will Table 2: Performance Profile on a 256 processor Intel Touchstone Delta system (time in seconds) 1000 123.06(91%) 6.87(5%) 4.27(5%) 134.22 2000 413.95(92%) 18.60(4%) 16.13(4%) 448.69 3000 717.04(90%) 36.76(5%) 38.37(5%) 792.18 take most of the time [13]. 3.2 Implementation and Performance on the CM-5 The Thinking Machines CM-5 was introduced in 1991. The tests in this section were run on a processor CM-5 at the University of California at Berkeley. Each CM-5 node contains a with an FPU and 64 KB cache, four vector floating points units, and of memory. The front end is a 33 HMz Sparc with 32 MB of memory. With the vector units, the peak 64-bit floating point performance is 128 megaflops per node (32 megaflops per vector unit). See [53] for more details. Algorithm 1 was implemented in CM Fortran (CMF) version 2.1 - an implementation of Fortran 77 supplemented with array-processing extensions from the ANSI and ISO (draft) standard Fortran 90 [53]. CMF arrays come in two flavors. They can be distributed across CM processor memory (in some user defined layout) or allocated in normal column major fashion on the front end alone. When the front end computer executes a CM Fortran pro- gram, it performs serial operations on scalar data stored in its own memory, but sends any instructions for array operations to the CM. On receiving an instruction, each node executes it on its own data. When necessary, CM processors can access each other's memory by any of three communication mechanisms, but these are transparent to the CMF programmer [52]. We also used CMSSL version 3.2, [54], TMC's library of numerical linear algebra rou- tines. CMSSL provides data parallel implementations of many standard linear algebra routines, and is designed to be used with CMF and to exploit the vector units. CMSSL's QR factorization (available with or without pivoting) uses standard Householder transformations. Column blocking can be performed at the user's discretion to improve load balance and increase parallelism. Scaling is available to avoid situations when a column norm is close to underflow or overflow, but this is an expensive "insurance policy". Scaling is not used in our current CM-5 code, but should perhaps be made available in our toolbox for the informed user. The QR with pivoting (QRP) factorization routine, which we shall use to reveal rank, is about half as fast as QR without pivoting. This is due in part to the elimination of blocking techniques when pivoting, as columns must be processed sequentially. Gaussian elimination with or without partial pivoting is available to compute LU factorizations and perform back substitution to solve a system of equations. Matrix inversion is matrix order time CMSSL 3.2 on 32 PEs with VUs CM-5 (timing in second) GEMM INV QRP matrix order mflops per node CMSSL 3.2 on 32 PEs with VUs CM-5 (Mflops per node) GEMM INV QRP Figure 4: Performance of some CMSSL 3.2 subroutines on 32 PEs with VUs CM-5 performed by solving the system I . The LU factors can be obtained separately - to support Balzer's and Byers' scaling schemes to accelerate the convergence of Newton, and which require a determinant computation - and there is a routine for estimating kA from the LU factors to detect ill-conditioning of intermediate matrices in the Newton iter- ation. Both the factorization and inversion routines balance load by permuting the matrix, and blocking (as specified by the user) is used to improve performance. The LU, QR and Matrix multiplication routines all have "out-of-core" counterparts to support matrices/systems that are too large to fit in main memory. Our current CM5 implementation of the SDC algorithms does not use any of the out-of-core routines, but in principle our algorithms will permit out-of-core solutions to be used. Figure 4 summarizes the performance of the CMSSL routines underlying this implementation Algorithm 1. We tested the Newton-Schulz iteration based algorithm for computing the spectral decomposition along the pure imaginary axis, since matrix multiplication can be twice as fast as matrix inversion; see Figure 4. The entries of random test matrices were uniformly distributed on [\Gamma1; 1]. We use the inequality kA n as switching criterion from the Newton iteration (2.5) to the Newton-Schulz iteration (2.6), i.e., we relaxed the convergence condition for the Newton-Schulz iteration to because this optimized performance over the test cases we ran. Table 3 shows the measured results of the backward accuracy, total CPU time and megaflops rate. The second column of the table is the backward error, the number of Newton iterations and the number of the Newton-Schulz iterations, respectively. From the table, we see that by comparing to CMSSL 3.2 matrix multiplication performance, we obtain 32% to 45% efficiency with the matrices sizes from 512 to 2048, even faster than the CMSSL 3.2 matrix inverse subroutine. We profiled the total CPU time on each phase of the algorithm, and found that about 83% of total time is spent on the Newton iteration, 9% on the QR decomposition with pivot- 4Actual Predicted GEMM- Inverse- (iter1, iter2) (seconds) (seconds) (total) (per node) (per node) (per node) Table 3: Backward accuracy, timing in seconds and megaflops of the SDC algorithm with Newton-Schulz iteration on a 32 PEs with VUs CM-5. ing, and 7.5% on the matrix multiplication for the Newton-Schulz iteration and orthogonal transformations. Performance Model Our model is based on the actual operation counts of the ScaLAPACK implementation and the following problem parameters and (measured) machine parameters. Matrix size p Number of processors b Block size (in the 2D block cyclic matrix data layout) [20] lat Time required to send a zero length message from one processor to another. - band Time required to send one double word from one processor to another. - DGEMM Time required per BLAS3 floating point operation Models for each of the building blocks are given in Table 4. Each model was created by counting the actual operations in the critical path. The load imbalance cost represents the discrepancy between the amount of work which the busiest processor must perform and the amount of work which the other processors must perform. Each of the models for the building blocks were validated against the performance data shown in the appendix. The load imbalance increases as the block size increases. Because it is based on operation counts, we can not only predict performance, but also estimate the importance of various suggested modifications either to the algorithm, the implementation or the hardware. In general, predicting performance is risky because there are so many factors which control actual performance, including the compiler and various library routines. However, since the majority of the time spent in Algorithm 1 is spent in either the BLACS or the level 3 PB-BLAS[15] (which are in turn implemented as calls to the BLACS[25] and the BLAS[38, 23, 22]), as long as the performance of the BLACS and the BLAS Computation Communication Cost Load Imbalance Cost Task Cost latency bandwidth \Gamma1 computation bandwidth \Gamma1 TRI 4n 3 Matrix multiply p- lat (1+ lg p Householder application Table 4: Models for each of the building blocks inversions applications Computation cost \Theta n 3 Latency cost \Thetan- lat 160+20 lg p 3 lg p 160+23lg p Bandwidth cost \Theta n 2 Imbalanced computation cost \Theta bn 2 Imbalanced bandwidth cost \Thetabn- band 20+35 lg p 20+35 lg p Table 5: Model of Algorithm 1 are well understood and the input matrix is not too small, we can predict the performance of Algorithm 1 on any distributed memory parallel computer. In Table 5, the predicted running time of each of the steps of Algorithm 1 is displayed. Summing the times in Table 5 yields: Using the measured machine parameters given in Table 8 with equation (4.9) yields the predicted times in Table 7 and Table 3. To get Table 4 and Table 5 and hence equation (4.9), we have made a number of simplifying assumptions based on our empirical results. We assume that 20 Newton iterations are required. We assume that the time required to send a single message of d double words is - lat regardless of how many messages are being sent in the system. Although there are many patterns of communication in the ScaLAPACK implementation, the majority of the communication time is spent in collective communica- tions, i.e. broadcasts and reductions over rows or columns. We therefore choose - lat and band based on programs that measure the performance of collective communications. We assume a perfectly square p p-by- processor grid. These assumptions allow us to keep the model simple and understandable, but limit its accuracy somewhat. Table Performance of the Newton iteration based algorithm (Algorithm 1) for the spectral decomposition along the pure imaginary axis, all backward errors (sec) (total) (sec) (total) (sec) (total) 2000 3000 Table 7: Predicted performance of the Newton iteration based algorithm (Algorithm 1) for the spectral decomposition along the pure imaginary axis. actual predicted actual predicted actual predicted time 2000 502.57 444.3 448.69 362.3 336.34 310.8 3000 1037.03 994.7 792.18 756.8 576.68 610.4 As Tables 6 and 7 show, our model underestimates the actual time on the Delta by no more than 20% for the machine and problem sizes that we timed. Table 3 shows that our model matches the performance on the CM5 to within 25% for all problem sizes except the smallest, i.e. The main sources of error in our model are: 1. uncounted operations, such as small BLAS1 and BLAS2 calls, data copying and norm computations, 2. non-square processor configurations, 3. differing numbers of Newton iterations required 4. communications costs which do not fit our linear model, 5. matrix multiply costs which do not fit our constant cost/flop model, and 6. the higher cost of QR decomposition with pivoting. We believe that uncounted operations account for the main error in our model for small n. The actual number of Newton iterations varies between exactly 20 Newton iterations are needed. Non-square processor configurations are slightly less efficient than square ones. Actual communication costs do not fit a linear model and depend upon the details such as how many processors are sending data simultaneously and to which processors they are sending. Actual matrix multiply costs depend upon the matrix Model Performance measured values -s Parameter Description limited by CM5 - DGEMM BLAS3 peak flop rate 1/90. 1/34. lat message latency comm. software 150 157 Table 8: Machine parameters sizes involved, the leading dimensions and the actual starting locations of the matrices. The cost of any individual call to the BLACS or to the BLAS may differ from the model by 20% or more. However, these differences tend to average out over the entire execution. Data layout, i.e. the number of processor rows and processor columns and the block size, is critical to the performance of this algorithm. We assume an efficient data layout. Specifically that means a roughly square processor configuration and a fairly large block size (say 16 to 32). The cost of redistributing the data on input to this routine would be tiny, O((n 2 =p)- band ), compared to the total cost of the algorithm. The optimal data layout for LU decomposition is different from the optimal data layout for computing U . The former prefers slightly more processor rows than columns while the latter prefers slightly more processor columns than rows. In addition, LU decomposition works best with a small block size, 6 on the Delta for example, whereas computing U best done with a large block size, 30 on the Delta for example. The difference is significant enough that we believe a slight overall performance gain, maybe 5% to 10%, could be achieved by redistributing the data between these two phases, even though this redistribution would have to be done twice for each Newton step. Table 3 shows that except for n ! 512 our model estimates the performance Algorithm 1 based on CMSSL reasonably well. Note that this table is for a Newton-Shultz iteration scheme which is slightly more efficient on the CM5 than the Newton based iteration. This introduces another small error. The fact that our model matches the performance of the CMSSL based routine, whose internals we have not examined, indicates to us that the implementation of matrix inversion on the CM5 probably requires roughly the same operation counts as the ScaLAPACK implementation. The performance figures in Table 8 are all measured by an independent program, except for the CM5 BLAS3 performance. The communication performance figures for the Delta in Table 8 are from a report by Littlefield 1 [43]. The communication performance figures for the CM5 are as measured by Whaley 2 [58]. The computation performance for the Delta is from the Linpack benchmark[21] for a 1 processor Delta. There is no entry for a 1 processor CM5 in the Linpack benchmark, so - DGEMM in Table 8 above is chosen from our own experience. Graphical User Interface to SDC To take advantage of the graphical nature of the spectral decomposition process, a graphical user interface (GUI) has been implemented for SDC. Written in C and based on X11R5's 1 The BLACS use protocol 2, and the communication pattern most closely resembles the "shift" timings.- lat is from Table 8 in[58] and -band is from Table 5. CODE XI CODE USER PARALLEL EXECUTION Interface of 7 routines Figure 5: The X11 Interface (XI) and SDC standard Xlib library, the Xt toolkit and MIT's Athena widget set, it has been nicknamed XI for "X11 Interface". When XI is paired with code implementing SDC we call the union XSDC. The programmer's interface to XI consists of seven subroutines designed independently of any specific SDC implementation. Thus XI can be attached to any SDC code. At present, it is in use with the CM-5 CMF/CMSSL implementation and the Fortran 77 version of our algorithm (both of which use real arithmetic only). Figure 1 shows the coupling of the SDC code and the XI library of subroutines. Basically, the SDC code calls an XI routine which handles all interaction with the user and returns only when it has the next request for a parallel computation. The SDC code processes this request on the parallel engine, and if necessary calls another XI routine to inform the user of the computational results. If the user had selected to split the spectrum, then at this point the size of the highlighted region, and the error bound on the computation (along with some performance information) is reported, and the user is given the choice of confirming or refusing the split. Appropriate action is taken depending on the choice. This process is repeated until the user decides to terminate the program. All data structures pertaining to the matrix decomposition process are managed by XI. A binary tree records the size and status (solved/not solved) of each diagonal block corresponding to a spectral region, the error bounds of each split, and other information. Having the X11 interface manage the decomposition data frees the SDC programmer of these responsibilities and encapsulates the decomposition process. The SDC programmer obtains any useful information via the interface subroutines. Figure 6 pictures a sample session of xsdc on the CM-5 with a 500 \Theta 500 matrix. The large, central window (called the "spectrum window") represents the region of the complex plane indicated by the axes. Its title - "xsdc :: Eigenvalues and Schur Vectors" - indicates that the task is to compute eigenvalues and Schur vectors for the matrix under analysis. Figure sample xsdc session The lines on the spectrum window (other than the axes) are the result of spectral divide and conquer, while the shading indicates that the "bowtie" region of the complex plane is currently selected for further analysis. The other windows (which can be raised/lowered at the user's request) show the details of the process and will be described later. The buttons at the top control I/O, the appearance of the spectrum window, and algorithmic choices: ffl File lets one save the matrix one is working on, start on a new matrix, or quit. ffl Zoom lets one navigate around the complex plane by zooming in or out on part of the spectrum window. Toggle turns on or off the features of the spectrum window (for example the axes, Gershgorin disks, eigenvalues). ffl Function lets one modify the algorithm, or display more or less detail about the progress being made. The buttons at the bottom are used in splitting the spectrum. For example clicking on Right halfplane and then clicking at any point on the spectrum window will split the spectrum into two halfplanes at that point, with the right halfplane selected for further division. This would signal the SDC code to decompose the matrix A to k A 11 A 12 where the k eigenvalues of A 11 are the eigenvalues of A in the right halfplane, and the eigenvalues of A 22 are the eigenvalues of A in the left halfplane. The button Left Halfplane works similarly, except that the left halfplane would then be selected for further processing and the roles of A 11 and A 22 would be reversed. In the same manner, Inside Circle and Outside Circle divide the complex plane at the boundary of a circle, while East-West Crosslines and North-South Crosslines split the spectrum with lines at 45 degrees to the real axis (described below). The Split Information window in the lower right corner of Figure 2 keeps track of the matrix splitting process. It reports the two splits performed to arrive at this current (shaded) spectral region. The first, an East-West Crossline split at the point 1.5 on the real axis, divided the entire complex plane into four sectors by drawing two lines at \Sigma 45 degrees through the point 1.5 on the real axis. SDC decomposed the starting matrix into: 260 A 11 A 12 where the East and West sectors correspond to the A 11 block while the North and South sectors correspond to the A 22 block. Continuing in the East-West sectors as indicated by the previous split, that region is divided into two sub-regions separated by the boundary of the circle of radius 4 centered at the origin. The circle is drawn, making sure that its boundary only intersects the East and West sectors, and the matrix is reduced to:B @ 106 154 240 106 A 11 A 12 A 13 A The shading indicates that the "bowtie" region (corresponding to the interior of the circle, and the A 11 block) is currently selected for further analysis. In the upper right corner of Figure 2 the Matrix Information window displays the status of the matrix decomposition process. Each of the three entries corresponds to a spectral region and a square diagonal block of the 3 \Theta 3 block upper triangular matrix, and informs us of the block's size, whether its eigenvalues (eigenvectors, Schur vectors) have been computed or not, and the maximum error bound encountered along this path of the decomposition process. The highlighted entry corresponds to the shaded region and reports that the A 11 block contains 106 eigenvalues, has been solved, and is in error by up to 1:44 \Theta 10 \Gamma13 . The eigenvalues - listed in the window overlapping the Matrix Information window - can be plotted on the spectrum at the user's request. The user may select any region of the complex plane (and hence any sub-matrix on the diagonal) for further decomposition by clicking the pointer in the desired region. A click at the point 10 on the imaginary axis for example, would unhighlight the current region and shade the North and South sectors. Since this region corresponds to the A 33 block, the third entry in the Matrix-Information window would be highlighted. The Split-Information window would also be updated to detail the single split performed in arriving at this region of the spectrum. Once a block is small enough, the user may choose to solve it (via the Function button at the top of the spectrum window). In this case the eigenvalues, and Schur vectors for that block would be computed using QR (as per the user's request) and the eigenvalues plotted on the spectrum. The current XI code supports real SDC only. It will be extended to handle the complex case as implementations of complex SDC become available. 6 Conclusions and Future work We have written codes that solve one of the hardest problems of numerical linear algebra: spectral decomposition of nonsymmetric matrices. Our implementation uses only highly efficient matrix computation kernels, which are available in the public domain and from distributed memory parallel computer vendors. The performance attained is encouraging. This approach merits consideration for other numerical algorithms. The object oriented user interface XI developed in this paper provides a paradigm for us in the future to design a more user friendly interface in the massively parallel computing environment. We note that all the approaches discussed here can be extended to compute the both right and left deflating subspaces of a regular matrix pencil A \Gamma -B. See [4, 7] for more details. As the spectrum is repeatedly partitioned in a divide-and-conquer fashion, there is obviously task parallelism available because of the independent submatrices that arise, as well as the data parallel-like matrix operations considered in this paper. Analysis in [13] indicates that this task parallelism can contribute at most a small constant factor speedup, since most of the work is at the root of the divide-and-conquer tree. This can simplify the implementation. Our future work will include the implementation and performance evaluation of the based algorithm, comparison with parallel QR, the extension of the algorithms to the generalized spectral decomposition problem, and the integration of the 3-step approach (see section 2.3) to an object oriented user interface. Acknowledgements Bai and Demmel were supported in part by the ARPA grant DM28E04120 via a subcontract from Argonne National Laboratory. Demmel and Petitet were supported in part by NSF grant ASC-9005933, Demmel, Dongarra and Robinson were supported in part by ARPA contact DAAL03-91-C-0047 administered by the Army Research Office. Ken Stanley was supported by an NSF graduate student fellowship. Dongarra was also supported in part by the Office of Scientific Computing, U.S. Department of Energy, under Contract DE-AC05- 84OR21400. This work was performed in part using the Intel Touchstone Delta System operated by the California Institute of Technology on behalf of the Concurrent Supercomputing Consortium. Access to this facility was provided through the Center for Research on Parallel Computing. --R Generalized QR factorization and its applictions. on parallelizable eigensolvers. Design of a parallel nonsymmetric eigenroutine toolbox on a block implementation of Hessenberg multishift QR it- eration Design of a parallel nonsymmetric eigenroutine toolbox Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems. A computational method for eigenvalue and eigen-vectors of a matrix with real eigenvalues A divide and conquer method for tridiagonalizing symmetric matrices with repeated eigenvalues. Circular dichotomy of the spectrum of a matrix. Numerical stability and instability in matrix sign function based algorithms. Solving the algebraic Riccati equation with the matrix sign function. on the benefit of mixed parallelism. Rank revealing QR factorizations. PB-BLAS: A set of Parallel Block Basic Linear Algebra Subprograms. PUMMA: Parallel universal matrix multiplication algorithms on distributed memory concurrent computers. A note on the homotopy method for linear algebraic eigenvalue problems. A parallel nonlinear least-squares solver: Theoretical analysis and numerical results Trading off parallelism and numerical stability. Parallel numerical linear algebra. Performance of various computers using standard linear equations soft- ware A set of Level 3 Basic Linear Algebra Subprograms. An Extended Set of FORTRAN Basic Linear Algebra Subroutines. A parallel algorithm for the non-symmetric eigenvalue problem A Users' Guide to the BLACS. The design of linear algebra libraries for high performance computers. The multishift QR algorithm: is it worth the trouble? on the Schur decomposition of a matrix for parallel computation. Finding eigenvalues and eigenvectors of unsymmetric matrices using a distributed memory multiprocessor. Problem of the dichotomy of the spectrum of a matrix. An efficient algorithm for computing a rank-revealing QR de- composition Parallelizing the QR algorithm for the unsymmetric algebraic eigenvalue problem: myths and reality. Computing the polar decomposition - with applications The rank revealing QR and SVD. The sign matrix and the separation of matrix eigenvalues. A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices. Rational iteration methods for the matrix sign function. Basic Linear Algebra Subprograms for Fortran usage. Solving eigenvalue problems of nonsymmetric matrices with real homotopies. The Touchstone A parallel algorithm for computing the eigenvalues of an unsymmetric matrix on an SIMD mesh of processors. Characterizing and tuning communication performance on the Touchstone DELTA and iPSC/860. Parallel algorithm for solving some spectral problems of linear algebra. Some aspects of generalized QR factorization. Linear model reduction and solution of the algebraic Riccati equation. on Jacobi and Jacobi-like algorithms for a parallel computer A parallel algorithm for the eigenvalues and eigenvectors of a general complex matrix. A Jacobi-like algorithm for computing the Schur decomposition of a non-Hermitian matrix A parallel implementation of the QR algorithm. Updating a rank-revealing ULV decomposition CM Fortran Reference Manual The Connection Machine CM-5 Technical Summary CMSSL for CM Fortran: CM-5 Edition Implementing the QR Algorithm on an Array of Processors. Efficient parallel implementation of the nonsymmetric QR algorithm. Shifting strategies for the parallel QR algorithm. Basic linear algebra communication subroutines: Analysis and implementation across multiple parallel architectures. --TR --CTR Peter Benner , Enrique S. Quintana-Ort , Gregorio Quintana-Ort, State-space truncation methods for parallel model reduction of large-scale systems, Parallel Computing, v.29 n.11-12, p.1701-1722, November/December Peter Benner , Ralph Byers , Rafael Mayo , Enrique S. Quintana-Ort , Vicente Hernndez, Parallel algorithms for LQ optimal control of discrete-time periodic linear systems, Journal of Parallel and Distributed Computing, v.62 n.2, p.306-325, February 2002 Leo Chin Sim , Graham Leedham , Leo Chin Jian , Heiko Schroder, Fast solution of large N N matrix equations in an MIMD-SIMD hybrid system, Parallel Computing, v.29 n.11-12, p.1669-1684, November/December Peter Benner , Maribel Castillo , Enrique S. Quintana-Ort , Vicente Hernndez, Parallel Partial Stabilizing Algorithms for Large Linear Control Systems, The Journal of Supercomputing, v.15 n.2, p.193-206, Feb.1.2000

spectral divide-and-conquer;invariant subspaces;nonsymmetric matrices;ScaLAPACK;parallelizable;eigenvalue problem;spectral decomposition

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The Influence of Interface Conditions on Convergence of Krylov-Schwarz Domain Decomposition for the Advection-Diffusion Equation.

Several variants of Schwarz domain decomposition, which differ in the choice of interface conditions, are studied in a finite volume context. Krylov subspace acceleration, GMRES in this paper, is used to accelerate convergence. Using a detailed investigation of the systems involved, we can minimize the memory requirements of GMRES acceleration. It is shown how Krylov subspace acceleration can be easily built on top of an already implemented Schwarz domain decomposition iteration, which makes Krylov-Schwarz algorithms easy to use in practice. The convergence rate is investigated both theoretically and experimentally. It is observed that the Krylov subspace accelerated algorithm is quite insensitive to the type of interface conditions employed.

Introduction We consider domain decomposition for the two-dimensional advection-diffusion equation with application to a boundary conforming finite volume incompressible Navier-Stokes solver in mind, see [27, 6, 20]. Therefore, our interests are more practical than theoretical. An advantage of the boundary conforming approach is that the structures of the systems of equations that arise are known beforehand. This enables us to develop efficient iterative solvers that can be vectorized with relative ease. A disadvantage is that the boundary conforming approach is not suitable for domains not topologically rectangular. Domain decomposition is used to overcome this problem. The name Krylov-Schwarz refers to methods in which a Schwarz type domain decomposition iteration is accelerated by a Krylov subspace method (such as CG or GMRES), see the preface of [17]. Equivalently, Krylov-Schwarz means that Schwarz domain decomposition is used as preconditioner for a Krylov subspace method. In this paper we use the GMRES method because of non-symmetry of the discretized advection-diffusion equation. Applied Analysis Group, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands, ([email protected]) y Applied Analysis Group, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands, ([email protected]) Schwarz domain decomposition considered here uses a minimal overlap and no coarse grid correction. Algebraically, this algorithm is formulated as a block iteration applied to a (possibly augmented) matrix, see [14, page 372] and [25]. Theory [28] and experiments [9] show that both a constant overlap in physical space and a coarse grid correction are needed to obtain constant iteration count when the mesh is refined and the number of subdomains increases. Examples of constant overlap in physical space can be found in [15, 22, 30]. However, despite that large overlap gives better convergence rates, minimal overlap typically leads to lower computing times, see [13, 11, 9], even for large and ill-conditioned problems. The reason is that with minimal overlap, there is less duplication of work in overlap regions. Methods with small overlap are also much easier to implement for practical complicated problems. A coarse grid correction [18, 8, 1, 2] can be quite effective for improving convergence of domain decomposition. However, as noted in [23, pp. 188], for small numbers of subdomains the additional cost in forming and solving the coarse grid problem outweighs the reduction of the number of iterations it gives. Moreover, in large codes used for engineering computations coarse grid correction is very difficult to implement. Similar to [12] and [25, 24], we study the influence of interface conditions on convergence rate. A unified treatment of nonoverlapping Neumann-Dirichlet and Schwarz methods with minimal overlap is not straightforward. An example at the analytic level can be found in [23, page 207]. Examples at the algebraic level are in [25, 24], where complicated augmentations of the discretization matrix are needed. This paper unifies Neumann-Dirichlet and Schwarz methods in a simpler way: by premultiplying the discretization matrix with a so-called influence matrix, which has a very simple structure. Our acceleration method differs from [12] in that we do not use relaxation but GMRES ac- celeration. In [25, 24] good convergence was obtained with the unaccelerated Schwarz algorithm by using optimized interface conditions. Instead, we do not use optimization, but investigate the effect of some simple strategies for choosing interface conditions on the convergence rate of the Krylov-Schwarz domain decomposition algorithm. We find that the accelerated method is quite insensitive to the types of interface conditions used. It is well-known that GMRES requires much storage if the vector length is large. In standard Krylov-Schwarz algorithms, the full vector length is used. We will show, that, if subdomain problems are solved accurately, one can reduce the vector length in GMRES. In fact, GMRES then solves a reduced system, which consists of equations with only unknowns on or near the interfaces. Discretization We consider the 2-D advection-diffusion equation written in general coordinates:X @ a i @u Equation (1) is obtained after a boundary-fitted coordinate transformation to a rectangular are interested in (1) because it is a realistic model for the momentum equations occurring in computational fluid dynamics for incompressible flows. We discretize (1) using either cell-centered or vertex-centered finite volumes with a central approximation of the advection terms on a uniform grid with mesh sizes h 1 and h 2 , which leads to a 9-point discretization molecule. The resulting system of discretized equations is denoted by 3 Domain decomposition In practice, complicated flow domains are used and one needs complete freedom in decomposing the domain into subdomains. However, for the present model study we investigate only a rectan- gular it into a rectangular array of (nonoverlapping) subdomains, see Figure 1. The difference between cell-centered and vertex-centered finite-volumes is that in the latter case there are nodal points on the interfaces between subdomains. Following [25], the unknowns at these nodal points are copied into each subdomain, see Figure 2.b. This means that most nodal points on the interface are doubled into a left and right unknown. At cross points, four copies are present. The respective equation is repeated for each copied unknown, adding zeroes in the discretization matrix at positions associated with copies from other domains. This is in contrast to [25] where the coefficients of the discretization are also adapted. The latter mechanism leads to subdomain problems with the desired types of boundary conditions on the interfaces. As opposed to [25], we use premultiplication with the influence matrix, see further on. For the analysis and description of the algorithm we restrict ourselves to two non-overlapping Some details on the multi-domain case will be given. \Omega 9 \Omega 6 Figure 1: Decomposition of a domain\Omega into 3 \Theta 3 domains of each 6 \Theta 6 cells. The global grid is uniform We find the following way to describe domain decomposition convenient. Replace (2) by The matrix M in (3) is called the influence matrix and is used to obtain various types of coupling conditions on the interfaces. Section 3.1 describes M in detail. The use of the matrix M leads to a 12 point molecule for the first layer of points near the interface, see Figure 2. Define the disjoint index sets I j such that i 2 I j if u i This definition of I j is also valid for the vertex-centered case because the unknowns at the interface are doubled in that case. Both A and - f are partitioned into blocks according to these index sets I 1 and I 2 . Eq. (3) then becomes A 11 A 12 A 21 A 22 Schwarz domain decomposition is a block Gauss-Seidel or Jacobi iteration applied to (4), with A 22 A 21 gives block Gauss-Seidel and Jacobi. The algorithm (5) is a Schwarz domain decomposition algorithm with a minimal (O(h)) overlap. Following [14, p. 370], we classify the algorithm as non-overlapping. The Gauss-Seidel variant corresponds to multiplicative Schwarz and the Jacobi variant, which is suitable for parallelization, corresponds to additive Schwarz. Section 3.3 describes Krylov subspace acceleration of this Schwarz algorithm. 3.1 The influence matrix The purpose of the influence matrix M , introduced in (3), is to give a unified framework to treat both classical Schwarz and Neumann-Dirichlet algorithms. The different types of interface conditions are obtained by varying parameters in M . This study only uses a one-parameter coupling, but extensions to more parameters are also possible. (a) (b) Figure 2: Interface variables: cell-centered (a) and vertex-centered (b) Let I denote the index set of all variables and let J be the subset of I containing the indices of two rows of variables near the interface, see Figure 2. These variables are called interface variables and play an important role in the next sections. Let K denote the subset of J containing the indices of variables in the first layer on either side of the interface \Gamma. The influence matrix M takes linear combinations of discretization molecules associated with unknowns in K, and thus influences the discretization at points in K. The influence matrix is defined as follows: 1. 2. do not correspond to the same subdomain. Furthermore, we restrict M by allowing nonzero M ij only at points i and j directly opposite each other with respect to the interface. This means that we omit any tangential dependencies in M and consider only a single normal dependencies. This leads to a 1-parameter coupling. For the situation of two subdomains in Figure 2, the matrix M has at most two non-zero entries at each row i 2 K: one at M ii = 1 and the other at . The subscript i shows that - may vary along the interface. The parameter - ij depends on the coupling strategy used. Of course, we also have M so that Invertibility of the influence matrix M is ensured by the condition In the general multi-domain case, cross-points can occur. At cross-points (corner points), the influence matrix M has at most three non-zero entries: M at one interface and M ij 2 at the other. Some interesting choices for - are: Neumann Dirichlet is the normal mesh P'eclet number defined by with the coordinate direction corresponding to the normal direction. For the present model study, we have, for vertical interfaces, with depending on which subdomain i corresponds to. This means that the normal mesh P'eclet numbers as seen from the different blocks always have opposite signs. In [7], the choices listed in (9) are worked out in detail for the constant coefficients case with In that case, the discretized subdomain problems are identical to the ones obtained by imposing Neumann (- N ij ) or Dirichlet (- D conditions at the interface and by discretizing using the finite volume method. At 0, the use of a Dirichlet condition is required to obtain well-posed subdomain problems. However, the corresponding - D ij is singular for \Gamma2. Therefore, we have omitted a further study of this choice and use - S ij instead. In the vertex-centered case, we can avoid this problem by taking This leads to a relation between the first layer of left and right unknowns only, which is equivalent to a direct Dirichlet update. In this way, we get a vertex-centered finite volume version of the Neumann-Dirichlet algorithm studied in [19], by taking In general one wants to vary the type of interface condition depending on the local flow parameters We choose parameter that can vary along the interface. The basic Schwarz iteration is obtained with The choice (14) is referred to as the Schwarz-Schwarz (S-S) algorithm. Neumann-Schwarz (N-S) is obtained with ae In this case the transition is discontinuous. The Robin-Schwarz (R-S) method uses a smooth transition, with This paper takes p 2. Note that the three strategies for domain decomposition (14), (15) and (16) all reduce to Schwarz iteration when applied to the Poisson equation. 3.2 The interface equations The Krylov-Schwarz algorithm corresponds to a Krylov subspace method for solving the following preconditioned system where N is the block lower triangular or block diagonal matrix from (6). We will show that we can reduce the system (17) to a smaller system concerning only interface unknowns u i with see Figure 2. Let denote the vector of interface variables and let the remaining variables. the following injection operators: I the trivial injection operator from w into u defined by else (18) I the trivial injection operator from v into u defined by ae The following theorem provides useful information about the structure of a matrix. Theorem 1 If the matrix A satisfies the following property I \Theta J; (20) then after ordering u such that , the system (17) becomes I P f Proof: or - f . By premultipling with N \Gamma1 , we get and by premultiplying the result with \Theta P Q , we get f f by ordering the components of u as pre- scribed. 2 Since the influence matrix has nonzero entries only for the first layer of unknowns near the interface, and the discretization is a 9-point molecule, it can be shown that the matrix N \Gamma A can only have nonzero elements at positions (i; I \Theta J ; the proof of this is omitted. Therefore, the matrices N and - A of this paper satisfy the conditions of Theorem 1. The block form of (21) shows that v can, in principle, be solved for independently of w, by solving the system of interface equations A numerical example of (24) can be found in [16] for a small one-dimensional Poisson problem. Note that we assume accurate subdomain solution: that is corresponding to the same domain. With inaccurate subdomain solution, we can get I 2 J which violates the assumption of Theorem 1. From (21) we see that the matrix N A in (17) has an eigenvalue equal to the number of non-interface variables and that all the other eigenvalues are shared with the interface equations (24). This property means that the interface equations (24) have the same spectrum as the preconditioned system (17), apart from - = 1. Hence, (24) does not need to be preconditioned further. The approach is strongly related to the Schur complement method, typical of finite element methods, in which subdomain problems are solved exactly and domain decomposition also amounts to solving an interface problem. In [4, 10], it is shown that the multiplicative overlapping Schwarz method is equivalent to a Schur complement method with appropriate block preconditioner. In [29], a different proof of this is given for the nonoverlapping method of the present paper. The interface problem is somewhat different from that arising in Schur complement methods because in our terminology, the interface unknowns do not (all) reside on the interface. Furthermore, our method is more general because it can be applied to finite volume/difference methods as well. 3.3 Krylov acceleration of the Schwarz method Since domain decomposition methods in general tend to converge slowly if at all, an acceleration technique is needed. We use the GMRES [21] Krylov subspace method to solve the interface equations (24). To solve (24), all that is required is a method to compute the interface matrix-vector product Q v. Similar to methods based on Schur's complement, see for instance [3], it is not necessary to form the matrix of the interface equations explicitly. It turns out that a single iteration of the unaccelerated algorithm (5) can be used to obtain the interface matrix-vector product for Q AQ. This enables a step-wise implementation of accelerated domain decomposition, which is of major importance for complicated CFD codes. Also, the required vector length in GMRES is quite small because only a small system of interface equations must be solved. This makes the approach practical for large problems. Given the implementation of unaccelerated domain decomposition (5), we can compute - f if u is given. Because of the property that I \Theta J , we have if f . Furthermore, if we introduce - u, we get f The injection Q and restriction Q T are easily implemented as subroutines, so that computation of given v is easy. If we define f . The problem to be solved is rewritten as Given the initial guess v 0 , GMRES acceleration proceeds as follows: 1. Compute the right-hand side 2. Solve the problem Q using GMRES with initial guess product is computed from Q 3. The final inner subdomain solutions collected in the vector u are computed by doing a last domain decomposition iteration with the computed interface solution v: f . Theoretical background Some theoretical analysis is possible under simplified conditions. In equation (1) we assume field with a 1 ; a 2 - 0. The boundary conditions are periodic in the y-direction. On the left boundary (inflow) we prescribe a Dirichlet condition and on the right boundary (outflow) a homogeneous Neumann condition. We take h and we split the domain in two parts with a vertical interface in the middle. To obtain convergence estimates for GMRES accelerated domain decomposition, we use theorem 5 of [21], from which it follows that for GMRES without restart for some K ? is the number of eigenvalues of B with non-positive real parts and the other eigenvalues are enclosed in a circle centered at C ? 0 with radius R ! C. In practice, formula (26) may give only a crude upper bound on convergence, especially if the spectrum is not evenly distributed but consists of a few clusters of eigenvalues, see [26]. However, it turns out that for our problems, (26) is a good estimate. Using Fourier analysis in the y-direction we can obtain the eigenvalues of the iteration matrix A of the multiplicative (block Gauss-Seidel) algorithm. Some straightforward but tedious calculations give the eigenvalues of E in closed, but difficult to analyze form, see [7, Appendix A]. For brevity we omit the details. The eigenvalues of the matrix coincide with those of N (apart from the multiple eigenvalue leading to an estimate of ae in formula (26). To compare the S-S, N-S, R-S and N-D (only vertex-centered) algorithms, we compute the theoretical convergence rates for different ranges of flow magnitudes and flow angles. The flow magnitude is given by the dimensionless mesh-P'eclet number with jaj the magnitude of velocity, the diffusion coefficient. The flow angle is given by ff, so ae a h pmax sin ff; (27) with flow normal to the interface and flow tangential to the interface. The average theoretical convergence rates 1 over (p are listed in Tables 1 and 2. The cell-centered results from Table 1 show that P'eclet range S-S N-S R-S Table 1: Average theoretical convergence rates and corresponding number of iterations (in brack- ets) for the multiplicative algorithm to solve with relative accuracy of 10 \Gamma4 , for different mesh P'eclet ranges averaged over all flow angles. Cell-centered discretization. for small flow magnitudes, the Neumann-Schwarz algorithm is approximately twice as fast as the Schwarz-Schwarz and Robin-Schwarz algorithms. The advantages of the Neumann-Schwarz algorithms are much smaller for larger flow magnitudes. The results indicate that Neumann- Schwarz is the best choice for the Poisson equation. However, because of symmetry of that equation, it is not certain at what sides of the interfaces, Neumann conditions must be imposed. This problem becomes important for small flow magnitudes when p ij changes sign along the interface. P'eclet range S-S N-S R-S N-D Table 2: Average theoretical convergence rates and corresponding number of iterations (in brack- ets) for the multiplicative algorithm to solve with relative accuracy of 10 \Gamma4 , for different mesh P'eclet ranges averaged over all flow angles. Vertex-centered discretization. The vertex-centered results from Table 2 show much smaller differences between the different methods as the cell-centered results from Table 1. With vertex-centered discretization, the differences are also small for small flow magnitudes. The Neumann-Dirichlet method from [19] has similar convergence as the Neumann-Schwarz method at low flow magnitudes. However, at larger flow magnitudes, the Neumann-Dirichlet has a worse convergence rate. The next section compares the above theoretical results with experiments. 5 Numerical experiments This section presents some numerical experiments and compares the influence of interface conditions (Schwarz-Schwarz, Neumann-Schwarz, Robin-Schwarz, Neumann-Dirichlet) on convergence of Krylov subspace accelerated Schwarz domain decomposition. 1 The N-S algorithm was modified so that also for flow tangential to the interface, Neumann and Schwarz conditions were used instead of the S-S algorithm We use a relative stopping criterion with r m the residual after m iterations defined by The experimental convergence rate ae(m) is computed from Some care must be taken interpreting convergence rates. Large differences in convergence rates do not necessarily indicate large differences in the number of iterations. For instance, to reach a relative accuracy of 10 \Gamma4 with ae = 0:1 we need 4 iterations while with ae = 0:2 we need only iterations more. On the other hand, with ae close to 1, small differences are very important: the difference between ae = 0:98 and ae = 0:99 is a factor of two in iteration count. All experiments take the initial guess v We use the Sparse 2 solver for solving the sub-domain problems. All results in this section are for the multiplicative method, which, in our experience, converges about twice as fast as the additive method. Restarted GMRES(20) was used for Krylov subspace acceleration of Schwarz domain decomposition. 5.1 Convergence as a function of flow magnitude and angle Results are given for a divided into two blocks by a vertical interface at . A uniform mesh of 40 \Theta 40 cells on\Omega is used. In the numerical results, a Dirichlet condition is enforced on the left and lower boundaries of\Omega and a Neumann condition is enforced on the right and upper boundaries. The right-hand side is f = 1. All coefficients in (1) are assumed to be constant with k As in Section 4, we compute the (experimental) convergence rate as a function of flow magnitude pmax and angle ff. Similar to Section 4, we have modified the Neumann-Schwarz method so that Neumann and Schwarz conditions are always used for block 1 and 2 respectively, even for flow tangential to the interface. This is different from the description of Neumann-Schwarz (15) but enables a comparison of the effect of different coupling conditions on convergence. To demonstrate the quality of the theoretical prediction of convergence rates, Figure 3 shows, as an example, a comparison of experimental and theoretical convergence rates for the multiplicative Neumann-Schwarz algorithm. The theoretical convergence rates agree well with the experimental ones. Note that the convergence rate is zero along the curve pmax cos 2. This is not a property of the domain decomposition algorithm but of the discretization. Since a central discretization is used for the advective terms, the discretization reduces to a first order upwind discretization for mesh P'eclet numbers equal to 2. To compare the S-S, N-S, R-S and N-D methods again, we use the same averaging procedure as described in Section 4. The average convergence rate is computed over some ranges of flow magnitude and over all flow angles. Tables 3 and 4 show the results. The results are similar to the theoretical results of Section 4. Also, the Neumann-Schwarz method performs best for low flow magnitudes in comparison to the Schwarz-Schwarz and Neumann-Schwarz methods for both cell-centered and vertex-centered. For larger flow magnitudes, the differences are almost negligible. Note that the differences between the methods are even smaller than in the theoretical analysis. This is because the weak periodic boundary conditions in the theoretical analysis were replaced by angle angle R/C experimental theoretical Figure 3: Experimental and theoretical convergence factors for the GMRES accelerated Neumann- Schwarz algorithm P'eclet range S-S N-S R-S Table 3: Average experimental convergence rates and corresponding number of iterations (in brackets) to solve with relative accuracy of 10 \Gamma4 , for different mesh P'eclet ranges averaged over all flow angles. Cell-centered discretization. stronger boundary conditions in the experiments. Similar to the theoretical results of Section 4, the Neumann-Dirichlet method of [19] (vertex-centered) shows almost identical convergence rate as the Neumann-Schwarz method for low flow magnitudes and shows a slightly worse convergence behavior for larger flow magnitudes. 5.2 Recirculating flow As an example, we investigate a uniform flow problem, for which domain decomposition in general converges slower than for simple uniform flow problems. The problem is defined by k a 2 (x; 2 Sparse is a public domain direct solver available from [email protected] P'eclet range S-S N-S R-S N-D Table 4: Average theoretical convergence rates and corresponding number of iterations (in brack- ets) to solve with relative accuracy of 10 \Gamma4 , for different mesh P'eclet ranges averaged over all flow angles. Vertex-centered discretization. on (x; y) controls the angle of flow across the interfaces. Two decompositions of the domain are considered: the first in only two blocks with a vertical interface at and the second into 2 \Theta 2 blocks with interfaces at uniform grid of on\Omega is used combined with a cell-centered discretization. Table 5 lists the results for the accelerated algorithm. Good convergence factors are obtained and, the algorithm is quite insensitive to the direction of the flow on the interface and to the type of coupling condition. The Robin-Schwarz method provides only a small improvement with respect to Neumann-Schwarz, but this difference is so small that it does not always show up in the iteration count. blocks blocks Table 5: Experimental convergence rate ae(m) (iteration counts in brackets) for the recirculation problem for increasing obliqueness, cell-centered discretization. The differences in number of iterations (work) are very small for the three coupling strate- gies. The relative differences in the number of iterations are even smaller when the number of subdomains is increased from 2 to 4. 5.3 Further remarks on robustness Further numerical experiments in [7] have shown that the GMRES accelerated algorithm is quite insensitive to the coupling strategy used. The experiments in [7] also investigate the influence on convergence rate of the types of external boundary conditions and of adding cross diffusion terms k 12 . Furthermore, some experiments investigate the effect of variations in the ordering of blocks and refinement within the subdomains. All these experiments show that the accelerated algorithm is rather insensitive to these factors. In particular with respect to refinement this is promising for applications to complicated flow problems. 6 Conclusions We have investigated three domain decomposition methods, namely: the Schwarz-Schwarz, Neumann- Schwarz and Robin-Schwarz algorithms. The algorithms were accelerated by a GMRES Krylov subspace method. Assuming accurate solution of subdomain problems, the dimension of the vector length in GMRES was reduced significantly by introducing the interface equations. This makes the overhead of Krylov subspace acceleration negligible and enables the solution of large complex CFD problems. The GMRES Krylov subspace acceleration procedure can be implemented easily on top of an already implemented unaccelerated domain decomposition algorithm, by repeatedly calling the subroutine that performs a single Schwarz domain decomposition iteration with given initial guess. This is of major importance for the implementation in complex CFD codes. The theoretical and experimental convergence rates agree reasonably well. The experiments show that for low flow magnitudes, the Neumann-Schwarz methods can provide a reduction in the number of iterations of at most a factor 2. For large flow magnitudes, the differences between the methods are less significant. The Robin-Schwarz and Schwarz-Schwarz methods are comparable in convergence rates for both low and high flow magnitudes. The Neumann-Dirichlet method of [19], has convergence rate similar to the Neumann-Schwarz method except for large flow magnitudes for which it requires more iterations. The differences in convergence rates found experimentally are typically less than the predicted convergence rates. This effect is even stronger when non-uniform flow fields are used. For the recirculating flow problem, the Robin-Schwarz (R-S) method has a slightly better convergence rate than than the Neumann-Schwarz (N-S) and Schwarz-Schwarz (S-S) algorithms. The differences in number of iterations (amount of work) between the S-S, N-S and R-S methods are very small: in the number of iterations the difference is not significant at all. The differences between the S-S, N-S, R-S methods are even less when the number of subdomains is increased from 2 to 4. Further numerical experiments in [7] show these conclusions to be true for a larger number of test problems: the method is reasonably robust with respect to coupling conditions, grid refinement, velocity field and external boundary conditions. Our experiments show that varying the type of interface conditions, depending of flow magnitude and angle, in general gives only a moderate reduction in iteration count. In the experiments, at most a reduction factor 2 was observed. Such limited reduction factors in general have a small effect on total computing time. This is because, especially with complex CFD applications, solving the system of equations may take only a small portion of the total computing time. Possibly, a more detailed study of interface conditions will lead to more significant reductions in iteration counts. In particular, this is interesting for limiting cases, such as low-speed (Stokes) or high-speed (Euler) flows. For example, in [25, 24] good convergence for the unaccelerated algorithm is obtained for such problems by optimizing interface conditions. Further research is necessary to determine whether such a conclusion also holds for the accelerated algorithm. A disadvantage of the methods described in [25, 24] is that convergence seems to depend sensitively on the coupling parameters. Another problem is that these methods are in general difficult to implement, especially in complex CFD codes. Besides that, one can imagine that the complexity of such methods increases with the complexity of the CFD code: for instance, extensions from two to three dimensions, adding new models and turbulence modeling. This property makes the CFD software more difficult to maintain. These are the reasons that we have omitted optimization of interface conditions and have chosen to start with a simple Schwarz-Schwarz domain decomposition method for the incompressible Navier-Stokes equations. We intend to study the parallel aspects of the present method and to generalize the method to the full incompressible Navier-Stokes equations in general coordinates on staggered grids, see [6, 5]. Acknowledgements The authors would like to thank P. Wesseling for many valuable discussions on this work. --R Domain decomposition algorithms of Schwarz type Iterative methods for the solution of elliptic problems on regions partitioned into substructures To overlap or not to overlap: a note on a domain decomposition method for elliptic problems A parallel domain decomposition algorithm for the incompressible Navier-Stokes equations Schwarz domain decomposition for the incompress- sible Navier-Stokes equations in general coordinates A domain decomposition method for the advection-diffusion equation The construction of preconditioners for elliptic problems by substructuring I A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems On the relationship between overlapping and nonoverlapping domain decomposition methods Some recent results on Schwarz type domain decomposition algorithms An iterative procedure with interface relaxation for domain decomposition methods Experiences with domain decomposition in three di- mensions: Overlapping Schwarz methods Iterative solution of large sparse systems of equations Multigrid on composite meshes Aerodynamics applications of Newton-Krylov-Schwarz solvers Iterative solution of elliptic equations with refinement: The two-level case A relaxation procedure for domain decomposition methods using finite elements Benchmark solutions for the incompressible Navier-Stokes equations in general coordinates on staggered grids GMRES: a generalized minimal residual algorithm for solving non-symmetric linear systems A domain decomposition method for incompressible flow Domain decomposition methods in computational mechanics Local coupling in domain decomposition The superlinear convergence behaviour of GMRES Some Schwarz methods for symmetric and nonsymmetric elliptic problems Schwarz and Schur: a note on finite-volume domain decomposition for advection-diffusion A pressure-based composite grid method for the Navier-Stokes equations --TR GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems The construction of preconditioners for elliptic problems by substructuring. I Iterative methods for the solution of elliptic problems on regions partitioned into substructures Multigrid on composite meshes To overlap or not to overlap: a note on a domain decomposition method for elliptic problems A relaxation procedure for domain decomposition methods using finite elements Generalized Schwarz splittings On the relationship between overlapping and nonoverlapping domain decomposition methods A domain decomposition method for incompressible viscous flow A pressure-based composite grid method for the Navier-Stokes equations The superlinear convergence behaviour of GMRES

krylov subspace method;advection-diffusion equation;neumann-dirichlet method;schwarz alternating method;domain decomposition;krylov-schwarz algorithm

271759

A Parallel Grid Modification and Domain Decomposition Algorithm for Local Phenomena Capturing and Load Balancing.

Lion's nonoverlapping Schwarz domain decomposition method based on a finite difference discretization is applied to problems with fronts or layers. For the purpose of getting accurate approximation of the solution by solving small linear systems, grid refinement is made on subdomains that contain fronts and layers and uniform coarse grids are applied on subdomains in which the solution changes slowly and smoothly. In order to balance loads among different processors, we employ small subdomains with fine grids for rapidly-changing-solution areas, and big subdomains with coarse grids for slowly-changing-solution areas. Numerical implementations in the SPMD mode on an nCUBE2 machine are conducted to show the efficiency and accuracy of the method.

Introduction Grid refinement methods have been proved to be essential and efficient in solving large-scale problems with localized phenomena, such as boundary layers or wave fronts. For many engineering problems, however, this still leads to large-size linear systems of algebraic equations, which can not be solved easily even on today's largest computing systems. Domain decomposition methods have been extensively investigated recently since they provide a mechanism for dividing a large problem into a collection of smaller problems; each of them can be solved sequentially or in parallel, on a workstation or a processor with moderate computing capability. In this work, we investigate the possibility of combining domain decomposition methods and grid refinement techniques. We first divide the original physical domain into a collection of subdo- mains, and then apply fine grids in subdomains that contain local phenomena and coarse grids in This paper appeared in: Journal of Scientific Computing, 12(1997), 99-117. y Department of Mathematics, Wayne State University, Detroit, Michigan 48202. Email: [email protected], [email protected] subdomains in which the solution changes slowly. For the purpose of load balancing among different processors, when solved on a parallel computer or a distributed computing system, the size of the subdomains that contain local phenomena should be kept small, while the subdomains in which the solution changes smoothly should be large. In our implementation, we try to keep approximately the same number of degrees of freedom for each subdomain. Our results are experimental and for one-dimensional problems. Multi-dimensional problems can be considered similarly but with more complexity. In section 2, we will describe the domain decomposition method with grid modification, and in section 3, we will conduct some numerical experiments. domain decomposition method with local grid refine- ment In this section, we describe a domain decomposition method which will be combined with grid refinement techniques. 2.1 The differential problem Consider the model problem in 1-D: @x @x @x (1) @- (2) @- where - is the outward normal, and - are given constants. Let the decomposed into K nonoverlapping subdomains satisfying Then the system (1)-(3) is equivalent to the following split problem: @x @x @x @-m @- @- is the outward normal to the boundary of subdomain\Omega k . In [5], Lions proposed a non-overlapping Schwarz algorithm which can be heuristically stated as follows. Choose the initial guess k satisfying the given boundary condition (2)-(3). For the sequence u n k is constructed such that \Gammaq @u n+1 @-m @- where - km is a parameter which can be chosen to speed up the convergence, and L is the given elliptic operator. Despr'es [1], Douglas et al [2], and Kim [4] have considered discretizations of this procedure by the hybridized mixed finite element method and the finite difference method and applied the procedure to wave equations. Mu and Rice [7] considered collaborating PDE solvers in the context of domain decompositions. 2.2 The finite difference discretization Assume that uniform grid with size h k is used in subdomain\Omega k and that there are d k grid points in \Omega k with the i-th grid point being denoted by x k;i , where the interface point between and\Omega k+1 is x k;d k (= x k+1;1 1: The endpoints of\Omega are x We denote, for example, In the following, we describe two schemes based on finite difference discretization, which differ from each other in the way of discretizing the convection term. Scheme 1: For using the exterior point x we have the second order finite difference scheme and the first order interface condition at x k;1 (see figure 1) \Gammaq k;1 k\Gamma1;d k\Gamma1;d Note that q only if q is continuous at x k;1 (= x k\Gamma1;d x x Thus (16) represents a general form of continuity of function value and flux. x k\Gamma1;1 a x k\Gamma1;2 a a x k\Gamma1;d \Omega x k;1 Figure 1: Grid points in subdomain\Omega k , and interface between and\Omega Eliminating k;0 from (15) and (16) we have the equation for the first point in subdomain\Omega k . For interior points of subdomain\Omega k , we apply the second order finite difference scheme. For the last point of subdomain\Omega k , we have Similarly, in (17)-(18) the point x k;d k is an imaginary one. The value u n k;d k +1 in (17)-(18) will be eliminated, under the assumption that q k;d k Thus we have the following linear system for k\Gamma1;d where for the first the equation (19) should be changed to and for the last subdomain\Omega K , the equation (21) should be changed to K;d K Here we assumed that h k has been chosen such that It is easy to see that when the sequence u n k;j converges as n !1, the limit, denoted by u k;j , will satisfy the standard global finite difference scheme for problem (1)-(3), and thus u O(h) in the L 1 norm, where g. Scheme 2: In this scheme we employ the first order finite difference approximation to the convection term on each interface. To be more specific, applying the first order finite difference to the convection term in (15), we have Similarly we replace (17) by Then the equation (19) should be changed to k\Gamma1;d k\Gamma1;d And the equation (21) should be changed to All other equations remain unchanged. Thus Scheme 2 consists of equations (27), (20), (28), (22), and (23). For this scheme, the assumption (24) is not required. 2.3 The matrix form We now rewrite Scheme 1 into a matrix form which may be helpful when implementing the scheme. Define the tridiagonal matrices, . ff k;d k ff K;d and define the block tridiagonal matrix where Then the system (19)-(21) can be rewritten into K;d K Scheme 2 can also be put in the matrix form (39), but with different definitions of G and H. 2.4 Convergence analysis and the relaxation parameter Let ae(A) denote the spectral radius of a matrix A. The relaxation parameters - k;m will be chosen such that ae(G Let the elements of D \Gamma1 k be denoted by y k i;j , that is, denote D \Gamma1 (y We will show that such - k;m can be obtained by the following relation: y if we require that - Note that the formula (41) is valid for both Scheme 1 and Scheme 2. Following Tang [8], we give a proof of (41) in case of three subdomains, that is, deleting all the columns except the eight nonzero ones and corresponding rows of G \Gamma1 H, we see that ae(G y (2) y (2) y (2) y (2) Let Then y (2) z z y (2) where z z z By virtue of (43) we have that ae(G z z In order to let ae(G choose z 0: In view of (35)-(38) for Scheme 1 and similar formulas for Scheme 2, we have y (1) y (1) y (2) y (2) This finishes the proof of (41) in case In order to find the optimal parameters by (41), we need to know the ratio of the two elements of the inverse of the matrix D k . Assume that - k\Gamma1;k has been found, the ratio can be determined from the relation y y Indeed, an incomplete (omitting the last step) LU-factorization to D k leads to : D L has diagonal elements 1, and U has elements U (k) Now the (d k \Gamma 1)\Gammath row of (46) gives us that U y In view of (41) we have U Thus optimal parameters - k;k+1 can be found recursively from (47) by incomplete LU-decompositions so that the iterative matrix of the system (39) has spectral radius 0. This also gives the convergence for both Scheme 1 and Scheme 2. Note that the matrix D k is tridiagonal. Thus the ratio U can be obtained by a single (not nested) loop (with less than 3d k floating point operations), and an incomplete LU decomposition is not needed. In our implementation, each processor (except the last one) computes the parameter at the right-hand boundary of the subdomain. For constant coefficients, Tang [8] used (41) to find optimal parameters by directly computing the inverse of the matrices D k . Note that Tang's analysis in case of minimum subdomain overlapping is equivalent to the nonoverlapping case presented here. Kim [4] applied the procedure (47) to wave problems. The analysis here extends Tang and Kim's to nonuniform grids with an explicit treatment of discontinuous diffusion coefficients. 3 Numerical experiments In this section, we implement our grid refinement and domain decomposition strategy on the nCUBE2, a MIMD parallel computer with distributed memory, located in Department of Computer Science at Purdue. It is observed that Scheme 2 gives a little more accuracy than Scheme 1 and does not have the restriction as described in (24). Thus we just report our results for Scheme 2. It should be noted that Scheme 1 gives second order approximation to the convection term as well the diffusion term and provides a natural approach to discretizing the differential equation. 3.1 Hypercube machines We now describe briefly hypercube machines and their properties that have influence on our imple- mentation. By definition, an r-dimensional hypercube machine has nodes and r2 r\Gamma1 edges. Each node corresponds to an r-bit binary string, and two nodes are linked with an edge if and only if their binary strings differ in precisely one bit. See Figure 2. As a consequence, each node is incident to r = log N other nodes, one for each bit position. In addition to a simple recursive structure, the hypercube also has low diameter (log N) and high bisection width (N=2). At each iteration of our algorithm, each node computes an error between the current solution and the solution at the previous iteration on its subdomain. The maximum among the errors on all the nodes need be found to decide if the iterative process should be stopped. In our implementation, we first let each of the nodes whose binary string have the form i its value to the node whose binary string is then let the node i 1 compute and keep the maximum of the two values. After this operation, we have only half of the values to be dealt with, stored in nodes By the recursive structure of the hypercube machine, we need to continue only log N steps to get the maximum value among all the subdomain errors. The maximum is thus finally found on node 0. If this maximum is less than a prescribed small number, node 0 then sends a message to every other node to stop. Otherwise the iterative process continues until, at some later iteration, the maximum error is small. Figure 2: Examples of interprocessor connection and data network for hypercube machines. For example, consider a 2-dimensional hypercube, corresponding to Figure 2. At the first step, nodes 10 and 11 send their values to nodes 00 and 01, respectively. Then node 01 computes and stores the maximum value among nodes 01 and 11, and node 00 computes and stores the maximum value among nodes 00 and 10. At the second step, node 01 sends its updated value to node 00, and node 00 computes and stores the maximum. This procedure can be efficiently implemented using bit masks provided by the C programming language. 3.2 Parallel implementation We will adopt the following stopping criterion for the iterative procedure: k1 denotes the discrete L 1 norm. However, subdomain problems are solved by Gauss elimination with banded storage. In all the computations below, we choose the initial guess to be zero. When the stopping criterion (48) is satisfied, the procedure is stopped and the relative errors in the L 1 norm between the iterative solution and the true solution are computed. To test the accuracy of the method, consider Example 1: du dx The exact solution to the problem is where the dimensionless quantity is the Peclet number. When B ?? 1, u(x) exhibits a boundary layer of thickness O(B 1. In this example, we keep different values for q. Table 1: Numbers of iterations and L 1 norms of the errors between the true solution and the domain decomposition solution for Example 1 with different Peclet numbers and grid sizes, implemented on processors with subdomains. Number of iterations 21 20 20 20 of GMDD 3.9E-2 4.7E-2 3.2E-2 3.4E-2 of FDM1 8.7E-2 2.5E-1 4.5E-1 5.8E-1 of FDM2 3.4E-2 3.5E-2 3.5E-2 3.5E-2 The domain is decomposed into 16 subdomains. Since the boundary layer is near the point apply coarse grid h 1for the left 10 subdomains, fine grid h is the Peclet number, for the rightmost 4 subdomains, and ffor the middle two (the 11-th and 12-th, if counting from the left) subdomains. The sizes of the subdomains are arranged such that each subdomain contains approximately the same number of degrees of freedom. Let H c ; H f ; and Hm be the size of coarse, fine, and medium grid subdomains, respectively. Then Solving the system we obtain the size of each subdomain. We denote this Grid Modification and Domain Decomposition method by GMDD. To compare its accuracy, we also employ the second order finite difference method without domain decomposition with 1. (denoted by FDM1) uniform grid such that the total number of unknowns is equal to that of GMDD. 2. (denoted by FDM2) uniform fine grid with . The total number of unknowns for FDM2 is much bigger than that for GMDD. The results in Table 1 show that the domain decomposition solution is more accurate than the global method with the same number of degrees of freedom, and as accurate as the global method with a much larger number of degrees of freedom. For example, for the our grid modification and domain decomposition method has only 80 degrees of freedom while the global finite difference method with fine grid (FDM2 in Table 1) has 600 degrees of freedom. But the accuracies of the two methods are about the same. We now consider the following non-selfadjoint problem with variable coefficients. Example 2: @x @x @x Table 2: Numerical results in the L 1 norm for Example 2 with different numbers of processors; each processor takes care of one subdomain. Number of processors 8 Number of iterations Errors in L 1 norm 7.21E-2 6.26E-2 6.27E-2 5.19E-2 Table 3: Numerical results for Example 3 with discontinuous diffusion coefficient and different grid sizes, in the case of 16 subdomains. Number of iterations 78 79 79 The function f on the right hand side is chosen such that the exact solution to the problem is The true solution has a boundary layer near the right boundary. The domain is decomposed into different numbers of subdomains, with the right most three subdomains being discretized with grid size 1, and the rest of the subdomains having grid size 1. The number of subdomains is equal to the number of the processors in the implementation. The results for this example with different numbers of processors are shown in Table 2. From Table 2 we know that the number of iterations required for the procedure to stop is bigger than for problems with constant coefficients, see Table 1. It is interesting to observe that the accuracy improves as the number of subdomains increases. This agrees with our computing experience in [9]. Then, we see how well this algorithm performs with discontinuous diffusion coefficients and consider Example 3: @x @x @x where and the exact solution is chosen as: Table 4: Numerical results for Example 4 with different numbers of processors; each processor takes care of one subdomain. Number of processors 8 Number of iterations 29 55 104 197 Errors in L 1 norm 2.99E-2 2.39E-2 2.37E-2 2.32E-2 We decompose the domain into 16 subdomains. The accuracy and number of iterations with different grid sizes for Example 3 are shown in Table 3. From Table 3 we see that the number of iterations and the errors are larger for discontinuous diffusion coefficients than for continuous coefficients. Finally, we consider another convection-diffusion problem Example 4: @x @x @x The function f on the right hand side is chosen such that the exact solution to the problem is The true solution has a boundary layer near the right boundary. The domain is decomposed into different numbers of subdomains, with one third of the subdomains on the right hand side being discretized with grid size h f = 1, two thirds of the subdomains on the left hand side having grid size h c = 1, and one middle subdomain between them having grid size (h c )=2. The length of each subdomain is chosen as in Example 1 such that each subdomain has approximately the same number of degrees of freedom. The number of subdomains is still equal to the number of the processors in the implementation. Numerical results for this example are shown in Table 4. Concluding Remarks We have implemented a non-overlapping Schwarz domain decomposition method with grid modification for elliptic problems involving localized phenomena, such as fronts or layers. In order to capture local phenomena and save computational work, we apply fine grids in subdomains that contain fronts or layers, and coarse grids in other subdomains. However, when implemented on parallel computing systems, the subdomain problems must have approximately the same computational complexity so that loads on different processors or workstations can be balanced. This is important for synchronization and achievement of a good speedup. Though our implementation is for one-dimensional and steady problems, the methodology applies to time-dependent and multi-dimensional problems. Time-dependent problems are usually first discretized in time to elliptic problems at each time step, and then domain decomposition algorithms can be applied. However, for advection-dominated transport problems, discretization in space and time should be coordinated. For example, unwinding techniques can be incorporated in the discretization process. Nonoverlapping Schwarz domain decomposition methods have recently received a lot of atten- tion, since its efficiency and elegant simplicity in implementation, great savings in computer storage (in 3D, even small overlapping of subdomains could cause a lot more storage), and direct applicability to transmission problems. In this direction, several other types of domain decomposition have been considered [6, 11, 10, 12]. Their modifications can apply directly to domain decompositions with cross points and with long and narrow subdomains. In [3], we successfully implemented a variant of the methods [12] for selfadjoint and non-selfadjoint elliptic partial differential equations with variable coefficients and full diffusion tensor, in the case of 100 subdomains with 81 cross points in 2-D. The variant also works well for long and narrow subdomains with length of a subdomain being times as large as the width of the subdomain. In [9], the author considered a dynamic domain decomposition method based on finite element discretization for two-dimensional problems. The dynamic change of domain decompositions can provide a mechanism for capturing moving layers and fronts, and for load balancing on different processors. Acknowledgement : The author would like to thank Professor John R. Rice for his valuable advice and for providing the computing facility in Department of Computer Science at Purdue. He also feels grateful to Professor Jim Douglas, Jr. and Dr. S. Kim for helpful discussions. The referee's suggestions also improved the quality of the paper. --R Domain decomposition method and the Helmholz problem Finite difference domain decomposition procedures for solving scalar waves On the Schwarz alternating method III: a variant for nonoverlapping subdomains A relaxation procedure for domain decomposition methods using finite elements Modeling with collaborating PDE solvers: theory and practice Different domain decompositions at different times for capturing moving local phenomena A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems A parallel iterative domain decomposition algorithm for elliptic problems A parallel iterative nonoverlapping domain decomposition method for elliptic interface problems --TR A relaxation procedure for domain decomposition methods using finite elements Generalized Schwarz splittings Different domain decompositions at different times for capturing moving local phenomena --CTR Daoqi Yang, Finite elements for elliptic problems with wild coefficients, Computational science, mathematics and software, Purdue University Press, West Lafayette, IN, 2002 J. R. Rice , P. Tsompanopoulou , E. Vavalis, Fine tuning interface relaxation methods for elliptic differential equations, Applied Numerical Mathematics, v.43 n.4, p.459-481, December 2002

finite difference method;grid modification;parallel computing;domain decomposition method

271778

Constructing compact models of concurrent Java programs.

Finite-state verification technology (e.g., model checking) provides a powerful means to detect concurrency errors, which are often subtle and difficult to reproduce. Nevertheless, widespread use of this technology by developers is unlikely until tools provide automated support for extracting the required finite-state models directly from program source. In this paper, we explore the extraction of compact concurrency models from Java code. In particular, we show how static pointer analysis, which has traditionally been used for computing alias information in optimizers, can be used to greatly reduce the size of finite-state models of concurrent Java programs.

Introduction Finite-state analysis tools (e.g., model checkers) can automatically detect concurrency errors, which are often subtle and difficult to reproduce. Before such tools can be applied to software, a finite-state model of the program must be constructed. This model must be accurate enough to verify the requirements and yet abstract enough to make the analysis tractable. In this paper, we consider the problem of constructing such models for concurrent Java programs. We consider Java because, with the explosion of internet applications, Java stands to become the dominant language for writing concurrent software. A new generation of programmers is now writing concurrent applications for the first time and encountering subtle concurrency errors that have heretofore plagued mostly operating system and telephony switch developers. Java uses a monitor-like mechanism for thread synchronization that, while simple to describe, can be difficult to use correctly (a colleague teaching concurrent Copyright c fl1998 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Java programming found that more than half of the students wrote programs with nested monitor deadlocks). Ideally, an analysis tool could extract a model from a program and use the model to verify some property of the program (e.g., freedom from deadlock). In practice, extracting concurrency models is difficult to automate completely. In order to obtain a model small enough for a tractable analysis, an analyst must assist most existing tools by specifying what aspects of the program to model. In particular, the representation of certain variables is often necessary to make the model sufficiently accurate, but these variables must often be abstracted or restricted to make the analysis tractable. Although a model that restricts the range of a variable does not represent all possible behaviors of the program and thus cannot technically be used to verify the program has a property, the conventional wisdom is that most concurrency errors are present in small versions of a system[6, 9], thus these models can still be useful for finding errors (testing). Most previous work on concurrency analysis of software has used Ada [7, 13, 12, 2, 3, 8]. Although some aspects of these methods can also be applied to Java programs, the Java language presents several new chal- lenges/opportunities: 1. Due to the object-oriented style of typical Java pro- grams, most of the variables that need to be represented are fields of heap allocated objects, not stack or statically allocated variables as is common in Ada. 2. Java threads must be created dynamically, thus is it impossible (in general) to determine how many threads a program will create. Although Ada tasks may be created dynamically, many concurrent Ada programs contain only statically allocated tasks. 3. Java has a locking mechanism to synchronize access to shared data. This can be exploited to reduce the size of the model. The main contribution of this paper is to show how static pointer analysis can be used to reduce the size of finite-state models of concurrent Java programs. The method employs virtual coarsening [1], a well-known technique for reducing the size of concurrency models by collapsing invisible actions (e.g., updates to variables that are local or protected by a lock) into adjacent visible actions. The static pointer analysis is used to construct an approximation of the run-time structure of the heap at each statement. This information can be used to identify which heap objects are actually local to a thread, and which locks guard access to which variables. This paper is organized as follows. We first provide a brief overview of Java's concurrency features in Section 2. Section 3 defines our formal model (transition systems) and Section 4 explains how the size of such models can be reduced with virtual coarsening given certain information on run-time heap structure is available. We then explain how to collect this information using static pointer analysis in Section 5. Section 6 shows how to use the heap structure information to apply the reductions. Finally, Section 7 concludes Concurrency in Java Java's essential concurrency features are illustrated by the familiar bounded buffer system shown in Figure 1. In Java, threads are instances of the class Thread (or a subclass thereof) and are created using an allocator (i.e., new). The constructor for Thread takes as a parameter any object implementing the interface Runnable, which essentially means the object has a method run(). Once a thread is started by calling its start() method, the thread executes the run() method of this object. Although threads may be assigned priorities to control scheduling, in this paper we assume all (modeled) threads have equal priority and are scheduled ar- this captures all possible executions. In the example, the program begins with the execution of the static method main() by the main thread. This creates an instance of an IntBuffer, creates instances of Producer and Consumer that point to this IntBuffer, creates instances of Thread that point to the Producer and Consumer, and starts these threads, which then execute the run() methods of the producer and consumer. The producer and consumer threads put/get integers from the shared buffer. There are two types of synchronization in the bounded buffer problem. First, access to the buffer should be mutually exclusive. Every Java object has an implicit lock. When a thread executes a synchronized statement, it must acquire the lock of the object named by the expression before executing the body of the statement, releasing the lock when the body is exited. If the lock is unavailable, the thread will block until the lock is released. Acquiring the lock of the current object (this) during a method body is a common idiom and may be abbreviated by simply placing the keyword synchronized in the method's signature. The second type of synchronization involves waiting: callers of put() must wait until there is space in the buffer, callers of get() must wait until the buffer is nonempty. On entry, the precondition for the operation is checked, and, if false, the thread blocks itself on the object by executing the wait() method, which releases the lock. When a method changes the state of the object in such a way that a pre-condition might now be true, it executes the notifyAll() method, which wakes up all threads waiting on the object (these threads must reacquire the object's lock before returning from wait()). 3 Formal Model We model a concurrent Java program with a finite-state transition system. Each state of the transition system is an abstraction of the state the Java program and each transition represents the execution of code transforming this abstract state. Formally, a transition system is a pair (S; T ) where Heap Structure: Thread Thread Consumer Producer IntBuffer int[] data buf buf public class IntBuffer - protected int [] data; protected int count = 0; protected int public IntBuffer(int capacity) - new int[capacity]; // allocate array // data.length == size of array (capacity) public void put(int x) - synchronized (this) - while (count == data.length) buffer not full data[(front if (count == 1) // buffer not empty public int get() - synchronized (this) - while (count == wait(); // wait until buffer not empty int if (count == data.length - 1) notifyAll(); // buffer not full return x; public class Producer implements Runnable - protected int next = 0; // next int to produce protected IntBuffer buf; public void run() - while (true) - public class Consumer implements Runnable - protected IntBuffer buf; public void run() - while (true) - int public class Main - public static void main(String [] args) - new new Thread(new Producer(buf)).start(); new Thread(new Consumer(buf)).start(); Figure 1: Bounded Buffer Example Transformations: State Variables: loc1; loc2: location of thread 1,2 lock: state of lock (0 is free, 1 is taken) x: value of program variable x (initially 1) Thread 1: 1: Lock 2: 3: Unlock 4: . Thread 2: 5: Lock . State Space (fragment): state = (loc1; loc2; lock; x) Figure 2: Example of Transition System Dn is a set of states. A state is an assignment of values to a finite set of state variables ranges over a finite domain is a transition relation. T is defined by a set of guarded transformations of the state variables, where called the guard, is a boolean predicate on states, and S, called the transformation, is a map from states to states: When g i (s) is true, we sometimes write t i A trace of a transition system is a sequence of transitions: such that (s The method of constructing the transition system representing a Java program is similar to the method presented in [5] for constructing the (untimed) transition system representing an Ada program. State variables are used to record the current control location of each thread, the values of key program variables, and any run-time information necessary to implement the concurrent semantics (e.g., whether each thread is ready, running, or blocked on some object). Each transformation represents the execution of a byte-code instruction for some thread. A depth-first search of the state space can be used to enumerate the reachable states for anal- ysis; at each state, a successor is generated for each ready thread, representing that thread's execution. The small example in Figure 2 gives the flavor of the translation. The Java heap must also be represented. We bound the number of states in the model by limiting the number of instances of each class (including Thread) that may exist simultaneously. For this paper, we assume these limits are provided by the analyst. If class C has instance limit kC , when a program attempts to allocate an instance of class C at a point where kC instances are still accessible (Java uses garbage collection), the transition system goes to a special trap state-the model does not represent the behavior of the program beyond this point. As discussed in the introduc- tion, such a restricted model can still be useful for finding errors. Consider again the bounded buffer example in Figure 1. We could generate a restricted model of this program by representing all the variables but restricting their ranges. By restricting the variables representing the contents of the buffer to f0; 1g and the variables representing the size of the buffer to f0; 1; 2g, we would obtain a very restricted but interesting model of the program (i.e., one that would likely contain any concurrency errors). Reductions The transition system (S; T ) produced by the method sketched in Section 3 is much larger than required for most analyses and is often too large to construct. Instead, we construct a reduced transition system (S and use this for the analysis. We reduce the size of the transition system using virtual coarsening [1], a well-known technique for reducing the size of concurrency models by amalgamating transitions. Since we are using an interleaving model of concurrency, reducing the number of transitions in each thread greatly reduces the number of possible states by eliminating the interleavings of the collapsed transition sequences. The reduced transition system is constructed by classifying each transformation defining T as visible or invisible and then composing each (maximal) sequence of invisible transformations in a given thread into the visible transformation following that sequence. The transitions and states generated by these composed transformations form (S ; T ). For example, in Figure 2, we might replace transformations t2 and t3 with a single transformation t2 ffi t3 that updates x and releases the lock; we could then eliminate control location 3 from the domain of loc1. We assume the requirement to be verified (tested) is specified as a stuttering-invariant formula f in linear temporal logic (LTL) [11], the atomic propositions of which are of the is a state variable and d i 2 D i . Statement s is true in state s of transition system (S; T ). To be useful, the reduced transition system should: 1. Be equivalent to the original transition system for the purpose of verification. Specifically, for all s 2 S s 2. Be constructible directly from the program, without first constructing (S; T ). Note that the reduced model constructed is specific to the thus the reduction must be repeated for each property verified. We classify a transformation as invisible and compose it with its successor transformation(s) only if we can show that this cannot change the truth value of f . An is constructed by applying temporal operators to state predicates, which are boolean combinations of atomic propo- sitions. Let pm be the state predicates of f . An f-observation in a state s denoted Pf (s) is a vector of m booleans giving the value of in s. A transformation f-observable if it can change an f-observation: Each trace (s0 defines a sequence of f - observations which we reduce by combining consecutive identical (i.e., stuttered) f-observations. It is easy to show that the set of these reduced f-observation sequences determines the truth value of f in s0 . Therefore, to satisfy condition 1 above, we construct the reduced transition system such that it has the same set of reduced f-observation sequences as the original transition system. To satisfy condition 2, we must do this without constructing (S; T ); we must classify transformations as vis- ible/invisible based on information obtained from the program code. Below, we give two cases in which transformations representing Java code can be made invisible. In both cases, we need information about the structure of the heap at run-time to apply the reduction. We show how to collect this information in Section 5. 4.1 Local Variable Reduction Some state variables are accessed only when a particular thread is running. For example, some program variables are locally scoped to a particular thread by the language seman- tics. Also, the state variable recording the control location of a thread is accessed only by that thread. Transformations that access exclusively such state variables may be made invisible provided they are not f-observable. To understand why, consider transformation t in Figure 3. Assume t is not f-observable and accesses only variables local to the thread whose code it represents. Let t 0 represent code in this same thread that can be executed immediately following the code represented by t. We can replace t and t 0 with t ffi t 0 (if there are multiple successors to t, say t replace t and t 0 To prove that this reduction does not change the truth value of f , we must show that the resulting transition system has the same set of reduced f-observation sequences as the original transition system. For any state s1 in which t is enabled, there may be one or more sequences of transformations representing the execution of code from other threads (i.e., not the thread of t). Combining t and t 0 eliminates traces in which occurs between t and t 0 . This does not eliminate any reduced f-observation sequences, however, since executing must produce the same reduced f - observation sequence as executing t before t accesses only variables that t1 independent of commutes: for any state s1 in which both t and are enabled, t is not f-observable, the trace obtained by executing t; must have the same reduced f-observation sequence as the trace obtained by executing To use this technique, we would like to determine what variables are local to a particular Java thread (i.e., can only be referenced by that thread). A program variable is local to a thread if: 1. The variable is stack allocated (i.e., is declared in a method body or as a formal parameter). 2. The variable is statically allocated and referenced by at most one thread. 3. The variable is heap allocated (i.e., an instance variable of an object) and the object is accessible only from one thread. For example, the variable next of class Before reduction After reduction Figure 3: Reduce by combining t and t 0 Producer in Figure 1 is accessible only by the producer thread. Case 1 is trivial to detect. Case 2 is more difficult due to the dynamic nature of thread creation, though the following conservative approximation is reasonable: a static variable may be considered local if it is accessed only by code reachable 1 from main(), or only by code reachable from a single run() method of a class that is passed to a Thread allocator at most once (the allocator is outside any loop or recursive procedure). For example, if the variable next were a static member of class Producer, then since that variable is accessed only by code reachable from the Producer's run() method, and since there is only one instance of Producer created, this analysis could determine next is local to the producer thread. Case 3 is the most difficult. Clearly if the object containing the variable is accessible only from stack or statically allocated variables that themselves are local to a specific thread (cases 1 and 2), then the heap allocated variable is also local to that thread, but determining this requires information about the accessibility of heap objects at run-time. 4.2 Lock Reduction We propose another technique for virtual coarsening based on Java's locking mechanism. A transformation that updates a variable x of an instance of class C may be made invisible provided it is not f-observable and there exists an object 'x such that any thread accessing x is holding the lock of 'x ('x may be the instance of C containing x). We say the lock on 'x protects x. The intuition behind this reduction is that, even though other threads may access x, they cannot do so until the current thread releases the lock on 'x , thus any changes to x need not be visible until that lock is released. The correctness of this reduction can be shown using the diagram in Figure 3. The reasoning is similar to that for the local variable reduction. Assume the only non-local variables t accesses are those that are protected by locks. The thread whose code t represents must hold the locks for these variables at s1 . Therefore, although there exist transformations representing code in other threads that accesses these variables, such transformations cannot be in the sequence since the other thread would block before reaching such transformations. Assuming f does not reference the state of a shared ob- ject, this reduction allow us to represent complex updates to such objects with two transformations. In the bounded statement s is reachable from a statement s 0 if there exists a path in the program's control flow graph from s 0 to s (i.e., a thread might execute s after executing s 0 ). Heap Structure: Object Programmer Object salaryLock hoursLock public class Programmer - protected long hours = 80; protected double salary = 50000.0; protected Object hoursLock = new Object(); protected Object new Object(); public void updateHours(long newHours) - synchronized public void updateSalary(double newSalary) - synchronized Figure 4: Example of Splitting Locks buffer example, each execution of put() or get() updates several variables, yet we can represent each call with a transformation that acquires the lock and a transformation that atomically updates the state of the buffer and releases the lock. In order to apply these reductions, we need to determine which locks protect which variables. Clearly if an instance variable of a class is only accessed within synchronized methods of that class, then the variable is protected by the lock of the object in which it is contained. Nevertheless, it is common for variables to be protected by locks in other objects. For instance, in the bounded buffer example, the array object referenced by instance variable data is protected by the lock on the enclosing IntBuffer object. This very common design pattern is known as containment [10]: an object X is conceptually 2 nested in an object Y by placing a reference to X in Y and accessing X only within the methods of Y. Another common design pattern in which locks protect variables in other objects is splitting locks [10]. A class might contain independent sets of instance variables that may be updated concurrently. In this case, acquiring a lock on the entire instance would excessively limit potential parallelism. Instead, each such set of instance variables has its own lock, usually an instance of the root class Object. An example is given in Figure 4; two threads could concurrently update a Programmer's hours and salary. In general, determining which locks protect which variables requires information about the structure of the heap at run-time. Collecting this information is the topic of the next section. 5 Reference Analysis In this section, we describe a static analysis algorithm that constructs an approximation of the run-time heap struc- ture, from which we can collect the information needed for the reductions. Understanding run-time heap structure is an important problem in compiler optimization since accurate knowledge of aliasing can improve many standard Java does not allow physical nesting of objects. optimizations. One common approach is to construct a directed graph for each program statement that represents a finite conservative approximation of the heap structure for all control paths ending at the statement. Several different algorithms have been proposed, differing in the method of approximation. Our algorithm is an extension of the simple algorithm given by Chase et al [4], which uses this basic approach. We extend Chase's algorithm in three ways. First, we handle Chase's algorithm is for sequential code. Second, we distinguish current and summary heap nodes; this allows us to collect information on one-to-one relationships between objects. Third, we handle arrays. 5.1 The Program For the reference analysis, we represent a multi-threaded program as a set of control flow graphs (CFGs) whose nodes represent statements and whose arcs represent possible control steps. There is one CFG for each thread: one CFG for the main() method and kC identical CFGs for each run() method of class C (recall kC is the instance limit for class C). In this paper, we do not handle interprocedural anal- ysis. We assume all procedure (method) calls have been inlined; this limits the analysis to programs with statically bounded recursion. Polymorphic calls can be inlined using a switch statement that branches based on the object's type tag; since this tag is not modeled in our analysis, all methods to which the call might dispatch will be explored. In our algorithm, we require the concept of a loop block. For each statement s, let loop(s) be the innermost enclosing loop statement s is nested within (or null if s is not in any loop). The set fs 0 jloop(s loop(s)g is called the loop block of s. Our analysis models only reference variables and values. References are pointers to heap objects. A heap object contains a fixed number of fields, which are references to other heap objects (we do not model fields not having a reference type). For class instances, the number of fields equals the number of instance variables with a reference type (possibly zero). For arrays, the number of fields equals zero (for an array of a primitive type) or one (for an array of references); in the latter case all array elements are represented by a single field named []. In Java, references can be manipulated only in four ways: the new allocator returns a unique new reference, a field can be selected, a field can be updated, and references can be checked for equality (this last operation is irrelevant to the analysis). 5.2 The Storage Structure Graph A storage structure graph (SSG) is a finite conservative approximation of all possible pointer paths through the heap at a particular statement s. There are two types of nodes in an SSG: variable nodes and heap nodes. There is one variable node for each statically allocated reference variable and for each stack allocated reference variable in scope at s. There are one or two heap nodes for each allocator A (e.g., new C()) in the program, depending on the location of statement s in relation to A. If s is within the loop block of A or in a different thread/CFG than A, the SSG for s contains a current node for A, which represents the current instance of class C-the instance allocated by A in the current iteration of A's loop block. For all statements s, the SSG for s contains a summary node for A, which represents x f y f z f 1:C 2:C 2:C* class C { // class with two fields { C x,y,z; // stack variables // Method body while (.) { 1: 2: 3: 4: 5: Figure 5: SSG for Statement 4 the summarized instances of class C-all instances allocated by A in completed iterations of A's loop block. Each heap node has a fixed number of fields from which edges may be directed. Each edge in the SSG for a statement s represents a possible reference value at s. Edges are directed from variable nodes and fields of heap nodes towards heap nodes. In general, more than one edge may leave a variable node or heap node field since different paths to s may result in different values for that reference. Even if there is only one path to s, there may be multiple edges leaving a summary node or array field since such nodes represent multiple variables at run-time. An example SSG is shown in Figure 5. We elide parts of the code not relevant to the analysis with . and prepend line numbers to simple statements for identification. Variable nodes are shown as circles, heap nodes as rectangles with a slot for each field. Heap nodes are labeled with the name of the class prefixed with the statement number of the allocator. Summary nodes are suffixed with an asterisk(*). Thus 2:C* represents the summary node for the allocator of class C at statement 2. We often omit disconnected nodes (e.g., the summary node for an allocator that is not in a loop). Note that the linked list is represented with a self loop on node 2:C*. Like Chase et al [4], we distinguish objects of the same class that were allocated by different allocators. This heuristic is based on the observation that objects allocated by a given allocator tend to be treated similarly. For example, both Employee and Meeting objects might contain a nested Date object allocated in their respective constructors (i.e., there are two Date allocators). By distinguishing the two kinds of Date objects, the analysis could determine that a Date inside of an Employee cannot be affected when the Date inside of a Meeting is updated. A conservative SSG for a statement s contains the following information about the structure of the heap at run-time: 1. If there exists an edge from the node for variable X to a heap node for allocator A, then after some execution path ending at s (i.e., s has just been executed by the thread of its CFG), X may point to an object allocated by A. Otherwise, X cannot point to any object allocated by A. 2. If there exists an edge from field F of the current heap node for allocator B to a heap node for allocator A, then after some execution path ending at s, the F field for the current instance allocated by B may point to an object allocated by A. Otherwise, the F field for the current instance allocated by B cannot point to any object allocated by A. 3. If there exists an edge from field F of the summary heap node for allocator B to a heap node for allocator A, then after some execution path ending at s, the F field for some summarized instance allocated by B may point to an object allocated by A. Otherwise, there is no summarized instance allocated by B whose F field points to any object allocated by A. 4. For each of the above three cases, if the heap node for allocator A is the current node, then the reference must be to the current instance allocated by A, otherwise the reference is to some summarized instance allocated by A. Note that the useful information is the lack of an edge. One graph is more precise than another if it has a strict subset of its edges. 5.3 The Algorithm We use a modified dataflow algorithm to compute, for each statement, a conservative SSG with as few edges as possi- ble. Initially, each statement has an SSG with no edges. A worklist is initialized to contain the start statement of main(). On each step, a statement is removed from the head of the worklist and processed, possibly updating the SSGs for that statement and all statements in other CFGs. If any edges are added to the statement's SSG, the successors of the statement in its CFG and any dependent statements in other CFGs are added to the tail of the worklist. One statement is dependent on another if they may reference the same variable at run-time: they select the same static variable or instance variable. The algorithm terminates when the worklist is empty. To process a statement, we employ three operations on SSGs: join, step, and summarize. First, we compute the pre- SSG for the statement by joining the SSGs of all immediate predecessors in its CFG. SSGs are joined by taking the union of their edge sets (this is an any-paths analysis). The pre- SSG is then updated by the step operation (discussed below) in a manner reflecting the semantics of the statement to produce the post-SSG. Finally, if the statement is the last statement of a loop block, the post-SSG is summarized to produce the new version of the statement's SSG, otherwise the post-SSG is the new version. We summarize an SSG by redirecting all edges to/from the current nodes of allocators within the loop block to their corresponding summary nodes (see the SSGs for statement 6 in Figure 6). The step operation uses abstract interpretation to up-date the SSG (an abstract representation of the run-time heap) according to the statement's semantics. Only assignments to reference variables need be considered; other statements cannot add edges to the SSG (i.e., the post-SSG equals the pre-SSG). Each pointer expression has an l-value and an r-value, defined as follows. The l-value of a variable is the variable's node. The l-value of a field selector expression x.f is the set of f fields of the nodes in the r-value of x. The r-value of an expression is the set of heap nodes pointed to from the expression's l-value, or, in the case of an allocator, the current node for that allocator. The semantics of an assignment whether the left hand side is a stack variable or a local static variable. If e1 is either a stack variable or a local static vari- able, we perform a strong update by removing all edges out of the node in l-value(e1) and then adding an edge from the node in l-value(e1) to each node in r-value(e2 ). Otherwise, we perform a weak update by simply adding an edge from each node/field in l-value(e1) to each node in r-value(e2 ). Any edges added to a statement's SSG (for a step or summarize operation) are also added to the SSGs for all statements in other CFGs; we assume threads may be scheduled arbitrarily, thus any statement in another thread may witness this reference value. The execution of a thread allocator new Thread(x) is treated as an assignment of x to a special field runnable in the Thread object (this reflects the inlining of the constructor for Thread). Let X be the set of classes to which the object referenced by x might belong (i.e., all subclasses of the type of x). When the allocator is processed, we add to the worklist the start statement of every CFG for a run() method of a class in X (i.e., the start statement of a CFG is implicitly dependent on every thread allocator that might start 3 the thread). When a CFG for a run() method of a class C accesses an instance variable of the current object this (e.g., the expression next in the Producer's run() method of Figure 1) the r-value of this is the set of heap nodes for class C pointed to by runnable fields of heap nodes for class Thread (i.e., we do not associate a given thread/CFG with a specific thread allocator). 5.4 Computing One-to-One Relationships The summarized information gathered by the above analysis is not sufficient for the lock reduction. An SSG edge from the summary node for an allocator A to the summary node for allocator B indicates that objects allocated by A may point to objects allocated by B. We need to know if each object allocated by A points to a different object allocated by B; only then would holding the lock of an A object protect a variable access in the nested B object. We can conservatively estimate this information when SSGs are summarized and updated as follows. An edge from the summary node for A to the summary node for B is marked one-to-one if each A points to a different B at run-time. If A and B are in the same loop block, then an edge from some field of the summary node of A to the summary node of B, when first added to an SSG by a summarize operation, is marked one-to-one. If the field of the summary node of A is subsequently updated by a step operation in such a way that another edge to the summary node of B would have been added, then the edge is no longer marked one-to-one. This method is based on the observation that nested objects are almost always allocated in the same loop block as their enclosing object (often in the enclosing object's con- structor). Given a constructor or loop body that allocates an object, allocates one or more nested objects, and links these objects together, the one-to-one relationships between the objects are recorded in the SSG as arcs between the current nodes of the allocators. When these nodes are summarized at the end of the loop block, this information is then preserved as annotations on the arcs between the summary nodes. In fact, this is the motivation for distinguishing 3 Technically, the thread is started when its start() method is called, but since we are not using any thread scheduling informa- tion, assuming the thread starts when allocated produces the same SSGs. current from summarized instances/nodes. 5.5 Example Consider the Java source in Figure 6. The first SSG in Figure 6 is the post-SSG for statement 6 the first time it is processed (i.e., before any summary information exists). The second SSG is the result of summarizing this SSG. Note that, since nodes 3:B and 5:A are summarized together, the arc from field a2 of 3:B* to 5:A* is labeled as one-to-one (1-1), but since 2:A is a current node, there is no one-to-one relationship between field a1 of 3:B* and 2:A (nor would there be if a loop were added around this code and 2:A was summarized). The last SSG is the final SSG for statement 9 (the end of the method). After statement 7, the Thread allocated there may have access to the A allocated by statement 2, while after statement 8, the a1 field of some B may point to some A allocated at statement 5. Note that stack variable b is out of scope at statement 9 and thus can be removed from the SSG. The arc from 2:A to 0:A is added by statement 0, which is placed on the worklist when statement 7 is processed. Although we have not shown the final SSGs for statements 1-8, all these SSGs would contain this arc, even though the reference value it represents cannot appear until after statement 7; no thread scheduling information is considered. 5.6 Complexity Given a program with S statements and V variables and allocators, our algorithm must construct S SSGs each containing O(V ) nodes and up to O(V 2 ) edges. The running time to process a statement is (at worst) proportional to the total number of edges in all SSGs, as is the number of times a statement can be processed before a fixpoint is reached. Thus the worst case running time is O(S 2 V 4 ). Here, S is the number of statements after inlining all procedure calls, which could produce an exponential blowup in the number of statements. Despite this complexity, we do not anticipate the cost of the reference analysis to be prohibitive. First, based on the application of the algorithm to several small examples, we believe the average complexity to be much lower. SSGs are generally sparse; many edges in a typical SSG would violate Java's type system and could not be generated by the analysis. Also, very few edges are added to a statement's SSG after it has been processed once, thus each statement is typically processed only a few times. Second, S and V refer to the number of modeled statements and variables-in a typical analysis, only a fraction of the program will be mod- eled. The reference analysis does not model variables having primitive (i.e., non-reference) types, nor need it model statements manipulating such variables exclusively. Also, a program requirement might involve only a small subset of the program's classes; the rest of the program need not be represented. 6 Applying the Reductions In this section, we explain how to use the information collected by the reference analysis to apply the local variable and lock reductions. class A implements Runnable - A a3; void run() - 0: new A(); class A a1; A a2; new A(); class Main - static void main(.) - 2: A a = new A(); while (.) - 3: new B(a); inlined constructor 4: 5: new A(); if (.) 7: (new Thread(a)).start(); else 8: a x 5:A a x a x 5:A a3 0:A a3 0:A a3 0:A 2:A 5:A* 9: (final) 2:A 5:A* (before summary, first iteration) 2:A 3:B a3 a3 a3 a3 a3 3:B 3:B* (after summary) 7:Thread 3:B* a3 a3 Figure Reference Analysis Example 6.1 Local Variable Reduction Applying the local variable reduction is straightforward. The set of heap nodes in an SSG that are local to a given thread are those that are accessible only from stack or static variables local to the thread. All heap variables are accessed with expressions of the form ref.id where ref is a reference expression and id is the name of the instance variable. The variable accessed by such an expression is local to the thread if the nodes in the r-value of ref are local to the thread in the pre-SSG for the statement. Note that heap variables may be local for some statements and non-local for others. A common idiom is for an object to be allocated, initialized, and then made available to other threads (e.g., the IntBuffer object of the example in Figure 1). The reference analysis can determine that the instance variables of such an object are local until the object is made available to other threads. 6.2 Lock Reduction Applying the lock reduction is more complex. We need to determine whether a variable is protected by a lock. In general, the relationship between a variable and the lock that protects it may be too elaborate to determine with static analysis. Here, we propose a heuristic that we believe is widely applicable and, in particular, works for the locking design patterns given in [10]. The heuristic assumes that the relationship between the object containing the variable and the object containing the lock matches the following general pattern: either the lock object is accessible from the variable object, or vice versa, or both are accessible from a third object, or the lock and variable are in the same object. This pattern can be expressed in terms of three roles: the root, the lock, and the variable. The lock object contains the lock, the variable object contains the variable, and from the root object, the other two objects are accessible. Each role must be played by exactly one object, but one object may play multiple roles. For the expression data[i] in the bounded buffer example, the IntBuffer object is both the root and the lock object, while the int array referenced by data is the variable object. For the expression count, the IntBuffer object plays all three roles. For the expression salary in the splitting locks example of Figure 4, the Programmer object plays the root and variable roles, while the Object referenced by salaryLock plays the role of lock. We consider all static variables to be fields of a special environment object called env, which can play the roles of variable and root, but not the role of lock. This generalizes the pattern to include the case where the lock object or the variable object are accessible from static variables, and the case where the variable is static. Also, we fully qualify all expressions by prepending this to expressions accessing variables in the current instance, and by prepending env to all static variable accesses. For each static/heap variable, we want to determine whether there exists a lock that protects the variable (i.e., any thread accessing the variable must be holding the lock). Static variables are represented by variable nodes, heap variables by fields of heap nodes, and locks by heap nodes in the SSG. Essentially, we use the expressions accessing the variable and lock to identify the lock object; we can interpret the expressions (abstractly) using their SSGs. Formally, for each static/heap variable v, we want to compute Protect(v): the set of locks protecting v. For each such v, let Access(v) be the set of program expressions that may access v; these sets can be constructed during the reference analysis. For each expression Ev in Access(v), we compute Protect(v; Ev ): the set of locks the thread is holding at Ev protecting v. Since a lock must protect a variable everywhere: If the lock is a summary node, then the variable must be a field of a summary node; the interpretation is that each variable object is protected by a unique lock object. Given an expression Ev accessing v, we compute is a lock expression at Ev if it is the argument to some enclosing synchronized statement. For each E ' , we define a triple (Er is the root expression, which is the common prefix of E ' and Ev , S ' is the lock selector, which is the part of E ' not in , and Sv is the variable selector, which is the part of Ev not in Er and with the final selector removed (i.e., Er:Sv is a reference to the object containing v, not v itself). For exam- ple, consider the expression hours in method updateHours in Figure 4. The fully qualified expression 4 accessing the variable is this.hours, a lock expression is this.hoursLock, and this pair yields the triple (this; hoursLock; -). Note that indicates the lock and root objects are the same, while indicates that the variable and root objects are the same. Given Ev and (Er ; S ' ; Sv ), we identify a candidate lock ' in the SSG as follows. For an SSG node n and a selector S, n:S is the set of nodes reached from n by following S, while is a field of an object in n:Sg is the set of SSG nodes such that applying selector S to these nodes may reach the object containing variable v. First, in the pre-SSG for Ev , we compute the set of possible root objects for Ev 's access to v: If R contains exactly one node, then this node is the candidate root r and we compute the set of possible locks in the pre-SSG of E ' . If L contains exactly one node ', then this node is the candidate lock. We include ' in Protect(v; Ev ) if we can deduce from the SSGs that, for each instance of v at run-time, there is a unique instance of ' held by the thread. Note that this does not follow immediately since r, ', and the SSG nodes on the paths from r to ' and from r to v might represent multiple objects at run-time. Nevertheless, we can still conclude that for each variable represented by v at run-time there is a unique lock represented by ' if both of the following are true: 1. For each variable represented by v at run-time, there is a unique root object represented by r. This holds r is a current node, or all arcs on the path selected by Sv are one-to-one arcs between summary nodes. 2. For each root object represented by r at run-time, there is a unique lock object represented by '. This 4 In our analysis, the method will have been inlined and the this variable replaced with a new temporary holding this implicit param- eter. In addition, a simple propagation analysis can be used to allow recognition of the pattern even if multiple selectors are decomposed into a series of assignments (e.g., x.f.g expressed as holds provided that is a current node, or all arcs on the path selected by S ' are one-to-one arcs between summary nodes. A variable v is protected if Protect(v) is nonempty. A transformation may be made invisible if it is not f-observable and all variables it might access are protected or local. Note that inaccuracy in the reference analysis leads to a larger model, not an incorrect model. If we cannot determine that a variable is local or protected by a lock, then a transformation accessing that variable will be visible; the transition system will have more states, but will still be represent all behaviors of the (possibly restricted) program. 6.3 Example Consider the bounded buffer example in Figure 1. The SSGs for all statements in the producer and consumer run() methods are isomorphic to the heap structure shown at the top of the figure (there would also be nodes for the stack variables). From these SSGs, we can deduce that variable next in the Producer object is local to the producer thread. Thus, for a formula f that does not depend on next, the transformation incrementing next may be invisible. Also in the bounded buffer example, the variable data[.] in the array object and all the instance variables of the IntBuffer class are protected by the lock of the IntBuffer object. Thus, for a formula that does not depend on these variables, the sequence of transformations representing the methods put() and get() may be combined into two transformations: one to acquire the lock, the other to update the variables and release the lock. Although a complete program is not shown for the splitting locks example of Figure 4, each allocator for Programmer would produce an SSG subgraph isomorphic to the heap structure shown at the top of the figure. The arcs from a summary Programmer node to its Object nodes would be one-to-one. The analysis could determine that each instance variable hours is protected by the Object accessible via field hoursLock. 7 Conclusion We have proposed a method for using static pointer analysis to reduce the size of finite-state models of concurrent Java programs. Our method exploits two common design patterns in Java code: data accessible by only one thread, and encapsulated data protected by a lock. The process of extracting models from source code must, to some degree, be depended on the source language. Although our presentation was restricted to Java, many aspects of our method are more widely applicable and could be used to reduce finite-state models of programs with heap data and/or a monitor-like synchronization primitive (e.g., Ada's protected types). The method is currently being implemented as part of a tool intended to provide automated support for extracting finite-state models from Java source code. Although we have no empirical data on the method's performance at this time, the effectiveness of virtual coarsening for reducing concurrency models is well known, and the manual application of the method to several small examples suggests that many transitions can be made invisible for a typical formula. With the arrival of Java, concurrent programming has entered the mainstream. Finite-state verification technology offers a powerful means to find concurrency errors, which are often subtle and difficult to reproduce. Unfortunately, extracting the finite-state model of a program required by existing verifiers is tedious and error-prone. As a result, widespread use of this technology is unlikely until the extraction of compact mathematical models from real software artifacts is largely automated. Methods like the one described here will be essential to support such extraction. Acknowledgements Thanks are due to George Avrunin for helpful comments on a draft of this paper. --R Formalization of properties of parallel programs. Automated analysis of concurrent systems with the constrained expression toolset. Compositional verification by model checking for counter examples. Analysis of pointers and structures. Timing analysis of Ada tasking pro- grams Protocol verification as a hardware design aid. Using state space reduction methods for deadlock analysis in Ada tasking. Data flow analysis for verifying properties of concurrent programs. Elements of style: Analyzing a software design feature with a counterexample detector. Concurrent Programming in Java: Design Principles and Patterns. Checking that finite state concurrent programs satisfy their linear specifica- tions Static infinite wait anomaly detection in polynomial time. --TR Integrated concurrency analysis in a software development enviornment Analysis of pointers and structures Automated Analysis of Concurrent Systems with the Constrained Expression Toolset Using state space reduction methods for deadlock analysis in Ada tasking Data flow analysis for verifying properties of concurrent programs Compositional verification by model checking for counter-examples Elements of style Timing Analysis of Ada Tasking Programs Checking that finite state concurrent programs satisfy their linear specification Concurrent Programming in Java Protocol Verification as a Hardware Design Aid --CTR James C. Corbett , Matthew B. Dwyer , John Hatcliff , Shawn Laubach , Corina S. Psreanu , Robby , Hongjun Zheng, Bandera: extracting finite-state models from Java source code, Proceedings of the 22nd international conference on Software engineering, p.439-448, June 04-11, 2000, Limerick, Ireland Gleb Naumovich , George S. Avrunin , Lori A. Clarke, Data flow analysis for checking properties of concurrent Java programs, Proceedings of the 21st international conference on Software engineering, p.399-410, May 16-22, 1999, Los Angeles, California, United States Klaus Havelund , Mike Lowry , John Penix, Formal Analysis of a Space-Craft Controller Using SPIN, IEEE Transactions on Software Engineering, v.27 n.8, p.749-765, August 2001 Gleb Naumovich , George S. Avrunin , Lori A. Clarke, An efficient algorithm for computing MHP Jonathan Aldrich , Emin Gn Sirer , Craig Chambers , Susan J. Eggers, Comprehensive synchronization elimination for Java, Science of Computer Programming, v.47 n.2-3, p.91-120, May Jong-Deok Choi , Manish Gupta , Mauricio Serrano , Vugranam C. Sreedhar , Sam Midkiff, Escape analysis for Java, ACM SIGPLAN Notices, v.34 n.10, p.1-19, Oct. 1999 Pramod V. Koppol , Richard H. Carver , Kuo-Chung Tai, Incremental Integration Testing of Concurrent Programs, IEEE Transactions on Software Engineering, v.28 n.6, p.607-623, June 2002 James C. Corbett, Using shape analysis to reduce finite-state models of concurrent Java programs, ACM Transactions on Software Engineering and Methodology (TOSEM), v.9 n.1, p.51-93, Jan. 2000 John Whaley , Martin Rinard, Compositional pointer and escape analysis for Java programs, ACM SIGPLAN Notices, v.34 n.10, p.187-206, Oct. 1999 Jong-Deok Choi , Manish Gupta , Mauricio J. Serrano , Vugranam C. Sreedhar , Samuel P. Midkiff, Stack allocation and synchronization optimizations for Java using escape analysis, ACM Transactions on Programming Languages and Systems (TOPLAS), v.25 n.6, p.876-910, November Premkumar T. Devanbu , Stuart Stubblebine, Software engineering for security: a roadmap, Proceedings of the Conference on The Future of Software Engineering, p.227-239, June 04-11, 2000, Limerick, Ireland John Penix , Willem Visser , Seungjoon Park , Corina Pasareanu , Eric Engstrom , Aaron Larson , Nicholas Weininger, Verifying Time Partitioning in the DEOS Scheduling Kernel, Formal Methods in System Design, v.26 n.2, p.103-135, March 2005

model extraction;finite-state verification;static analysis

271798

Improving efficiency of symbolic model checking for state-based system requirements.

We present various techniques for improving the time and space efficiency of symbolic model checking for system requirements specified as synchronous finite state machines. We used these techniques in our analysis of the system requirements specification of TCAS II, a complex aircraft collision avoidance system. They together reduce the time and space complexities by orders of magnitude, making feasible some analysis that was previously intractable. The TCAS II requirements were written in RSML, a dialect of state-charts.

Introduction Formal verification based on state exploration can be considered an extreme form of simulation: every possible behavior of the system is checked for correctness. Symbolic model checking [?] using binary decision diagrams (BDDs) [?] is an efficient state-exploration technique for finite state sys- tems; it has been successful on verifying (and falsifying) many industry-scale hardware systems. Its application to non-trivial software or process-control systems is far less mature, but is increasingly promising [?, ?, ?, ?]. For ex- ample, we obtained encouraging results from applying symbolic model checking to a portion of a preliminary version of the system requirements specification of TCAS II, a complex software avionics system for collision avoidance [?]. The full requirements, comprising about four hundred pages, were written in the Requirements State Machine Language This work was supported in part by National Science Foundation grant CCR-970670. W. Chan was supported in part by a Microsoft graduate fellowship. (RSML) [?], a hierarchical state-machine language variant of statecharts [?]. By representing state sets and relations implicitly as BDDs for symbolic model checking, the sheer number of reachable states is no longer the obstacle to analysis. Instead, the limitation is the size of the BDDs, which depend on the structure of the system analyzed. Considerable effort on hardware formal verification has been focused on controlling the BDD size for typical circuits. However, transferring this technology to new domains may require alternative techniques and heuristics to combat the BDD-blowup problem. In this pa- per, we present modifications to the algorithms implemented in a symbolic model checker (SMV [?]), modifications to the model, as well as a simple abstraction technique, to improve the time and space efficiency of the TCAS II analy- sis. Experimental results show that the techniques together reduce the time and space complexities by orders of magni- these improvements have made feasible some analysis that was previously intractable. The specific techniques we discuss in the paper are: ffl Short-circuiting to reduce the number of BDDs generated by stopping the iterations before a fixed point is reached. Managing forward and backward traversals, to reduce the size of the BDD generated at each iteration. Notably, we improve backward traversals by making certain invariants (in particular, that some events are mutually ex- clusive) explicit in the search. ffl More sophisticated conjunctive partitioning of the transition relation and applying disjunctive partitioning in an unusual way, to reduce the size of the intermediate BDDs at each iteration. Further improvements were made by combining the two techniques to obtain DNF partitioning. ffl Abstraction to decrease the number of BDD variables. Given a property to check, we perform a simple dependency analysis to generate a reduced model that is guaranteed to give the same results as with the full model. Techniques like short-circuiting and abstraction are conceptually straightforward and applicable to many systems. Most other techniques were designed to exploit the simple synchronization patterns of TCAS II (for example, most events are mutually exclusive, and most state machines are not enabled simultaneously), and we believe they can also help analyze other statecharts machines with simple synchronization patterns. We provide experimental results showing how each of these techniques affected the performance of the TCAS II analysis. PSfrag replacements A w[:a]=y w[a]=x Figure 1: An example of statecharts The effects of combinations of the improvements are shown in addition to the individual effects. We focus on reachability problems, because most properties of TCAS II we were interested in fall into this class. However, in principle, all of the techniques should benefit general temporal-logic model checking as well. We conclude the paper with discussion on some related techniques. Background In this section, we give a brief overview of statecharts and RSML. We then turn our attention to symbolic model check- ing. Finally, we review how we applied symbolic model checking to the TCAS II requirements. 2.1 RSML and Statecharts The TCAS II requirements were written in RSML, a language based on statecharts. Like other variants of statecharts, RSML extends ordinary state-machine diagrams with state hierarchies; that is, every state can contain orthogonal or mutually exclusive child states. However, this feature does not concern us in this paper (the state hierarchy in the portion of TCAS II that we analyzed is shallow and does not incur special difficulties in model checking). Instead, we can think of the system consisting of a number of parallel state machines, communicating and executing in a synchronous way. Figure ?? above gives a simple example with two parallel state machines A and B. If A is in local state 0, we say that the system is in state A.0. State machines are synchronized using events. Arrows without sources indicate the start local states. Other arrows represent local transitions, which are labeled with the form u[c]=v where u is a trigger event, c is the guarding condition and v is an action event. The guarding condition is simply a predicate on local states of other states machines and/or inputs to the system; for exam- ple, a guarding condition may say that the system is in B.0 and an input altitude is at least 1 000 meters. (In RSML, the guarding condition is specified separate from the diagram in a tabular form called AND/OR table, but we use the simpler statecharts notation instead.) The guarding condition and the action are optional. The general idea is that, if event u occurs and the guarding condition c either is absent or evaluates to true, then the transition is enabled. Initially some external events along with some (possibly nu- meric) inputs from the environment arrive, marking the beginning of a step. The events may enable some transitions as described above. A maximal set of enabled transitions, collectively called a microstep, is taken-the system leaves the source local states, enters the target local states, and generates the action events (if any). All events are broadcast to the entire system, so these generated events may enable more transitions. The events disappear after one microstep, unless they are regenerated by other transitions. The step is finished when no transitions are enabled. The semantics of RSML assume the synchrony hypothesis: During a step, the values of the inputs do not change and no new external events may ar- rive; in other words, the system is assumed to be infinitely faster than the environment. In Figure ??, assume that w is the only external event, a is a Boolean input, and the system is currently in A.0 and B.0. When w arrives, if the input a is false, then the event y is generated. The step is finished since no new transitions are enabled. If instead a is true when w arrives, the transitions from A.0 to A.1 and from B.0 to B.1 are simultaneously taken and event x is generated, completing one microstep. Then a second microstep starts; notice that because of the synchrony hypothesis, the input a must be true as before and the external event w cannot occur. So only the transition from B.1 to B.2 is enabled and taken, generating event z and finishing the step. Subtle but important semantic differences exist among variants of statecharts. The semantics of STATEMATE [?], another major variant of statecharts, are close to those of RSML. STATEMATE does not enforce the synchrony hypothesis in the semantics, but provides it as an option in the simulator. RSML and STATEMATE also have a richer set of synchronization primitives and provide some sort of variable assignments; however, these features are not important for this paper. 2.2 Symbolic Model Checking We now switch gears to discuss ordinary finite-state transition systems (without state hierarchies, the synchrony hy- pothesis, etc.) and model checking. The goal of model checking is to determine whether a given state transition system satisfies a property given as a temporal logic formula, and if not, try to give a counterexample (a sequence of states that falsifies the formula). Example properties include that a (bad) state is never reached, and that a (good) state is always reached infinitely often. In "explicit" model check- ing, the answer is determined in a graph-theoretic manner by traversing and labeling the vertices in the state graph [?]. The method is impractical for many large systems because of the state explosion problem. Much more efficient for large state spaces is symbolic model checking, in which the model checker visits sets of states instead of individual states. For illustration, we focus on the reachability problem, the simplest and the most common kind of temporal property checked in practice. Let Q be the finite set of system states, Q the state transition relation, I ' Q the set of initial states, and E ' Q a set of error states. The reachability problem asks whether the system always stays away from the error states E, and if not, demands a counterexample, that is, a sequence of states q 0 , q We define to compute the pre-image (or the Start with Y iteratively compute reaching a fixed point. PSfrag replacements Y Backward Traversal fixed point Figure 2: An algorithm for computing Pre (E) 1. Let Q 0 be any nonempty subset of Pre (E) " I. Iteratively compute Q reaching E. PSfrag replacements Forward Traversal 2. Start with some qm 2 Qm " E and iteratively pick some to obtain a counterexample q 0 , PSfrag replacements Qm Figure 3: An algorithm for counterexample search weakest pre-condition) of a set of states under the transition relation R: Intuitively, it is the set of states that may reach some state in S in one transition. Then we can characterize the decision problem of reachability in a set-theoretic manner using fixed points: Determine whether I " Pre (E) is empty, where Pre (E) is the set of states that may eventually reach an error state. More specifically, it is the smallest state set Y that satisfies Its existence is guaranteed by the finiteness of Q and the monotonicity of Pre. Figure ?? shows an iterative algorithm for computing this fixed point. The set Y i is the states that may reach an error state in at most i transitions. Many other temporal properties can be similarly defined and computed using (possibly multiple or nested) fixed points [?]. If the intersection of Pre (E) and the initial states I is empty, then the set E is not reachable and we are done. Otherwise, we would like to find a counterexample. We first define Post post-images: In other words, Post(S) is the set of states reachable from S in one transition. Figure ?? shows a counterexample search algorithm. The set Q 0 can be any nonempty subset of the intersection, but it is convenient to choose Q 0 to be an arbitrary singleton set. The set Q i is the states that are reachable from Q 0 in at most i transitions. We obtain a counterexample by tracing backward from Qm " E. (We will improve this algorithm later.) The crucial factor for efficiency is the representation for state sets. Notice that the state space Q can be represented by a finite set of variables X , such that each state in Q corresponds to a valuation for the variables and no two states correspond to the same valuation. For finite state systems, we can assume without loss of generality that each variable is Boolean. A set of states S is then symbolically represented as a Boolean function S(X) such that a state is in the set if and only if it makes the function true. The transition relation of states can be similarly represented as a Boolean function is a copy of X and represents the next state. Intersection, union and complementation on sets or relations respectively becomes conjunction, disjunction and negation on Boolean functions. Now the problem of representation of state sets is reduced to that of Boolean functions. Empirically, the most efficient representation for Boolean functions is BDDs [?]. They are canonical, with efficient implementation for Boolean operations. For example, the time and space complexities of computing the conjunction or disjunction of two BDDs are at most the product of their sizes; usually the complexities observed in practice are still lower. Negation and equivalence checking can be done in constant time. BDDs are often succinct, but this relies critically on a chosen linear variable order of the variables in X . We can now represent a state set S and the transition relation R as BDDs and compute the pre-image and post-image of S as follows: The notation 9X refers to existentially quantifying out all the variables in X . In addition to Boolean operations and equivalence checking, operations like existential quantification and variable substitution can also be performed, so the algorithms in Figures ?? and ?? (and similar algorithms for many temporal logics such as CTL [?]) can be implemented using BDDs. Thanks to the succinctness of BDDs and the efficiency of their algorithms, some systems with over 10 120 states can be analyzed [?]. 2.3 Symbolic Model Checking for TCAS II We analyzed the TCAS II requirements using a symbolic model checker SMV (Version 2.4.4). SMV uses algorithms similar to those in Figures ?? and ??. A notable difference is that in Figure ??, instead of computingY uses the equivalent recurrence Y with the advantage that Y usually requires a much smaller BDD than Y i does, resulting in faster pre-image com- putation. (In fact, it is sufficient to compute the pre-image of any Z with Y apply to the computation of each Q i in Figure ??. Because SMV does not support hierarchical states and other RSML features directly, we had to translate the requirements into an ordinary finite-state transition system in the SMV language. The requirements consist of two main parts, Own-Aircraft and Other-Aircraft, which occupy about 30% and 70% of the document respectively. In our initial study, we translated Own-Aircraft quite faithfully to the SMV lan- guage, and abstracted Other-Aircraft as a mostly nondeterministic state machine. The details of the translation, including how the transitions, the state hierarchy and the synchrony hypothesis were handled, as well as the properties analyzed, were given in a previous paper [?]. Certain details about the system model are relevant to this paper: ffl An RSML microstep corresponds to a transition in the SMV program, and thus a step corresponds to a sequence of transitions. ffl We encode each RSML event as a Boolean variable, which is true if and only if the event has just occurred. ffl We assume each numeric input to be discrete and bounded, and encode each bit as a Boolean variable. ffl To maintain the synchrony hypothesis, we prevent the inputs from changing and the external events from arriving when some of the variables that encode events are true. ffl We analyze one instance of TCAS II only, so the asynchrony among multiple instances of the system is not an issue. A major source of complexity of the analysis was the tran- sitions' guarding conditions, some of which occupy many pages of description. They contain predicates of local states and of the input variables, and may involve complex arith- metic. While many other researchers conservatively encode each arithmetic predicate as an independent Boolean variable [?,?, ?], we encode each input bit as a Boolean variable, resulting in more accurate analysis at the expense of more Boolean variables. In addition, a guarding condition can refer to any part of the system, so the interdependencies between the BDD variables are high. These all imply relatively large BDDs for guarding conditions. On the plus side, the control flow of Own-Aircraft is simple, and concurrency among the state machines in Own-Aircraft is minimal. As we will see, some of the techniques presented later attempt to exploit these simple synchronization patterns. Short-Circuiting It is easy to see that in Figure ??, we do not need to compute a fixed point when the error states are reachable-we can stop once the intersection of some Y i and I is not empty, because all we need is an element in the intersection. This short-circuiting technique may substantially reduce the time and space used when a short counterexample exists. More generally, short-circuiting can be applied to the outermost temporal operator in temporal-logic model checking (however, the reduction obtained is probably less than Start with some q iteratively pick some to obtain a counterexample q 0 , q 1 , PSfrag replacements Y I Figure 4: A simplified algorithm for counterexample search in reachability analysis, because only one of the many fixed points can be stopped prematurely.) 4 Forward vs. Backward Traversals Fixed-point computation or counterexample search can be done either forward or backward. In this section we elaborate on their performance difference in our analysis. In short, backward traversals generate smaller BDDs and are a big win for our system. They can be further improved by incorporating certain invariants to prune the searches. 4.1 Improved Counterexample Search During the analysis of TCAS II, we found that when a property was disproved in a few minutes, finding a counterexample might take hours. A coauthor of a previous paper subsequently simplified the counterexample search algorithm, resulting in substantial speedup [?]. This is the only technique described here that was used in that study. The forward traversal in the first part of Figure ?? is the bot- tleneck. For our system, the sequence of post-images requires large BDDs. However, we can eliminate this step if we remember every Y i computed in Figure ?? (our actual implementation stores the difference Y Our modification, illustrated in Figure ??, is by no means innovative and should be considered natural. 1 A disadvantage of the algorithm is the use of additional memory to store the state sets, which is wasted in case the error states are not reachable. Nevertheless, the dramatic speedup made possible far outweighs the modest additional memory requirements An important question remains: Why is the backward traversal in Figure ?? much more efficient than the forward traversal in Figure ??? The inefficiency of forward traversals is also witnessed by SMV's inability to compute the set of reachable states of the system. Finding the reachable state set by searching forward from the initial states is a common technique in hardware verification; the set can be used to help analyze other temporal properties and synthesize the Indeed, if we search forward to find the reachable state set, SMV can optionally use a similar counterexample search algorithm, but it is not used with the default backward traversal PSfrag replacements A x[a]=y x[b]=y x[b]=y x[a]=y Figure 5: A state machine with local invariants circuit. A backward traversal often takes fewer iterations to reach a fixed point than a forward traversal, because the set of error states is usually more general than the set of initial states. However, the problem here is not the number of iterations, but rather, the size of the BDDs generated. In general, we observe that in backward traversals, the BDDs usually have between hundreds to at most tens of thousands of BDD nodes, while in forward traversals, they can be two or more orders of magnitude larger. Nevertheless, the verification of many hardware systems tends to benefit, rather than suffer, from forward traversals. For example, Iwashita et al. report significant speedup in CTL model checking for their hardware benchmarks when forward instead of backward traversals are used [?]. Partly inspired by Hu and Dill [?], we believe that the inefficiency is mainly due to the complex invariants of TCAS II, which are maintained by forward but not backward traver- sals. As an example, consider the state machine in Figure ??. If event y is only generated in A, then an invariant of the system is that, whenever event y has just occurred, the machine is in A.0 if and only if condition a is true. If the BDD for a is large, so will the BDD for the invariant. Even if they are small, there are likely to be many such implicit invariants in the system, and their conjunction may have a large BDD representation. In addition, invariants may globally relate different state machines, also likely to result in large BDDs. Forward traversals maintain all such invariants, so intuitively the BDDs for forward traversals tend to blow up in size. In low-level hardware verification, the BDDs often remain small, because each invariant is usually localized and involves only a small number of state variables. This is however not the case in TCAS II. For backward traversals, the situation is quite different. For example, there are no counterparts of the invariant mentioned above when backward traversals are used, because the truth value of a does not imply the state of the system before the microstep. Certainly, some different (backward) invariants are maintained in backward traversals, but they tend to depend on the states from which the search starts, and their BDDs tend to be smaller for our system. 4.2 Improved Backward Traversals Using Invariants Interestingly, the main disadvantage of backward traversals is also that (forward) invariants are not maintained. Some in- variants, particularly those with small BDDs, can help simplify the BDDs of state sets, and can speed up backward traversals if they are incorporated into the search. In the context of statecharts, many systems have simple synchronization patterns, which are lost in backward traversals. A particular invariant that we find useful to rectify this prob- PSfrag replacements A u[a]=v v[b]=w w[c]=x Figure system with a linear structure lem is the mutual exclusion of certain events. We illustrate this idea with an example. Consider the system in Figure ??. Assuming u is the only external event, there is no concurrency in the system-at most one local transition can be enabled at any time. Forward traversals do not explore concurrent executions of the state machines. However, in backward traversals, the analysis may be fooled to consider many concurrent executions, which are not reachable. Suppose we want to check whether the system can be in B.1 and C.1 simultaneously. Traversing back- ward, we find that in the previous microstep, the system may be in (B.0;C.1), (B.1;C.0), or (B.0;C.0). The last case, however, is not possible, because events v and w cannot occur at the same time. (Notice that this is true only if we assume the synchrony hypothesis.) Tracing more iterations, we can see that the search considers not only concurrent executions but also many unreachable interleaving ones. The BDDs thus may blow up if the guarding conditions are complex Fortunately, we can greatly simplify the search by observing that all the events are mutually exclusive. This invariant can be incorporated into the traversals by either intersecting it with the state sets or using it as a care-set to simplify the BDDs [?]. To find out such a set of mutually exclusive events, we may perform a conservative static analysis on the causality of the events. Alternatively, the designer may know which events are mutually exclusive, because the synchronization patterns should have been designed under careful consideration. To confirm the mutual exclusion, we may verify, using model checking or other static analysis techniques, that the states with are not reachable, where S is the set of state variables encoding the events under consideration. In the case of TCAS II, a large part of our model behaves similarly to the machine in Figure ??, and the set of mutually exclusive events was evident. Partitioned Transition Relation Apart from the BDD size for state sets, another bottleneck of model checking is the BDD size for the transition relation, which can be reduced by conjunctive or disjunctive partitioning [?]. The former can be used naturally for TCAS II, and we have modified SMV to partition the transition relation more effectively. We also apply disjunctive partition- ing, which is normally used only for asynchronous systems. Combining the two techniques, we obtain DNF partitioning. As we will see, the issues in this section are not only the BDD size for the transition relation, but also the size of the intermediate BDDs generated for each image computation. 5.1 Background In this subsection, we review the idea of conjunctive and disjunctive partitioning, described in Burch et al. The transition relation R is sometimes given as a disjunction and the BDD for R can be huge even though each disjunct has a small BDD. So instead of computing a monolithic BDD for R, we can keep the disjuncts separate. The image computations can be easily modified by distributing the existential quantification over the disjunc- tion. For pre-image computation, we thus have So we can compute the pre-image without ever building the BDD for R. Post-image computation is symmetric. If, however, R is given as a conjunction C 1 we can still keep the conjuncts separate as above, but image computations become more complicated. The problem is that existential quantification does not distribute over con- junctions, so it appears that we have to compute the BDD for R anyway before we can quantify out the variables. A trick to avoid this is early quantification. Define X 0 to be the disjoint subsets of X 0 such that their union is X 0 and the conjunctC i does not depend on any variable in X p for any p ! i. Consider again pre-image computation. We compute The intuition is to quantify out variables as early as possi- ble, and hope that each intermediate c i for remains small. The effectiveness of the procedure depends critically on the choice and ordering of the conjuncts C 1 , C 5.2 Determining Conjunctive Partition We could not construct the monolithic BDD for the transition relation R for our model of TCAS II in hours of CPU time, but R is naturally specified as a conjunction, so we can use conjunctive partitioning. Although SMV supports this fea- ture, it determines the partition in a simplistic way: An SMV program consists of a list of parallel assignments, whose conjunction forms the transition relation. SMV constructs the BDDs for all assignments, and incrementally builds their conjunction in the (reverse) order they appear in the pro- gram. In this process, whenever the BDD size exceeds a user-specified threshold, it creates a new conjunct in the par- tition. So the partition is solely determined by the syntax, and no heuristic or semantic information is used. To better determine the partition, we changed SMV to allow the user to specify the partition manually. We also implemented in SMV a variant of the heuristics by Geist and Beer [?] and by Ranjan et al. [?] to automatically determine the partition. The central idea behind the heuristics is to select conjuncts that allow early quantification of more variables while introducing fewer variables that cannot be quantified out. Our implementation of the heuristics worked quite well; the partitions generated compared favorably with, and sometimes outperformed, the manual partitions that we tried. 5.3 Disjunctive Partitioning for Statecharts Disjunctive partitioning is superior to conjunctive partitioning in the sense that ordering the disjuncts is less critical, and that each intermediate BDD is a function of X (instead of thus tends to be smaller. (Another advantage that we have not exploited is the possibility of parallelizing the image computation by constructing the intermediate BDDs concurrently.) Unfortunately, when the transition relation R is a conjunc- tion, in general there are no simple methods for converting it to a small set of small disjuncts. If we define a cover disjunction is the tautology, then we can indeed disjunctively partition R by distributing R over the cover: But for most choices of covers, each D i is still large. For TCAS II and many other statecharts, however, we can again exploit the mutual exclusion of certain events, say u a PSfrag replacements A Figure 7: Event x triggers two state machines. a In other words, a i corresponds to the states in which only u i has just occurred, a j , none of the events have, and a j+1 , at least two of the events have. They clearly form a cover. We made two observations. First, we can drop a j+1 , which is a contradiction because of the mutual exclusion assumption. Second, most of the parallel assignments in our SMV program are guarded by conditions on the events; for example, an assignment that models a state transition requires the occurrence of the trigger event. If the event is, say u then the BDD for the assignment is applicable only to the disjunct D i , and all the other disjuncts of the transition relation are unaffected. So each disjunct may remain small. Notice that to apply this technique, we have to find a set of provably mutually exclusive events, which can be done as described in Section ??. 5.4 DNF Partitioning and Serialization A disadvantage of partitioning R based on events is that the sizes of the disjuncts are often skewed. In particular, if a single event may trigger a number of complex transitions, its corresponding disjunct could be large. Figure ?? shows an example in which an event x triggers two state machines. If all the guarding conditions are complex, the BDD for the disjunct corresponding to x may be large. One solution to this problem is to apply conjunctive partitioning to large disjuncts, resulting in what we call DNF partitioning. It uses both BDD size (as in conjunctive par- titioning) and structural information (as in disjunctive parti- tioning) to partition the transition relation, and may perform better than relying on either alone. Alternatively, we may serialize the complicated microstep into cascading microsteps to reduce the BDD size. Figure ?? illustrates this idea. We have "inserted" a new event u after x. Note that the resulting machine has more microsteps in a step. So although this method is effective in reducing the BDD size, it often increases the number of iterations to reach a fixed point. Also, the transformation may not preserve the behavior of the system and the property analyzed. A sufficient condition is that the guarding conditions in the PSfrag replacements A Figure 8: The serialized machine machine B do not refer to machine A's local states, x is mutually exclusive with all other events, and we are checking a reachability property that does not explicitly mention any of the state machines, transitions or events involved in the 6 Abstraction In this section, we give a simple algorithm to remove part of the system from the model that is guaranteed not to interfere with the property being checked. For example, a state machine may have a number of outputs (which may be local states or events). If we are verifying only one of them, the logic that produces other outputs may be abstracted away, provided these outputs are not fed back to the system. The abstraction obtained is exact with respect to the property, in the sense that the particular property holds in the abstracted model if and only if it holds in the original model. 6.1 Dependency Analysis We determine the abstraction by a simple dependency analysis on the statecharts description. Initially, only the local states, events, transitions, or inputs that are explicitly mentioned in the property are considered relevant to the analysis. Then the following rules are applied recursively: ffl If an event is relevant, then so are all the transitions that may generate the event. ffl If a transition is relevant, then so are its trigger event, its source local state, and everything that appears in its guarding condition. ffl If a local state is relevant, then so are all the transitions out of or into it, and so is its parent state in the state hierarchy. These rules are repeated until a fixed point is reached. Es- sentially, this is a search in the dependency graph, and the time complexity is linear in the size of the graph. It should be evident that everything not determined relevant by these rules can be removed without affecting the analysis result. 2 The same criterion can be applied to arbitrary CTL for- mulas, provided we do not use the the next-time operator X, which can count the number of microsteps. In other words, under the assumptions, the transformation preserves equivalence under stuttering bisimulation [?]. PSfrag replacements w[:b]=x w[b]=y w[a]=y w[:a]=x x[:d]=y x[d]=y x[c]=y x[:c]=y Figure 9: False dependency: Event y does not depends on any guarding condition. 6.2 False Dependency Similar dependency analyses could also be performed by model checkers (such as VIS [?]) on the Boolean model of the statecharts machine. However, a straightforward implementation would not be effective. The reason is that in the model, an input would appear to depend on every event because of the way we encoded the synchrony hypothesis (Sec- tion ??). On the other hand, carrying out dependency analysis on the high-level statecharts description does not fall prey to such false dependencies. Other forms of false dependencies are possible, however. Suppose we are given the system in Figure ?? from the previous section. From the syntax, the event u appears to depend on both conditions a and a 0 , but in fact it does not, because regardless of the truth values a and a 0 , event u will be generated as a result of event x. To detect such false dependencies, one can check whether the disjunction of the guarding conditions of the transitions out of a local state with the same trigger and action events is a tautology. This can sometimes be checked efficiently using BDDs [?]. However, the syntax of RSML and STATEMATE allows easy detection of most false dependencies of this kind. Notice that the self-loops in Figure ?? are solely for synchronization-they make sure that u is generated regardless whether there has been a local state change. To improve the visual presentation, RSML and STATEMATE allow specifying the generation of such events separate from the state diagram using identity transitions and static reactions respectively. (Actually, their semantics are slightly different from self-loops, but the distinctions are not important here.) Some false dependencies are harder to detect automatically. For example, maybe the guarding conditions involved do not form a tautology, but in all reachable states, one of the guarding conditions holds whenever the trigger event occurs. As another example, in Figure ??, the event y does not depends on any of the guarding conditions, because it is always generated one or two microsteps after w. 3 In practice, the synchronization of the system should be evident to the designer, who may specify the suspected false dependencies in temporal logic formulas, which can be verified using model check- ing. If the results indeed show no real dependencies, this in- 3 However, if the next-time operator X is used, then y may be considered conservatively to be dependent on a and b. formation can be used in the dependency analysis to obtain a smaller abstracted model of the system. In our TCAS II anal- ysis, the synchronization of Own-Aircraft is simple enough that false dependencies can be easily detected. However, this method may be used for analyzing the rest of TCAS II or other systems. 7 Experimental Results The table above summarizes the results of applying the techniques mentioned to our model of TCAS II. It shows the resources (time in seconds and number of BDD nodes used in thousands) for building the BDDs for the transition relation R as well as the resources for evaluating six properties. Note that the latter excludes the time spent on building the transition relation or the resources for finding the counterex- amples. The counterexample search took about one to two seconds per state in the counterexample and was never a bottleneck thanks to the algorithm in Figure ??. That algorithm was used in all the checks, because without it, none of the counterexamples could be found in less than one hour. The table also shows the number of iterations needed to reach fixed points and the length of the (shortest) counterexamples. We performed the experiments on a Sun SPARCstation 10 with 128MB of main memory. Most successful checks used less than 30MB of main memory. Properties P1 through P4 refers to the properties Increase- Descent Inhibition, Function Consistency, Transition Con- sistency, and Output Agreement explained in a previous paper [?]. Property P5 refers to an assertion in Britt [?, p. 49] that Own-Aircraft should never be in two local states Corrective-Climb . Yes and Corrective-Descend. Yes simultaneously (comments in our version of the TCAS II re- quirements, however, explicitly say that the two local states are not mutually exclusive). Property P6 is somewhat con- trived: It is simply the conjunction of P3 and P4. Since searching simultaneously from two unrelated sets of states tends to blow up the BDDs, checking this property provides an easy way of scaling up the BDD size. It also mimics checking properties involving a large part of the system. All six properties are reachability, and are violated by the model. An entry with - in the table indicates timeout after one hour. We emphasize that the purpose of the data is to investigate the general effects of the techniques on our model of TCAS II. They are not for picking a clear winner among the techniques, since the BDD algorithms are very sensitive to the various parameters chosen and to the model analyzed. Note also that the results shown here should not be compared directly with out earlier results [?], because the models, the parameters, and the model checking algorithms used were different. Full Model The first part of the table shows the results for the full model with 227 Boolean state variables. Row 1 gives the results for the base analysis: Two properties could not be completed using the conjunctive partitioning as implemented in SMV. (Actually, we implemented a small improvement that was used in all results including the base analysis. As explained in Section ??, an image computation step involves a conjunction and an existential quantification. The two operations can be carried out simultaneously to avoid building the usually large conjunction explicitly [?]. SMV performs this optimization except when conjunctive partitioning is used. Building P1 P2 P3 P4 P5 P6 BDDs for R Full Model (227 variables) No. of fixpoint iterations 24 29 29 38 26 26 Counterexample length 15 15 11 24 17 11 Optimizations time nodes time nodes time nodes time nodes time nodes time nodes time nodes Mistranslated Model Optimizations time nodes time nodes time nodes time nodes time nodes time nodes time nodes Serialized Model (231 variables) No. of fixpoint iterations 36 41 45 54 38 38 Counterexample length 23 23 19 36 25 19 Optimizations time nodes time nodes time nodes time nodes time nodes time nodes time nodes Abstracted Models No. of variables 142 142 150 142 150 171 Optimizations time nodes time nodes time nodes time nodes time nodes time nodes time nodes SC: short-circuiting MX: mutual exclusion of events CP: improved conjunctive partitioning DP: disjunctive partitioning No. of fixpoint iterations and counterexample lengths are identical to those of the full model. Table 1: Resources used in the analysis We simply changed SMV to do this optimization with conjunctive partitioning.) As expected, short-circuiting (SC) gave savings, because the number of iterations needed became the length of the shortest counterexample. Incorporating the mutual exclusion of certain events into backward traversals (MX) generally gave an order of magnitude time and space reduction. In addition, we could now easily disprove P5 and P6 (in particular, Britt's claim mentioned above is provably not true in our version of the requirements). For improved conjunctive partitioning (CP), as mentioned in Section ??, we used a heuristic to produce a partition, which was effective in reducing time and space used. Disjunctive partitioning (DP), which must be combined with mutual exclusion of events, appeared to be inefficient (Row 5). The reason is that one of the disjuncts of the transition relation was large, with over 10 5 BDD nodes, at least an order of magnitude larger than other disjuncts; this is reflected in the table by the large number of BDD nodes needed to construct the transition relation. We conjunctively partitioned large disjuncts, leading to DNF partitioning (in- dicated on Row 7 by marking both CP and DP). It performs marginally better than pure conjunctive partitioning in terms of time, but the space requirements were consistently lower. Combining it with the other two optimizations, we observed orders of magnitude improvements in time and space (Row 8). Mistranslated Model To further illustrate the differences between conjunctive and DNF partitioning, we looked at a version of the model that contains a translation error from the RSML machine to the SMV program. It was a real bug we made early in the study, although we soon discovered it by inspection. The mistake was omitting some self-loops similar to those in Figure ??. BDDs for faulty systems are often larger than those for the corrected versions, because bugs tend to make the system behavior less "regular". Therefore, investigating the performance of BDD algorithms on faulty designs is meaningful. Interestingly, the particular partition generated by the the heuristic performed poorly for this model (Row 10). DNF partitioning continued to give significant time and space savings (Row 11). Serialized Model As mentioned above, the disjunctive partition contains a disproportionally large BDD. We serialized a microstep in the full model to break the large disjunct into four BDDs of sizes about a hundred times smaller. As expected, disjunctive partitioning now performed better (Rows 5 vs. 14). However, since the number of microsteps in a step increased, all partitioning techniques suffered from the larger number of iterations needed to reach fixed points. They all ended up performing about the same, with disjunctive and DNF partitioning having the slight edge. The data suggest that serializing the microstep in order to use disjunctive partitioning is not advantageous for this model. In general, we find the effects of serializing and its dual, collapsing microsteps, difficult to predict. It represents a trade-off between the complexity of image computations and the number of search iterations. Abstracted Models The last part of the table shows the performance of analyzing the abstracted models obtained by the dependency analysis in Section ??. The number of variables abstracted away is quite large. Recall that in our full model, we omitted most of the details in Other-Aircraft. Many of the outputs of Own-Aircraft that are inputs to Other- Aircraft thus become irrelevant, unless we explicitly mention them in the property. This explains the relatively large reduction obtained. 8 Discussion and Related Work We first summarize some differences between symbolic model checking for hardware circuits and for TCAS II. A major focus of hardware verification is on concurrent systems with complex control paths and often subtle concurrency bugs, but their data paths are relatively simple. Forward traversals usually perform much better, because the BDDs tend to be small in their reachable state spaces. In contrast, the major complexity of the TCAS II requirements lies not in the concurrency among components, but in the intricate influence of data values on the control paths. The BDD for the transition relation tends to be huge and forward traversals inefficient. Backward traversals usually perform better by focusing on the property analyzed, and can be further improved by exploiting the simple synchronization patterns Our method of pruning backward traversals using invariants is similar in spirit to the work on hardware verification by Cabodi et al., who propose doing an approximate forward traversal to compute a superset of the reachable states, which is then used to prune backward traversals [?]. Their method is more automatic, while the invariants we suggest take advantage of the simple synchronization of the system. They also independently propose disjunctive partitioning for synchronous circuits [?]. Their method requires the designer to come up with a partition manually, and we again exploit mutually exclusive events. In work also independent of ours, Heimdahl and Whalen [?] use a dependency analysis technique similar to the one described Section ??, but their motivation is to facilitate manual review of the TCAS II requirements, rather than automatic verification. As noted before, we gained relatively large reduction because Other-Aircraft was not fully mod- eled, and we suspect that in a complete system, the reduction obtained by this exact analysis could be limited. However, more reduction can be obtained if we forsake exactness. For example, localization reduction [?] is one such technique, which aggressively generates an abstracted model that may not satisfy the property while the full model does. If the model checker finds in the abstracted model a counterexample that does not exist in the full model, it will automatically refine the abstraction and iterate the process until either a correct counterexample is found or the property is verified. It would be interesting to see how well the techniques in this paper scale with the system complexity. The natural way is to try applying them to the rest of TCAS II. Unfortunately, that part contains arithmetic operations, such as multiplica- tion, that provably cannot be represented by small BDDs [?]. In a recent paper, we suggest coupling a decision procedure for nonlinear arithmetic constraints with BDD-based model checking to attack the problem [?]. More research is needed to see whether this technique scales to large systems. Acknowledgments We thank Steve Burns, who observed the inefficiency of the algorithm in Figure ?? and implemented the one in Figure ?? in SMV. --R Efficient implementation of a BDD package. Case study: Applying formal methods to the Traffic Alert and Collision Avoidance System (TCAS) II. Characterizing finite Kripke structures in propositional temporal logic. Efficient state space pruning in symbolic backward traversal. Combining constraint solving and symbolic model checking for a class of systems with non-linear con- straints Automatic verification of finite-state concurrent systems using temporal logic specifications Verification of the Futurebus Verification of synchronous sequential machines based on symbolic execution. Model checking graphical user interfaces using abstractions. Statecharts: A visual formalism for complex systems. The STATEMATE semantics of statecharts. Completeness and consistency analysis of state-based require- ments Reducing BDD size by exploiting functional dependencies. New techniques for efficient verification with implicitly conjoined BDDs. model checking based on forward state traversal. Requirements specification for process-control systems Symbolic Model Checking. Formal verification of the Gigamax cache consistency protocol. Automatic verification of a hydroelectric power plant. Dynamic variable ordering for ordered binary decision diagrams. Feasibility of model checking software requirements: A case study. --TR Automatic verification of finite-state concurrent systems using temporal logic specifications Graph-based algorithms for Boolean function manipulation Statecharts: A visual formalism for complex systems Characterizing finite Kripke structures in propositional temporal logic Verification of synchronous sequential machines based on symbolic execution On the Complexity of VLSI Implementations and Graph Representations of Boolean Functions with Application to Integer Multiplication Reducing BDD size by exploiting functional dependencies Requirements Specification for Process-Control Systems Computer-aided verification of coordinating processes Completeness and Consistency in Hierarchical State-Based Requirements The STATEMATE semantics of statecharts Model checking large software specifications model checking based on forward state traversal Disjunctive partitioning and partial iterative squaring Model checking graphical user interfaces using abstractions Reduction and slicing of hierarchical state machines Symbolic Model Checking Efficient State Space Pruning in Symbolic Backward Traversal Automatic Verification of a Hydroelectric Power Plant Combining Constraint Solving and Symbolic Model Checking for a Class of a Systems with Non-linear Constraints Efficient Model Checking by Automated Ordering of Transition Relation Partitions VIS --CTR Gleb Naumovich, A conservative algorithm for computing the flow of permissions in Java programs, ACM SIGSOFT Software Engineering Notes, v.27 n.4, July 2002 Jamieson M. Cobleigh , Lori A. Clarke , Leon J. Osterweil, The right algorithm at the right time: comparing data flow analysis algorithms for finite state verification, Proceedings of the 23rd International Conference on Software Engineering, p.37-46, May 12-19, 2001, Toronto, Ontario, Canada Jin Yang , Andreas Tiemeyer, Lazy symbolic model checking, Proceedings of the 37th conference on Design automation, p.35-38, June 05-09, 2000, Los Angeles, California, United States Ji Y. Lee , Hye J. Kim , Kyo C. Kang, A real world object modeling method for creating simulation environment of real-time systems, ACM SIGPLAN Notices, v.35 n.10, p.93-104, Oct. 2000 C. Michael Overstreet, Improving the model development process: model testing: is it only a special case of software testing?, Proceedings of the 34th conference on Winter simulation: exploring new frontiers, December 08-11, 2002, San Diego, California Ofer Strichman, Accelerating Bounded Model Checking of Safety Properties, Formal Methods in System Design, v.24 n.1, p.5-24, January 2004 Jonathan Whittle, Formal approaches to systems analysis using UML: an overview, Advanced topics in database research vol. 1, Chan , Richard J. Anderson , Paul Beame , David H. Jones , David Notkin , William E. Warner, Decoupling synchronization from local control for efficient symbolic model checking of statecharts, Proceedings of the 21st international conference on Software engineering, p.142-151, May 16-22, 1999, Los Angeles, California, United States Chan , Richard J. Anderson , Paul Beame , David Notkin , David H. Jones , William E. Warner, Optimizing Symbolic Model Checking for Statecharts, IEEE Transactions on Software Engineering, v.27 n.2, p.170-190, February 2001 Guoqing , Shu Fengdi , Wang Min , Chen Weiqing, Requirements specifications checking of embedded real-time software, Journal of Computer Science and Technology, v.17 n.1, p.56-63, January 2002 Shoham Ben-David , Cindy Eisner , Daniel Geist , Yaron Wolfsthal, Model Checking at IBM, Formal Methods in System Design, v.22 n.2, p.101-108, March Chan , Richard J. Anderson , Paul Beame , Steve Burns , Francesmary Modugno , David Notkin , Jon D. Reese, Model Checking Large Software Specifications, IEEE Transactions on Software Engineering, v.24 n.7, p.498-520, July 1998

formal verification;reachability analysis;abstraction;system requirements specification;TCAS II;partitioned transition relation;binary decision diagrams;statecharts;symbolic model checking;RSML

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On the limit of control flow analysis for regression test selection.

Automated analyses for regression test selection (RTS) attempt to determine if a modified program, when run on a test t, will have the same behavior as an old version of the program run on t, but without running the new program on t. RTS analyses must confront a price/performance tradeoff: a more precise analysis might be able to eliminate more tests, but could take much longer to run.We focus on the application of control flow analysis and control flow coverage, relatively inexpensive analyses, to the RTS problem, considering how the precision of RTS algorithms can be affected by the type of coverage information collected. We define a strong optimality condition (edge-optimality) for RTS algorithms based on edge coverage that precisely captures when such an algorithm will report that re-testing is needed, when, in actuality, it is not. We reformulate Rothermel and Harrold's RTS algorithm and present three new algorithms that improve on it, culminating in an edge-optimal algorithm. Finally, we consider how path coverage can be used to improve the precision of RTS algorithms.

Introduction The goal of regression test selection (RTS) analysis is to answer the following question as inexpensively as possible: Given test input t and programs old and new, does new(t) have the same observable behavior as old(t)? To appear, 1998 ACM/SIGSOFT International Symposium on Software Testing and Analysis Of course, it is desired to answer this question without running program new on test t. RTS analysis uses static analysis of programs old and new in combination with dynamic information (such as coverage collected about the execution old(t) in order to make this determination. An RTS algorithm either selects a test for re-testing or eliminates the test. Static analyses for RTS come in many varieties: some examine the syntactic structure of a program [6]; others use control flow or control dependence information [11, 12]; more ambitious analyses examine the def-use chains or flow dependences of a program [9, 5]. Typically, each of these analyses is more precise than the previous, but at a greater cost. A safe (conservative) RTS analysis never eliminates a test t if new(t) has different behavior than old(t). A safe algorithm may select some test when it could have been eliminated. We focus on the application of control flow analysis to safe regression testing (from now on we will use CRTS to refer to "Control- flow-based RTS"). Previous work has improved the precision of CRTS analysis but left open the question of what the limit of such analyses are. CRTS can be improved in two ways: by increasing the precision of the analysis applied to the control flow graph representations of programs old and new, or by increasing the precision of the dynamic information recorded about the execution old(t). We will address both issues and the interactions between them. Our results are threefold: ffl (Section 3) Building on recent work in CRTS by Rothermel and Harrold [12], we show a strong relationship between CRTS, deterministic finite state automata, and the intersection of regular languages. We define the intersection graph of two control flow graphs, which precisely captures the goal of CRTS and forms the basis for a family of CRTS algorithms, parameterized by what dynamic information is collected about old(t). ffl (Section 4) We consider the power of CRTS when the dynamic information recorded about old(t) is edge coverage (i.e., whether or not each edge of old's control flow graph was executed). We define a strong optimality condition (edge- optimality) for CRTS algorithms based on edge coverage. We then reformulate Rothermel and Harrold's CRTS algorithm in terms of the intersection graph and present three new algorithms that improve on it, culminating in an edge-optimal algorithm. The first algorithm eliminates a test whenever the Rothermel/Harrold algorithm does, and safely eliminates more tests in general, at the same cost. The next two algorithms are even more precise, but at greater computational cost. ffl (Section 5) By recording path coverage information about rather than edge coverage, we can improve upon edge- optimal algorithms. However, if path profiling is limited to tracking paths of a bounded length (which is motivated by concerns of efficiency), then an adversary will always be able to choose a program new that will cause any CRTS algorithm based on path coverage to fail. Section 6 reviews related work and Section 7 summarizes the paper. Background We assume a standard imperative language such as C, C++, or Java in which the control flow graph of a procedure P is completely determined at compile time. In P's control flow graph G, each vertex represents a basic block of instructions and each edge represents a control transition between blocks. The translation of an abstract syntax tree representation of a procedure into its control flow graph representation is well known [1]. Since G is an executable representation of P, we will talk about executing both P and G on a test t. We now define some graph terminology that will be useful in the sequel. s; x) be a directed control flow graph with vertices a unique entry vertex s, from which all vertices are reachable, and exit vertex x, which has no successors and is reachable from all vertices. A vertex v is labelled with BB(v), the code of the basic block it contains. Two different vertices may have identical labels. It is often convenient to refer to a vertex by its label and we will often do so, distinguishing vertices with identical labels when necessary. An edge source vertex v to target vertex w via a directed edge labelled l. The outgoing edges of each vertex are uniquely labeled. Labels are values (typically, true or false for boolean predicates) that determine where control will transfer next after execution of BB(v). 2 If a vertex v has only one outgoing edge, its label is e, which is not shown in the figures. Since the outgoing edges of a vertex are uniquely labelled, an edge may also be represented by a pair (v; l), which we call a control transition or transition, for short. The vertex succ(v; l) denotes the vertex that is the l-successor of vertex v (if A path in G is a sequence of edges where the target vertex of e i is the source vertex of e i+1 for 1 - path may be represented equivalently by an alternating sequence of vertices and edge labels is the source vertex of edge e i (for 1 - i - n), v n+1 is the target vertex of e n , and l i is the label of edge e Given a path p of n edges (and n [i] be the i th vertex [i] be the i th edge label (1 - i - n). 2 The number of outgoing edges of vertex v and the labels on these edges are uniquely defined by BB(v). Thus, different vertices that have identical basic blocks will have the same number of out-going edges with identical labels. Paths beginning at a designated vertex (for our purposes, the entry vertex s) are equivalently represented by a sequence of basic blocks and labels (rather than a sequence of edges or vertices and labels): A complete path is a path from s to x. Figure 1 shows two programs P and P 0 and their corresponding control flow graphs G and G 0 . For both G and G 0 , the entry vertex is and the exit vertex is . The label of a vertex v denotes its basic block BB(v). Graph G has one occurrence of basic block C while graph G 0 has two occurrences of C. The graph G 00 is the intersection graph of G and G 0 and is discussed next. 3 CRTS and the Intersection Graph In control flow analysis, the graphical structure of a program is an- alyzed, but the semantics of the statements in a program are not, except to say whether or not two statements are textually identical. This implies that CRTS algorithms must assume that every complete path through a graph is potentially executable (even though there may be unexecutable paths). Unexecutable paths cannot affect the safety of CRTS algorithms, but may decrease their preci- sion, just as they do in compiler optimization. CRTS algorithms must be able to determine if two basic blocks are semantically equivalent. Of course, this is undecidable in gen- eral. Following Rothermel and Harrold, we use textual equivalence of the code as a conservative approximation to semantic equiva- lence, which is captured in the definition of equivalent vertices: Two vertices v and w (from potentially different graphs) are equivalent if the code of BB(v) is lexicographically identical to BB(w). Let Equiv(v;w) be true iff v is equivalent to w. 3 Once we have equivalent vertices, we can extend equivalence to paths, as follows: paths p and q are identical if p and q are the same length, Equiv(p v [i]; q v [i]) is true for all i, and p l all i. That is, p and q are identical words (over an alphabet of basic blocks and labels). The following simple definition (a restatement of that found in [11]) precisely captures the power of CRTS: If graph G run on input t (denoted by G(t)) traverses complete path p and graph G 0 contains complete path p 0 identical to p, then G 0 (t) will traverse path p 0 and have the same observable behavior as G(t). The above definition translates trivially into the most precise and computationally expensive CRTS algorithm: record the complete execution path of G(t) (via code instrumentation that traces the path [4]) and compare it to the control flow graph of G 0 to determine if the path exists there. We will see later in Section 5 that any algorithm that does not record the complete execution path of G(t) can be forced, by an adversary choosing an appropriate graph G 0 , to select a test that could have been eliminated. We observe that a control flow graph G may be viewed as a deterministic finite automaton (DFA) with start state s and final state x that accepts the language L(G), the set of all complete paths in G. More precisely, a control flow graph G has a straightforward interpretation as a DFA in which each vertex v in V corresponds to two 3 The exit vertex x can only be equivalent to other exit vertices (i.e., vertices with no successors). U,U f f f G" A,A reject accept if A { { U } if C { Y } else { Z } } else { if C { W } U f f Y Z G' A if A { { U } if C { W } U f f f G A Figure 1: Example programs P and P 0 , their corresponding control flow graphs G and G 0 , and the intersection graph G 00 of G and G 0 . states, v 1 and v 2 . These states are connected by a state transition labelled by BB(v). Edges in E are also interpreted as state transitions: an edge v ! l w is interpreted as a state transition . The alphabet of the DFA is the union of all basic blocks and all edge labels, s 1 is the start state, and x 2 is the final state. The recognizes precisely the complete paths of G. Rather than represent the control flow graph in this more verbose fashion, we choose to present it in its traditional form but keep its DFA interpretation in mind. Given this insight, the CRTS question reduces to: Is a complete path p from L(G) also in L(G 0 )? ), the paths for which re-testing is not needed, and let D(G;G 0 ), the paths for which re-testing is needed. A CRTS algorithm is optimal if, given any path p in I(G;G 0 ), the algorithm reports that p is in I(G;G 0 ). A CRTS algorithm is safe if, given any path p in D(G;G 0 ), the algorithm reports that p is in To help reason about I(G;G 0 ) and D(G;G 0 ), we define a new graph the intersection graph of which also has a straightforward interpretation as a DFA. 4 This graph can be efficiently constructed from G and G 0 . The vertex set V 00 of G 00 is simply the cross product of V and V 0 , with two additional vertices: We use the following relation to help define is essentially an optimized version of a product automaton of G and G 0 [7]. The edge set E 00 is defined in terms of ,! l and the Equiv relation. not ) is the entry vertex of G 00 . We will restrict the vertex and edge sets of G 00 to be the vertices and edges reachable from ). If no vertices (other than (s; s 0 or edges are reachable from are not equivalent. A pair (v; v 0 ) is reachable from (s; s 0 there is a path p in G from s to v that is a prefix of a path in I(G;G 0 reject represents the reject state, which corresponds to paths in D(G;G 0 represents the accept state, which corresponds to paths in I(G;G 0 ). Figure 1 shows the intersection graph G 00 of the graphs G and G 0 in the figure. We can see that there are two paths in I(G;G 0 ), corresponding to the paths: and in G 00 . The corresponding paths in G are: [A; f ;C; f Graph G 00 also shows that any path that begins with the transition (A;t) is in D(G;G 0 Two straightforward results about the intersection graph G 00 will inform the rest of the paper: A path p is in I(G;G 0 ) iff it is represented by a path from (s; s 0 ) to accept in G 00 ; a path p is in D(G;G 0 ) iff it is represented by a path from (s; s 0 ) to reject in G 00 . Of course, every complete path p in G is either in I(G;G 0 ) or D(G;G 0 ). More Theorem 1 Let G 00 be the intersection graph of graphs G and G 0 . Path from G is in I(G;G 0 ) iff is in G 00 . Theorem 2 Let G 00 be the intersection graph of graphs G and G 0 . Path from G is in D(G;G 0 ) iff there exists n) such that is in G 00 . Figure shows how the intersection graph of graphs G and G 0 is computed via a synchronous depth-first search of both graphs. The procedure DFS is always called with equivalent vertices v and v 0 . If (v; v 0 ) is already in V 00 , this pair has been visited before and the procedure returns. Otherwise, (v; v 0 ) is inserted into V 00 and each its corresponding edge is considered in turn. 5 Edges are appropriately inserted into E 00 to reflect whether or not vertices w and w 0 are equivalent, and whether or not w is the exit vertex of G. The algorithm recurses only when w and w 0 are equivalent and w is not the exit vertex of G. The algorithm also computes the set of vertices V 00 accept from which accept is reachable in G 00 , which will be used later. The worst-case time complexity of the algorithm is O(jEj \Delta jE 0 j). Note that it is not necessary to store the relation E 00 explicitly, since it can be derived on demand from V 00 , E and E 0 . Thus, the space complexity for storing the intersection graph (as well as V 00 accept ) is in the worst case. 4 CRTS Using Edge Coverage What is the limit of CRTS given that the dynamic information collected about G(t) is edge coverage? Consider a complete path p representing the execution path of G(t) and the set of edges E p of G that it covers. There may be another complete path q in G, distinct from p, such that E represent the set of paths (including p) whose edge sets are identical to E p . To determine whether or not G 0 needs retesting, a CRTS algorithm using edge coverage must consider (at least implicitly) all the paths in P p . If all of these paths are members of I(G;G 0 ) then the CRTS algorithm can and should eliminate the test that generated path p. However, if even one of the paths in P p is in D(G;G 0 ) then the algorithm must select the test in order to be safe. Given this insight, we can now define what it means for a CRTS algorithm to be edge-optimal: A CRTS algorithm is edge-optimal if for any path p such that P p ' I(G;G 0 ), the algorithm reports that p is in I(G;G 0 5 Note that if v and v 0 are equivalent then w must be defined since BB(v 0 ) is identical to BB(v). 6 Note that no such paths can exist if G is acyclic. In this case, each complete path has a different set of edges than all other complete paths. accept := facceptg procedure begin for each edge do else else accept then accept [f(v;v 0 )g ni od Figure 2: Constructing the intersection graph of G and G 0 via a synchronous depth-first search of the two graphs. The algorithm also determines the set of vertices V 00 accept from which accept is reachable. accept reject accept reject accept reject Rothermel/Harrold algorithm Partial-reachability algorithm Full-reachability algorithm accept reject Valid-reachability algorithm Figure 3: The four edge-based CRTS algorithms, summarized pictorially with the intersection graph. The dotted outline represents V 00 accept , the vertices of G 00 from which accept is reachable. Algorithm Time Space Precision Edge-optimal? Rothermel/Harrold O(jEj \Delta (jE Partial-reachability O(jEj \Delta (jE Full-reachability O(jEj \Delta jE Table 1: Comparison of four edge-based CRTS algorithms. We first present the Rothermel/Harrold (RH) algorithm, restated in terms of the intersection graph. We then present three new algo- rithms, culminating in an edge-optimal algorithm. Figure 3 illustrates what the RH algorithm and each of the four algorithms does, using the intersection graph. 7 Each picture shows the start vertex states reject and accept. The dotted outline represents accept , the vertices of G 00 from which accept is reachable. ffl The RH algorithm detects whether or not E p covers an edge incident to reject. If it does not, then path p must be in I(G;G 0 ffl The partial-reachability algorithm detects whether or not E p covers a path in the intersection graph from an edge leaving accept to the reject vertex. Again, if no such path exists then p is in I(G;G 0 ). A surprising result is that partial-reachability of reject can be determined with time and space complexity equivalent to the RH algorithm. This algorithm is more precise than the RH algorithm since it may be the case that E p contains an edge incident to reject but does not cover a partial path from a vertex in V 00 accept to reject. ffl The full-reachability algorithm determines whether or not E p covers a path from (s; s 0 ) to reject. If not, then p is in I(G;G 0 ). This algorithm is more precise than the partial-reachability al- gorithm, but at a greater cost. However, it is still not edge- optimal. ffl The valid-reachability algorithm makes use of a partial order v on edges in G to rule out certain "invalid" paths. We show that if P cannot cover a valid reaching path to reject from (s; s ), yielding an edge-optimal algorithm. Table 1 summarizes the time and space complexity for the four al- gorithms. T represents the set of tests on which G has been run. All edge-based CRTS algorithms incur a storage cost of O(jEj \Delta jT for the edge coverage information stored for each test in T , which we factor out when discussing the space complexity of these algorithms 7 If s is not equivalent to s 0 , then I(G;G 0 ) is empty. We assume that all four algorithms initially check this simple condition before proceeding. 4.1 The Rothermel-Harrold Algorithm We now present the RH algorithm in terms of the intersection graph The RH algorithm first computes the set D of control transitions incident to reject (using a synchronous depth-first search of graphs G and G 0 similar to that in Figure 2): Given D and an edge set E p , the RH algorithm then operates as must be in I(G;G 0 since it contains no transition from D, which is required for p to be in D(G;G 0 Otherwise, conservatively assume that p is in D(G;G 0 Consider the intersection graph of Figure 1. For this graph, )g. Since every path from A to X in graph G contains one of these transitions, the RH algorithm will require all tests to be rerun on G 0 . However, in this example, for any path p in I(G;G 0 ), so the RH algorithm is not edge-optimal. Consider such a path The transitions of G 00 covered by E p are shown as bold edges in Figure 1. There is no complete path other than p that covers exactly the transitions (A; f ), (C;t) and (W;e). The time and space complexity to compute D is clearly the same as that for the depth-first search algorithm of Figure 2. To compute, for all tests t in a set of tests T , whether or not the set of edges covered by G(t) contains a transition from D, takes O(jEj \Delta jT time. Thus, the RH algorithm has an overall running time of O(jEj \Delta (jE and space complexity of O(jV j \Delta jV 0 j). Rothermel and Harrold show that if G and G 0 do not have a "multiply-visited vertex" then their algorithm will never report that p is in D(G;G 0 actually is in I(G;G 0 ). This means that their algorithm is optimal (and thus edge-optimal) for this class of graphs. Stated in terms of the intersection graph G 00 , a vertex v in G is a "multiply-visited vertex" if: So in Figure vertex C of graph G is a multiply-visited vertex. Rothermel and Harrold ran their algorithm on a set of seven small A,A U,U f f f f G" reject accept A U f f Y G' U f f f G A if A { { U } if C { Y } } else { if C { W } if A { { U } if C { W } Figure 4: An example that shows that the partial-reachability algorithm is not edge-optimal. programs (141-512 lines of code, 132 modified versions) and one larger program (49,000 lines of code, 5 modified versions), and found that the multiply-visited vertex condition did not occur for these programs and their versions [12]. Further experimentation is clearly needed on larger and more diverse sets of programs to see how often this condition arises. 4.2 The Partial-reachability Algorithm Let us reconsider the example of Figure 1. The dotted outline in graph G 00 shows the set V 00 accept . The only transition leaving this set is (A;t). Any path leading to reject must include this transition. Thus, if this transition is not in E p then p must be in I(G;G 0 ), as is the case with path which has E (C;t);(W;e)g. Consider the projection of E p onto the edge set of G and the graph G 00 results (the edges of E 00 are shown in bold in Figure 1). It is straightforward to see that, in general, for any edge v 00 ! w 00 in G 00 reject must be reachable from w 00 in G 00 . Therefore, for an edge in V 00 accept and w 00 is not in V 00 accept it must be the case that reject is reachable from w 00 . This observation leads to the partial-reachability algorithm which has time and space complexity identical to that of the RH algorithm, yet is more precise. This algorithm does not require construction of G 00 p , but is able to determine whether or not reject is partially- reachable from an edge leaving V 00 accept . Similar to the RH algorithm, this algorithm first computes a set D reject of transitions in G using the intersection accept g The set D reject contains transitions in G that transfer control out of accept . The algorithm then operates as follows: If /then p is in I(G;G 0 ), since p must contain a transition from D reject in order to be in D(G;G 0 ). Otherwise, conservatively assume that p is in D(G;G 0 It is easy to see that the partial-reachability algorithm subsumes the RH algorithm, since whenever reject is not empty, will not be empty. Stated another way, whenever the RH algorithm reports that p is in I(G;G 0 ), the partial-reachability algorithm will report the same. As shown in Figure 2, the set V 00 accept can be determined during construction of the intersection graph, in O(jEj \Delta jE 0 space. To compute D reject takes O(jEj simply requires visiting every edge e 00 in E 00 to determine if e 00 leaves accept . If so, then the transition e in G corresponding to e 00 is added to D reject . Once D reject has been computed, the rest of the algorithm is identical to the RH algorithm: for each test in T , check whether or not the set of edges covered by the test has an edge in D reject . Thus, the time and space complexity of this algorithm is identical to the RH algorithm. 4.3 The Full-reachability Algorithm Figure 4 shows that the partial-reachability algorithm is not edge- optimal. In this example, the intersection graph G 00 has Thus, for the path which is in I(G;G 0 and for which P p ' I(G;G 0 both the RH algorithm and partial- reachability algorithm will fail to report that p is in I(G;G 0 since transition (C;t) is covered by path p. Note, however, that in G 00 p the reject vertex is not reachable from (s; s 0 ). In general, either reject or accept must be reachable from (s; s 0 ) in G 00 . The full-reachability algorithm is simple: If reject is not reachable in G 00 then p is in I(G;G 0 Otherwise, conservatively assume that p is in D(G;G 0 Consider graph G in Figure 4. Any complete path in G containing U while A { } G A f U U if A { if A { while A { } } else { Y } f f U,U f f f A 3 G' Y U reject accept Figure 5: An example that shows that the full-reachability algorithm is not edge-optimal. the transition additionally, does not contain the transition (A;t). Therefore, for any such path p, vertex reject is not reachable from vertex (A;A) in G 00 . The DFS algorithm in Figure 2 can be easily modified to compute the reachability of reject in G 00 p , but must be run for each test in T , resulting in an overall running time of O(jEj \Delta jE j). The space complexity remains the same as before. 4.4 The Valid-reachability Algorithm: An Edge-optimal Algorithm As shown in Figure 5, the full-reachability algorithm is not edge- optimal. Consider the path in graph G, which is in I(G;G 0 ) and has coverage )g. Every path in G that covers exactly these transitions is in I(G;G 0 ). Nonetheless, the projection of E p onto G 00 yields a graph in which reject is reachable from (U;U) via the path However, notice that for any path in graph G that includes both the transitions (A;t) and (A; f ), the first occurrence of the transition (A;t) in the path must occur before the first occurrence of (A; f ). Therefore, all paths in P p must have this property, since by definition they cover (A;t), (A; f ), and (U;e). While the set of transitions in the path by which reject is reachable in G 00 includes (U;e) and does not include (A;t) before (A; f ). So, this path cannot be in P p and should be ignored. The problem then is that the full-reachability algorithm considers paths that are not in P p but reach reject in G 00 . By refining the notion of reachability, we arrive at an edge-optimal algorithm. We define a partial order on the edges of graph G as follows: containing both edges e and f , the first instance of e in p precedes the first instance of f in p. We leave it to the reader to prove that v is indeed a partial order (it is anti-symmetric, transitive, and reflexive). An equivalent but constructive definition of v follows: dominates f 8 or ( f is reachable from e, and e is not reachable from f ). The v relation for graph G in Figure 5 is (U;e) v (A;t) v (A; f ). The valid-reachability algorithm is based on the following observa- tion: If a path q contains a transition f 2 E p but does not contain a transition e 2 E p such that e v f in G, then any path with q as a prefix cannot be a member of P p . We say that such a path does not respect v. The valid-reachability algorithm first checks if reject is reachable from (s; s 0 ) in G 00 . If not, then p is in I(G;G 0 ), as before. If (s; s 0 ) is reachable, the algorithm computes R 00 , the set of transitions in G 00 that are reachable from (s; s 0 ) and from which reject is reachable. It also computes the projection R of these transitions onto G. That is, R is a subset of E p . If E p contains edges e and f such that e 62 R, f 2 R and e v f , then the algorithm outputs that p is in ). Otherwise, the algorithm conservatively assumes that p is in D(G;G 0 It is straightforward to show that the valid-reachability algorithm is safe. The following theorem shows that it is also edge-optimal: Theorem 3 Given graphs G and G 0 and their intersection graph G 00 . If P p ' I(G;G 0 ) for any complete path p in G, then either ffl reject is not reachable from (s; s 0 p , or Proof: If reject is not reachable in G 00 then we are done. Instead, suppose that reject is reachable from (s; s 0 ) in G 00 . Furthermore, assume that for all f 2 R and e Given these assumptions, we will show that there is a complete path q in ) such that E contradicting our initial assumption that all paths with edge coverage equal to E p are in I(G;G 0 There are two parts to the proof: 1. show that there is a path q 1 in G from entry to v that covers only transitions from R, respects v and 8 An edge e dominates edge f in graph G if every path from s to f in G contains e. U while A { } G A f U f f f G" U,U f f f A 3 U U if A { if A { while A { } V2reject accept Figure An example for which any CRTS algorithm based on edge coverage cannot distinguish a path in I(G;G 0 ) from a path in D(G;G ). induces a path in G 00 from (s; s ) to reject; 2. show that there is a path q 2 from v to x in G that covers the transitions in does not cover a transition outside E p . The concatenation of paths yields a path q in D(G;G 0 ) such that E The existence of path q 1 follows from the closure property of R with respect to v (if f 2 R, e 2 E p and e v f then e 2 R), and the fact that R is the projection of R 00 , the transitions by which reject is reachable from (s; s 0 ) in G 00 . We now show the existence of path q 2 . Let e be the last edge in path q 1 . Since E p is the edge coverage of a complete path p, it follows that for all edges e and f in E p , either f is reachable from e in G via transitions in E p or e is reachable from f via transitions in . Since the path q 1 respects v, it also follows that for all edges f in cannot be related by v. In the former case, f is reachable from e via transitions in E p . In the latter case, edges e and f are not related by v, so it follows that e and f must both be reachable from the other via transitions in E p , completing our proof. The time complexity of the valid-reachability algorithm is O(jEj \Delta j). The algorithm requires, for each test in T , the construction of G 00 p and the set R, which takes time O(jEj \Delta jE 0 j), dominating all other steps in the algorithm. Using an extended version of the Lengauer/Tarjan immediate dominator algorithm [8], the immediate v relation for G can be computed in near-linear time and space in the size of G. To determine whether or not the set of edges R is closed with respect to E p and v requires the following steps: 1. projecting to create v p , an O(E) operation; 2. visiting each immediate relation e v p f to check if e 62 R and f 2 R. As two constant-time set membership operations are performed for each immediate v p relation, of which there are O(E), this step takes O(E) time. The space complexity of the valid-reachability algorithm remains at O(jV j \Delta jV 0 j). 5 CRTS Using Path Coverage Figure 6 shows that any CRTS algorithm based on edge coverage can be forced to make an incorrect (but safe) decision. It presents two programs, their graphs G and G 0 , and their intersection graph is in I(G;G 0 ). The path p has )g. This is exactly the same set of edges covered by any path in D(G;G 0 ), such as: Thus, it is impossible to determine whether a path in I(G;G ) or in produced the edge set E p . We consider how the path profiling technique of Am- mons/Ball/Larus (ABL) [2] applied to the graphs in Figure 6 can separate the paths p and q. The ABL algorithm decomposes a control flow graph into acyclic paths based on the backedges identified by a depth-first search from s. Suppose that v ! w is a backedge. The ABL decomposition yields four classes of paths: (1) A path from s to x. (2) A path from s to v, ending with backedge v ! w. (3) A path from w to v (after execution of backedge v !w) ending with execution of backedge v ! w. After execution of backedge v ! w, a path from w to x. Graph G has backedge A ! t A. Applying the ABL decomposition to graph G in Figure 6 yields a total of four paths (corresponding to the four types listed above): The ABL algorithm inserts instrumentation into program P to track whether or not each of these four paths is covered in an execution. Recall the paths p and q that got edge-based CRTS into trouble. Path p is composed of the paths p 2 followed by p 4 , so ABL will record that only these two paths are covered when p executes. On the other hand, the path q is composed of p 2 , followed by p 3 , followed by p 4 . Thus, for this example where edge coverage could not distinguish the two paths, the ABL path coverage does. As mentioned in the introduction, an adversary can create a graph G 0 such that any control-flow-based RTS algorithm that records less than the complete path executed through G will be unable to distinguish a path in I(G;G 0 ) from a path in D(G;G 0 ). This is only true if G contains cycles, as it does in our example In the example from Figure 6, we can defeat the ABL path coverage by adding another if-then conditional (with basic block the outermost conditional in program P 0 . Now, the path is in I(G;G 0 ) and the path in which (A;t) occurs one more time, is in D(G;G 0 ). However, both these paths cover exactly the set of they will not be distinguished unless longer paths are tracked. For any cutoff chosen, we can add another level of nesting and achieve the same effect. 6 Related Work Rothermel and Harrold define a framework for comparing different regression test selection methods [11], based on four characteristics ffl Inclusiveness, the ability to choose modification revealing tests (paths in D(G;G 0 ffl Precision, the ability to eliminate or exclude tests that will not reveal behavioral differences (paths in I(G;G 0 ffl Efficiency, the space and time requirements of the method, and ffl Generality, the applicability of the method to different classes of languages, modifications, etc. Our approach shares many similarities with the RH algorithm. The three reachability algorithms are based on control flow analysis and edge coverage. The partial-reachability algorithm is just as inclusive as the RH algorithm but is more precise with equivalent effi- ciency. The full-reachability and valid-reachability algorithms are even more precise, but at a greater cost. We have not yet considered how to generalize our algorithms to handle interprocedural control flow, as they have done. Rothermel shows that the problem of determining whether or not a new program is "modification-traversing" with respect to an old program and a test t is PSPACE-hard [10]. Intuitively, this is because the problem involves tracing the paths that the programs execute and the paths can have size exponential in the input program size (or worse). Of course, given a complete path through an old program and a new program, it is a linear-time decision procedure to determine if the new program contains the path. However, this defines away the real problem: that the size of the path can be un- bounded. We have considered the best a CRTS algorithm can do when the amount of information recorded about a program's execution is O(E) (edge coverage) or exponential in the number of edges (ABL path coverage). Summary We have formalized control-flow-based regression test selection using finite automata theory and the intersection graph. The partial- reachability algorithm has time and space complexity equivalent to the best previously known algorithm, but is more precise. In ad- dition, we defined a strong optimality condition for edge-based regression test selection algorithms and demonstrated an algorithm (valid-reachability) that is edge-optimal. Finally, we considered how path coverage can be used to further improve regression test selection. A crucial question on which the practical relevance of our work hinges is whether or not the "multiply-visited" vertex condition defined by Rothermel and Harrold occurs in practice. For versions of programs that do not have this condition, the RH algorithm is op- timal. When this condition does occur, as we have shown, the RH algorithm is not even edge-optimal. We plan to analyze the extensive version control repositories of systems in Lucent [3] to address this question. Acknowledgements Thanks to Mooly Sagiv and Patrice Godefroid for their suggestions pertaining to finite state theory. Thanks also to Glenn Bruns, Mary Jean Harrold, Gregg Rothermel, Mike Siff, Mark Staskauskas and Peter Mataga for their comments. --R Exploiting hardware performance counters with flow and context sensitive profiling. If your version control system could talk. Optimally profiling and tracing pro- grams Incremental program testing using program dependence graphs. A system for selective regression testing. Introduction to Automata The- ory A fast algorithm for finding dominators in a flow graph. Using data flow analysis for regression testing. Efficient, Effective Regression Testing Using Safe Test Selection Techniques. Analyzing regression test selection techniques. --TR Compilers: principles, techniques, and tools Incremental program testing using program dependence graphs Optimally profiling and tracing programs Analyzing Regression Test Selection Techniques A safe, efficient regression test selection technique TestTube Exploiting hardware performance counters with flow and context sensitive profiling A fast algorithm for finding dominators in a flowgraph Introduction To Automata Theory, Languages, And Computation Efficient, effective regression testing using safe test selection techniques --CTR Amitabh Srivastava , Jay Thiagarajan, Effectively prioritizing tests in development environment, ACM SIGSOFT Software Engineering Notes, v.27 n.4, July 2002 Guoqing Xu, A regression tests selection technique for aspect-oriented programs, Proceedings of the 2nd workshop on Testing aspect-oriented programs, p.15-20, July 20-20, 2006, Portland, Maine Mary Jean Harrold , Gregg Rothermel , Rui Wu , Liu Yi, An empirical investigation of program spectra, ACM SIGPLAN Notices, v.33 n.7, p.83-90, July 1998 Alessandro Orso , Nanjuan Shi , Mary Jean Harrold, Scaling regression testing to large software systems, ACM SIGSOFT Software Engineering Notes, v.29 n.6, November 2004 Gregg Rothermel , Roland J. 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profiling;coverage;control flow analysis;regression testing

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All-du-path coverage for parallel programs.

One significant challenge in bringing the power of parallel machines to application programmers is providing them with a suite of software tools similar to the tools that sequential programmers currently utilize. In particular, automatic or semi-automatic testing tools for parallel programs are lacking. This paper describes our work in automatic generation of all-du-paths for testing parallel programs. Our goal is to demonstrate that, with some extension, sequential test data adequacy criteria are still applicable to parallel program testing. The concepts and algorithms in this paper have been incorporated as the foundation of our DELaware PArallel Software Testing Aid, della pasta.

Introduction Recent trends in computer architecture and computer networks suggest that parallelism will pervade workstations, personal comput- ers, and network clusters, causing parallelism to become available to more than just the users of traditional supercomputers. Experience with using parallelizing compilers and automatic parallelization tools has shown that these tools are often limited by the underlying sequential nature of the original program; explicit parallel programming by the user replacing sequential algorithms by parallel algorithms is often needed to take utmost advantage of these modern systems. A major obstacle to users in ensuring the correctness and reliability of their parallel software is the current lack of software testing tools for this paradigm of programming. Researchers have studied issues regarding the analysis and testing of concurrent programs that use rendezvous communication. A known hurdle for applying traditional testing approaches to testing parallel Prepared through collaborative participation in the Advanced Telecommunications/Information Distribution Research Program (ATIRP) Consortium sponsored by the U.S. Army Research Laboratory under Cooperative Agreement DAAL01-96-2-0002. programs is the nondeterministic nature of these programs. Some researchers have focused on solving this problem[13, 15], while others propose state-oriented program testing criteria for testing concurrent programs[14, 10]. Our hypothesis is that, with some extension, sequential test data adequacy criteria are still applicable to parallel program testing of various models of communication. Although many new parallel programming languages and libraries have been proposed to generate and manage multiple processes executing simultaneously on multiple processors, they can be categorized by their synchronization and communication mechanisms. Message passing parallel programming accomplishes communication and synchronization through explicit sending and receiving of messages between processes. Message passing operations can be blocking or nonblocking. Shared memory parallel programming uses shared variables for communication, and event synchronization operations. In this paper, we focus on the applicability of one of the major testing criteria, all-du-path testing[16], to both shared memory and message passing parallel programming. In particular, we examine the problem of finding all-du-path coverage for testing a parallel program. The ultimate goal is to be able to generate test cases automatically for testing programs adequately according to the all-du-path criteria. Based on this criterion, all define-use associations in a program will be covered by at least one test case. The general procedure for finding a du-pair coverage begins with finding du-pairs in a program. For each du-pair, a path is then generated to cover the specific du- Finally, test data for testing the path is produced [2][7]. This testing procedure has been well established for sequential programs; however, there is currently no known method for determining the all-du-path coverage for parallel programs. Moreover, the issues to be addressed toward developing such algorithms are not well defined. We present our algorithms for shared memory parallel programs, and then discuss the modifications necessary for the message passing paradigm. We have been building a testing tool for parallel software, the Delaware Parallel Software Testing Aid, called della pasta, to illustrate the effectiveness and usefulness of our tech- niques. della pasta takes a shared memory parallel program as input, and interactively allows the user to visually examine the all- du-path test coverages, pose queries about the various test coverages, and modify the test coverage paths as desired. In our earlier paper, we focused strictly on the all-du-path finding algorithm[18]. We begin with a description of the graph representation of a parallel program used in our work. We then describe our testing paradigm and how we cope with the nondeterministic nature of parallel programs during the testing process. We discuss the major problems in providing all-du-path coverage for shared memory parallel pro- grams, and a set of conditions to be used in judging the effectiveness of all-du-path testing algorithms. Current approaches to all-du-path coverage for sequential programs of closest relevance to our work are then discussed. We present our algorithm for finding an all-du-path coverage for shared memory parallel programs, which combines and extends previous methods for sequential pro- grams. Modification of our data structures and algorithms for other parallel paradigms is discussed followed by the description of the della pasta tool. Finally, a summary of contributions and future directions are stated. Model and Notation The parallel program model that we use in this paper consists of multiple threads of control that can be executed simultaneously. A thread is an independent sequence of execution within a parallel program, (i.e., a subprocess of the parallel process, where a process is a program in execution). The communication between two threads is achieved through shared variables; the synchronization between two threads is achieved by calling post and wait system calls; and thread creation is achieved by calling the pthread create system call. We assume that the execution environment supports maximum par- allelism. In other words, each thread is executed in parallel independently until a wait node is reached. This thread halts until a matching post is executed. The execution of post always succeeds without waiting for any other program statements. Formally, a shared memory parallel program can be defined as threads. Moreover, T 1 is defined as the manager thread while all other threads are defined as worker threads, which are created when a pthread create() system call is issued.T post wait pthread_ create loop loop begin begin d:m=x+y u:z=3*m Figure 1: Example of a PPFG To represent the control flow of a parallel program, a Parallel Program Flow Graph (PPFG) is defined to be a graph which V is the set of nodes representing statements in the program, and E consists of three sets of edges . The set E I consists of intra-thread control flow edges (m i , n are nodes in thread T i . The set E S consists of synchronization edges (post post i is a post statement in thread T i , wait j is a wait statement in thread T j , and i 6= j. The set E T consists of thread creation edges (n i , n j ), where n i is a call statement in thread T i to the pthread create() function, and n j is the first statement in thread We define a path P i (n i u 1 simply P i , within a thread T i to be an alternating sequence of nodes and intra-thread edges or simply a sequence of nodes n i , where uw is the unique node index in a unique numbering of the nodes and edges in the control flow graph of the thread T i (e.g., a reverse postorder numbering). Figure 1 illustrates a PPFG. All solid edges are intra-thread edges. Edges in E S and E T are represented by dotted edges. This diagram also shows a define node d of variable m, i.e., y, and a use node u, i.e., m. The sequence begin \Gamma is a path. A du-pair is a triplet ( var, is the u th node in thread T i in the unique numbering of the nodes in thread T i , and the program variable var is defined in the statement represented by node while the program variable var is referenced in the v th node in the unique ordering of nodes in thread T j . In a sequential program or a single thread T i of a parallel program, we say that a node is covered by a path, denoted n 2 p P, if there exists a node n s in the path such that We say that a node in a parallel program is covered by a set of paths respectively, or simply We represent the set of matching posts of a wait node as and the set of matching waits of a post node as g. We use the symbol "OE" to represent the relation between the completion times of instances of two statement nodes. We say a if an instance of the node a completes execution before an instance of the node b. Finally, the problem of finding all-du-path coverage for testing a shared memory parallel program can be stated as: Given a shared memory parallel program, v ), in PROG, find a set of paths in threads T 1 , that covers the du-pair ( var, v ), such that n u OE n v . 1 3 Nondeterminism and the Testing Process Nondeterminism is demonstrated by running the same program with the same input and observing different behaviors, i.e., different sequences of statements being executed. Nondeterminism makes it difficult to reproduce a test run, or replay an execution for debugging. It also implies that a given test data set may not actually force the intended path to be covered during a particular testing run. One way to deal with nondeterminism is to perform a controlled execution of the program, by having a separate execution control We focus on finding du-pairs with the define and use in different threads; du-pairs within the same thread are a subcase. mechanism that ensures a given sequence of execution. We advocate controlled execution for reproducing a test when unexpected results are produced from a test, but we have not taken this approach to the problem of automatically generating and executing test cases to expose errors. Instead, we advocate temporal testing for this stage of testing. We briefly describe our temporal testing paradigm here, and refer the reader to [19] for a more detailed description. Temporal testing alters the scheduled execution time of program segments in order to detect synchronization errors. Formally, a program test case TC is a 2-tuple (PR OG, I ) where I is the input data to the program PR OG, whereas a temporal test case TTC is a 3-tuple (PR OG, I , D) where the third component, referred to as timing changes, is a parameter for altering the execution time of program segments. Based on D, the scheduled execution time of certain synchronization instructions n, represented as t(n), will be changed for each temporal test and the behavior of the program PR OG will be observed. Temporal testing is used in conjunction with path testing. For ex- ample, temporal all-du-path testing can be implemented by locating delay points along the du-paths being tested. The goal is to alter the scheduled execution time of all process creation and synchronization events along the du-paths. Delayed execution at these delay points is achieved by instrumenting the program with dummy computation statements. A testing tool is used to automatically generate and execute the temporal test cases. Similarly, new temporal testing criteria can be created by extending other structural testing criteria. With the temporal testing approach, the testing process is viewed as occurring as follows: (1) Generate all-du-paths statically. (2) Execute the program multiple times without considering any possible timing changes. (3) Examine the trace results. If the trace results indicate that different paths were in fact executed, it is a strong indication that a synchronization error has occurred and the du-path expected to be covered may provide some clue about the probable cause. Controlled execution may be used to reproduce the test. However, even if the same du-path was covered in multiple execution runs, temporal testing should still be performed. Generate temporal test cases with respect to the du-paths. Perform temporal testing automatically. Examine the results. In this paper, we focus on the first step, i.e., developing an algorithm to find all-du-paths for shared memory parallel programs. The results of this paper can be used for generating temporal test cases with respect to the all-du-path coverage criterion. It should be noted that it is possible that the path we want to cover is not executed during a testing run due to nondeterminism, because we are not using controlled execution; instead, we use automatic multiple executions with different temporal testings to decrease the chances that the intended path will not be covered. All-du-path Coverage In this section, we use some simplified examples to demonstrate some of the inherent problems to be addressed in finding all-du- paths in parallel programs. This list is not necessarily exhaustive, but instead meant to illustrate the complexity of the problem of automatically generating all-du-paths for parallel programs. Figure contains two threads, the manager thread and a worker PATH COVERAGE: 1: 3:loop 4:if 15:wait 14:y=x 13:wait 12:if 10: 16:end begin begin 2:pcreate 8:end 5:x=3 6:post 7:post manager worker Figure 2: Du-pair coverage may cause an infinite wait. thread. This figure demonstrates a path coverage that indeed covers the du-pair, but does not cover both the post and the wait of a matching post and wait. If the post is covered and not a matching wait, the program will execute to completion, despite the fact that the synchronization is not covered completely. However, if the wait is covered and not a matching post, then the program will hang with the particular test case. In this example, the worker thread may not complete execution, whereas the manager thread will terminate successfully. The generated path will cause the loop in the manager thread to iterate only once, while the loop in the worker thread will iterate twice. This shows how the inconsistency in the number of loop iterations may cause one thread to wait infinitely. In addition, branch selection at an if node can also influence whether or not all threads will terminate successfully. PATH COVERAGE: 15:post 2:pcreate worker 4:if 12:if 6:post 5:y=x+3 13:wait 3:loop 1:begin 8:end 16:end manager 7:wait Figure 3: Du-pair is incorrectly covered. In figure 3, the generated paths cover both the define (14) and the use (5) nodes, but the use node will be reached before the define node, that is, de f ine 6OE use. If the data flow information reveals that the definition of x in the worker thread should indeed be able to reach the use of x in the manager thread, then we should attempt to find a path coverage that will test this pair. The current path coverage does not accomplish this. 4.1 Test Coverage Classification The examples motivate a classification of all-du-path coverage. In particular, we classify each du-path coverage generated by an algorithm for producing all-du-path coverage of a parallel program as acceptable or unacceptable, and w-runnable or non-w-runnable. 4.1.1 Acceptability of a du-path coverage We call a set of paths PATH an acceptable du-path coverage, denoted as PATH a , for the du-pair (de f ine, use) in a parallel program free of infeasible paths of the sequential programming kind (see the later section on infeasible paths), if all of the following conditions are satisfied: 1. de f ine 2 p PATH; use 2 p PATH, 2. 8wait nodes w 2 p PATH, 9 a post node p 2 MP(w), such that 3. if 9(post;wait) 2 E S , such that de f ine OE post OE wait OE use, then post;wait 2 p PATH. 4. 8n 2 p PATH where (n These conditions ensure that the definition and use are included in the path, and that any (post,wait) edge between the threads containing the definition and use, and involved in the data flow from the definition to the use are included in the path. Moreover, for each sink of a thread creation edge, the associated source of the thread creation edge is also included in the path. If any of these conditions is violated, then the path coverage is considered to be unacceptable. For instance, if only the wait is covered in a path coverage and a matching post is not, the path coverage is not a PATH a . Figure 3, where the define and use are covered in reverse order, shows another instance that only satisfies the first two conditions, but fails to satisfy the third condition. 4.1.2 W-runnability of a du-path coverage We have seen through the examples that a parallel program may cause infinite wait under a given path coverage, even when the du-path coverage is acceptable. If a path coverage can be used to generate a test case that does not cause an infinite wait in any thread, we call the path coverage a w-runnable du-path coverage. When a PATH is w-runnable, we represent it as PATHw . Although we call a PATH w-runnable, we are not claiming that a PATHw is free of errors, such as race conditions, or synchronization errors. More formally, a PATH a is w-runnable if all of the following additional conditions are satisfied: 1. For each instance of a wait, w t (possibly represented by the same node n t 2 p PATH), 9 an instance of a post, p s instance of a wait or post is one execution of the wait or post; there may be multiple instances of the same wait or post in the program. 2. 6 9post nodes PATH such that The first condition ensures that, for each instance of a wait in the PATH, there is a matching instance of a post. However, it is not required that for every instance of post, a matching wait is covered. In other words, the following condition is not required: 8post nodes a wait node w 2 MW(p), such that w 2 p PATH. The second condition ensures that the generated path is free of deadlock. We can develop algorithms to find PATH a automatically. How- ever, we utilize user interaction in determining PATHw in the more difficult cases, and sometimes have to indicate to the user that we cannot guarantee that the execution will terminate on a given test case (i.e., path coverage). In this case, the program can still be run, but may not terminate. We will find a PATH a and report that this path coverage may cause an infinite wait. 4.2 Infeasible Paths Infeasible paths in a graph representation of a program are paths that will never be executed given any input data. In control flow graphs for a single thread, infeasible paths are due to data dependencies and conditionals. In interprocedural graph structures, infeasible paths are due to calling a function from multiple points. These kinds of infeasible paths can occur in sequential programs, and can also occur in parallel programs. In a parallel program, another kind of infeasible path can also occur due to synchronization dependencies. Infeasible paths due to synchronization dependencies can cause deadlock or infinite wait at run-time. Like most path finding algorithms, we assume that the paths we identify are feasible with respect to the first causes. With regard to infeasible paths due to synchronizations, our work uses a slightly different characterization of paths, PATH a and PATHw . Our du-path coverage algorithm finds paths in a way to guarantee that we will have matching synchronizations included in the final paths, that is, it finds paths that are PATH a . However, a deadlock situation could occur for a path coverage that is a PATH a , but not PATHw . To guarantee finding matching synchronizations, we currently assume that matching post and wait operations both appear in a program. If a program contains a post and no matching wait or vice versa, we expect that the compiler will report a warning message prior to the execution of our algorithm. 5 Related Work In the context of sequential programs, several researchers have examined the problems of generating test cases using path finding as well as finding minimum path coverage [3, 11, 1]. All of these methods for finding actual paths focus on programs without parallel programming features and, therefore, cannot be applied directly to finding all-du-path coverage for parallel programs. However, we have found that the depth-first search approach and the approach of using dominator and post-dominator trees can be used together with extension to provide all-du-path coverage for parallel programs. We first look at their limitations for providing all-du-path coverage for parallel programs when used in isolation. Gabow, Maheshwari, and Osterweil [3] showed how to use depth-first search (DFS) to find actual paths that connect two nodes in a sequential program. When applying DFS alone to parallel programs, we claim that it is not appropriate even for finding PATH a , not to mention PATHw . The reason is that although DFS can be applied to find a set of paths for covering a du-pair, this approach does not cope well with providing coverage for any intervening wait's, and the corresponding coverage of their matching post's as required to find PATH a . For example, consider a situation where there are more wait nodes to be included while completing the partial path for covering the use node. Since the first path is completed and a matching post is not included in the original path, the first path must be modified to include the post. This is not a straightforward task, and becomes a downfall of using DFS in isolation for providing all-du-path coverage for parallel programs. Bertolino and Marr- e have developed an algorithm (which we call DT-IT) that uses dominator trees (DT) and implied trees (IT) (i.e., post-dominator trees) to find a path coverage for all branches in a sequential program [1]. A dominator tree is a tree that represents the dominator relationship between nodes (or edges) in a control flow graph, where a node n dominates a node m in a control flow graph if every path from the entry node of the control flow graph to must pass through n. Similarly, a node m postdominates a node p if every path from p to the exit node of the control flow graph passes through m. The DT-IT approach finds all-branches coverage for sequential programs as follows. First, a DT and an IT are built for each sequential program. Edges in the intersection of the set of all leaves in DT and IT, defined as unconstrained edges, are used to find the minimum path coverage based on the claim that if all unconstrained edges are covered by at least one path, all edges are covered. The algorithm finds one path to cover each unconstrained edge. When one edge is selected, one sub-path is found in DT as well as in IT. When one node and its parent node in DT or IT are not adjacent to each other in the control flow graph of the program, users are allowed to define their own criteria for connecting these two nodes to make the path. The two sub-paths, one built using DT and the other built using IT, are then concatenated together to derive the final path coverage. If we try to run this algorithm to find all-du-path coverage for parallel programs, we need to find a path coverage for all du-pairs instead of all-edges, which is a minor modification. However, this approach will also run into the same problem as in DFS. That is, if some post or wait is reached when we are completing a path, we need to adjust the path just found to include the matching nodes. In addition, we will run into another problem regarding the order in which the define and use nodes are covered in the final path. For instance, in Figure 3, an incorrect path coverage will be generated using the DT-IT approach alone. The final path will have de f ine 6OE use. Thus, using this method alone cannot guarantee that we find a PATH a . Yang and Chung [20] proposed a model to represent the execution behavior of a concurrent program, and described a test execution strategy, testing process and a formal analysis of the effectiveness of applying path analysis to detect various faults in a concurrent program. An execution is viewed as involving a concurrent path (C-path), which contains the flow graph paths of all concurrent tasks. The synchronizations of the tasks are modelled as a concurrent route (C-route) to traverse the concurrent path in the execution, by building a rendezvous graph to represent the possible rendezvous conditions. The testing process examines the correctness of each concurrent route along all concurrent paths of concurrent programs. Their paper acknowledges the difficulty of C-path generation; how- ever, the actual methodologies for the selection of C-paths and C-routes are not presented in the paper. 6 A Hybrid Approach In this section, we describe our extended "hybrid" approach to find the actual path coverage of a particular du-pair in a parallel program. There are actually two disjoint sets of nodes in a path used to cover a du-pair in a parallel program: required nodes and optional nodes. The set of required nodes includes the pthread create() calls as well as the define node and use node to be covered, and the associated post and wait with which the partial order de f ine OE use is guaranteed. All other nodes on the path are optional nodes for which partial orders among them are not set by the requirements for a PATH a . However, if a wait is covered by the path, a matching post must be covered. For instance, in figure 6, the nodes 2, 4, 7, 25, and 26 are required nodes, whereas all other synchronization nodes are optional. Among the required nodes, the partial orders are uniquely identified, whereas the partial orders among the optional nodes are not. For example, it is acceptable to include either post 3 or post 4 first in a path coverage. We can even include wait 1 later than post 4 in a PATH a . The DFS approach is most useful for finding a path that connects two nodes whose partial order is known. The DT-IT approach is most appropriate for covering nodes whose partial order is not known in advance. Therefore, DFS is most useful for finding a path between the required nodes, whereas the DT-IT approach is most useful for ensuring that the optional nodes are covered. Our algorithm consists of two phases. During the first phase, called the annotate phase, the depth-first search (DFS) approach is employed to cover the required nodes in the PPFG. Then, the DT-IT approach is used to cover the optional nodes. After a path to cover a node is found, all nodes in the path are annotated with a traversal control number (TRN). In the second phase, called the path generation phase, the actual path coverage is generated using the traversal control annotations. We first describe the data structures utilized in the du-pair path finding algorithm, and then present the details of the algorithm. The algorithm assumes that the individual du-pairs of the parallel program have been found. Previous work computing reaching definitions for shared memory parallel programs has been done by Grunwald and Srinivasan [5]. 6.1 Data Structures The main data structures used in the hybrid algorithm are: (1) a PPFG, (2) a working queue per thread to store the post nodes that are required in the final path coverage, (3)a traversal control associated with every node used to decide which node must be included in the final path coverage and how many iterations are required for a path through a loop, (4) a reverse post-order number (RPO) for each node in the PPFG used in selecting a path at loop nodes, (5) a decision queue per if-node, and (6) one path queue per thread to store the resulting path. 6.2 The Du-path Finding Algorithm We describe the du-path finding algorithm with respect to finding du-pairs in which the define and use are located in different threads. The handling of du-pairs with the define and use in the same thread is a simplification of this algorithm. Figure 4 contains the annotate the graph() algorithm, which accomplishes the annotate phase. The traverse the graph() algorithm, shown in Figure 5, traverses the PPFG and generates the final du-path coverage. We describe each step of these algorithms in more detail here. Phase 1: Annotating the PPFG. Step 1. Initialize the working and decision queues to empty, and set TRN of each node to zero. Step 2. Use DFS to find a path from the pthread create of the thread containing the define node to the define node, and then from the 2. Find a path to cover pthread_create and define nodes using dfs; Algorithm annotate_the_graph() Output: Input: A DU-pair, and a PPFG Annotated PPFG 1. Initialize TRN's, decision queues, and working queues; From the define node, search for the use node using dfs; 3. Complete the two sub-paths using DT-IT. 5. /* process the synchronization nodes */ while ( any working queue not empty ) { For each thread, if working queue not empty { Remove one node from the working queue; if the node's TRN is zero { Find a path to cover this node 4. For each node in the complete paths: Increment TRN by one; If node is a WAIT, Add matching nodes into appropriate working queues, If node is an if-node, Add the successor node in the path into decision queue; For each node in the complete path: If node is a WAIT, If node is an if-node, Add matching nodes into appropriate working queues, Add the successor node in the path into decision queue; Increment TRN by one; Figure Phase 1: Annotate the graph. define node to the use node. When a post node is found in the path, a matching wait is placed as the next node to be traversed, and the search for the use node continues. Upon returning from each DFS() call after a wait is traversed, return to the matching post before continuing the search for the use node if not yet found. Step 3. Apply DT-IT to complete the sub-paths found in Step 2. To complete the sub-path in the thread containing the define node, use the dominator tree of the define node and the post-dominator tree of the post node that occurs after the define node in the sub-path just found. Similarly, to complete the sub-path in the thread containing the use node, use the dominator tree of a matching wait of the post node and the post-dominator tree of the use node. Step 4. For each node covered by either of these two paths, (1) increment the node's TRN by one to indicate that the node should be traversed at least once. (2) If the node is a wait, add a matching post into the working queue of the thread where the post is located. (3) If the node is an if-node, add the Reverse Post-Order Num- ber(RPO) of the successor node within the path into the if-node's decision queue to ensure the correct branch selection in phase 2. Step 5. While any working queue is not empty, remove one post node from a thread's working queue, and find a path to cover the node. Increment the TRN of the nodes in that path. In this way, the TRN identifies the instances of each node to be covered. This is particularly important in finding a path coverage for nodes inside loops, where it might be necessary to traverse some loop body nodes several times to ensure that branches inside the loop are covered appropriately. Process wait and if-nodes in this path as in Step 4. Algorithm traverse_the_graph() For all threads while ( current node's TRN > 0 and current is not the end node ) { add the current node to the result DU-path; decrement TRN of current node by one; { Output: A DU-path Input: An annotated PPFG node of the thread; if ( current is an if-node ) first node from decision queue; else delete the first node in the queue; if ( current is a loop node ) else smallest non-zero RPO; current = successor node of current; Figure 5: Phase 2: Generate the du-path coverage Phase 2: Generating a du-path. For each thread, perform the following steps: Step 1. Let n be the begin node of the thread. Step 2. While n's TRN > 0 and n is not the end node, Add n to the path queue, which contains the resulting path coverage, and decrement n's TRN. If n is an if-node, then let the new n be the node removed from n's decision queue. Otherwise, if n is a loop node, the successor with the smallest non-zero TRN is chosen to be the new n. If the children have the same TRN, then the child with the smallest RPO is chosen. Otherwise, if n is not an if-node or loop node, let new n be the successor of n. 6.3 Examples In this section, we use two examples to illustrate the hybrid ap- proach. The first example illustrates generating a PATHw , while the second example illustrates generating a (non-PATHw ) PATH a . Both examples cover the du-pair with the define of X at node 4 and the use of X at node 26 in Figure 6. Example 1 Generating a PATHw : During the second step of the first phase, the required nodes, including the pthread create, de f ine, post 2 , wait 2 , and the use nodes, are included in a partial path. The identified partial path is 2-3-4-5-7- 25-26. During the third step of the first phase, the two sub-paths are completed, using the DT-IT approach. The two identified complete paths are 1-2-3-4-5-7-8-9-3-11 for manager and 21-22-23-25-26- . The TRN for every node along the two paths equals 1 after step 4 except the loop node 22 for which the TRN is 2. When node 9 was reached during this traversal, nodes 28 and were put into the working queues for worker 1 and worker 2 , respectively. When node 28 is taken out of the working queue in step 5, it is found to have a nonzero TRN, and thus no more paths are added. When node 35 is taken out of the working queue, the TRN is zero. Hence, the path 31-32-33-34-35-32-36 is found to cover node 35. With the annotated PPFG as input, the second phase finds a final path of 1-2-3-4-5-7-8-9-3-11 for manager, 21- for worker 2 . worker worker 9: 1: 28:7: create POST POST WAIT POST 341 22beginbegin begin 34:n=5; 3:loop POST POST 25: 10: 30:end 2:pthread_ 36:end Figure Example of the path finding algorithm Example 2 Generating a non-PATHw : During the second step of the first phase, the required nodes, including the pthread create, de f ine, post 2 , wait 2 , and the use nodes, are included in a partial path identified as 2-3-4-5-7-25-26. During the third step of the first phase, the two sub-paths are completed, and found to be 1-2-3-4-5-7-8-9-3-11 for manager and 21-22-23-25-26- . The TRN for every node along the two paths equals 1 after step 4 except for the loop node 22; the TRN for node 22 equals 2. When node 9 was reached during this traversal, nodes 28 and 35 were put into the working queues. Similarly, during the fifth step of the first phase, two paths 21-22-23-25-26-27-28-22- and 31-32-33-34-35-32-36, are found to cover nodes 28 and 35, respectively. The final TRN's for this example label each node in Figure 6. The second phase finds final paths of 1-2-3-4-5-7-8-9-3-11 for manager, 21-22-23-25-26-27-29-22-23-25-26-27-28-22-30 for worker 1 and 31-32-33-34-35-32-36 for worker 2 . This set of paths is not w-runnable because worker 1 has an infinite wait (at node 25). It should be noted that regardless of the path constructed, the user will have to validate that the w property holds. 6.4 Correctness and Complexity Given a du-pair in a parallel program where the de f ine node and the use node are located in two different threads, we show that this algorithm indeed will terminate and find a PATH a . We first introduce some lemmas before we give the final proof. Lemma 1: TRN preserves the number of required traversals of each node within a loop body. During the first phase, the TRN of a node is incremented by one each time a path is generated that includes that node. Therefore, the number of traversals of each node in paths found during the first phase is preserved by the TRN. Although the number of traversals during the first phase is preserved, we are not claiming that these nodes will indeed be traversed during the second phase that same number of times. For nodes outside of a loop body, each node will be traversed at most as many times as its TRN. But a node may not need to be traversed that many times because the path generation phase may reach the End node before the TRN of all nodes becomes zero. Moreover, if there is no loop node in a program, only required nodes will be traversed as many times as the TRN indicates. Lemma 2: The decision queue and TRN of an if-node guarantee that the same sequence of branches selected during the first phase will be selected during the second phase. When an if-node is found in a path during the first phase, one branch is stored into the decision queue at that time. Hence, the number of branches in the decision queue of a given if-node is equal to the TRN of that if-node. Each time the if-node is traversed during the second phase, one node is taken out of the decision queue and the TRN of the if-node is decremented by one. Therefore, the sequence that a branch is selected is preserved. Lemma 3: DFS used during the first phase ensures de f ine OE post OE wait OE use in the final generated path. During the first phase, the required nodes will be marked by DFS prior to any other nodes in the graph. This ensures that necessary branches are stored in the decision queues first. By Lemma 2, these branches will be traversed first during the second phase. Hence, these nodes will be traversed in the correct order as given by the relationships above. Therefore, Lemma 3 is valid. Lemma 4: The working queues and TRN together guarantee the termination of the Du-path Finding Algorithm. We must show that both phases terminate. Phase 1 Termination: We use mathematical induction on m, where m represents the total number of pairs of synchronization nodes covered in a path coverage. Base case: there is only one pair of synchronization calls, the required ones, they will be included in the path generated by the DFS. The completion of the two partial paths will automatically terminate since there are no extra post or wait's involved. is an integer greater than 1. We need to show that Lemma 4 is also true when If the post and wait have been traversed previously, the TRN of these nodes will be greater than zero. Hence, they will not be included again during the first phase. When we generate a new path to cover this pair of post and wait nodes, if they currently have TRN=0, all other pairs of synchronization nodes will have been covered. (by the induction step) Hence, this new pair of synchronization calls will not trigger an unlimited number of actions. Therefore, the annotation phase will terminate. Phase 2 Termination: Since the TRN for each node must be traversed is a finite integer, and the TRN is decremented each time it is traversed during phase 2, the traversal during phase 2 will not iterate forever. Whenever a node with zero TRN or the End node is reached, the path generation phase terminates. Finally, we show the proof of the following theorem. Theorem 1: Given a du-pair in a shared memory, parallel program, the hybrid approach terminates and finds a PATH a . Proof: (1) By Lemma 4, the hybrid approach terminates. (2) To show that a PATH a is generated, we must show that the conditions described in the definition of PATH a are satisfied. By Lemma 1, Lemma 2, and Lemma 3, we can conclude that the de f ine, use, the required post, and wait nodes will be covered in the correct order. Step 1 of Phase 1 ensures that all appropriate pthread create calls are covered. Step 5 ensures that a matching post node regarding each wait node included in the path is also covered. Therefore, all conditions for a PATH a are satisfied. Q.E.D. The running time of the hybrid approach includes the time spent searching for the required nodes and time spent generating the final path coverage. We assume that the dominator/implied trees and the du-pairs have been provided by an optimizing compiler. Theorem 2: For a given G = (V;E), and a du-pair (d, u), the total running time of the du-path finding algorithm is equal to O(2 k jEj)), where the total number of post or the wait calls is denoted by k. Proof: The running time for searching for the required nodes is equal to O(jV To complete the two partial paths, the running time is equal to O(2 k (jV where the total number of post or the wait calls is denoted by k. Finally the second phase takes time O(2 k (jV j + jEj)) to finish. Hence, the total running time is equal to O(2 k (jV jEj)). For a given graph, usually the number of edges is greater than that of the nodes. Then, the running time is equal to O(2 k jEj). Q.E.D. 7 Other Parallel Paradigms 7.1 Rendezvous communication Among other researchers, Long and Clarke developed a data flow analysis technique for concurrent programs[9]. After their data flow analysis is performed, we can apply a modified version of our algorithm to find all-du-path coverage for a concurrent program with rendezvous communication. In particular, we need to modify the following: (1) construction of the PPFG, (2) definition of path acceptability, and (3) the all-du-path finding algorithm. First, to accommodate the request and accept operations in concurrent programs to achieve rendezvous communication, the PPFG needs to include a directed edge from a request to an accept node. Secondly, since the execution of a request is synchronous, the second condition in the definition of PATH a must be replaced by the following two conditions: 2a. 8accept nodes a 2 p PATH, 9 a request node r 2 MR(a), such that 2b. 8request nodes r 2 p PATH, 9 a accept node a 2 MA(r), such that a 2 p PATH. The set MR(a) is defined as the set of matching requests for the accept node a; the set MA(r) is defined as the set of matching accepts for the request node r. Finally, during the first phase of the algorithm, whenever a request or an accept is found, the matching node must be added into the working queue. 7.2 Message Passing Programs For analyzing message passing programs, a data flow analysis similar to interprocedural analysis for sequential programs is needed to compute the define-use pairs across processes. Several researchers have developed interprocedural reaching definitions data flow analysis techniques, even in the presence of aliasing in C programs [12, 6]. Although this analysis may find define-use pairs that may not actually occur during each execution of the program, the reaching definition information is sufficient for program testing. After this information is computed, we can apply our algorithm to find all-du- path coverage for a given C program with message passing library calls. The Message Passing Interface(MPI)[4] standard is a library of routines to achieve various types of inter-process communication, i.e., synchronous or asynchronous send/receive operations. To find all-du-path coverage for message passing programs, we need to identify the type of send or receive operations first, i.e., synchronous or asynchronous. If the send operation is synchronous, the definition of a PATH a must be modified to include both the send and the matching receive in the path coverage similar to the change made for supporting rendezvous-communication parallel programs. If the send is asynchronous, we only need to replace post by send in this paper. For each synchronous receive operation, we need to replace the wait by a receive in our algorithms and definitions. 8 The della pasta Tool The algorithm described in this paper has been incorporated into della pasta, the prototype tool that we are building for parallel software testing. The objective is to demonstrate that the process of test data generation can be partially automated, and that the same tool can provide valuable information in response to programmer queries regarding testing. The current major functions of this tool are: (1) finding all du-pairs in the parallel program, (2) finding all-du-path coverage to cover du-pairs specified by the user, (3) displaying all- du-path coverage in the graphic or text mode as specified by the user, and (4) adjusting a path coverage when desired by the user. della pasta consists of two major components: the static analyzer which accepts a file name and finds all du-pairs as well as the all-du-path coverage for each du-pair, and the path handler which interacts with the user to display the PPFG, a path coverage, and accept commands for displaying individual du-pair coverages and for modifying a path. The static analyzer uses a modified version of the Grunwald and Srinivasan algorithm[5] to find du-pairs in parallel programs of this model, and is implemented using the compiler optimizer generating tool called nsharlit, which is part of the SUIF compiler infrastructure [8]. The path handler is built on top of dflo which is a data-flow equation visualizing tool developed at Oregon Graduate Institute. 2 The user interface of della pasta is illustrated in figure 7. On the left of the screen, the PPFG is illustrated; on the right, the corresponding textual source code is shown. A user can resize the data flow graph as desired. The currently selected def-use pair is shown at the top of the screen. The corresponding du-pair path coverage is depicted in the PPFG as well as in the text as highlighted nodes and statements, respectively. Clicking on any node in the PPFG will pop up an extra window with some information about the node, and allow the user to modify a path coverage. In this example, a reader/writer program is illustrated in which the main thread creates three additional threads: two readers and one writer. The main thread then acts as one writer itself and communicates with one of the two readers just created. These two pairs of readers/writers will work independently in parallel. The du-pair coverage shown in this example only involves two of the 4 threads in the program. We are currently extending della pasta to use the du-pair coverage This tool can be downloaded from the Internet. Refer to the web site http://www.cse.ogi.edu:80/Sparse/dflo.html for details. Figure 7: della pasta user interface information already available through our static analyzer to answer queries of the following kind: Will the test case execute successfully without infinite wait caused by the path coverage? What other du-pairs does a particular path coverage cover? We are also incorporating our temporal testing techniques [17] into the tool in order to provide testing aid for delayed execution in addition to the traditional all-du-path testing. 9 Summary and Future Work To our knowledge, this is the first effort to apply a sequential testing criterion to shared memory or message passing parallel programs. Our contributions include sorting out the problems of providing all-du-path coverage for parallel programs, classifying coverages, identifying the limitations of current path coverage techniques in the realm of parallel programs, developing an algorithm that successfully finds all-du-path coverage for shared memory parallel programs, showing that it can be modified for message passing and rendezvous communication, and demonstrating its effectiveness through implementation of a testing tool. The all-du-path coverage algorithm presented in this paper has some limitations. The all-du-path algorithm requires that a PPFG be constructed statically. If a PPFG cannot be constructed statically to represent the execution model of a program, the analysis that constructs the du-pairs may not produce meaningful du-pairs. Thus, the number of worker threads is currently assumed to be known at static analysis time. In the case where a clear operation is used to clear an event before or after the wait is issued, our analysis will report more du-pairs than needed. In testing, this only implies that we indicate more test cases than really needed. We are in the process of examining these limitations, while experimentally analyzing the effectiveness of fault detection for parallel programs using the all-du-paths criterion with della pasta, and investigating other structural testing criteria for testing parallel programs Acknowledgements We would like to thank Barbara Ryder for her helpful comments in preparing the final paper. "The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expr essed or implied, of the Army Research Laboratory or the U.S. Government." --R A system to generate test data and symbolically execute programs. On two problems in the generation of program test paths. Using MPI: Portable Parallel Programming with the Message Passing Interface. Data flow equations for explicitly parallel programs. Efficient computation of interprocedural definition-use chains Automated software test data generation. Introduction to the SUIF compiler system. Data flow analysis of concurrent systems that use the rendezvous model of synchronization. On path cover problems in digraphs and applications to program testing. Interprocedural def-use associations for C systems with single level pointers Testing of concurrent software. Structural testing of concurrent programs. A formal framework for studying concurrent program testing. The evaluation of program-based software test data adequacy criteria The challenges in automated testing of multithreaded programs. An algorithm for all-du-path testing coverage of shared memory parallel programs Path analysis testing of concurrent programs. --TR The evaluation of program-based software test data adequacy criteria Automated Software Test Data Generation Path analysis testing of concurrent programs Structural Testing of Concurrent Programs Data flow equations for explicitly parallel programs Efficient computation of interprocedural definition-use chains Automatic Generation of Path Covers Based on the Control Flow Analysis of Computer Programs Using MPI Interprocedural Def-Use Associations for C Systems with Single Level Pointers An Algorithm for All-du-path Testing Coverage of Shared Memory Parallel Programs --CTR Arkady Bron , Eitan Farchi , Yonit Magid , Yarden Nir , Shmuel Ur, Applications of synchronization coverage, Proceedings of the tenth ACM SIGPLAN symposium on Principles and practice of parallel programming, June 15-17, 2005, Chicago, IL, USA C. Michael Overstreet, Improving the model development process: model testing: is it only a special case of software testing?, Proceedings of the 34th conference on Winter simulation: exploring new frontiers, December 08-11, 2002, San Diego, California John Penix , Willem Visser , Seungjoon Park , Corina Pasareanu , Eric Engstrom , Aaron Larson , Nicholas Weininger, Verifying Time Partitioning in the DEOS Scheduling Kernel, Formal Methods in System Design, v.26 n.2, p.103-135, March 2005

parallel programming;all-du-path coverage;testing tool

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The Static Parallelization of Loops and Recursions.

We demonstrate approaches to the static parallelization of loops and recursions on the example of the polynomial product. Phrased as a loop nest, the polynomial product can be parallelized automatically by applying a space-time mapping technique based on linear algebra and linear programming. One can choose a parallel program that is optimal with respect to some objective function like the number of execution steps, processors, channels, etc. However,at best,linear execution time complexity can be atained. Through phrasing the polynomial product as a divide-and-conquer recursion, one can obtain a parallel program with sublinear execution time. In this case, the target program is not derived by an automatic search but given as a program skeleton, which can be deduced by a sequence of equational program transformations. We discuss the use of such skeletons, compare and assess the models in which loops and divide-and-conquer resursions are parallelized and comment on the performance properties of the resulting parallel implementations.

Introduction We give an overview of several approaches to the static parallelization of loops and recursions 1 , which we pursue at the University of Passau. Our emphasis in this paper is on divide-and- conquer recursions. Static parallelization has the following benefits: 1. Efficiency of the target code. One avoids the overhead caused by discovering parallelism at run time and minimizes the overhead caused by administering parallelism at run time. 2. Precise performance analysis. Because the question of parallelism is settled at compile time, one can predict the performance of the program more accurately. 3. Optimizing compilation. One can compile for specific parallel architectures. One limitation of static parallelization is that methods which identify large amounts of parallelism usually must exploit some regular structure in the source program. Mainly, this 1 We can equate loops with tail recursions, as is done in systolic design [28, 36]. structure concerns the dependences between program steps, because the dependences impose an execution order. Still, after a source program has been "adapted" to satisfy the requirements of the parallelization method, the programmer need think no more about parallelism but may simply state his/her priorities in resource consumption and let the method make all the choices. We illustrate static parallelization methods for recursive programs on the example of the polynomial product. We proceed in four steps: Sect. 2. We provide a specification of the polynomial product. This specification can be executed with dynamic parallelism. The drawback is that we have no explicit control over the use of resources. Sect. 3. We refine the specification to a double loop nest with additional dependences. We parallelize this loop nest with the space-time mapping method based on the polytope model [29]. This method searches automatically a large number of possible parallel implementations, optimizing with respect to some objective function like the number of execution steps, processors, communication channels, etc. Sect. 4. We refine the specification to a divide-and-conquer (D&C) algorithm which has fewer dependences than the loop nest. This is the central section of the paper. For D&C, parallelization methods are not as well understood as for nested loops. Thus, one derives parallel implementations by hand, albeit formally, with equational reasoning. However, most of the parallelization process is problem-independent. The starting point is a program schema called a skeleton [9]. We discuss two D&C skeletons, instantiated to the polynomial product, and their parallelizations: Subsect. 4.1. The first is a skeleton for call-balanced fixed-degree D&C, which we parallelize with an adapted space-time mapping method based on the method for nested loops [25]. The target is again a parallel loop nest, which can also be represented as an SPMD program. Subsect. 4.2. The second skeleton is a bit less general. It is parallelized based on the algebraic properties of its constituents [20]. It is used to generate coarser-grained parallelism in the form of an SPMD program. In this paper, we are mainly comparing and evaluating. The references cited in the individual sections contain the full details of the respective technical development. Our comparison is concerned with the models and methods used in the parallelization and with the asymptotic performance of the respective parallel implementations. 2 The polynomial product Our illustrating example is the product of two polynomials A and B of degree n, specified in the quantifier notation of Dijkstra [14]: Let us name the product polynomial C : Note that this specification does not prescribe a particular order of computation for the cumulative sums which define the coefficients of the product polynomial. We can make this specification executable without having to think any further. A simple switch of syntax to the programming language Haskell [40] yields: c a b Haskell will reduce the sums in some total order, given by its sequential semantics, or in some partial order, given by its parallel semantics. The programmer pays for the benefit of not having to choose the order with a lack of control over the use of resources in the computation. The main resources are time (the length of execution) and space (the number of processors), others are the number of communication channels, the memory requirements, etc. 3 A nested loop program 3.1 From the source program to the target program To apply loop parallelization, one must first impose a total order on the reductions in the specification. This means adding dependences to the dependence graph of the specification. The choice of order may influence the potential for parallelism, so one has to be careful. We choose to count the subscript of A up and that of B down; automatic methods can help in exploring the search space for this choice [3]. Another change we make is that we convert the updates of c to single-assignment form, which gives rise a doubly indexed variable - c; there are also automatic techniques for this kind of conversion [15]. This leads to the following program, in which the elements contain the final values of the coefficients of the product polynomial: for to n do do The dependence graph of a loop nest with affine bounds and index expressions forms a polytope, in which each loop represents one dimension. The vertices of graphs of this form can be partitioned into temporal and spatial components. This is done by linear algebra and integer linear programming. In our example, the choices are fairly obvious. Consider Fig. 1 for the polynomial product. The upper left shows the polytope on the integer lattice; each dot represents one loop step. The upper right is the dependence graph; only the dependences on - c are shown, since only - c is updated. The lines on the lower left represent "time slices": all points on a line can be executed in parallel. This choice is minimal with respect to execution time. Note that dependence arrows must not be covered by these lines! The lines in the lower right represent processors: a line contains the sequence of loop steps executed by a fixed processor. These lines may not be parallel to the temporal s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R @ @R source polytope data dependences- i s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ temporal partitioning spatial partitioning Figure 1: The polytope and its partitionings- t 2\Deltan s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s synchronous program: seqfor t := 0 to n do parfor p := t to t +n do asynchronous program: parfor p := 0 to 2 n do seqfor t := max(0; to min(n; p) do Figure 2: The target polytope and the target programs lines. We have chosen them to cover the dependences, i.e., we have minimized the number of communication channels between processor (to zero). The partitionings in time and space can be combined to a space-time mapping, an affine transformation to a coordinate system in which each axis is exclusively devoted to time (i.e., represents a sequential loop) or space, i.e. represents a parallel loop. This is like adjusting a "polarizing filter" on the source polytope to make time and space explicit. Fig. 2 depicts the target polytope and two target programs derived from it. Again, we have a choice of order, namely the order of the loops in the nest. If the outer loop is in time and the inner loop in space, we have a synchronous program with a global clock, typical in the SIMD style. To enforce the clock tick, we need a global synchronization after each step of the time loop. If we order the loops vice versa, we have an asynchronous program, with a private clock for each processor, typical in the SPMD style. Here, we need communication statements to enforce the data dependences, but we have chosen the spatial partitioning such that no communications are required-at the expense of twice the number of processors necessary. In both programs, the code of the loop body is the same as in the source program, only the indices change because the inverse of the space-time mapping must be applied: (i 3.2 Complexity considerations The most interesting performance criteria-at least the ones we want to consider here-are execution time, number of processors and overall cost. The cost is defined as the product of execution time and number of processors. One is interested in cost maintenance, i.e., in the property that the parallelization does not increase the cost complexity of the program. With the polytope model, the best time complexity one can achieve is linearity in the problem size: 2 at least one loop must enumerate time, i.e., be sequential. In the pure version of the model, one can usually get away with just one sequential loop [5]. The remaining loops enumerate space, i.e., are parallel. This requires a polynomial amount of processors since the loops bounds are affine expressions. The cost is not affected by the parallelization: one simply trades between time and space, the product of time and space stays the same. 3.3 Evaluation Let us sum up the essential properties of the polytope model for the purpose of a comparison: 1. The dependence graph can be embedded into a higher-dimensional space. The dimensionality is fixed: it equals the number of loops in the loop nest. 2. The extent of the dependence graph in each dimension is usually variable: it depends on the problem size. However, a very important property of the polytope model is that the complexity of the optimizing search is independent of the problem size. 3. Each vertex in the dependence graph represents roughly the same amount of work. More precisely, we can put a constant bound on the amount of work performed by any one vertex. 2 The only exception is the trivial case of no dependences at all, in which all iterations can be executed in one common step. 4. There is a large choice of problem-dependent affine dependences. Thus, given a loop nest with its individual dependences, an automatic optimizing search is performed to maximize parallelism. 5. One can save processors by "trading" one spatial dimension to time, i.e., emulating a set of processors by a single processor. 6. The end result is a nest of sequential and parallel loops with affine bounds and dependences 7. If one loop is sequential and the rest are parallel (which can always be achieved) [5], one obtains an execution time linear in the problem size. But, to save processors, one can trade space to time at the price of an increased execution time complexity. In particular, one can make the number of processors independent of the problem size by partitioning the resulting processor array into fixed-size "tiles" [13, 39]. 4 A divide-and-conquer algorithm Rather than enforcing a total order on the cumulative summation in specification (2) of the coefficients of the product polynomial, we can accumulate the summands with a D&C algorithm by exploiting the associativity of polynomial addition \Phi on the left side of the following We can write this more explicitly, showing the degrees and the variable of the polynomials. Let m=n div 2 and assume for the rest of this paper, for simplicity, that n is a power of 2: a The suffix l stands for "lower part", h for "higher part" of the polynomial; a and b are the input polynomials. The dependence graph of this algorithm is depicted in Fig. 3; c, d , e, and f , the resulting polynomials of the four subproblems. This is our starting point for a parallelization which will give us sublinear execution time. The fundamental difference in the parallelization of D&C as opposed to nested loops is that there is no choice of dependences: the dependence graph is always the same as the call graph. Since we have no problem-dependent dependences, we have no need for an automatic parallelization based on them. Instead, we can take the skeleton approach [9]: we can provide a program schema, a so-called algorithmic skeleton, for D&C which is to be filled in with further program pieces-we call them customizing functions-in order to obtain a D&C ap- plication. Then, our task is to offer for the D&C skeleton one or several high-quality parallel implementations, we call these architectural skeletons. This may, again, involve a search, but the search space is problem-independent and, thus, need not be redone for every application. For the user at the application end, the only challenge that remains is to cast the problem in the form of the algorithmic skeleton. Alternatively, the user might develop an architectural skeleton with even better performance by exploiting problem-specific properties of his/her application. combine divide recursion ah bh ah bl al bh al ch cl dh dl eh el fh fl ch cl fh fl bl dh dl el eh al bh bl Figure 3: Call graph of the D&C polynomial product Research on the parallelization of D&C is at an earlier stage than that of nested loops. Many different algorithmic skeletons-and even more architectural skeletons-can be envi- sioned. No common yardstick by which to evaluate them has been found as of yet. We discuss two algorithmic skeletons and the respective approaches to their parallelization. 4.1 Space-time mapping D&C 4.1.1 From the Source Program to the Target Program The call graph in Fig. 3 matches an algorithmic skeleton for call-balanced fixed-degree D&C which we have developed. The skeleton is phrased as a higher-order function in Haskell [25]: divcon divcon k basic divide where solve indata = if length indata j 1 then map basic indata else let solve (transpose x in (-l :map fst l ++ map snd l ) (map combine y) else error "list length" right Let us comment on a few functions which you will see again in this paper: map op xs applies a unary operator op to a list xs of values and returns the list of the results; map is one main source of parallelism in higher-order programs. xs++ys returns the concatenation of list xs with list ys. zip xs ys "zips" the lists xs and ys to a list of corresponding pairs of elements. The further details of the body of divcon are irrelevant to the points we want to make in this paper-and indeed irrelevant to the user of the skeleton. All that matters is that it is a higher-order specification, which specifies a generalized version of the schema depicted in Fig. 3, and whose parameters k , basic, divide and combine the caller fills with the division degree and with appropriate functions for computing in the basic case and dividing and combining in the recursive case. The example below shows, using the names in Fig. 3, how to express the polynomial product in terms of the divcon skeleton. The polynomials to be multiplied have to be represented as lists of their coefficients in order: x where preadapt x postadapt fst z ++ map snd z divide (ah; bh) (al combine [(ch; cl); (dh; dl); (eh; el); (fh ; fl )] The skeleton takes as input and delivers as output lists of size n. The operands and result of the polynomial product have to be formatted accordingly: function preadapt zips both input polynomials to a single list, and function postadapt unpacks the zipped higher and lower parts for the result again. Given unlimited resources, it is clear without a search what the temporal and the spatial partitioning should be: each horizontal layer of the call graph should be one time slice. This seems to suggest a two-dimensional geometrical model with one temporal axis (pointing down) and one spatial axis (pointing sideways). However, it pays to convert the call graph to a higher-dimensional structure. The reason is that the vertices in the graph represent grossly unequal amounts of work. In other words, the amount of work of any one vertex cannot be capped by a constant: because of the binary division of data, a node in some fixed layer of the divide phase represents double the amount of work as a node in the layer below, and the reverse applies for the combine phase. This behaviour holds for all algorithms, which fit into this skeleton. We obtain a graph in which the work a node represents is bounded by a constant if we split a node which works on aggregate data into a set of nodes each of which works on atomic data only. This fragmentation of nodes is spread across additional dimensions, yielding the higher-dimensional graph of Fig. 4. Time now points into depth and, for the given size of the call graph, each time slice is two-dimensional, not just one-dimensional. With increasing problem size, further spatial dimensions are added. A parallel loop program, which scans this graph in a similar manner as it would scan a polytope, can be derived [25]; here, r is the number of recursive calls log n), the elements reading input data divide in dim 0 solve basic cases divide in dim 1 combine in dim 0 result, depth dimension combine in dim 1 Figure 4: Higher-dimensional call graph of the D&C polynomial product of AB are pairs of input coefficients, and the elements of C are pairs of output coefficients of the higher and lower result polynomial: seqfor to r do parfor parfor seqfor parfor The program consists of a sequence of three loop nests, for the divide, sequential, and combine phase. The loops on d enumerate the levels of the graph and are therefore sequential, whereas the loops on q are parallel because they enumerate the spatial dimensions. The spatial dimensions are indexed by the digits of q in radix k representation. This allows us to describe iterations across an arbitrary number of dimensions by a single loop, which makes the text of the program independent of the problem size. q (k) denotes the vector of the digits of q in radix k . q (k) [d ] selects the dth digit. In accesses of the values of the points whose index differs only in dimension d , we use the notation (q (k) the number, which one obtains from q by replacing the dth digit by i . This representation differs from target programs in the polytope model, in which each dimension corresponds to a separate loop. The data is indexed with a time component d and a space component q . The divide and combine functions are given the actual coordinate as a parameter in order to select the appropriate functionality for the particular subproblem for divide resp. data partition for combine. Such loop programs can also be derived formally by equational reasoning [26]. This program is data-parallel. Therefore, it can be implemented directly on SIMD ma- chines. Additionally, the program can be transformed easily to an SPMD program for parallel machines with distributed memory using message passing. The two-dimensional arrays now become one-dimensional, because the space-component has been projected out by selecting a particular processor. The all-to-all communications are restricted to groups of k processors. Processor q executes the following: seqfor to r do all-to-all (list of values received by all-to-all) ; seqfor all-to-all (list of values received by all-to-all) One is interested in transforming the computation domain in Fig. 4 further in two ways: 1. As in loop parallelization, this approach has the potential of very fine-grained paral- lelism. As in loop parallelization, spatial dimensions can be moved to time to save processors. This is more urgent here, since the demand for processors grows faster with increasing problem size. 2. If the spatial part of the computation domain remains of higher dimensionality after this, its dimensionality can be reduced as depicted in Fig. 5. This is done, e.g., if the target processor topology is a mesh. It works because the extent of each dimension is fixed. x x y z Figure 5: Dimensionality reduction of the computation domain 4.1.2 Complexity Considerations For the implementation, it is not very efficient to assign each basic problem to a separate physical processor. Instead, spatial dimensions are mapped to time. The result is s slightly modified SPMD program. In brief, the operations on single elements are replaced by operations on segments of data. Let n be the size of the polynomials, and P the number of processors. In the basic phase on segments, the work is equally distributed among the processors, i.e., each processor is responsible for n=2 log P= log elements. The time for computing the basic phase on segments is therefore O(n 2 =P). The computation in the divide and combine phase takes O(log n) steps. In each step, a segment of size n= P has to be divided or combined in parallel. The entire computation time for both phases is therefore in O(log n \Deltan = if the dimensionality of the target mesh equals the number of dimensions mapped to space, the total time is in: O The execution time is sublinear if the number of processors is asymptotically greater than the problem size, and it maintains the cost of O(n 2 ) if the number of processors is asymptotically not greater than O(n n). The best execution time, which can be achieved under cost maintenance is O(log 2 n). If the dimensionality of the target topology is taken into account, our calculations have revealed the following: 1. Sublinear execution time can only be achieved if the dimensionality is at least 3. 2. The computation is sublinear and cost-maintaining for a cubic three-dimensional mesh if the number of processors is asymptotically between n and n 1:2 . There is an algorithm for polynomial product with a sequential time complexity of O(n log 3 ), the so-called Karatsuba algorithm [1, Sect. 2.6]. It has a division degree of 3 and can be expressed with our skeleton, with whose parallel implementation [25] the algorithm is cost- maintaining for reasonable problem sizes. Our experiments have shown that the sequential version of the Karatsuba algorithm beats the sequential version of our conventional algorithm if both polynomials have a size of at least 16. Since its subproblems are slightly more computation-intensive, the parallel version of the Karatsuba algorithm (with our skeleton) is a bit slower than the parallel version of the conventional algorithm, but one saves processors (precisely, 4.1.3 Evaluation What are the properties of this space-time mapping model, compared with the polytope model of the previous section? 1. The dimensionality of the call graph is variable: it equals the number of layers of the divide phase, which depends on the problem size. 2. The extent of the dependence graph in each spatial dimension is fixed: it is the degree of the problem division. 3. Each vertex in the call graph represents the same amount of work. 4. There is no choice of dependences and no search for parallelism is necessary. 5. The only variety in parallelism is given by the option to trade off spatial dimensions to time. 6. The end result is, again, a nest of sequential and parallel loops. 7. The upper bound of the temporal loop is logarithmic in the problem size, and the upper bound of the spatial loop is exponential in the upper bound of the temporal loop, i.e., polynomial in the problem size. When looking at the computation domain (Fig. 4), the extent of each spatial dimension is constant, but the number of spatial dimensions grows with the problem size. Sublinear execution time (in a root of the problem size) are possible on mesh topologies, but the conditions for maintaining cost-optimality in this case are very restrictive. 4.2 Homomorphisms A very simple D&C skeleton is the homomorphism. It does not capture all D&C situations, and it is defined most often for lists [7, 37], although it can also be defined for other data structures, e.g., trees [17] and arrays [33]. 4.2.1 From the source program to the target program Unary function h is a list homomorphism [7] iff its value on a concatenation of two lists can be computed by combining the values of h on the two parts with some operation fi : The significance of homomorphisms for parallelization is given by the promotion property, a version of which is as follows: red dist (6) This equality is also proved by equational reasoning. In the literature, one has used the Bird- Meertens formalism (BMF) [7], an equational theory for functional programs in which red and map are the basic functions on lists: red reduces a list of values with a binary operator (which, in our case, inherits associativity from list concatenation) and returns the result value, and map we have seen in the previous subsection. Both red and map have a high potential for parallelism: red can be performed in time logarithmic in the length of the list and map can be performed in constant time, given as many processors as there are list elements. The third function appearing in the promotion property, dist (for distribute), partitions an argument list into a list of sublists; it is the right inverse of reduction with concatenation: red (++) Equality (6) reveals that every homomorphism h can be computed in three stages: (1) an input list is distributed, (2) function h is computed on all sublists independently, (3) the results are combined with operator fi . The efficiency of this parallel implementation depends largely on the form of operation fi . E.g., there is a specialization of the homomorphism skeleton, called DH (for distributable homomorphism), for which a family of practical, efficient implementations exists [19, 21]. The similarity of (3) and (5) is obvious: h should be the polynomial product fi, and operation fi should be polynomial addition \Phi. However, there are two mismatches: 1. Operations fi and \Phi are defined on polynomials of possibly different degree. Thus, the list representation of a polynomial needs to be refined to a pair (int ; list ), where int is the power of the polynomial and list is the list of the coefficients. For simplicity, we ignore this subtlety in the remainder of this paper. 2. A more serious departure from the given homomorphism skeleton is that polynomial product fi, our equivalent of h, requires two arguments, not one. To match this with the skeleton, we might give h a list of coefficient pairs, but this destroys the homomorphism property: in the format provided by h, the product of two polynomials cannot be constructed from the products of the polynomials' halves. It is just as well that we cannot fit the (binary) polynomial product into the (unary) homomorphism skeleton. The second argument gives us an additional dimension of parallelism which the unary homomorphism cannot offer: because we have two arguments each of whom is to be divided, we obtain four partial results to be combined rather than two, as prescribed by the skeleton. To exploit the quadratic parallelism, we use a different, binary homomorphism skeleton: Now, the polynomial product fits perfectly. The resulting promoted, two-dimensional skeleton does everything twice-once for each dimension: dividing (with dist ), computing in parallel (with map) and combining (with red ); we write map 2 f for map (map f ) and zip 2 for where dist \Theta map (copy) ffi dist) (8) The complication of distribute is due to the fact that list portions must be spread out across two dimensions now. Note also that we have not provided a definition of the two-dimensional reduction: red 2 (fi ) can be computed in two orders: row-major or column-major. Actually, each of the two choices leads to an equal amount of parallelism. However, by a problem-specific optimization of the combine stage, we can do even bet- ter. Fig. 6 depicts this optimized solution on a virtual square of processors (we make no assumptions about the physical topology). Exploiting the commutativity of the customizing operator \Phi, we can reduce first along the diagonals-the corresponding polynomials have equal power-and then reduce the partial results located at the northern and eastern bor- ders. The latter step can be improved further to just pairwise exchange between neighbouring processors if we allow for the result polynomial product to be distributed blockwise along the border processors. The three-stage format of the promoted homomorphism skeleton suggests an SPMD program which also has three stages. For the binary homomorphism, optimized for the polynomial product, every processor q executes the following MPI-like program: distribute compute combine Figure Three stages of the two-dimensional promoted homomorphism skeleton broadcast (A) in row ; broadcast (B) in column ; reduce (\Phi) in diagonal(q) ; exchange-neighbours This program does not show loops explicitly-but, of course, they are there. The outer, spatial loops are implicit in the SPMD model and the inner, sequential loop is hidden in the call of function PolyProd which is a sequential implementation of fi. With MPI-like communications, this SPMD program could be the point where the programmer stops the refinement and the machine vendor takes over. Alternatively, the user can him/herself program broadcasts, reductions and exchanges with neighbours and define a suitable physical processor topology, e.g., a mesh of trees [18]. 4.2.2 Complexity considerations We consider multiplying two polynomials of degree n on a virtual square of p 2 processors. The time complexity with pipelined broadcasting and reduction is [18]: \Deltam (n; p) where m(n; p+1)). The value of p can be chosen between 1 and n. If p =n= log n then t =O(logn), which yields the optimal, logarithmic time complexity on processors. The cost is O(n 2 ), and is thus maintained. Other interesting cases are: so the parallelization has not worsened the sequential cost; =O(log n) on n 2 processors; this solution yields the optimal time but is clearly not cost-maintaining; log n: a cost-maintaining solution with t =O(log 2 n) on (n= log n) 2 processors; n: t =O(n) on n processors: equal to the systolic solution of Sect. 3 and cost- maintaining. In practice, the processor number is an arbitrary fixed value and the problem size n is relatively large. Then the term (n=p) 2 dominates in the expression of the time complexity which guarantees a so-called scaled linear speed-up [35]. This term can be improved to O((n=p) log 3 ) or to O((n=p)\Delta log(n=p)) by applying the Karatsuba or FFT-based algorithm, respectively, in the processors at the compute stage. Whether the Karatsuba algorithm can be phrased as a homomorphism is an open question. 4.2.3 Evaluation Let us review the parallelization process in the homomorphism approach. Actually, it is not very different from the skeleton approach of the previous section. One just uses a different skeleton and is led in the parallelization by algebra rather than by geometry. 1. The homomorphism skeleton is more restrictive than the Haskell skeleton in the previous section, and also more restrictive than the earliest D&C skeleton [34], in which it corresponds to a postmorphism (see [21] for a classification of D&C skeletons). The strength of the homomorphism is its direct correspondence with a straight-forward three-stage SPMD program. For the polynomial product, it yields a parallel implementation which is both time-optimal and cost-maintaining. Time optimality cannot be achieved on a mesh topology with constant dimensional- ity. In the homomorphism approach, we obtain a topology-independent program with MPI-like communication primitives. The best implementations of these primitives on topologies such as the hypercube and the mesh of trees are time-optimal [27]. 2. As many other skeletons, the homomorphism skeleton can come in many different varieties: for unary, binary, ternary operations, for lists and other data structures. At the present state of our understanding, all these versions are developed separately. 3. Even if all these skeletons are available to the user, an adaptation problem remains. This is true for the previous approaches as well. E.g., in loop parallelization, a dependence which does not satisfy the restrictions of the model is replaced by a set of dependences which do [29]. In the previous subsection, we format the input and the output of polynomial product with adaptation functions to make it match with our Haskell skeleton. The homomorphic form of a problem may exist but be not immediately clear. An example is scan [20]. Other algorithms can be turned into a D&C and, further, into a homomorphic form with the aid of auxiliary functions [10, 38]. 4. The application of the promotion property gives us a parametrized granularity of parallelism which is controlled by the size of the chunks in which distribution function dist splits the list. Depending on the available number of processors, input data can be distributed more coarsely or finely, down to a single list element per processor. The only requirement on the number of processors in the case of the two-dimensional homomorphism is that it be a square, which is not as restrictive as in the skeleton of the previous subsection, where a power of the division degree k is required. Moreover, homomorphic solution is not restricted to polynomials whose length is a power of 2. 5. Note that the promotion property is only applied once-as opposed to the previous subsection, in which we parallelize at each level of the call graph. This decreases the amount of necessary communication. The number of parallelized levels depends on the choice of granularity; all remaining levels are captured by a call of the sequential implementation PolyProd of h2. This enables an additional optimization: processors can call an optimal sequential algorithm for the given problem, rather than the algorithm which was chosen for the parallelization. In our case, PolyProd can be the more efficient Karatsuba or FFT-based algorithm for the polynomial product. Summary Let us summarize the present state of the art of the static parallelization of loops and recur- sions, as illustrated with the polynomial product. Static parallelization works best for programs which exhibit some regular structure. The structural requirements can be captured by restrictions on the form of the program, but many applications will not satisfy these restrictions immediately. Thus, static parallelization is definitely not for "dusty decks". However, many algorithms can be put into the required form and parallelized. In partic- ular, certain computation-intensive application domains like image, signal, text and speech processing, numerical and graph algorithms are candidates for a static parallelization. Dense data structures hold more promise of regular dependences, but sparse data structures might also be amenable to processing with while loop nests or with less regular forms of parallel D&C. 5.1 Loop parallelization Loop parallelization is much better understood than the parallelization of divide-and-conquer. The polytope model has been extended significantly recently: 1. Dependences and space-time mappings may be piecewise affine (the number of affine pieces must be constant, i.e., independent of the problem size) [16]. 2. Loop nests may be imperfect (i.e., not all computations must be in the innermost loop) [16]. 3. Upper loop bounds may be arbitary and, indeed, unknown at compile time [23]. A consequence of (3) is that while loops can be handled [12, 22, 30]. This entails a serious departure from the polytope model. The space-time mapping of loops is becoming a viable component of parallelizing compilation [31]. Loop parallelizers that are based on the polytope model include Bouclettes [8], LooPo [24], OPERA [32], Feautrier's PAF, and PIPS [2]. However, recent sophisticated techniques of space-time mapping have not yet filtered through to commercial compilers. In particular, automatic methods for partitioning and projecting (i.e., trading space for time) need to be carried through to the code generation stage. Most large academic parallelizing compilation projects involve also loop parallelization techniques that are not phrased in (or even based on) the polytope model. Links to some of them are provided in the Web pages cited here. A good, unhurried introduction to loop parallelization with an emphasis on the polytope model is the book series of Banerjee [4, 5, 6]. 5.2 Divide-and-conquer parallelization For the parallelization of D&C, there is not yet a unified model, in which different choices of parallelization can be evaluated with a common yardstick and compared with each other. The empirical approach taken presently uses skeletons: algorithm patterns with a high potential for parallelism are linked with semantically equivalent architectural patterns which provide efficient implementations of these algorithms on available parallel machines. This approach makes fewer demands on compiler technology. However, it expects the support of a "systems programming" community which provides architectural skeletons for existing parallel machines. The application programmer can then simply look for the schema in a given skeleton library, and adapt his/her application to this schema. In the last couple of years, the development and study of skeletons has received an increasing amount of attention and a research community has been forming [11]. The skeleton approach can become a viable paradigm for parallel programming if 1. the parallel programming community manages to agree on a set of algorithmic skeletons which capture a large number of applications and are relatively easy to fill in, and 2. the parallel machine vendors community, or some application sector supporting it, succeeds in providing efficient implementations of these skeletons on their products. One can attempt to classify the best parallel implementations of some skeleton, which represents a popular programming paradigm, by tabulating special cases. We have done this for the paradigm of linear recursion [41]. The interesting special cases are copy , red and scan. Compositions of these cases can be optimized further. 5.3 Conclusions Ultimately, one will have to wait and see whether the static or some dynamic approach to parallelism will win the upper hand. Since parallelism stands for performance, the lack of overhead and the precision of the performance analysis are two things in favor of static as opposed to dynamic parallelism-for problems which lend themselves to a static parallelization. Acknowledgements This work received financial support from the DFG (projects RecuR and RecuR2 ) and from the DAAD (ARC and PROCOPE exchange programs). Thanks to J.-F. Collard for a reading and comments. The Parsytec GCel 1024 of the Paderborn Center for Parallel Computing used for performance measurements. --R The Design and Analysis of Computer Algorithms. PIPS: A framework for building interprocedural compilers Efficient exploration of nonuniform space-time transformations for optimal systolic array synthesis Loop Transformations for Restructuring Compilers: The Foundations. Loop Parallelization. Dependence Analysis. Lectures on constructive functional programming. Reference manual of the Bouclettes parallelizer. Algorithmic Skeletons: Structured Management of Parallel Computation. Parallel programming with list homomorphisms. Theory and practice of higher-order parallel programming Automatic parallelization of while-loops using speculative execution Regular partitioning for synthesizing fixed-size systolic arrays Predicate Calculus and Program Semantics. Array expansion. Automatic parallelization in the polytope model. Upwards and downwards accumulations on trees. From transformations to methodology in parallel program development: a case study. Systematic efficient parallelization of scan and other list homomorphisms. Systematic extraction and implementation of divide-and-conquer paral- lelism Formal derivation of divide-and-conquer programs: A case study in the multidimensional FFT's The Mechanical Parallelization of Loop Nests Containing while Loops. Classifying loops for space-time mapping The loop parallelizer LooPo. On the space-time mapping of a class of divide-and- conquer recursions Parallelization of divide-and-conquer by translation to nested loops Introduction to Parallel Computing: Design and Analysis of Algorithms. A view of systolic design. Loop parallelization in the polytope model. On the parallelization of loop nests containing while loops. Loop parallelization. OPERA: A toolbox for loop parallelization. A Constructive Theory of Multidimensional Arrays. An algebraic model for divide-and-conquer algorithms and its parallelism Parallel Computing. Systolic Algorithms and Architectures. Foundations of Parallel Programming. Applications of a strategy for designing divide-and-conquer algorithms Control generation in the design of processor arrays. Parallel implementations of combinations of broadcast --TR Applications of a strategy for designing divide-and-conquer algorithms Algorithms Array expansion Scans as Primitive Parallel Operations Predicate calculus and program semantics Control generation in the design of processor arrays Regular partitioning for synthesizing fixed-size systolic arrays Algorithmic skeletons Introduction to parallel computing Automatic parallelization of <italic>while</italic>-loops using speculative execution Foundations of parallel programming From transformations to methodology in parallel program development Haskell Dependence Analysis Loop Parallelization Loop Transformations for Restructuring Compilers The Design and Analysis of Computer Algorithms Systematic Extraction and Implementation of Divide-and-Conquer Parallelism Transformation of Divide MYAMPERSANDamp; Conquer to Nested Parallel Loops Classifying Loops for Space-Time Mapping Systematic Efficient Parallelization of Scan and Other List Homomorphisms Loop Parallelization in the Polytope Model Automatic Parallelization in the Polytope Model Upwards and Downwards Accumulations on Trees OPERA On the parallelization of loop nests containing while loops Parallel Implementations of Combinations of Broadcast, Reduction and Scan

polytope model;divide-and-conquer;higher-order function;SPMD;loop nest;homomorphism;parallelization;skeletons

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Approximate Inverse Techniques for Block-Partitioned Matrices.

This paper proposes some preconditioning options when the system matrix is in block-partitioned form. This form may arise naturally, for example, from the incompressible Navier--Stokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner. The storage requirements for these preconditioners may be much less than for incomplete LU factorization (ILU) preconditioners for tough, large-scale computational fluid dynamics (CFD) problems. The numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite.

Introduction Consider the block partitioning of a matrix A, in the form (1) where the blocking naturally occurs due the ordering of the equations and the variables. Matrices of this form arise in many applications, such as in the incompressible Navier-Stokes equations, where the scalar momentum equations and the continuity condition form separate blocks of equations. In the 2-D case, this is a system of the form A =B @ pC A =B @ f u f pC A (2) where u and v represent the velocity components, and p represents the pressure. Here, the B submatrix is a convection-diffusion operator, the F submatrices are pressure gradient operators, and the E submatrices are velocity divergence operators. Traditional techniques such as the Uzawa algorithm have been used for these problems, often because the linear systems that must be solved are much smaller, or because there are zeros or small values on the diagonal of the fully-coupled system. These so-called segregated approaches, however, suffer from slow convergence rates when compared to aggregated, or fully-coupled solution techniques. Another source of partitioned matrices of the form (1) is the class of domain decomposition methods. In these methods the interior nodes of a subdomain are ordered consecutively, subdomain after subdomain, followed by the interface nodes ordered at the end. This ordering of the unknowns gives rise to matrices which have the following structure: 0 Typically, the linear systems associated with the B matrix produced by this reordering are easy to solve, being the result of restricting the original PDE problem into a set of independent and similar PDE problems on much smaller meshes. One of the motivations for this approach is parallelism. This approach ultimately requires solution methods for the Schur complement S. There is a danger, however, that for general matrices, B may be singular after the reordering. Much work has been done on exploiting some form of blocking in conjunction with preconditioning. In one of the earlier papers on the subject, Concus, Golub, and Meurant introduce the idea of block preconditioning, designed for block-tridiagonal matrices whose diagonal blocks are tridiagonal. The inverses of tridiagonal matrices encountered in the approximations are themselves approximated by tridiagonal matrices, exploiting an exact formula for the inverse of a tridiagonal matrix. This was later extended to the more general case where the diagonal blocks are arbitrary [4, 17]. In many of these cases, the incomplete block factorizations are developed for matrices arising from the discretization of PDE's [2, 3, 7, 17, 19] and utilize approximate inverses when diagonal blocks need to be inverted. More recently, Elman and Silvester [13] proposed a few techniques for the specific case of the Stokes and Navier-Stokes problems. A number of variations of Block- preconditioners have also been developed [1, 9]. In these techniques the off-block diagonal terms are either neglected or an attempt is made to approximate their effect. This paper explores some preconditioning options when the matrix is expressed in block-partitioned form, either naturally or after some domain decomposition type re- ordering. The iterative method acts on the fully-coupled system, but the preconditioning has some similarity to segregated methods. This approach only requires preconditioning or approximate solves with submatrices, where the submatrices correspond to any combination of operators, such as reaction, diffusion, and convection. It is particularly advantageous to use the block-partitioned form if we know enough about the submatrices to apply specialized preconditioners, for example operator-splitting and semi-discretization, as well as lower-order discretizations. Block-partitioned techniques also require the sparse approximate solution to sparse linear systems. These solutions need to be sparse because they form the rows or columns of the preconditioner, or are used in further computations. Dense solutions here will cause the construction or the application of the preconditioner to be too expensive. This problem is ideally suited for sparse approximate inverse techniques. The approximate solution to the sparse system is found by using an iterative method implemented with sparse matrix-sparse vector and sparse vector-sparse vector operations. The intermediate and final solutions are forced to be sparse by numerically dropping elements in x with small magnitudes. If the right-hand- side b and the initial guess for x are sparse, this is a very economical method for computing a sparse approximate solution. We have used this technique to construct preconditioners based on approximating the inverse of A directly [6]. This paper is organized as follows. In Section 2 we describe the sparse approximate inverse algorithm and some techniques for finding sparse approximate solutions with the Schur complement. Section 3 describes how block-partitioned factorizations may be used as preconditioners. The most effective of these are the approximate block LU factorization and the approximate block Gauss-Seidel preconditioner. Section 4 reports the results of several numerical experiments, including the performance of the new preconditioners on problems arising from the incompressible Navier-Stokes equations. Sparse approximate inverses and their use It is common when developing preconditioners based on block techniques to face the need to compute an approximation to the inverse of a sparse matrix or an approximation to columns of the f in which both B and f are sparse. This is particularly the case for block preconditioners for block-tridiagonal matrices [7, 19]. For these algorithms to be practical, they must provide approximations that are sparse. A number of techniques have recently been developed to construct a sparse approximate inverse of a matrix, to be used as a preconditioner [5, 6, 8, 10, 15, 17, 18]. Many of these techniques approximate each row or column independently, focusing on (in the column-oriented case) the individual minimizations where e j is the j-th column of the identity matrix. Such a preconditioner is distinctly easier than most existing preconditioners to construct and apply on a massively parallel computer. Because they do not rely on matrix factorizations, these preconditioners often are complementary to ILU preconditioners [6, 22]. Previous approaches select a sparsity pattern for x and then minimize (4) in a least squares sense. In our approach, we minimize (4) with a method that reduces the residual norm at each step, such as Minimal Residual or FGMRES [20], beginning with a sparse initial guess. Sparsity is preserved by dropping elements in the search direction or current solution at each step based on their magnitude or criteria related to the residual norm reduction. The final number of nonzeros in each column is guaranteed to be not more than the parameter lfil. In the case of FGMRES, the Krylov basis is also kept sparse by dropping small elements. To keep the iterations economical, all computations are performed with sparse matrix-sparse vector or sparse vector-sparse vector operations. For our application here, we point out that the approximate inverse technique for each column may be generalized to find a sparse approximate solution to the sparse linear problem by minimizing possibly with an existing preconditioner M for A. 2.1 Approximate inverse algorithm We describe a modification of the technique reported in [6] that guarantees the reduction of the residual norm at each minimal residual step. Starting with a sparse initial guess, the fill-in is increased by one at each iteration. At the end of each iteration, it is possible to use a second stage that exchanges entries in the solution with new entries if this causes a reduction in the residual norm. Without the second stage, entries in the solution cannot be annihilated once they have been introduced. For the problems in this paper, however, this second stage has not been necessary. In the first stage, the search direction d is derived by dropping entries from the residual direction r. So that the sparsity pattern of the solution x is controlled, d is chosen to have the same sparsity pattern as x, plus one new entry, the largest entry in absolute value. Minimization is performed by choosing the steplength (Ad; Ad) and thus the residual norm for the new solution is guaranteed to be not more than the previous residual norm. The solution and the residual is updated at the end of this stage. If A is indefinite, the normal equations residual direction A T r may be used as the search direction, or simply to determine the location of the new fill-in. It is interesting to note that the largest entry in A T r gives the greatest residual norm reduction in a one-dimensional minimization. This explains why a transpose initial guess for the approximate inverse combined with self-preconditioning (preconditioning r with the current approximate inverse) is so effective for some problems [6]. There are many possibilities for the second stage. We choose to drop one entry in x and introduce one new entry in d if this causes a decrease in the residual norm. The candidate for dropping is the smallest absolute nonzero entry in x. The candidate to be added is the largest absolute entry in the previous search direction (at the beginning of stage 1) not already included in d. The previous direction is used so that the candidate may be determined in stage 1, and an additional search is not required. The steplength fi is chosen by minimizing the new residual norm where e i is the i-th coordinate vector, x s is the entry in x to be dropped at position s while fi is the entry to be added at position l (largest), and we have generalized the notation so that b is the right-hand-side vector, previously denoted m j . Let A j denote the j-th column of A. Then the minimization gives which just involves one sparse SAXPY since b \Gamma Ax is already available as r, and one sparse dot-product, since we may scale the columns of A to have unit 2-norm. It is guaranteed that s 6= l since l is chosen from among the entries not including s. The preconditioned version of the algorithm for minimizing kb \Gamma Axk 2 with explicit preconditioner M may be summarized as follows. A is assumed to be scaled so that its columns all have unit 2-norm. The number of inner iterations is usually chosen to be lfil or somewhat larger. Algorithm 2.1 Approximate inverse algorithm 1. Starting with some initial guess x, r := b \Gamma Ax 2. For do 3. t := Mr 4. Choose d to be t with the same pattern as x; one entry which is the largest remaining entry in absolute value 5. q := Ad 6. ff := (r;q) 7. r := r \Gamma ffq 8. x 9. s := index of smallest nonzero in abs(x) 10. l := index of largest nonzero in abs(t \Gamma d) 11. fi := (r 12. ~ r := r 13. If k~rk ! krk then 14. Set x s := 0 and x l := fi 15. r := ~ r 16. End if 17. End do 2.2 Sparse solutions with the Schur complement Sparse approximate solutions with the Schur complement are often required in the preconditioning for block-partitioned matrices. We will briefly describe three approaches in this section: (1) approximating S, (2) approximating S \Gamma1 , and (3) exploiting a partial approximate inverse of A. 2.2.1 Approximating S To approximate S with a sparse matrix, we can use ~ where Y is computed by the approximate inverse technique, possibly preconditioned with whatever we are using to solve with B. Since Y is sparse, ~ S computed this way is also sparse. Moreover, since S is usually relatively dense, solving with ~ S is an economical approach. Typically, a zero initial guess is used for Y . We remark that it is usually too expensive to form Y by solving B approximately and then dropping small elements, since it is rather costly to search for elements to drop. We also note that we can generate ~ column-by-column, and if necessary, compute a factorization of ~ S on a column-by-column basis as well. The linear systems with ~ S can be solved in any fashion, including with an iterative process with or without preconditioning. 2.2.2 Approximating S \Gamma1 Another method is to compute an approximation to S \Gamma1 using the idea of induced pre- conditioning. is the (2,2) block of \GammaS we can compute a sparse approximation to it by using the approximate inverse technique applied to the last block-column of A and then throwing away the upper block. In practice, the upper part of each column may be discarded before computing the next column. In our experiments, since the approximate inverse algorithm is applied to A, an indefinite matrix in most of the problems, the normal equations search direction A T r is used in the algorithm, with a scaled identity initial guess for the inverse. 2.2.3 Partial approximate inverse A drawback of the above approach is that the top submatrix of the last block-column is discarded, and that the resulting approximation of S \Gamma1 may actually contain very few nonzeros. A related technique is to compute the partial approximate inverse of A in the last block-row. This technique does not give an approximation to S \Gamma1 , but defines a simple preconditioning method itself. Writing the inverse of A in the form, we can then get an approximate solution to A y with f It is not necessary to solve accurately with B. Again, the normal equations search direction is used for the approximate inverse algorithm in the numerical experiments. Some results of this relatively inexpensive method will be given in Section 4. Block-partitioned factorizations of A We consider a sparse linear system which is put in the block form, y For now the only condition we require on this partitioning is that B be nonsingular. We use extensively the following block LU factorization of A, I in which S is the Schur complement, As is well-known, we can solve (12) by solving the reduced system, to compute y, and then back-substitute in the first block-row of the system (11) to obtain x, i.e., compute x by The above block structure can be exploited in several different ways to define preconditioners for A. Thus, the block preconditioners to be defined in this section combine one of the preconditioners for S seen in Section 2.2 and a choice of a block factorization. Next, we describe a few such options. 3.1 Solving the preconditioned reduced system A method that is often used is to solve the reduced system (14), possibly with the help of a certain preconditioner M S for the Schur complement matrix S. Although this does not involve any of the block factorizations discussed above, it is indirectly related to it and to other well-known algorithms. For example, the Uzawa method which is typically formulated on the full system, can be viewed as a Richardson (or fixed point) iteration applied to the reduced system. The matrix S need not be computed explicitly; instead, one can perform the matrix-vector product with the matrix S, via the following sequence of operations: 1. Compute 2. Solve 3. Compute If we wish to use a Krylov subspace technique such as GMRES on the preconditioned reduced system, we need to solve the systems in Step 2, exactly, i.e., by a direct solver or an iterative solver requiring a high accuracy. This is because the S matrix is the coefficient matrix of the system to be solved, and it must be constant throughout the GMRES iteration. We have experimented with this approach and found that this is a serious limitation. Convergence is reached in a number of steps which is typically comparable with that obtained with methods based on the full matrix. However, each step costs much more, unless a direct solution technique is used, in which case the initial LU factorization may be very expensive. Alternatively, a highly accurate ILU factorization can be employed for B, to reduce the cost of the many systems that must be solved with it in the successive outer steps. 3.2 Approximate block diagonal preconditioner One of the simplest block preconditioners for a matrix A partitioned as in (1) is the block-diagonal matrix in which MC is some preconditioning for the matrix C. If as is the case for the incompressible Navier-Stokes equations, then we can define I for example. An interesting particular case is when C is nonsingular and MC = C. This corresponds to a block-Jacobi iteration. In this case, we have the eigenvalues of which are the square roots of the eigenvalues of the matrix C Convergence will be fast if all these eigenvalues are small. 3.3 Approximate block LU factorization The block factorization (12) suggests using preconditioners based on the block LU factor- ization in which and to precondition A. Here M S is some preconditioner to the Schur complement matrix S. If we had a sparse approximation ~ S to the Schur complement S we could compute a preconditioning matrix M S to ~ S, for example, in the form of an approximate LU factorization. We must point out here that any preconditioner for S will induce a preconditioner for A. As was discussed in Section 3.1 a notable disadvantage of an approach based on solving the reduced system (14) by an iterative process is that the action of S on a vector must be computed very accurately in the Krylov acceleration part. In an approach based on the larger system (11) this is not necessary. In fact any iterative process can be used for solving with M S and B provided we use a flexible variant of GMRES such as FGMRES [20]. Systems involving B may be solved in many ways, depending on their difficulty and what we know about B. If B is known to be well-conditioned, then triangular solves with incomplete LU factors may be sufficient. For more difficult B matrices, the incomplete factors may be used as a preconditioner for an inner iterative process for B. Further, if the incomplete factors are unstable (see Section 4.2), an approximate inverse for B may be used, either directly or as a preconditioner. If B is an operator, an approximation to it may be used; its factors may again be used either directly or as a preconditioner. This kind of flexibility is typical of what is available for using iterative methods on block-partitioned matrices. An important observation is that if we solve exactly with B then the error in this block ILU factorization lies entirely in the (2,2) block since, One can raise the question as to whether this approach is any better than one based on solving the reduced system (14) preconditioned with M S . It is known that in fact the two approaches are mathematically equivalent if we start with the proper initial guesses. Specifically, the initial guess should make the x-part of the residual vector equal to 0 for the original system (11), i.e., the initial guess is with This result, due to Eisenstat and reported in [16], immediately follows from (16) which shows that the preconditioned matrix has the particular form, Thus, if the initial residual has its x-component equal to zero then all iterates will be vectors with y components only, and a GMRES iteration on the system will reduce to a GMRES iteration with the matrix M \Gamma1 involving only the y variable. There are many possible options for choosing the matrix M S . Among these we consider the following ones. ffl no preconditioning on S. ffl precondition with the C matrix if it is nonsingular. Alternatively we can precondition with an ILU factorization of C. construct a sparse approximation to S and use it as a preconditioner. In general, we only need to approximate the action of S on a vector, for example, with the methods described in Sections 2.2.1 and 2.2.2. The following algorithm applies one preconditioning step to to get y Algorithm 3.1 Approximate block LU preconditioning 1. x 2. y 3. x := We have experimented with a number of options for solving systems with M S in step 2 of the algorithm above. For example, M S may be approximated with ~ computed by the approximate inverse technique. If this approximation is used, it is possible to also use Y in place of B \Gamma1 F in step 3. 3.4 Approximate block Gauss-Seidel By ignoring the U factor of the approximate block LU factorization, we are led to a form of block Gauss-Seidel preconditioning, defined by The same remarks on the ways to solve systems with B and ways to define the preconditioning matrix M S apply here. The algorithm for this preconditioner is the same as Algorithm 3.1 without step 3. To analyze the preconditioner, we start by observing that showing that the only difference with the preconditioned matrix (17) is the additional in the (1,2) position. The iterates associated with the block form and those of the associated Schur complement approach M \Gamma1 are no longer simply related. However there are a few connections between (17) and (19). First, the spectra of the two matrices are identical. This does not mean, however, that the two matrices will require the same number of iterations to converge in general. Consider a GMRES iteration to solve the preconditioned system M Here, we take an initial guess of the form in which x 0 is arbitrary. With this we denote the preconditioned initial residual by Then GMRES will find a vector u of the form belonging to the Krylov subspace which will minimize kM . For an arbitrary u in the affine space the preconditioned residual is of the form and by (19) this becomes, As a result, ks Note that ks represents the preconditioned residual norm for the reduced system for the y obtained from the approximation of the large system. We have ks which implies that if the residual for the bigger system is less than ffl, then the residual obtained by using a full GMRES on the associated preconditioned reduced system will also be less than ffl. We observe in passing that the second term in the right-hand-side of (21) can always be reduced to zero by a post-processing step which consists of forcing the first part of the residual to be zero by changing ffi (only) into: Equivalently, once the current pair x; y is obtained, x can be recomputed by satisfying the first block equation, i.e., This post-processing step requires only one additional B solve. Assume now that we know something about the residual vector associated with m steps of GMRES applied to the preconditioned reduced system. Can we say something about the residual norm associated with the preconditioned unreduced system? We begin by establishing a simple lemma. Lemma 3.1 Let Then, the following equality holds Proof. First, it is easy to prove that in which Y We now multiply both members of the above equality I \Gamma Z to obtain, 'We now state the main result concerning the comparison between the two approaches. Theorem 3.1 Assume that the reduced system (14) is solved with GMRES using the preconditioner M S starting with an arbitrary initial guess y 0 and let s the preconditioned residual obtained at the m-th step. Then the preconditioned residual vector r m+1 obtained at the (m 1)-st step of GMRES for solving the block system (11) preconditioned with the matrix M of (18) and with an initial guess u in which x 0 is arbitrary satisfies the inequality In particular if s Proof. The preconditioned matrix for the unreduced system is of the form (22) with S. The residual vector s m of the m-th GMRES approximation associated with the reduced system is of the form, in which ae m is the m-th residual polynomial, which minimizes kp(G)s 0 k 2 among all polynomials p of degree m satisfying the constraint: Consider the polynomial of degree m+ 1 defined by It is clear that The residual of um+1 , the m+1-st approximate solution obtained by the GMRES algorithm for solving the preconditioned unreduced system minimizes p(Z)r 0 over all polynomials p of degree m+ 1 which are consistent, i.e., such that Therefore, Using the equality established in the lemma, we now observe that The first matrix in the right-hand-side of the last equality is nothing but I \Gamma Z. Hence, the residual vector r m+1 is such that which completes the proof. 2 It is also interesting to relate the convergence of this algorithm to that of the block-diagonal approach in the particular case when M case corresponds to a block Gauss-Seidel iteration. We can exploit Young and Frankel's theory for 2-cyclic matrices to compare the convergence rates of this and the block Jacobi approach. Indeed, in this case, we have from (19) that Therefore, the eigenvalues of this matrix are the squares of those of matrix I associated with the block-Jacobi preconditioner of Section 3.2. 4 Numerical Experiments This section is organized as follows. In Section 4.1 we describe the test problems and list the methods that we use. In Section 4.2, we illustrate for comparison purposes the difficulty of incomplete LU factorizations for solving these problems in a fully-coupled manner. In Section 4.3, we make some comments in regard to domain decomposition types of reorderings. In Section 4.4 we show some results of the new preconditioners on a simple PDE problem. Finally, in Sections 4.5 and 4.6, we present the results of the new preconditioners on more realistic problems arising from the incompressible Navier-Stokes equations. Linear systems were constructed so that the solution is a vector of all ones. A zero initial guess for right-preconditioned FGMRES [20] restarted every 20 iterations was used to solve the systems. The Tables show the number of iterations required to reduce the residual norm by 10 \Gamma7 . The iterations were stopped when 300 matrix-vector multiplications were reached, indicated by a dagger (y). The codes were written in FORTRAN 77 using many routines from SPARSKIT [23], and run in single precision on a Cray C90 supercomputer. 4.1 Test problems and methods The first set of test problems is a finite difference Laplace equation with Dirichlet boundary conditions. Three different sized grids were used. The matrices were reordered using a domain decomposition reordering with 4 subdomains. In the following tables, n is the order of the matrix, nnz is the number of nonzero entries, nB is the order of the B submatrix, and nC is the order of the C submatrix. The second set of test matrices were extracted from the example incompressible Navier-Stokes problems in the FIDAP [14] package. All problems with zero C submatrix were tested. In the case of transient problems, the matrices are the Jacobians when the Newton iterations had converged. The matrices are reordered so that the continuity equations are Grid n nnz nB nC by 48 by 48 2209 10857 2116 93 by 64 3969 19593 3844 125 Table 1: Laplacian test problems. ordered last. The scaling of many of the matrices are poor, since each matrix contains different types of equations. Thus, we scale each row to have unit 2-norm, and then scale each column the same way. The problems are all originally nonsymmetric except 4, 12, 14 and 32. Matrix n nnz nB nC Hamel flow EX12 3973 79078 2839 1134 Stokes flow Surface disturbance attenuation EX23 1409 42761 1008 401 Fountain flow coating EX26 2163 74465 1706 457 Driven thermal convection EX28 2603 77031 1853 750 Two merging liquids species deposition Radiation heat transfer EX36 3079 53099 2575 504 Chemical vapor deposition Table 2: FIDAP example matrices. The third set of test problems is from a finite-element discretization of the square lid- driven cavity problem. Rectangular elements were used, with biquadratic basis functions for velocities, and linear discontinuous basis functions for pressure. We will show our results for problems with Reynolds number 0, 500, and 1000. All matrices arise from a mesh of 20 by 20 elements, leading to matrices of size having nnz =138,187 nonzero entries. These matrices have 3363 velocity unknowns, and 1199 pressure un- knowns. The matrices are scaled the same way as for the FIDAP matrices-the problems are otherwise very difficult to solve. We will use the following names to denote the methods that we tested. ILUT(nfil) and ILUTP(nfil) Incomplete LU factorization with threshold of nfil nonzeros per row in each of the L and U factors. This preconditioner will be described in Section 4.2. PAR(lfil) Partial approximate inverse preconditioner described in Section 2.2.3, using lfil nonzeros per row in M 2 . ABJ Approximate block-Jacobi preconditioner described in Section 3.2. This preconditioner only applies when C 6= 0. ABLU(lfil) Approximate block LU factorization preconditioner described in Section 3.3. The approximation (6) to S with lfil nonzeros per column of Y was used. ABLU y(lfil) Same as above, but using Y whenever B needs to be applied in step 3 of Algorithm 3.1. ABLU s(lfil ) Approximate block LU factorization preconditioner, using (7) to approximate S \Gamma1 with lfil nonzeros per column when approximating the last block column of the inverse of A. ABGS(lfil) Approximate block Gauss-Seidel preconditioner described in Section 3.4. The approximation (6) to S with lfil nonzeros per column of Y was used. The storage requirements for each preconditioner are given in Table 3. The ILUT preconditioner to be described in the next subsection requires considerably more storage than the approximate block-partitioned factorizations, since its storage depends on n rather than nC . Because the approximation to S \Gamma1 discards the upper block, the storage for it is less than lfil \Thetan C . The storage required for ~ S is more difficult to estimate since it is at least the product of two sparse matrices. It is generally less than 2 \Theta lfil \Theta nC ; Table 11 in Section 4.5 gives the exact number of nonzeros in ~ S for the FIDAP problems. 4.2 ILU for the fully-coupled system We wish to compare our new preconditioners with the most general, and in our experi- ence, one of the most effective general-purpose preconditioners for solving the fully-coupled system. In particular, we show results for ILUT, a dual-threshold, incomplete LU factorization preconditioner based on a drop-tolerance and the maximum number of new fill-in elements allowed per row in each L and U factor. This latter threshold allows the storage for the preconditioner to be known beforehand. Drop-tolerance ILU rather than level-fill ILU is often more effective for indefinite problems where numerical values play a much more important role. A variant that performs column pivoting, called ILUTP, is even more suitable for highly indefinite problems. Matrices Matrix locations ABJ none none S less than 2 \Theta lfil \Theta nC ABLU y(lfil) ~ lfil \Theta nC S less than 2 \Theta lfil \Theta nC Table 3: Storage requirements for each preconditioner. We use a small modification that we have found to often perform better and rarely worse on matrices that have a wide ranging number of elements per row or column. This arises for various reasons, including the fact that the matrix contains the discretization of different equations. Instead of counting the number of new fill-ins, we keep the nonzeros in each row of L and U fixed at nfil , regardless of the number of original nonzeros in that row. We also found better performance when keeping nfil constant rather than having it increase or decrease as the factorization progresses. If A is highly indefinite or has large nonsymmetric parts, an ILU factorization often produces unstable L and U factors, i.e., k(LU) \Gamma1 k can be extremely large, caused by the long recurrences in the forward and backward triangular solves [11]. To illustrate this point, we computed for a number of factorizations the rough lower bound where e is a vector of all ones. For the FIDAP example matrix EX07 modeling natural convection with order 1633 and 46626 nonzeros, we see in Table 4 that the norm bound increases dramatically as nfil is decreased in the incomplete factorization. GMRES could not solve the linear systems with these factorizations as the preconditioner. This matrix we chose is a striking example because it can be solved without preconditioning. log Table 4: Estimate of k(LU) \Gamma1 k1 from ILUT factors for EX07. To illustrate the difficulty of solving the FIDAP problems with ILUTP, we progressively allowed more fill-in until the problem could be solved, incrementing nfil in multiples of 10, with no drop tolerance. The results are shown in Table 5. For these types of prob- lems, it is typical that very large amounts of fill-in must be used for the factorizations to be successful. An iterative solution was not attempted if the LU condition lower bound was greater than 10 . If a zero pivot must be used, ILUT and ILUTP attempt to complete the factorization by using a small value proportional to the norm of the row. The matrices were taken in their original banded ordering, where the degrees of freedom of a node or element are numbered together. As discussed in the next subsection, this type of ordering having low bandwidth is often essential for an ILU-type preconditioning-many problems including these cannot be solved otherwise. Matrix nfil EX06 50 Table 5: nfil required to solve FIDAP problems with ILUTP. We should note that ILUTP is occasionally worse than ILUT. This can be alleviated somewhat by using a low value of mbloc, a parameter in ILUTP that determines how far to search for a pivot. In summary, indefinite problems such as these arising from the incompressible Navier-Stokes equations may be very tough for ILU-type preconditioners. 4.3 Domain decomposition reordering considerations Graph partitioners subdivide a domain into a number of pieces and can be used to give the domain decomposition reordering described in Section 1. This is a technique to impose a block-partitioned structure on the matrix, and adapts it for parallel processing, since B is now a block-diagonal matrix. This technique is also useful if B is highly indefinite and produces an unstable LU factorization; by limiting the size of the factorization, the instability cannot grow beyond a point for which the factorization is not useful. For general, nonsymmetric matrices, the partitioner may be applied to a symmetrized graph. In Table 6 we show some results of ILUT(40) on the Driven cavity problem with different matrix reorderings. We used the original unblocked ordering where the degrees of freedom of the elements are ordered together, the blocked ordering where the continuity equations are ordered last, and a domain decomposition reordering found using a simple automatic recursive dissection procedure with four subdomains. This latter ordering found nodes internal to the subdomains, and 882 interface nodes. Re. Unblocked Blocked DD ordered 1000 78 y 51 Table Effect of ordering on ILUT for Cavity problems. The poorer quality of the incomplete factorization for the Driven cavity problems in block-partitioned form is due to the poor ordering rather than instability of the L and U factors; in fact, zero pivots are not encountered. For the problem with Reynolds number 0, the unblocked format produces 745,187 nonzeros in the strictly lower-triangular part during the incomplete factorization (which is then dropped down to less than n\Theta nonzeros) while the block-partitioned format produces 2,195,688 nonzeros, almost three times more. The factorization for the domain decomposition reordered matrices encounters many zero pivots when it reaches the (2,2) block. These latter orderings do not necessarily cause ILUT to fill-in zeros on the diagonal. Nevertheless, the substitution of a small pivot described above seems to be effective here. The domain decomposition reordering also reduces the amount of fill-in because of the shape of the matrix (a downward pointing arrow). Combined with its tendency to limit the growth of instability, the results show this reordering is advantageous even on serial computers. In Table 7 we compare the difficulty of solving the B and ~ S subsystems for the blocked and domain decomposition reorderings of the Driven cavity problems. ~ S was computed as ~ computed using the approximate inverse technique with lfil of 30. Here we used ILUT(30) and only solved the linear systems to a tolerance of 10 \Gamma5 . Solves with these submatrices in the block-partitioned preconditioners usually need to be much less accurate. In most of the experiments that follow, we used unpreconditioned iterations to a tolerance of 10 \Gamma1 or 100 matrix-vector multiplications to solve with B and ~ S. Other methods would be necessary depending on the difficulty of the problems. The table gives an idea of how difficult it is to solve with B and ~ S, and again shows the advantage of using domain decomposition reorderings for hard problems. Re. Blocked DD ordered 1000 y y 7 Table 7: Solving with B and ~ S for different orderings of A. 4.4 Test results for the Laplacian problem In Tables 8 and 9 we present the results for the Laplacian problem with three different grid sizes, using no preconditioning, approximate block diagonal, partial approximate inverse, approximate block LU, and approximate block Gauss-Seidel preconditioners. Note that in Table 9, an lfil of zero for the approximate block LU and Gauss-Seidel preconditioners respectively indicate the preconditioners I and Grid NOPRE ABJ PAR by 48 by 48 367 50 29 21 19 17 by 64 532 57 36 33 25 20 Table 8: Test results for the Laplacian problem. Grid ABLU ABGS by 48 by 48 17 by 64 19 20 Table 9: Test results for the Laplacian problem. 4.5 Test results for the FIDAP problems For the block-partitioned factorization preconditioners, unpreconditioned GMRES, restarted every 20 iterations, was used to approximately solve the inner systems involving B and ~ by reducing the initial residual norm by a factor of 0.1, or using up to 100 matrix-vector multiplications. Solves with the matrix B are usually not too difficult because for most problems, it is positive definite. A zero initial guess for these solves was used. The results for a number of the preconditioners with various options are shown in Table 10. The best preconditioner appears to be ABLU using Y for B \Gamma1 F is better than solving a system with B very inaccurately. The number of nonzeros in ~ S is small, as illustrated by Table 11 for two values of lfil. Matrix Table 10: Test results for the FIDAP problems. 4.6 Test results for the Driven cavity problems The driven cavity problems are much more challenging because the B block is no longer positive definite, and in fact, acquires larger and larger negative eigenvalues as the Reynolds number increases. For these problems, the unpreconditioned GMRES iterations with B were done to a tolerance of 10 \Gamma3 or a maximum of 100 matrix-vector multiplications. Again, ABLU y appears to be the best preconditioner. The results are shown in Table lfil EX26 13395 21468 EX36 13621 21063 Table 11: Number of nonzeros in ~ S. ABLU ABLU y ABGS 1000 y y 164 118 y y Table 12: Test results for the Driven cavity problems. Conclusions We have presented a few preconditioners which are defined by combining two ingredients: (1) a sparse approximate inverse technique for obtaining a preconditioner for the Schur complement or a part of the inverse of A, and (2) a block factorization for the full sys- tem. The Schur complement S which appears in the block factorization is approximated by its preconditioner. Approximate inverse techniques [6] are used in different ways to approximate either S directly or a part of A \Gamma1 . As can be seen by comparing Tables 5 and 10, we can solve more problems with the block approach than with a standard ILU factorization. In addition, this is typically achieved with a far smaller memory requirement than ILUT or a direct solver. The better robustness of these methods is due to the fact that solves are only performed for small matrices. In effect, we are implicitly using the power of the divide-and-conquer strategy which is characteristic of domain decomposition methods. The smaller matrices obtained from the block partitioning can be preconditioned with a standard ILUT approach. The larger matrices use a block-ILU, and the glue between the two is the preconditioning of the Schur complement. Acknowledgements The authors wish to acknowledge the support of the Minnesota Supercomputer Institute which provided the computer facilities and an excellent environment to conduct this research. --R Incomplete block matrix factorization preconditioning methods. Iterative Solution Methods. On approximate factorization methods for block matrices suitable for vector and parallel processors. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Approximate inverse preconditioners for general sparse matri- ces Block preconditioning for the conjugate gradient method. Approximate inverse preconditioning for sparse linear systems. Parallelizable block diagonal preconditioners for the compressible Navier-Stokes equations A new approach to parallel preconditioning with sparse approximate inverses. A stability analysis of incomplete LU factorizations. Multigrid and Krylov subspace methods for the discrete Stokes equa- tions Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations FIDAP: Examples Manual Parallel preconditioning and approximate inverses on the Connection Machine. A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation. On a family of two-level preconditionings of the incomplete block factorization type Factorized sparse approximate inverse precon- ditionings I Incomplete block factorizations as preconditioners for sparse SPD matrices. A flexible inner-outer preconditioned GMRES algorithm ILUT: A dual threshold incomplete LU factorization. Preconditioned Krylov subspace methods for CFD applications SPARSKIT: a basic tool kit for sparse matrix computations --TR --CTR N. Guessous , O. Souhar, Multilevel block ILU preconditioner for sparse nonsymmetric M-matrices, Journal of Computational and Applied Mathematics, v.162 n.1, p.231-246, 1 January 2004 Kai Wang , Jun Zhang, Multigrid treatment and robustness enhancement for factored sparse approximate inverse preconditioning, Applied Numerical Mathematics, v.43 n.4, p.483-500, December 2002 Prasanth B. Nair , Arindam Choudhury , Andy J. Keane, Some greedy learning algorithms for sparse regression and classification with mercer kernels, The Journal of Machine Learning Research, 3, 3/1/2003 Edmond Chow , Michael A. Heroux, An object-oriented framework for block preconditioning, ACM Transactions on Mathematical Software (TOMS), v.24 n.2, p.159-183, June 1998 Howard C. Elman , Victoria E. Howle , John N. Shadid , Ray S. Tuminaro, A parallel block multi-level preconditioner for the 3D incompressible Navier--Stokes equations, Journal of Computational Physics, v.187 n.2, p.504-523, 20 May Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

sparse approximate inverse;block-partitioned matrix;navier-stokes;schur complement;preconditioning

272879

Circuit Retiming Applied to Decomposed Software Pipelining.

AbstractThis paper elaborates on a new view on software pipelining, called decomposed software pipelining, and introduced by Gasperoni and Schwiegelshohn, and by Wang, Eisenbeis, Jourdan, and Su. The approach is to decouple the problem into resource constraints and dependence constraints. Resource constraints management amounts to scheduling an acyclic graph subject to resource constraints for which an efficiency bound is known, resulting in a bound for loop scheduling. The acyclic graph is obtained by cutting some particular edges of the (cyclic) dependence graph. In this paper, we cut edges in a different way, using circuit retiming algorithms, so as to minimize both the longest dependence path in the acyclic graph, and the number of edges in the acyclic graph. With this technique, we improve the efficiency bound given for Gasperoni and Schwiegelshohn algorithm, and we reduce the constraints that remain for the acyclic problem. We believe this framework to be of interest because it brings a new insight into the software problem by establishing its deep link with the circuit retiming problem.

Introduction OFTWARE PIPELINING is an instruction-level loop scheduling technique for achieving high performance on processors such as superscalar or VLIW (Very Long Instruction Word) architectures. The main problem is to cope with both data dependences and resource constraints which make the problem NP-complete in general. The software pipelining problem has motivated a great amount of research. Since the pioneering work of Rau and Glaeser [1], several authors have proposed various heuristics [2], [3], [4], [5], [6] in various frameworks. An extended survey on software pipelining is provided in [7]. Recently, a novel approach for software pipelining, called decomposed software pipelining, has been proposed simultaneously by Gasperoni and Schwiegelshohn [8], and by Wang, Eisenbeis, Jourdan, and Su [9]. The idea is to decompose the NP-complete software pipelining problem into two subproblems: a loop scheduling problem ignoring resource constraints, and an acyclic graph scheduling prob- lem, for which efficient techniques (such as list scheduling for example) are well known. Although splitting the problem in two subproblems is clearly not an optimal strat- egy, Wang, Eisenbeis, Jourdan and Su have demonstrated, through an experimental evaluation on a few loops from the Livermore Benchmark Kernels, that such an approach P.-Y. Calland is supported by a grant of R'egion Rh"one-Alpes. A. Darte and Y. Robert are supported by the CNRS-ENS Lyon-INRIA project ReMaP. The authors are with Laboratoire LIP, URA CNRS 1398, Ecole Normale Sup'erieure de Lyon, F-69364 LYON Cedex 07, e-mail: [Pierre-Yves.Calland,Alain.Darte,Yves.Robert]@ens-lyon.fr is very promising with respect to time efficiency and space efficiency. In both approaches, the technique is to pre-process the loop data dependence graph (which may include cycles) by cutting some dependence edges. After the pre-processing, the modified graph becomes acyclic and classical scheduling techniques can be applied on it to generate the "pattern" (or "kernel") of a software pipelining loop. However, the way edges are cut is ad-hoc in both cases, and no general framework is given that explains which edges could and/or should be cut. The main contribution of this paper is to establish that this pre-processing of the data dependence graph has a deep link with the circuit retiming problem. The paper is organized as follows: in Section II, we described more precisely our software pipelining model. In Section III, we recall the main idea of decomposed software pipelining: we illustrate this novel technique through Gasperoni and Schwiegelshohn algorithm, and we show that decomposed software pipelining can be re-formulated in terms of retiming algorithms that are exactly the tools needed to perform the desired edges cut. Then, we demonstrate the interest of our framework by addressing two optimization problems: ffl In Section IV, we show how to cut edges so that the length of the longest path in the acyclic graph is minimized. With this technique, we improve the performance bound given for Gasperoni and Schwiegelshohn algorithm. ffl In Section V, we show how to cut the maximal number of edges, so as to minimize the number of constraints remaining when processing the acyclic graph. This criteria is not taken into account, neither in Gasperoni and Schwiegelshohn algorithm, nor in Wang, Eisenbeis, Jourdan and Su algorithm. Finally, we discuss some extensions in Section VI. We summarize our results and give some perspectives in Section VII. II. A simplified model for the software pipelining problem We first present our assumptions before discussing their motivations. A. Problem formulation In this paper, we consider a loop composed of several operations that are to be executed a large number of times on a fine-grain architecture. We assume both resource constraints and dependence constraints, but, compared to more general frameworks, we make the following simplifying hypotheses: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. XX, NO. Y, MONTH 1996 Resources. The architecture consists in p identical, non pipelined, resources. The constraint is that, in each cy- cle, the same resource cannot be used more than once. Dependences. Dependences between operations are captured by a doubly weighted graph ffi). V is the set of vertices of G, one vertex per operation in the loop. E is the set of edges in G. For each edge e in E, d(e) is a nonnegative integer, called the dependence distance. For each vertex u in V , ffi(u) is a nonnegative integer, called the delay. ffi and d model the fact that for each edge the operation v at iteration i has to be issued at least ffi(u) cycles after the start of the operation u at iteration We assume that the sum of dependence distances along any cycle is positive. These hypotheses are those given by Gasperoni and Schwiegelshohn in the non pipelined case. Before discussing the limitations of this simplified model, let us illustrate the notion of operations, iterations, delays, and dependence distance, with the following example. We will work on this example throughout the paper. ENDDO The loop has 6 operations (A, B, C, D, E, and F ) and N iterations, each operation is executed N times. N is a parameter, of unknown value, possibly very large. The associated graph G is given in Figure 1. Delays are depicted in square boxes, for example the delay of operation F is 10 times greater than the delay of operation A. Dependence distances express the fact that some computations must be executed in a specified order so as to preserve the semantics of the loop. For example, operation A at iteration k writes a(k), hence it must precede computation B at iteration which reads this value. This constraint is captured by the label equal to 2 associated to the edge (A; B) in the dependence graph of Figure 1. A F 121414Fig. 1. An example of dependence graph G The software pipelining problem is to find a schedule oe that assigns an issue time oe(u; for each operation instance operation u at iteration k). Each edge in the graph gives rise to a constraint for scheduling: Valid schedules are those schedules satisfying both dependence constraints (expressed by Equation 1) and resource constraints. Because of the regular structure of the software pipelining problem, we usually search for a cyclic (or modulo) scheduling oe: we aim at finding a nonnegative integer (called the initiation interval of oe) and constants c u such that oe(u; Because the input loop is supposed to execute many iterations (N is large), we focus on the asymptotic behavior of oe. The initiation interval is a natural performance estimator of oe, as 1= measures oe's throughput. Note that if the reduced dependence graph G is acyclic and if the target machine has enough processors, then can be zero (this type of schedule has infinite throughput). A variant consists in searching for a nonnegative rational a=b and to let oe(u; (with rational constants c u ). This amounts to unroll the input loop by a factor b. We come back on this variant in Section VI. Note also that rational cyclic schedules are dominant in the case of unlimited resources [4]. B. Limitations of the model Compared to more sophisticated models, where more general programs may be handled (such as programs with conditionals) and where more accurate architecture description can be given, our framework is very simple and may seem unrealistic. We (partly) agree but we would like to raise the following arguments: Delays. In many frameworks, the delay is defined on edges and not on vertices as in our model. We chose the latter for two reasons. First, we want to compare our technique to Gasperoni and Schwiegelshohn algorithm which uses delays on vertices. Second, we use graph retiming techniques that are also commonly defined with delays on vertices. How- ever, we point out that retiming in a more general model is possible, but technically more complex. See for example [10, Section 9]. Resources. Our architecture model, with non pipelined and identical resources, is very simple. The main reason for this restricted hypothesis is that we want to demonstrate, from a theoretical point of view, that our technique allows to derive an efficiency bound, as for Gasperoni and Schwiegelshohn algorithm. From a practical point of view, our technique can still be used, even for more sophisticated resource models. Indeed, the retiming technique that we use is independent of the architecture model, and can be seen as a pre-loop transfor- mation. Resource constraints are taken into account only in the second phase of the algorithm: additional features on the architecture can then be considered when scheduling the acyclic graph obtained after retiming. In particular, such a technique can be easily integrated, regardless of the architecture details, in a compiler that has an instruction scheduler. Extensions of the model. Decomposed software pipelining is still a recent approach for software pipelining. It has thus been studied first from a theoretical point of view, and only on restricted models. This is also the case in this paper. The whole problem is (not yet) well understood enough P.-Y. CALLAND, A. DARTE AND Y. ROBERT: CIRCUIT RETIMING APPLIED TO DECOMPOSED SOFTWARE PIPELINING 3 to allow more general architecture features be taken into account. However, we believe that our new view on the problem, in particular the use of retiming for controlling the structure of the acyclic graph, will lead in the future to more accurate heuristics on more sophisticated architecture models. III. Going from cyclic scheduling to acyclic scheduling Before going into the details of Gasperoni and Schwiegelshohn heuristic (GS for short), we recall some properties of cyclic schedules, and the main idea of decomposed software pipelining, so as to make the rest of the presentation clearer. A. Some properties of cyclic scheduling Consider a dependence graph d) and a cyclic schedule oe, oe(u; that satisfies both dependence constraints and resource constraints. Such a cyclic schedule is periodic, with period : the computation scheme is reproduced every units of time. More pre- cisely, if instance (u; is assigned to begin at time t, then instance will begin at time t + . Therefore, we only need to study a slice of clock cycles to know the behavior of the whole cyclic schedule in steady state. Let us observe such a slice, e.g. the slice SK from clock cycle K up to clock cycle (K enough so that the steady state is reached. Figure 2 depicts the steady state of a schedule, for the graph of Figure 1, with initiation interval time resources A,k F,k initiation interval Fig. 2. Successive slices of a schedule for graph G. Boxes in grey represent the operation initiated in the same slice SK . Now, for each perform the Euclidean division of c u by : c This means that one and only one instance of operation u is initiated within the slice SK : it is instance issued r u clock cycles after the beginning of the slice. The quantities r u and q u are similar to the row and column numbers introduced in [9]. If the schedule is valid, both resource constraints and dependence constraints are satisfied. Dependence constraints can be separated in two types depending on how they are either two dependent operation instances are initiated in the same slice SK (type 1) or they are initiated in two different slices (type 2). Of course, the partial dependence graph induced by type 1 constraints is acyclic, because type 1 dependences impose a partial order on the operations, according to the order in which they appear within the slice. In Figure 2, arrows represent pendences. All other dependences (not depicted in the fig- are type 2 dependences. The main idea of Gasperoni and Schwiegelshohn algorithm (GS), and more generally of decomposed software pipelining, is the following. Assume that we have a valid cyclic schedule of period 0 for a given number p 0 of pro- cessors, and that we want to deduce a valid schedule for a smaller number p of processors. A way of building the new schedule is to keep the same slice structure, i.e. to keep the same operation instances within a given slice. Of course we might need to increase the slice length to cope with the reduction of resources. In other words, we have to stretch the rectangle of size 0 \Theta p 0 to build a rectangle of size \Theta p. Using this idea, type 2 dependences will still be satisfied if we choose large enough. Only type 1 dependences have to be taken into account for the internal reorganization of the slice (see Figure 3). But since the corresponding partial dependence graph is acyclic, we are brought back to a standard acyclic scheduling problem for which many theoretical results are known. In particular, a simple list scheduling technique provides a performance bound (and the shorter the longest path in the graph, the more accurate the performance bound). time resources A,k F,k time A,k Fig. 3. Two different allocations of a slice of graph G (p Once this main principle is settled, there remain several open questions: 1. How to choose the initial scheduling? For which 0 ? 2. How to choose the reference slice? There is no reason a priori to choose a slice beginning at a clock cycle congruent to 0 modulus 0 . 3. How to decide that an edge is of type 1, hence to be considered in the acyclic problem? These three questions are of course linked together. Intu- itively, it seems important to (try to) minimize both ffl the length of the longest path in the acyclic graph, which should be as small as possible as it is tightly linked to the performance bound obtained for list scheduling, and ffl the number of edges in the acyclic graph, so as to reduce the dependence constraints for the acyclic scheduling problem. 4 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. XX, NO. Y, MONTH 1996 We will give a precise formulation to these questions and give a solution. Beforehand, we review the choices of GS. B. The heuristic of Gasperoni and Schwiegelshohn In this section we explain with full details the GS heuristic [8]. The main idea is as outlined in the previous section. The choice of GS for the initial scheduling is to consider the optimal cyclic scheduling for an infinite number of processors constraints. B.1 Optimal schedule for unlimited resources Consider the cyclic scheduling problem d) without resource constraints Let be a nonnegative integer. Define from G an edge-weighted graph G 0 ffl Vertices of G 0 to V a new vertex s: ffl Edges of G 0 an edge from s to all other ver- tices: ffl Weight of edges of G 0 We have the following well-known result: Lemma 1: is a valid initiation interval , G 0 has no cycle of positive weight. Furthermore, if G 0 has no cycle of positive weight, and if t(s; u) denotes the length of the longest path, in G 0 , from s to u, then oe(u; is a valid cyclic schedule. Lemma 1 has two important consequences: ffl First, given an integer , it is easy to determine if is a valid initiation interval and if yes, to build a corresponding cyclic schedule by applying Bellman-Ford algorithm [11] on . ffl The optimal initiation interval 1 is the smallest non-negative integer such that G 0 has no positive cy- cle. Therefore, maxfd C cycle of Gg otherwise. Furthermore, a binary search combined with Bellman-Ford algorithm computes 1 in polynomial time. B.2 Algorithm GS for p resources As said before, in the case of p identical processors, the algorithm consists in the conversion of the dependence graph G into an acyclic graph G a . G a is obtained by deleting some edges of G. As initial scheduling, GS takes the optimal scheduling with unlimited resources As reference slice, GS takes a slice starting at a clock cycle congruent to 0 modulus 1 , i.e. a slice from clock cycle K1 up to clock cycle (K This amounts to decomposing t(s; u) into In other words r Consider an edge In the reference slice, the operation instance , the operation instance of v which is performed within the reference slice, namely (v; started before the end of the operation (u; K \Gamma q u ). Hence this operation instance not the one that depends upon completion of In other words, K \Gamma q The two operations in dependence through edge e are not initiated in the same slice. Edge e can be safely considered as a type 2 edge, and thus can be deleted from G. This is the way edges are cut in GS heuristic 1 . We are led to the following algorithm: Algorithm 1: (Algorithm GS) 1. Compute the optimal cyclic schedule oe 1 for unlimited resources. 2. Let be an edge of G. Then e will be deleted from G if and only if This provides the acyclic graph G a . 3. (a) Consider the acyclic graph G a where vertices are weighted by ffi and edges represent task dependences, and perform a list scheduling oe a on the p processors. (b) Let be the makespan (i.e. the total execution time) of the schedule for G a . 4. For all t(s; u) is a valid cyclic schedule. The correctness of Algorithm GS can be found in [8]. It can also be deduced from the correctness of Algorithm CDR (see Section IV-B.1). B.3 Performances of Algorithm GS GS gives an upper bound to the initiation interval obtained by Algorithm 1. Let opt be the optimal (smallest) initiation interval with p processors. The following inequality is established: where \Phi is the length of the longest path in G a . Moreover, owing to the strategy for cutting edges, \Phi where in [8]). This implies which leads to opt opt GS is the first guaranteed algorithm. We see from equation (2) that the bound directly depends upon \Phi, the length of the longest path in G a . Example. We go back to our example. Assume processors. The graph G is the graph of Figure 1. We have In Figure 4(a), we depict the graph G 0 12 . The different values t(s; u) for all are given in circles on the figure. The schedule oe 1 (u; already represented in Figure 2: 4 processors are needed. 1 However, this is not the best way to determine type 2 edges. See Section III-C. P.-Y. CALLAND, A. DARTE AND Y. ROBERT: CIRCUIT RETIMING APPLIED TO DECOMPOSED SOFTWARE PIPELINING 5 Figure 4(b) shows the acyclic graph G a obtained by cutting edges r Finally, Figure 4(c) shows a possible schedule of operations provided by a list scheduling. The initiation interval of this solution is A F A A,k time resources (a) (b) (c) Fig. 4. (a): The graph G 0(b): The acyclic graph Ga corresponding list scheduling allocation: C. Cutting edges by retiming Let us summarize Algorithm GS as follows: first compute the values t(s; u) in G 0 1 to provide the optimal scheduling without resource constraints oe 1 . Then take a reference slice starting at clock cycle 0 Finally, delete from G some edges that necessarily correspond to dependences between different slices: only those edges are removed by algorithm GS. However, edges that correspond to dependences between different slices are those such that q u within the reference slice, the scheduled computation instances are Therefore, the computation (v; depends upon performed in the same slice iff wise, it is performed in a subsequent slice, and in this case Let us check this mathematically, for an arbitrary slice. Consider a valid cyclic scheduling oe(u; (with 6= 0), and let c where t 0 is given. For each edge the dependence constraint is satisfied, thus r u r Finally, dividing by , we get Furthermore, if q v then the dependence constraints directly writes r u . We thus have: ae Therefore, the condition for cutting edges corresponding to dependences between different slices (i.e. those we called Furthermore, if an edge is cut by GS, then it is also cut by our new rule. We are led to a modified version of GS which we call mGS. Since we cut more edges in mGS than in GS, the acyclic graph mG a obtained by mGS contains a subset of the edges of the acyclic graph G a . See Figure 5 to illustrate this fact. A A,k time resources (a) (b) Fig. 5. (a): The acyclic graph provided by Algorithm mGS (b): A corresponding list scheduling allocation: Actually, we need neither an initial ordering nor a reference slice any longer. What we only need is to determine a function Then, we define the acyclic graph mG a as follows: an edge in mG a iff q(v) Clearly, mG a is acyclic (assume there is a cycle, and sum up the quantities q(v) q(u) on this cycle to get a contradiction). Finally, given mG a , we list schedule it as a DAG whose vertices are weighted by the initial ffi function. Such a function q is called a retiming in the context of synchronous VLSI circuits [10]. Given a graph d), q performs a transformation of G into a new graph G d q ) where d q is defined as follows: if is an edge of E then d q This transformation can be interpreted as follows: if d(e) represents the number of "registers" on edge e, a retiming q amounts to suppress q(u) registers to each edge leaving u, and to add q(v) registers to each edge entering v. A retiming is said valid if for each edge e of E, d q (e) 0 (at least one register per edge in G q , see Equation 3). Edges such that d q are edges "with no register". Note that we assumed that the sum of the d(e) on any cycle of G is positive: using VLSI terminology, we say G is synchronous. We are now ready to formulate the problem. Recall that our goal was to answer the following two questions: ffl How to cut edges so as to obtain an acyclic graph G a whose longest path has minimal length? ffl How to cut as many edges as possible so that the number of dependence constraints to be satisfied by the list- scheduling of G a is minimized? 6 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. XX, NO. Y, MONTH 1996 using our new formulation, we can state our objectives more precisely in terms of retiming: Objective 1 Find a retiming q that minimizes the longest path in mG a , i.e. in terms of retiming, that minimizes the clock period \Phi of the retimed graph. Objective 2 Find a retiming q so that the number of edges in mG a is minimal, i.e. distribute registers so as to leave as few edges with no register as possible. In Section IV, we show how to achieve the first objective. There are several possible solutions, and in Section V, we show how to select the best one with respect to the second objective, and we state our final algorithm. We improve upon GS for two reasons: first we have a better bound, and second we cut more edges, hence more freedom for the list scheduling. IV. Minimizing the longest path of the acyclic graph There are well-known retiming algorithms that can be used to minimize the clock period of a VLSI circuit, i.e. the maximal weight (in terms of delay) of a path with no register. We first recall these algorithms, due to Leiserson and Saxe [10], then we show how they can be applied to decomposed software pipelining. A. Retiming algorithms We first need some definitions. We denote by u P a path P of G from u to v, by d(P e2P d(e) the sum of the dependences of the edges of P , and by ffi(P v2P ffi(v) the sum of the delays of the vertices of P . We define D and \Delta as follows: D and \Delta are computed by solving an all-pairs shortest-path algorithm on G where edge u e weighted with the pair (d(e); \Gammaffi(u)). Finally, let path of G; d(P \Phi(G) is the length of the longest path of null weight in G (and is called the clock period of G in VLSI terminology). Theorem 1: (Theorem 7 in [10]) Let d) be a synchronous circuit, let be an arbitrary positive real number, and let q be a function from V to the integers. Then q is a legal retiming of G such that \Phi(G q ) if and only if 1. for every edge u e 2. that \Delta(u; v) ? . Theorem 1 provides the basic tool to establish the following algorithm (Algorithm 2) that determines a retiming such that the clock period of the retimed graph is minimized Algorithm 2: (Algorithm OPT1 in [10]) 1. Compute D and \Delta (see Algorithm WD in [10]). 2. Sort the elements in the range of \Delta. 3. Binary search among the elements \Delta(u; v) for the minimum achievable clock period. To test whether each potential clock period is feasible, apply the Bellman-Ford algorithm to determine whether the conditions in Theorem 1 can be satisfied. 4. For the minimum achievable clock period found in step 3, use the values for the q(v) found by the Bellman-Ford algorithm as the optimal retiming. This algorithm runs in O(jV j 3 log jV j), but there is a more efficient algorithm whose complexity is O(jV jjEj log jV j), which is a significant improvement for sparse graphs. It runs as the previous algorithm except in step 3 where the Bellman-Ford algorithm is replaced by the following algorithm: Algorithm 3: (Algorithm FEAS in [10]) Given a synchronous d) and a desired clock period , this algorithm produces a retiming q of G such that G q is a synchronous circuit with clock period \Phi , if such a retiming exists. 1. For each vertex set q(v) to 0. 2. Repeat the following (a) Compute graph G q with the existing values for q. (b) for any vertex v 2 V compute \Delta 0 (v) the maximum sum ffi(P ) of vertex delays along any zero-weight directed path P in G leading to v. This can be done in O(jEj). (c) For each vertex v such that \Delta 0 (v) ? , set q(v) to 3. Run the same algorithm used for step 2(b) to compute \Phi. If \Phi ? then no feasible retiming exists. Otherwise, q is the desired retiming. B. A new scheduling algorithm: Algorithm CDR We can now give our new algorithm and prove that both resource and dependence constraints are met. Algorithm 4: (Algorithm CDR) Let d) be a dependence graph. 1. Find a retiming q that minimizes the length \Phi of the longest path of null weight in G q (use Algorithm 2 with the improved algorithm for step 3). 2. Delete edges of positive weight, or equivalently keep edges edges with no register). By this way, we obtain an acyclic graph G a . 3. Perform a list scheduling oe a on G a and compute 4. Define the cyclic schedule oe by: Note that the complexity of Algorithm CDR is determined by Step 1 whose complexity is O(jV jjEj log(jV j)). In comparison, the complexity of Algorithm GS is O(jV jjEj log(jV jffi max )). The difference comes from the fact that \Phi opt can be searched among the jV j 2 values \Delta(u; v) whereas 1 is searched among all values between 0 and algorithms have similar complexities. P.-Y. CALLAND, A. DARTE AND Y. ROBERT: CIRCUIT RETIMING APPLIED TO DECOMPOSED SOFTWARE PIPELINING 7 B.1 Correctness of Algorithm CDR Theorem 2: The schedule oe obtained with Algorithm CDR meets both dependence and resource constraints. Proof: Resource constraints are obviously met because of the list scheduling and the definition of , which ensures that slices do not overlap. To show that dependence constraints are satisfied for each of E, we need to verify , oe a (u) , oe a On one hand, suppose that e is not deleted, i.e. e 2 G a . It is equivalent to say that the weight of e after the retiming is equal to zero: q(v) since oe a is a schedule for G a : oe a (u) Thus, inequality (4) is satisfied. On the other hand, if e is deleted, then . But, by definition of we have oe a (u) Thus, inequality (4) is satisfied. B.2 Performances of Algorithm CDR Now, we use the same technique as in [8] in order to show that our algorithm is also guaranteed and we give an upper bound for the initiation interval that is smaller than the bound given for Algorithm GS. Theorem 3: Let G be a dependence graph, \Phi opt the minimum achievable clock period for G, the initiation interval of the schedule generated by Algorithm CDR when p processors are available, and opt the best possible initiation interval for this case. Then opt '' \Phi opt opt Proof: By construction, \Phi opt is the length of the longest path in G a , thus with the same proof technique as in [8], i.e a list scheduling technique, we can prove that which leads to the desired inequality. Now we show that the bound obtained for Algorithm CDR (Theorem 3) is always better than the bound for Algorithm GS (see Equation 2). This is a consequence of the following lemma: Lemma 2: 1 \Phi opt Proof: Let us apply Algorithm CDR with unlimited resources. For that, we define a retiming q such that and we define the graph G a by deleting from G all edges e such that d q (e) ? 0. Then, we define a schedule for G a with unlimited resources by oe a P path of G a leading to ug. The makespan of oe a is \Phi opt by construction. Finally, we get a schedule for G by defining tion, the smallest initiation interval for Now, consider an optimal cyclic schedule oe for unlimited resources, oe(u; as defined in Section III- B.1. Let As proved in Section III-C, q defines a valid retiming for G, i.e. for all edges G a by deleting from G all edges e such that d q (e) ? 0 (as in Algorithm mGS). Let P be any path in G a , Summing up these By construction, \Phi(G q ) is the length of the longest path in G a , thus \Phi(G q Finally, we have \Phi opt \Phi(G q ), hence the result. Theorem 4: The performance upper bound given for Algorithm CDR is better than the performance upper bound given for Algorithm GS. Proof: This is easily derived from the fact that \Phi opt shown by Lemma 2. Note: this bound is a worst case upper bound for the initiation interval. It does not prove however that CDR is always better than GS. Example. We can now apply Algorithm CDR to our key example (assume again available processors). \Phi 14 and the retiming q that achieves this clock period is obtained in two steps by Algorithm 3: q Figures 6(a), 6(b) and 6(c) show the successive retimed graphs. Figure 6(d) shows the corresponding acyclic graph G a and finally, Figure 6(e) shows a possible schedule of operations provided by a list scheduling technique, whose initiation interval is This is better than what we found with Algorithm mGS (see Figure 5(b)), and with Algorithm GS (see Figure 4(c)). B.3 Link between 1 and \Phi opt As shown in Lemma 2, 1 and \Phi opt are very close. How- ever, the retiming that can be derived from the schedule with initiation interval 1 does not permit to define an acyclic graph with longest path \Phi opt . In other words, looking for 1 is not the right approach to minimizing the period of the graph. In this section, we investigate more deeply this fact, by recalling another formulation of the retiming problem given by Leiserson and Saxe [10]. Lemma 3: (Lemma 9 in [10]) Let d) be a synchronous circuit, and let c be a positive real number. 8 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. XX, NO. Y, MONTH 1996 A F F F (b) (c) A A,k B,k C,k time resources (d) (e) Fig. 6. (a): Initial dependence graph G (b) and (c): Successive steps of retiming used in CDR Corresponding acyclic graph corresponding list scheduling allocation: Then there exists a retiming q of G such that \Phi(G q ) if and only if there exists an assignment of a real value s(v) and an integer value q(v) to each vertex v 2 V such that the following conditions are satisfied: \Gammas(v) \Gammaffi(v) for every vertex s(v) for every vertex such that q(u) \Gamma By letting every vertex u, inequalities 5 are equivalent to: such that q(u) \Gamma This last system permits to better understand all the techniques that we developed previously: Optimal schedule for unlimited resources. As seen in Lemma 2, the schedule oe(u; System 6 with except the second inequality. We do have r(v) but not necessarily r(v) and in this case, proof). Algorithm CDR for unlimited resources. By construc- tion, with the retiming such that \Phi(G q and System 6 is satisfied with the smallest value for . Therefore, this technique leads to the better cyclic schedule with unlimited resources for which the slices do not overlap (because of the second inequality). It is not always possible to find 1 this way. Algorithms CDR and GS for p resources. The schedule obtained satisfies System 6 with the makespan of oe a . For CDR, q is the retiming that achieves the optimal period, whereas for GS, q is the retiming defined from 1 1 c). For CDR, the fourth inequality is satisfied exactly for all edges However, for GS, oe is required to satisfy the fourth inequality for more edges than necessary (actually for all edges such that r(u)+ ffi(u) r(v)). Note that for both algorithms, there are additional conditions imposed by the resource constraints that do not appear in System 6. V. Minimizing the number of edges of the acyclic graph Our purpose in this section is to find a retimed graph with the minimum number of null weight edges among all retimed graphs whose longest path has the best possible length \Phi opt . Removing edges of non null weight will give an acyclic graph that matches both objectives stated at the end of Section III-C. Example. Consider step 1 of Algorithm CDR in which we use the retiming algorithm of Leiserson and Saxe [10]. The final retiming does minimize the length \Phi of the longest path of null weight, but it does not necessarily minimize the number of null weight edges. See again our key example, Figure 6 (c), for which 14. We can apply yet another retiming to obtain the graph of Figure 7(a): 0, and A F (c) A B,k time resources Fig. 7. (a): The final retimed graph (b): The corresponding acyclic graph corresponding list scheduling allocation: The length of the longest path of null weight is still 14, but the total number of null weight edges is smaller. This implies that the corresponding acyclic graph G a (see Figure contains fewer edges than the acyclic graph of Figure 6(d) and therefore, is likely to produce a smaller initiation interval 2 . That is the case in our example: we find an initiation interval equal to 19 (see Figure 7(c)). It turns out that = 19 is the best possible integer initiation List scheduling a graph which is a subset of another graph will not always produce a smaller execution time. But intuition shows that it will in most practical cases (the fewer constraints, the more freedom). P.-Y. CALLAND, A. DARTE AND Y. ROBERT: CIRCUIT RETIMING APPLIED TO DECOMPOSED SOFTWARE PIPELINING 9 interval with processors: the sum of all operation delays is 37, and d 37 Recall that a retiming q such that \Phi(G q integral solution to the following system (see formulation of Theorem 1): such that \Delta(u; v) ? \Phi opt Among these retimings, we want to select one particular retiming q for which the number of null weight edges in G q is minimized. This can be done as follows: Lemma 4: Let d) be a synchronous circuit. A retiming q such that \Phi(G q and such that the number of null weight edges in G q is minimized can be found in polynomial time by solving the following integer linear program: min such that \Delta(u; v) ? \Phi opt Proof: Consider an optimal integer solution (q; v) to System 8. q defines a retiming for G with \Phi(G q since system 7 is satisfied: indeed q(v) \Gamma q(u)+d(e)+v(e) Note that each v(e) is constrained by only one equation: 1. There are two cases: ffl The weight of e in G q is null, i.e. is the only possibility. ffl The weight of e in G q is positive, i.e. q(v)\Gammaq(u)+d(e) 1 (recall that q and d are integers). In this case, the minimal value for v is 0. Therefore, given a retiming q, e2E v(e) is minimal when it is equal to the number of null weight edges in G q . it remains to show that such an optimal integer solution can be found in polynomial time. For that, we System 8 in matrix form as minfcx j Ax bg: I d C I d where C is the transpose of the jV j\ThetajEj-incidence matrix of G, C 0 is the transpose of the incidence matrix of the graph whose edges are the pairs (u; v) such that \Delta(u; v) ? \Phi opt and I d is the jEj \Theta jEj identity matrix. An incidence matrix (such as C) is totally unimodular (see [12, page 274]). Then, it is easy to see that A is also totally unimodular. Therefore, solving the ILP Problem 8 is not NP-complete: System 8 considered as an LP problem has an integral optimum solution (Corollary 19.1a in [12]) and such an integral solution can be found in polynomial time (Theorem 16.2 in [12]). Let us summarize how this refinement can be incorporated into our software pipelining heuristic: first, we compute \Phi opt the minimum achievable clock period for G, then we solve System 8 and we obtain a retiming q. We define G a as the acyclic graph whose edges have null weight in the longest path in G a is minimized and the number of edges in G a is minimized. Finally, we schedule G a as in Algorithm CDR. We call this heuristic the modified CDR (or simply mCDR). Remark: Solving System 8 can be expensive although polynomial. An optimization that permits reducing the complexity is to pre-compute the strongly connected components G i of G and to solve the problem separately for each component G i . Then, a retiming that minimizes the number of null weight edges in G q is built by adding suitable constants to each retiming q i so that all edges that link different components have positive weights. Future work will try to find a pure graph-theoretic approach to the resolution of System 8, so that the practical complexity of our software pipelining heuristic is decreased. VI. Load balancing We have restricted so far initiation intervals to integer values. As mentioned in Section II, searching for rational initiation intervals might give better results, but at the price of an increase in complexity: searching for b can be achieved by unrolling the original loop nest by a factor of b, thereby processing an extended dependence graph with many more vertices and edges. In this section, we propose a simple heuristic to alleviate potential load imbalance between processors, and for which there is no need to unroll the graph. Remember the principle of the four previously described heuristics (GS, mGS, CDR and mCDR). First, an acyclic graph G a is built from G. Then, G a is scheduled by a list scheduling technique. This defines the schedule oe a inside each slice of length (the initiation interval). Finally, slices are concatenated, a slice being initiated just after the completion of the previous one. The main weakness of this principle is that slices do not overlap. Since the schedule in each slice has been defined by an As-Soon-As-Possible (ASAP) list scheduling, what usually happens is that many processors are idle during the last time steps of the slice. The idea to remedy this problem is to try to fill these "holes" in the schedule with the tasks of the next slice. For that, instead of scheduling the next slice with the same schedule oe a , we schedule it with an As-Late- As-Possible (ALAP) so that "holes" may appear in the first time steps of the slice. Then, between two successive slices, processors are permuted so that the computational load is (nearly) equally balanced when concatenating both slices. Of course dependences between both slices must now be taken into account. A precise formulation of this heuristic can be found in [13]. Here, we only illustrate it on our key example. Example. Figure 7(c) shows a possible allocation of an instance of G a provided by an ASAP list scheduling. Figure 8(a) shows an allocation provided by an ALAP list scheduling and Figure 8(b) the concatenation of these two instances. The initiation interval that we obtain is equal to 37 for two instances. i.e. which is better than the initiation interval obtained with Algorithm Figure 7(c)). This cannot be improved further: the two processors are always busy, as 2. (a) time resources B,k (b) B,k A,k D,k C,k F,k Fig. 8. (a): ALAP scheduling (b): Concatenation of two instances Another possibility is to schedule, in an acyclic manner, two (or more) slices instead of one, after the retiming as been chosen. This is equivalent to unroll the loop after the retiming has been performed. In this case, once the first slice has been processed, the second slice can be allocated the same way, taking into account dependence constraints coming from the acyclic graph, plus dependences between the two slices. In other words, the retiming can be seen as a pre-loop transformation, that consists in changing the structure of the sub-graph induced by loop independent edges. Once this retiming has been done, any software pipelining algorithm can still be applied. VII. Conclusion In this paper, we have presented a new heuristic for the software pipelining problem. We have built upon results of Gasperoni and Schwiegelshohn, and we have made clear the link between software pipelining and retiming. In the case of identical, non pipelined, resources, our new heuristic is guaranteed, with a better bound than that of [8]. Unfortunately, we cannot extend the guarantee to the case of many different resources, because list scheduling itself is not guaranteed in this case. We point out that our CDR heuristic has a reasonable complexity, similar to classical software pipelining algo- rithms. As for mCDR, further work will be aimed at deriving an algorithmic implementation that will not require the use of Integer Linear Programming (even though the particular instance of ILP invoked in mCDR is polynomial). Finally, note that all edge-cutting heuristics lead to cyclic schedules where slices do not overlap (by construction). Our final load-balancing technique is a first step to overcome this limitation. It would be very interesting to derive methods (more sophisticated than loop unrolling) to synthesize resource-constrained schedules where slices can overlap. Acknowledgments The authors would like to thank the anonymous referees for their careful reading and fruitful comments. --R "Some scheduling techniques and an easily schedulable horizontal architecture for high performance scientific computing," "Software pipelining; an effective scheduling technique for VLIW machines," "Perfect pipelining; a new loop optimization technique," "Cyclic scheduling on parallel proces- sors: an overview," "Fine-grain scheduling under resource con- straints," "A frame-work for resource-constrained, rate-optimal software pipelining," "Software pipelining," "Generating close to optimum loop schedules on parallel processors," "Decomposed software pipelining," "Retiming synchronous circuitry," Introduction to Algorithms Theory of Linear and Integer Program- ming "A new guaranteed heuristic for the software pipelining problem," --TR --CTR Han-Saem Yun , Jihong Kim , Soo-Mook Moon, Time optimal software pipelining of loops with control flows, International Journal of Parallel Programming, v.31 n.5, p.339-391, October Dongming Peng , Mi Lu, On exploring inter-iteration parallelism within rate-balanced multirate multidimensional DSP algorithms, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.13 n.1, p.106-125, January 2005 Timothy W. O'Neil , Edwin H.-M. Sha, Combining Extended Retiming and Unfolding for Rate-Optimal Graph Transformation, Journal of VLSI Signal Processing Systems, v.39 n.3, p.273-293, March 2005 Timothy W. O'Neil , Edwin H.-M. Sha, Combining extended retiming and unfolding for rate-optimal graph transformation, Journal of VLSI Signal Processing Systems, v.39 n.3, p.273-293, March 2005 Alain Darte , Guillaume Huard, Loop Shifting for Loop Compaction, International Journal of Parallel Programming, v.28 n.5, p.499-534, Oct. 2000 Greg Snider, Performance-constrained pipelining of software loops onto reconfigurable hardware, Proceedings of the 2002 ACM/SIGDA tenth international symposium on Field-programmable gate arrays, February 24-26, 2002, Monterey, California, USA Karam S. Chatha , Ranga Vemuri, Hardware-Software partitioning and pipelined scheduling of transformative applications, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.10 n.3, p.193-208, June 2002 R. Govindarajan , Guang R. Gao , Palash Desai, Minimizing Buffer Requirements under Rate-Optimal Schedule in Regular Dataflow Networks, Journal of VLSI Signal Processing Systems, v.31 n.3, p.207-229, July 2002

software pipelining;circuit retiming;list scheduling;cyclic scheduling;modulo scheduling

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Abstractions for Portable, Scalable Parallel Programming.

AbstractIn parallel programming, the need to manage communication, load imbalance, and irregularities in the computation puts substantial demands on the programmer. Key properties of the architecture, such as the number of processors and the cost of communication, must be exploited to achieve good performance, but coding these properties directly into a program compromises the portability and flexibility of the code because significant changes are then needed to port or enhance the program. We describe a parallel programming model that supports the concise, independent description of key aspects of a parallel programincluding data distribution, communication, and boundary conditionswithout reference to machine idiosyncrasies. The independence of such components improves portability by allowing the components of a program to be tuned independently, and encourages reuse by supporting the composition of existing components. The isolation of architecture-sensitive aspects of a computation simplifies the task of porting programs to new platforms. Moreover, the model is effective in exploiting both data parallelism and functional parallelism. This paper provides programming examples, compares this work to related languages, and presents performance results.

Introduction The diversity of parallel architectures puts the goals of performance and portability in conflict. Programmers are tempted to exploit machine details-such as the interconnection structure and the granularity of parallelism-to maximize performance. Yet software portability is needed to reduce the high cost of software development, so programmers are advised to avoid making machine-specific assumptions. The challenge, then, is to provide parallel languages that minimize the tradeoff between performance and portability. 1 Such languages must allow a programmer to write code that assumes no particular architecture, allow a compiler to optimize the resulting code in a machine-specific manner, and allow a programmer to perform architecture-specific performance tuning without making extensive modifications to the source code. In recent years, a parallel programming style has evolved that might be termed aggregate data-parallel computing. This style is characterized by: ffl Data parallelism. The program's parallelism comes from executing the same function on many elements of a collection. Data parallelism is attractive because it allows parallelism to grow-or scale-with the number of data elements and processors. SIMD architectures exploit this parallelism at a very fine grain. ffl Aggregate execution. The number of data elements typically exceeds the number of processors, so multiple elements are placed on a processor and manipulated sequentially. This is attractive because placing groups of interacting elements on the same processor vastly reduces communication costs. Moreover, this approach uses good sequential algorithms locally, which is often more efficient than simply multiplexing parallel algorithms. Another benefit is that data can be passed between processors in batches to amortize communication overhead. Finally, when a computation on one data element is delayed waiting for communication, other elements may be processed. ffl Loose synchrony. Although strict data parallelism applies the "same" function to every element, local variations in the nature or positioning of some elements can require different implementations of the same conceptual function. For instance, data elements on the boundary of a computational domain have no neighbors with which to communicate, but data parallelism normally assumes that interior and exterior elements be treated the same. By executing a different function on the boundaries, these exceptional cases can be easily handled. These features make the aggregate data-parallel style of programming attractive because it can yield efficient programs when executed on typical MIMD architectures. However, without linguistic support this style of programming promotes inflexible programs through the embedding of performance-critical features as constants, such as the number of processors, the number of data elements, boundary conditions, the processor interconnection, and system-specific communication syntax. If the machine, its size, or the problem size changes, significant program changes to these fixed quantities are generally required. As a consequence, 1 We consider a program to be portable with respect to a given machine if its performance is competitive with machine-specific programs solving the same problem [2]. several languages have been introduced to support key aspects of this style. However, unless all aspects of this style are supported, performance, scalability, portability, or development cost can suffer. For instance, good locality of reference is an important aspect of this programming style. Low-level approaches [25] allow programmers to hand-code data placement. The resulting code typically assumes one particular data decomposition, so if the program is ported to a platform that favors some other de- composition, extensive changes must be made or performance suffers. Other languages [4, 5, 15] give the programmer no control over data decomposition, leaving these issues to the compiler or hardware. But because good data decompositions can depend on characteristics of the application that are difficult to determine statically, compilers can make poor data placement decisions. Many recent languages [6, 22] provide support for data decompositions, but hide communication operations from the programmer and thus do not encourage locality at the algorithmic level. Consequently, there is a reliance on automated means of hiding latency-multithreaded hardware, multiple lightweight threads, caches, and compiler optimizations that overlap communication and computation-which cannot always hide all latency. The trend towards relatively faster processors and relatively slower memory access times only exacerbates the situation. Other languages provide inadequate control over the granularity of parallelism, requiring either one data point per process [21, 43], assuming some larger fixed granularity [14, 29], or including no notion of granularity at all, forcing the compiler or runtime system to choose the best granularity [15]. Given the diversity of parallel computers, no particular granularity can be best for all machines. Computers such as the CM-5 prefer coarse granularities; those such as the J Machine prefer finer granularity; and those such as the MIT Alewife and Tera computer benefit from having multiple threads per process. Also, few languages provide sufficient control over the algorithm that is applied to aggregate data, preferring instead to multiplex the parallel algorithm when there are multiple data points on a processor [43, 44]. Many language models do not adequately support loose synchrony. The boundaries of parallel computations often introduce irregularities that require significant coding effort. When all processes execute the same code, programs become riddled with conditionals, increasing code size and making them difficult to understand, hard to modify, and potentially inefficient. Programming in a typical MIMD-style language is not much cleaner. For instance, writing a slightly different function for each type of boundary process is problematic because a change to the algorithm is likely to require all versions to be changed. In this paper we describe language abstractions-a programmingmodel-that fully support the aggregate data-parallel programming style. This model can serve as a foundation for portable, scalable MIMD languages that preserve the performance available in the underlying machine. Our belief is that for many tasks programmers-and not compilers or runtime systems-can best handle the performance-sensitive aspects of a parallel program. This belief leads to three design principles. First, we provide abstractions that are efficiently implementable on all MIMD architectures, along with specific mechanisms to handle common types of parallelism, data distribution, and boundary conditions. Our model is based on a practical MIMD computing model called the Candidate Type Architecture (CTA) [45]. Second, the insignificant but diverse aspects of computer architectures are hidden. If exposed to the programmer, assumptions based on these characteristics can be sprinkled throughout a program, making portability difficult. Examples of characteristics that are hidden include the details of the machine's communication style and the processor (or memory) interconnection topology. For instance, one machine might provide shared memory and another message passing, but either can be implemented with the other in software. Third, architectural features that are essential to performance are exposed and parameterized in an architecture-independent fashion. A key characteristic is the speed, latency, and per-message overhead of communication relative to computation. As the cost of communication increases relative to computation, communication costs must be reduced by aggregating more processing onto a smaller number of processors, or by finding ways to increase the overlap of communication and computation. The result is the Phase Abstractions parallel programming model, which provides control over granularity of parallelism, control over data partitioning, and a hybrid data and function parallel construct that supports concise description of boundary conditions. The core of our solution is the ensemble construct that allows a global data structure to be defined and distributed over processes, and allows the granularity-and the location of data elements-to be controlled by load-time parameters. The ensemble also has a code form for describing what operations to execute on which elements and for handling boundary conditions. Likewise, interprocessor connections are described with a port ensemble that provides similar flexibility. By using ensembles for all three components of a global operation-data, code and communication-they can be scaled together with the same parameters. Because the three parts of an ensemble and the boundary conditions are specified independently, reusability is enhanced. The remainder of this paper is organized as follows. We first present our solution to the problem by describing our architectural model and the basic language model-the CTA and the Phase Abstractions. Section 3 then gives a detailed illustration of our abstractions, using the Jacobi Iteration as an example. To demonstrate the expressiveness and programmability of our abstractions, Section 4 shows how simple array language primitives can be built on top of our model. Section 5 discusses the advantages of our programming model with respect to performance and portability, and Section 6 presents experimental evidence that the Phase Abstractions support portable parallel programming. Finally, we compare Phase Abstractions with related languages and models, and close with a summary. Phase Abstractions In sequential computing, languages such as C, Pascal and Fortran have successfully combined efficiency with portability. What do these languages have in common that make them successful? All are based on a model where a sequence of operations manipulate some infinite random-access memory. This programming model succeeds because it preserves the characteristics of the von Neumann machine model, which itself has been a reasonably faithful representation of sequential computers. While these models are never literally implemented-unit-cost access to infinite memory is an illusion provided by virtual memory, caches and backing store-the model is accurate for the vast majority of programs. There are only rare cases, such as programs that perform extreme amounts of disk I/O, where the deviations from the model are costly to the programmer. It is critical that the von Neumann model capture machine features that are relevant to performance: If some essential machine features were ignored, better algorithms could be developed using a more accurate machine model. Together, the von Neumann machine model and its accompanying programming model allow languages such as C and Fortran to be both portable and efficient. In the parallel world, we propose that the Candidate Type Architecture (CTA) play the role of the von Neumann model, 2 and the Phase Abstractions the role of the programming model. Finally, the sequential languages are replaced by languages based on the Phase Abstractions, such as Orca C [32, 34]. The CTA. The CTA [45] is an asynchronous MIMD model. It consists of P von Neumann processors that execute independently. Each processor has its own local memory, and the processors communicate through some sparse but otherwise unspecified communication network. Here "sparse" means that the network has a constant degree of connectivity. The network topology is intentionally left unbound to provide maximum generality. Finally, the model includes a global controller that can communicate with all processors through a low bandwidth network. Logically, the controller provides synchronization and low bandwidth communication such as a broadcast of a single value. Although it is premature to claim that the CTA is as effective a model as the von Neumann model, it does appear to have the requisite characteristics: It is simple, makes minimal architectural assumptions, but captures enough significant features that it is useful for developing efficient algorithms. For example, the CTA's unbound topology does not bias the model towards any particular machine, and the topologies of existing parallel computers are typically not significant to performance. On the other hand, the distinction between global and local memory references is key, and this distinction is clear in the CTA model. Finally, the assumption of a sparse topology is realistic for all existing medium and large scale parallel computers. The Phase Abstractions extend the CTA in the same way that the sequential imperative programming 2 The more recent BSP [48] and LogP [8] models present a similar view of a parallel machine and for the most part suggest a similar way of programming parallel computers. model extends the von Neumann model. The main components of the Phase Abstractions are the XYZ levels of programming and ensembles [1, 19, 46]. 2.1 XYZ Programming Levels A programmer's problem-solving abilities can be improved by dividing a problem into small, manageable pieces-assuming the pieces are sufficiently independent to be considered separately. Additionally, these pieces can often be reused in other programs, saving time on future problems. One way to build a parallel program from smaller reusable pieces is to compose a sequence of independently implemented phases, each executing some parallel algorithm that contributes to the overall solution. At the next conceptual level, each such phase is comprised of a set of cooperating sequential processes that implements the desired parallel algorithm. Each sequential process may be developed separately. These levels of problem solving-program, phase, and process, also called the Z, Y, and X levels-have direct analogies in the CTA. The X level corresponds to the individual von Neumann processors of the CTA, and an X level program specifies the sequential code that executes in one process. Because the model is MIMD, each process can execute different code. The Y level is analogous to the set of von Neumann processors cooperating to compute a parallel algorithm, forming a phase. The Y-level may specify how the X-level programs are connected to each other for inter-process communication. Examples of phases include parallel implementations of the FFT, matrix multiplication, matrix transposition, sort, and global maximum. A phase has a characteristic communication structure induced by the data dependencies among the processes. For example, the FFT induces a butterfly, while Batcher's sort induces a hypercube [1]. Finally, the Z level corresponds to the actions of the CTA's global controller, where sequences of parallel phases are invoked and synchronized. A Z level program specifies the high level logic of the computation and the sequential invocation of phases (although their execution may overlap) that are needed to solve complex problems. For example, the Car-Parrinello molecular dynamics code simulates the behavior of a collection of atoms by iteratively invoking a series of phases that perform FFT's, matrix products, and other computations [49]. In Z-Y-X order, these three levels provide a top-down view of a parallel program. Example: XYZ Levels of the Jacobi Iteration. Figure 1 illustrates the XYZ levels of programming for the Jacobi Iteration. The Z level consists of a loop that invokes two phases, one called Jacobi(), which performs the over-relaxation, the other called Max(), which computes the maximum difference between iterations that is used to test for termination. Each Y level phase is of a collection of processes executing concurrently. Here, the two phases are graphically depicted with squares representing processes and arcs representing communication between processes. program Jacobi data := Load(); while (error>Tolerance) { error := Max(); Z Level X Level Y Level { for each (i,j) in local section new(i,j)=(old(i,j+1)+old(i,j-1)+ { local_max=Max(local_max, left_child); local_max=Max(local_max, right_child); Send local_max to parent; Figure 1: XYZ Illustration of the Jacobi Iteration The Jacobi phase uses a mesh interconnection topology, and the Max phase uses a binary tree. Other details of the Y level, such as the distribution of data, are not shown in this figure but will be explained in the next subsection. Finally, a sketch of the X level program for the two phases is shown at the right of Figure 1. The X level code for the Jacobi phase assigns to each data point the average of its four neighbors. The Max phase finds, for all data points, the largest difference between the current iteration and the previous iteration. 2 A Z level program is basically a sequential program that provides control flow for the overall computation. An X level program, in its most primitive form, can also be viewed as a sequential program with additional operations that allow it to communicate with other processes. Although parallelism is not explicitly specified at the X and Z levels, these two levels may still contain parallelism. For example, phase invocation may be pipelined, and the X level processes can execute on superscalar architectures to achieve instruction-level parallelism. It is the Y level that specifies scalable parallelism and most clearly departs from a sequential program. Ensembles support the definition and manipulation of this parallelism. 2.2 Ensembles The Phase Abstractions use the ensemble structure to describe data structures and their partitioning, process placement, and process interconnection. In particular, an ensemble is a partitioning of a set of elements- data, codes, or port connections-into disjoint sections. Each section represents a thread of execution, so the section is a unit of concurrency and the degree of parallelism is modulated by increasing or decreasing the number of sections. Because all three aspects of parallel computation-data, code and communication-are unified in the ensemble structure, all three components can be reconfigured and scaled in a coherent, concise fashion to provide flexibility and portability. A data ensemble is a data structure with a partitioning. At the Z level the data ensemble provides a logically global view of the data structure. At the X level a portion of the ensemble is mapped to each section and is viewed as a locally defined structure with local indexing. For example, the 6 \Theta 6 data ensemble in Figure 2 has a global view with indices [0 : 5] \Theta [0 : 5], and a local view of 3 \Theta 3 subarrays with indices 2]. The mapping of the global view to the local view is performed at the Y level and will be described in Section 3. The use of local indexing schemes allows an X level process to refer to generic array bounds rather than to global locations in the data space. Thus, the same X level source code can be used for multiple processes. Global View Figure 2: A 6\Theta6 Array (left) and its corresponding Data Ensemble for a 2\Theta2 array of sections. A code ensemble is a collection of procedures with a partitioning. The code ensemble gives a global view of the processes performing the parallel computation. When the procedures in the ensemble differ the model is MIMD; when the procedures are identical the model is SPMD. Figure 3 shows a code ensemble for the Jacobi phase in which all processes execute the xJacobi() function. Figure 3: Illustration of a Code Ensemble Finally, a port ensemble defines a logical communication structure by specifying a collection of port name pairs. Each pair of names represents a logical communication channel between two sections, and each of these port names is bound to a local port name used at the X level. Figure 4 depicts a port ensemble for the Jacobi phase. For example, the north port (N) of one process is bound to the south port (S) of its neighboring process. Figure 4: Illustration of a Port Ensemble A Y level phase is composed of three components: a code ensemble, a port ensemble that connects the code ensemble's processes, and data ensembles that provide arguments to the processes of the code ensemble. The sections of each ensemble are ordered numerically so that the i th section of a code ensemble is bound to the i th section of each data and port ensemble. This correspondence allows each section to be allocated to a processor for normal sequential execution: The process executes on that processor, the data can be stored in memory local to that processor, and the ports define connections for interprocessor communication. Consequently, the i th sections of all ensembles are assigned to the same processor to maintain locality across phases. If two phases share a data ensemble but require different partitionings for best performance, a separate phase may be used to move the data. The Z level logically stores ensembles in Z level variables, composes them into phases and stores their results. The phase invocation interface between the Z and X levels encourages modularity because the same level code can be invoked with different ensemble parameters in the same way that procedures are reused in sequential languages. The ensemble abstraction helps hide the diversity of parallel architectures. However, to map well to individual architectures the abstraction must be parameterized, for example, by the number of processors and the size of the problem. This parameterization is illustrated in the next section. 3 Ensemble Example: Jacobi To provide a better understanding of the ensembles and the Phase Abstractions, we now complete the description of the Jacobi program. We adopt notation from the proposed Orca C language [30, 32], but other languages based on the Phase Abstractions are possible (see Section 4). 3.1 #define Rows 1 /* Constants to define the shape */ #define Cols 2 /* of the logical processor array */ #define TwoD 3 program Jacobian (shape, Processors) switch (shape)f /* Configuration Computation */ case Rows: rows = Processors; break; case Cols: rows break; case TwoD: Partition2D(&rows, &cols, Processors); break; (rows, cols, Processors) /* Configuration Parameter List */ !data ensemble definitions?; /* Y Level */ !port ensemble definitions?; !code ensemble definitions?; !process definitions?; /* X Level */ begin /* Z Level */ while (tolerance ? delta) f and newP to prepare for */ the next iteration */ Figure 5: Overall Phase Abstraction Program Structure As shown in Figure 5, a Phase Abstractions program consists of X, Y, and Z descriptions, plus a list of configuration parameters that are used by the program to adapt to different execution environments. In this case, two runtime parameters are accepted: Processors and shape. The first parameter is the number of processors, while the second specifies the shape of the processor array. As will be discussed later, the program uses a 2D data decomposition, so by setting shape to Rows (Cols) we choose a horizontal strips (vertical strips) decomposition. (The function Partition2D() computes values of rows and cols such that (rows Processors and the difference between rows and cols is minimized.) With this configuration computation this program can, through the use of different load time parameters, adapt to different numbers of processors and assume three different data decompositions. The configuration computation is executed once at load time. 3.2 Z Level of Jacobi After the program is configured, the Z level program is executed, which initializes program variables, reads the input data, and then iteratively invokes the Jacobi and Max phases until convergence is reached, at which point an output phase is invoked. The data, processing, and communication components of the Jacobi and Max phases are specified by defining and composing code, data and port ensembles as described below. 3.3 Y Level: Data Ensembles This program uses two arrays to store floating point values at each point of a 2D grid. Parallelism is achieved by partitioning these arrays into contiguous 2D blocks: partition block[r][c] float p[rows][cols], float newP[rows][cols]; This declaration states that p and newP have dimensions (rows * cols) and are partitioned onto an (r * c) section array (process array). The keyword partition identifies p and newP as ensemble arrays, and block names this partitioning so that it can be reused to define other ensembles. This partitioning corresponds to the one in Figure 2 when rows=6, cols=6, 2, and this ensemble declaration belongs in the !data ensembles? meta-code of Figure 5. (Section 5 shows how an alternate decomposition is declared.) The values of r and c are assumed to be specified in the program's configuration parameter list. Each section is implicitly defined to be of size (s * r c . (If r does not divide rows evenly, some sections will have r e while others will have r Consequently, X level processes contain no assumptions about the data decomposition except the dimension of the subarrays, so these processes can scale in both the number of logical processors and in the problem size. 3.4 Jacobi Phase Ensemble. The Jacobi phase computes for each point the average of its four nearest neighbors, implying that each section will communicate with its four nearest neighbor sections (See Figure 4). The following Y level ensemble declaration defines the appropriate port ensemble: Jacobi.portnames !-? N, E, W, S /* North, East, West, South */ The first line declares the phase's port names so the following bindings can be specified. The second and third lines define a mesh connectivity between Y level port names. This port ensemble declaration does not specify connections for the ports that lie on the boundaries. In this case these unbound ports are bound to derivative functions, which compute boundary conditions using data local to the section. The following binds derivative functions to ports on the edges of Jacobi. Jacobi[0][i] .port.N receive !-? RowZero, where 0 receive !-? ColZero, where 0 != Jacobi[i][0] .port.W receive !-? ColZero, where 0 != Jacobi[r-1][i] .port.S receive !-? RowZero, where 0 RowZero and ColZero are defined as: double RowZero() static double row[1:t] /* default initialized to 0's */ return row; double ColZero() static double col[0][1:s] /* default initialized to 0's */ return col[0]; The values of s and t are determined by the process' X level function-in this case xJacobi(). In the absence of derivative functions, X level programs could check for the existence of neighbors, but such tests complicate the source code and increases the chance of introducing errors. As Section 5 shows, even modestly complicated boundary conditions can lead to a proliferation of special case code. Code Ensemble. To define the code ensemble for Jacobi, each of the r * c sections is assigned an instance of the xJacobi() code: Jacobi[i][j].code !-? xJacobi(); where 0 Because Jacobi contains heterogeneity only on the boundaries, which in this program is handled by derivative functions, all the functions are the same. In general, however, the only restriction is that the function's argument types and return type must match those of the phase invocation. Level. The X level code for Jacobi is shown in Figure 6. It first sends edge values to its four neighbors, it then receives boundary values from its neighbors, and finally it uses the four point stencil to compute the average for each interior point. Several features of the X level code are noteworthy: ffl parameters-The arguments to the X level code establish a correspondence between local variables and the sections of the ensembles. In this case, the local value array is bound to a block of ensemble values. ffl communication-Communication is specified using the transmit operator (!==), for which a port name on the left specifies a send of the righthand side, and a port on the right indicates a receive into the variable on the lefthand side. The semantics are that receive operations block, but sends do not. ffl uniformity-Because derivative functions are used, the xJacobi() function contains no tests for boundary conditions when sending or receiving neighbor values. ffl border values-The values s and t, used to define the bounds of the value array, are parameters derived from the section size of the data ensemble. To hold data from neighboring sections, value is declared to be one element wider on each side than the incoming array argument. This extra storage is explicitly specified by the difference between the local declaration, x[0:s+1][0:t+1], and the formal declaration, x[1:s][1:t], where the upper bounds of these array declarations are inclusive. ffl array slices-Slices provide a concise way to refer to an entire row (or in general, a d-dimensional block) of data. When slices are used in conjunction with the transmit operator (!==), the entire block is sent as a single message, thus reducing communication overhead. The Complete Phase. To summarize, the data ensemble, the port ensemble, and the code ensemble collectively define the Jacobi phase. Upon execution the sections declared by the configuration parameters are logically connected in a nearest-neighbor mesh, and each section of data is manipulated by one xJacobi() process. The end result is a parallel algorithm that computes one Jacobi iteration. 3.5 Max Phase The Max phase finds the maximum change of all grid points, and uses the same data ensemble as the Jacobi phase. The port ensemble is shown graphically in Figure 8 and is defined below. Max.portnames !-? P, L, R /* Parent, Left, Right */ xJacobi(value[1:s][1:t], new-value[1:s][1:t]) double value[0:s+1][0:t+1]; /* extra storage on all four sides */ double new-value[0:s+1][0:t+1]; port North, East, West, South; double new-value[0:s+1][0:t+1]; int i, j; /* Send neighbor values */ North !== value [1][1:t]; /* 1:t is an array slice */ East !== value[1:s][t]; West !== value[1:s][1]; South !== value[s][1:t]; Receive neighbor values */ for (i=1; i!=s; i++) for (j=1; i!=t; i++) for (i=1; i!=s; i++) for (j=1; i!=t; i++) Figure Level Code for the Jacobi Phase The derivative functions for this phase are bound so that a receive from a leaf section's Left or Right port will return the value computed by the Smallest Value() function, and a send from the root's unbound Parent port will be a no-op. Max[i].port.L receive !-? Smallest-Value() where r*c/2 Processors Max[i].port.R receive !-? Smallest-Value() where r*c/2 Processors Max[i].port.P send !-? No-Op() where The Smallest Value() derivative function simply returns the smallest value that can be represented on the architecture. The code ensemble for this phase is similar to the Jacobi phase, except that xMax() replaces xJacobi(). (See Figure 7.) xMax(value[1:s][1:t], new-value[1:s][1:t]) double value[1:s][1:t]; double new-value[1:s][1:t]; port Parent, Left, Right; int i, j; double local-max; double temp; /* Compute the local maximum */ for (i=1; i!=s; i++) for (j=1; j!=t; j++) /* Compute the global maximum */ temp !== Left; /* receive */ temp !== Right; /* receive */ Parent !== local-max; /* send */ Figure 7: X Level Code for the Max Phase With applications that are more complicated than Jacobi, the benefit of using ensembles increases while their cost is amortized over a larger program. The cost of using ensembles will also decrease as libraries of Figure 8: Illustration of a Tree Port Ensemble ensembles, phases, derivative functions and X level codes are built. For example, the Max phase of Jacobi is common to many computations and would not normally be defined by the programmer. 4 High Level Programming with the Phase Abstractions Phase Abstractions are not a programming language, but rather a foundation for the development of parallel programming languages that support the creation of efficient, scalable, portable programs. Orca C, used in the previous section, is a literal, textual instantiation of the Phase Abstractions. It clearly shows the power of the Phase Abstractions, but some may find it too low-level and tedious. In fact, a departure from the literal Orca C language is not required to achieve an elegant programming style. By adopting certain conventions, it is possible to build reusable abstractions directly on top of Orca C. By staying within the Orca C framework, this solution has the advantage that different sublanguages can be used together for a single large problem that requires diverse abstractions for good performance. As an example, consider the design of an APL-like array sublanguage for Orca C. 3 Recall that an X level procedure receives two kinds of parameters-global data passed as arguments and port connections-that support two basic activities: computations on data and communication. However, it is possible to constrain X level functions to perform just one of these two tasks-a local computation or a communication operation. That is, there could be separate computation phases and communication phases. For example, there can be X level computation functions for adding integers, computing the minimum of some values, or sorting some elements. There can be X level communication functions for shifting data cyclically in a ring, for broadcasting data, or for communicating up and down a tree structure. Reductions, which naturally combine both communication and computation, are notable exceptions where the separation of 3 Since the submission of this paper, an array sublanguage known as ZPL has been developed to support data parallel computations [35, 47, 31, 37]. While its syntax differs significantly from Orca C, ZPL remains true to the Phase Abstractions model. It provides a powerful Z level language that hides all of the X and Y level details from the user. communication from computation is not desirable. For such operations it suffices to define a communication-oriented phase that takes an additional function parameter for combining the results of communications. To illustrate, reconsider the Jacobi example. Rather than specify the entire Jacobi iteration in one X level process, each communication operation constitutes a separate phase and the results are combined by Z level add and divide phases. The convergence test is computed at the Z level by subtracting the old array from the new one and performing a maximum reduction on the differences. The program skeleton in Figure 9 illustrates this method, providing examples of X level functions for (referred to as operator+ in the syntactic style of C++), shift, and reduce; the Z level code shows how data ensembles are declared and how phase structures for add, left-shift and reduce are initialized. The divide and subtract phases are analogous to add, and the other shift functions are analogous to the left-shift. There are three consequences of this approach. First, the interface to a phase is substantially simplified. Second, some problems are harder to describe because it is not possible to combine computation and communication within a single X level function. Finally, X level functions (and the phases that they comprise) are smaller and are more likely to perform just one task, increasing their composability and reusability. Although the array sublanguage defined here is similar to APL, it has some salient differences. Most significantly, the Orca C functions operate on subarrays, rather than individual elements, which means that fast sequential algorithms can be applied to subarrays. So while this solution achieves some of the conciseness and reusability of APL, it does not sacrifice control over data decompositions or lose the ability to use separate global and local algorithms. This solution also has the advantage of embedding an array language in Orca C, allowing other programming styles to be used as they are needed. The power of the Phase Abstractions comes from the decomposition of parallel programs into X, Y and Z levels, the encoding of key architectural properties as simple parameters, and the concept of ensembles, which allows data, port and code decompositions to be specified and reused as individual components. The three types of ensembles work together to allow the problem and machine size to be scaled. In addition, derivative functions allow a single X level program to be used for multiple processes even in the presence of boundary conditions. This section discusses the Phase Abstractions with respect to performance and expressiveness. Portability and Scalability. When programs are moved from one platform to another they must adapt to the characteristics of their host machine if they are to obtain good performance. If such adaptation is automatic or requires only minor effort, portability is achieved. The Phase Abstractions support portability and scalability by encoding key architectural characteristics as ensemble parameters and by separating phase definitions into several independent components. xproc TYPE[1:s][1:t] operator+(TYPE x[1:s][1:t], TYPE y[1:s][1:t]) TYPE result[1:s][1:t]; int i, j; for (i=1; i!=s; i++) for (j=1; j!=t; i++) return result; xproc void shift(TYPE val[1:s][1:t]) port write-neighbor, int for (i=2; i!=t; i++) xproc int reduce(TYPE val[1:k], TYPE*()op) port Parent, int for (i=2; i!=k; i++) for (i=1; i!=n; i++) Parent !== accum; begin Z double X[1:J][1:K], OldX[1:J][1:K]; phase operator+; phase Left; phase Reduce; do Figure 9: Jacobi Written in an Array Style Using Orca C Changes to either the problem size or the number of processors are encapsulated in the data ensemble declaration. As in Section 3, we relate the size of a section (s * t), the overall problem size (rows * cols), and the number of sections (r * c) as follows: The problem size scales by changing the values of rows and cols, the machine size scales by changing the values of r and c, and the granularity of parallelism is controlled by altering either the number of processors or the number of sections in the ensemble declaration. This flexibility is an important aspect of portability because different architectures favor different granularities. While it is desirable to write programs without making assumptions about the underlying machine, knowledge of machine details can often be used to optimize program performance. Therefore, tuning may sometimes be necessary. For example, it may be beneficial for the logical communication graph to match the machine's communication structure. Consider embedding the binary tree of the Max phase onto a mesh architecture: Some logical edges must span multiple physical links. This edge dilation can be eliminated with a connectivity that allows comparisons along each row of processors and then along a single column (see Figure 10). Figure 10: Rows and Columns to Compute the Global Maximum To address the edge dilation problem the fixed binary tree presented in Section 3 can be replaced by a new port ensemble that uses a tree of variable degree. Such a solution is shown in Figure 11, where the child ports are represented by an array of ports. This new program can use either a binary tree or the "rows and columns" approach. The port ensemble declaration for the latter approach is shown below. /* Rows and Columns communication structure */ With the code suitably parameterized, this program can now execute efficiently on a variety of architectures by selecting the proper port ensemble. xMax(value[1:s][1:t], new[1:s][1:t], numChildren) double value[1:s][1:t]; double new-value[1:s][1:t]; port Parent, Child[numChildren]; int i, j; double local-max; double temp; /* Compute the local maximum */ for (i=1; i!=s; i++) for (j=1; i!=t; i++) /* Compute the global maximum */ for (i=0; i!numChildren; i++) temp !== Child[i]; /* receive */ Parent !== local-max; /* send */ Figure Parameterized X Level Code for the Max Phase Locality. The best data partitioning depends on factors such as the problem and machine size, the hard- ware's communication and computation characteristics, and the application's communication patterns. In the Phase Abstractions model, changes to the data partitioning are encapsulated by data ensembles. For example, to define a 2D block partitioning on P processors, the configuration code can define the number of sections to be P: If a 1D strip partitioning is desired, the number of sections can simply be defined to be P. This strip decomposition requires that each process have only East-West neighbors instead of the four neighbors used in the block decomposition. By using the port ensembles to bind derivative functions to unused ports-in this case the North and South ports-the program can easily accommodate this change in the number of neighbors. No other source level changes are required. The explicit dichotomy between local and non-local access encourages the use of different algorithms locally and globally. Batcher's sort, for example, benefits from this approach (see Section 1). This contrasts with most approaches in which the programmer or compiler identifies as much fine-grained parallelism as possible and the compiler aggregates this fine-grained parallelism to a granularity appropriate for the target machine. Boundary Conditions. Typically, processes on the edge of the problem space must be treated separately. 4 In the Jacobi Iteration, for example, a receive into the East port must be conditionally executed because processes on the East edge have no eastern neighbors. Isolated occurrences of these conditionals pose little problem, but in most realistic applications these lead to convoluted code. For example, SIMPLE can have up to nine different cases-depending on which portions of the boundaries are contained within a process-and these conditionals can lead to code that is dominated by the treatment of exceptional cases [18, 41]. For example, suppose a program with a block decomposition assumes in its conditional expression that a process is either a NorthEast, East, or SouthEast section, as shown below: if (NorthEast) /* special case 1 */ else if (East) /* special case 2 */ else if (SouthEast) /* special case 3 */ A problem arises if the programmer then decides that a vertical strips decomposition would be more efficient. 4 Although we discuss this problem in the context of a message passing language, shared memory programs must also deal with these special cases. The above code assumes that exactly one of the three boundary conditions holds. But in the vertical strips decomposition there is only one section on the Eastern edge, so all three conditions apply, not just one. Therefore, the change in data decomposition forces the programmer to rewrite the above boundary condition code. Our model attempts to insulate the port and code ensembles from changes in the data decomposition: Processes send and receive data through ports that in some cases involve interprocess communication and in other cases invoke derivative functions. The handling of boundary conditions has thus been decoupled from the X level source code. Instead of cluttering up the process code, special cases due to boundary conditions are handled at the problem level where they naturally belong. Reusability. The same characteristics that provide flexibility in the Phase Abstractions also encourage reusability. For example, the Car-Parrinello molecular dynamics program [49] consists of several phases, one of which is computed using the Modified Gram-Schmidt (MGS) method of solving QR factorization. Empirical results have shown that the MGS method performs best with a 2D data decomposition [36]. However, other phases of the Car-Parrinello computation require a 1D decomposition, so in this case a 1D decomposition for MGS yields the best performance since it avoids data movement between phases. This illustrates that a reusable component is most effective if it is flexible enough to accommodate a variety of execution environments. Irregular Problems. Until now this paper has only described statically defined array-based ensembles. However, this should not imply that Phase Abstractions are ill suited to dynamic or unstructured problems. In fact, to some extent LPAR [28], a set of language extensions for irregular scientific computations (see Section 7), can be described in terms of the Phase Abstractions. The key point is that an ensemble is a set with a partitioning; to support dynamic or irregular computations we can envision dynamic or irregular partitionings that are managed at runtime. Consider first a statically defined irregular problem such as finite element analysis. The programmer begins by defining a logical data ensemble that will be replaced by a physical ensemble at runtime. This logical definition includes the proper record formats and an array of port names, but not the actual data decomposition or the actual port ensemble. At runtime a phase is run that determines the partitioning and creates the data and port ensembles: The size and contents of the data ensemble are defined, the interconnection structure is determined, and the sections are mapped to physical processors. We assume that the code ensemble is SPMD since this obviates the need to assign different codes to different processes dynamically. Once this partitioning phase has completed the ensembles behave the same as statically defined phases. Dynamic computations could be generalized from the above idea. For example, a load balancing phase could move data between sections and also create revised data and port ensembles to represent the new partitioning. Technical difficulties remain before such dynamic ensembles can be supported, but the concepts do not change. Limits of the Non-Shared Memory Model. The non-shared memory model encourages good locality of reference by exposing data movement to the programmer, but the performance advantage for this model is small for applications that inherently have poor locality. For example, direct methods of performing sparse factorization have poor locality of reference because of the sparse and irregular nature of the input data. For certain solutions to this problem, a shared memory model performs better because the single address space leads to better load balance through the use of a work queue model [38]. The shared memory model also provides notational convenience, especially when pointer-based structures are involved. 6 Portability Results Experimental evidence suggests that the Phase Abstractions can provide portability across a diverse set of MIMD computers [32, 33]. This section summarizes these results for just one program, SIMPLE, but similar results were achieved for QR factorization and matrix multiplication [30]. Here we briefly describe SIMPLE, the machines on which this program was run, the manner in which this portable program was implemented, and the significant results. SIMPLE is a large computational fluid dynamics benchmark whose importance to high performance computing comes from the substantial body of literature already devoted to its study. It was introduced in 1977 as a sequential benchmark to evaluate new computers and Fortran compilers [7]. Since its creation it has been studied widely in both sequential and parallel forms [3, 9, 13, 16, 17, 23, 24, 40, 42]. Hardware. The portability of our parallel SIMPLE program was investigated on the iPSC/2 S, iPSC/2 F, nCUBE/7, Sequent Symmetry, BBN Butterfly GP1000, and a Transputer simulator. These machines are summarized in Table 1. The two Intel machines differ in that the iPSC/2 S has a slower Intel 80387 floating point coprocessor, while the other has the faster iPSC SX floating point accelerator. The simulator is a detailed Transputer-based non-shared memory machine. Using detailed information about arithmetic, logical and communication operators of the T800 [24], this simulator executes a program written in a Phase Abstraction language and estimates program execution time. Implementation. The SIMPLE program was written in Orca C. Since no compiler exists for any language based on the Phase Abstractions, this program was hand-compiled in a straight-forward fashion to C code that uses a message passing substrate to support the Phase Abstractions. The resulting C code is machine- Machine Sequent Intel Intel nCUBE BBN Transputer model Symmetry A iPSC/2 S iPSC/2 F nCUBE/7 Butterfly GP1000 simulator nodes 20 processors Intel 80386 Intel 80386 Intel 80386 custom Motorola 68020 T800 memory 32MB 4 MB/node 8 MB/node 512 KB/node 4 MB/node N/A cache 64KB 64 KB 64KB none none network bus hypercube hypercube hypercube omega mesh Table 1: Machine Characteristics independent except for process creation, which is dependent on each operating system's method of spawning processes. Number of 1680 points on a Transputer 1680 points on the Intel iPSC/2 1680 points on the Butterfly 1680 points on the NCUBE/7 1680 points on the Symmetry Number of 28 Pingali&Rogers Lin&Snyder Hiromoto et al. Figure 12: (a) SIMPLE Speedup on Various Machines (b) SIMPLE with 4096 points Figure 12(a) shows that similar speedups were achieved on all machines. Many hardware characteristics can affect speedup, and these can explain the differences among the curves. In this discussion we concentrate on communication costs relative to computational speed, the feature that best distinguishes these machines. For example, the iPSC/2 F and nCUBE/7 have identical interconnection topologies but the ratio of computation speed to communication speed is greater on the iPSC/2 [11, 12]. This has the effect of reducing speedup because it decreases the percentage of time spent computing and increases the fraction of time spent on non-computation overhead. Similarly, since message passing latency is lowest on the Sequent's shared bus, the Sequent shows the best speedup. This claim assumes little or no bus contention, which is a valid assumption considering the modest bandwidth required by SIMPLE. Figure 12(b) shows the SIMPLE results of Hiromoto et al. on a Denelcor HEP using 4096 data points [23], which indicate that our portable program is roughly competitive with machine-specific code. The many differences with our results-including different problem sizes, different architectures, and possibly even different problem specifications-make it difficult to draw any stronger conclusions. As another reference point, Figure 12(b) compares our results on the iPSC/2 S against those of Pingali and Rogers' parallelizing compiler for language [42]. Both experiments were run on iPSC/2's with 4MB of memory and 80387 floating point units. All other parameters appear to be identical. The largest potential difference lies in the performance of the sequential programs on which speedups are computed. Although these results are encouraging for proponents of functional languages, we point out that our results do not make use of a sophisticated compiler: The type of compiler technology developed by Pingali and Rogers can likely improve the performance of our programs as well. Even though the machines differ substantially-for example, in memory structure-the speedups fall roughly within the same range. Moreover, this version of SIMPLE compares favorably with machine-specific implementations. These results suggest, then, that portability has been achieved for this application running on these machines. 7 Related Work Many systems support a global view of parallel computation, SPMD execution, and data decompositions that are similar to various aspects of the Phase Abstractions. None, however, provide support for an X- level algorithm that is different from the Z-level parallel algorithm. Nor do any provide general support for handling boundary conditions or controlling granularity. This section discusses how some of these systems address scalability and portability in the aggregate data parallel programming style. Dataparallel C. Dataparallel C [21] (DPC) is a portable shared-memory SIMD-style language that has similarities to C++. Unlike the Phase Abstractions, DPC supports only point-wise parallelism. DPC has point-wise processor (poly) variables that are distributed across the processors of the machine. Unlike its predecessor C* [43], DPC supports data decompositions of its data to improve performance on coarse-grained architectures. However, because DPC only supports point-wise communication, the compiler or runtime system must detect when several point sends on a processor are destined for the same processor and bundle them. Also, to maintain performance of the SIMD model on a MIMD machine, extra compiler analysis is required to detect when the per-instruction SIMD synchronizations are not necessary and can be removed. Because each point-wise process is identical, edge effects must be coded as conditionals that determine which processes are on the edge of the computation. It is hard to reuse such code because the boundary conditions may change from problem to problem. Constant and variable boundary conditions can be supported by expanding the data space and leaving some processes idle. Dino. Dino [44] is a C-like, SPMD language. Like C*, it constructs distributed data structures by replicating structures over processors and executing a single procedure over each element of the data set. Dino provides a shared address space, but remote communication is specified by annotating accesses to non-local objects by the # symbol, and the default semantics are true message-passing. Parallel invocations of a procedure synchronize on exit of the procedure. Dino allows the mapping of data to processes to be specified by programmer-defined functions. To ensure fast reads to shared data, a partitioning can map an individual variable to multiple processors. Writes to such variables are broadcast to all copies. Dino handles edge effects in the same fashion as C*. Because Dino only supports point-wise communication, the compiler or runtime system must combine messages. Mehrotra and Rosendale. A system described by Mehrotra and Rosendale [39] is much like Dino in that it supports a small set of data distributions. However, this system provides no way to control or precisely determine which points are local to each other, so it is not possible to control communication costs or algorithm choice based on locality. On the other hand, this system does not require explicit marking of external memory references as in Dino. Instead, their system infers, when possible, which references are global and which are not. In algorithms where processes dynamically choose their "neighbors," this simplifies programming. Also, programs are more portable than those written in Dino. The communication structure of the processor is not visible to the programmer, but the programmer can change the partitioning clauses on the data aggregates. SPMD processing is allowed, but there are no special facilities for handling edge effects. Fortran Dialects. Recent languages such as Kali [26], Vienna Fortran [6], and HPF [22] focus on data decomposition as the expression of parallelism. Their data decompositions are similar to the Phase Abstractions notion of data ensembles, but the overall approach is fundamentally different. Phase Abstractions require more effort from the programmer, while this other approach relies on compiler technology to exploit loop level parallelism. This compiler-based approach can guarantee deterministic sequential semantics, but it has less potential for parallelism since there may be cases where compilers cannot transform a sequential algorithm into an optimal parallel one. Kali, Vienna Fortran and HPF depart from sequential languages primarily in their support for data though some of these languages do provide mechanisms for specifying parallel loops. Vienna Fortran provides no form of parallel loops, while the FORALL statement in HPF and Kali specifies that a loop has no loop carried dependencies. To ensure deterministic semantics of updates to common variables by different loop iterations, values are deterministically merged at the end of the loop. This construct is optional in HPF; the compiler may attempt to extract parallelism even where a FORALL is not used. HPF and Vienna Fortran allow arrays to be aligned with respect to an abstract partitioning. These are very powerful constructs. For example, arrays can be dynamically remapped, and procedures can define their own data distribution. Together these features are potentially very expensive because although the programmer helps in specifying the data distribution at various points of the program, the compiler must determine how to move the data. In addition to data distribution directives, Kali allows the programmer to control the assignment of loop iterations to processors through the use of the On clause, which can help in maintaining locality. LPAR. LPAR is a portable language extension that supports structured, irregular scientific parallel computations [28, 27]. In particular, LPAR provides mechanisms for describing non-rectangular distributed partitions of the data space to manage load-balancing and locality. These partitions are created through the union, intersection and set difference of arrays. Because support for irregular decompositions has a high cost, LPAR syntactically distinguishes irregular decompositions so that faster runtime support can be used for regular decompositions. 5 Computations are invoked on a group of arrays by the foreach operator, which executes its body in parallel on each array to yield coarse-grained parallelism. LPAR uses the overlapping indices of distributed subarrays to support sharing of data elements. Overlapping domains provide an elegant way of describing multilevel mesh algorithms and computations for boundary conditions. There is an operator for redistributing data elements, but LPAR depends on a routine written in the base language to compute what the new decomposition should be. The Phase Abstraction's potential to support dynamic, irregular decompositions is discussed in Section 5. For multigrid decompositions, a sublanguage supporting scaled partitionings and communication between scaled ensembles would be useful. The Phase Abstractions' support for loose synchrony naturally supports the use of refined grids in conjunction with the base grid. Split-C. Split-C is a shared-memory SPMD language with memory reference operations that support latency-hiding [10]. Split-C procedures are concurrently applied in an "owner-computes" fashion to the partitions of an aggregate data structure such as an array or pointer-based graph. A process reads data that it does not own with a global pointer (a Split-C data type). To hide latency, Split-C supports an asynchronous read-akin to an unsafe Multilisp future [20]-that initiates a read of a global pointer but does not wait for the data to arrive. A process can invoke the sync() operation to block until all outstanding reads 5 Scott Baden, Personal Communication. complete. There is a similar operation for global writes. These operations hide latency while providing a global namespace and reducing the copying of data in and out of message queues. (Copying may be necessary for bulk communication of non-contiguous data, such as the column of an array.) However, these operations can lead to complex programming errors because a misplaced reference or synchronization operation can lead to incorrect output but no immediate failure. Array distribution in Split-C is straightforward but somewhat limited; some number of higher order dimensions can be cyclically distributed while the remaining dimensions are distributed as blocks. Load balance, locality, and irregular decompositions may be difficult to achieve for some applications. Array distribution declarations are tied to a procedure's array parameter declarations, which can limit reusability and portability because these declarations and the code that depends on them must be modified when the distribution changes. This coupling can also incur a performance penalty because the benefit of an optimal array distribution for one procedure invocation may be offset by the cost of redistributing the array for other calculations that use the array. Split-C provides no special support for boundary conditions. The usual trick of creating an enlarged array is possible; otherwise, irregularities must be handled by conditional code in the body of the SPMD procedures. 8 Conclusion Parallelism offers the promise of great performance but thus far has been hampered by a lack of portability, scalability, and programming convenience that unacceptably increase the time and cost of developing efficient programs. Support is required for quickly programminga solution and easily moving it to new machines as old ones become obsolete. Rather than defining a new parallel programming paradigm, the Phase Abstractions model supports well-known techniques for achieving high-performance-computing sequentially on local aggregates of data elements and communicating large groups of data as a unit-by allowing the programmer to partition data across parallel machines in a scalable manner. Furthermore, by separating a program into reusable parts-X level, Y level, Z-level, ensemble declarations, and boundary conditions-the creation of subsequent programs can be significantly simplified. This approach provides machine-independent, low-level control of parallelism and allows programmers to write in an SPMD manner without sacrificing the efficiency of MIMD processing. Message passing has often been praised for its efficiency but condemned as being difficult to use. The contribution of the Phase Abstractions is a language model that focuses on efficiency while reducing the difficulty of non-shared memory programming. The programmability of this model is exemplified by the straight-forward solution of problems such as SIMPLE, as well as the ability to define specialized high-level array sublanguages. Because the Phase Abstractions model is designed to be structurally similar to MIMD architectures, it performs very well on a variety of MIMD processors. 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High Performance Fortran Forum. Experiences with the Denelcor HEP. Processor Element Architecture for Non-Shared Memory Parallel Computers iPSC/2 User's Guide. Compiling global name-space parallel loops for distributed execution Lattice parallelism: A parallel programming model for non-uniform An implementation of the LPAR parallel programming model for scientific computations. The Portability of Parallel Programs Across MIMD Computers. ZPL language reference manual. A portable implementation of SIMPLE. Portable parallel programming: Cross machine comparisons for SIMPLE. Data ensembles in Orca C. ZPL: An array sublanguage. Accommodating polymorphic data decompositions in explicitly parallel programs. SIMPLE performance results in ZPL. Towards a machine-independent solution of sparse cholesky factorization Compiling high level constructs to distributed memory architectures. Analysis of the SIMPLE code for dataflow computation. Experiences with Poker. Compiler parallelization of SIMPLE for a distributed memory machine. The Dino parallel programming language. Type architecture The XYZ abstraction levels of Poker-like languages A ZPL programming guide. A bridging model for parallel computation. A parallel implementation of the Car-Parrinello method --TR --CTR Marios D. Dikaiakos , Daphne Manoussaki , Calvin Lin , Diana E. Woodward, The portable parallel implementation of two novel mathematical biology algorithms in ZPL, Proceedings of the 9th international conference on Supercomputing, p.365-374, July 03-07, 1995, Barcelona, Spain Bradford L. Chamberlain , Sung-Eun Choi , E. Christopher Lewis , Calvin Lin , Lawrence Snyder , W. Derrick Weathersby, ZPL: A Machine Independent Programming Language for Parallel Computers, IEEE Transactions on Software Engineering, v.26 n.3, p.197-211, March 2000

portable;programming model;scalable;parallel;MIMD

273411

Automatic Determination of an Initial Trust Region in Nonlinear Programming.

This paper presents a simple but efficient way to find a good initial trust region radius (ITRR) in trust region methods for nonlinear optimization. The method consists of monitoring the agreement between the model and the objective function along the steepest descent direction, computed at the starting point. Further improvements for the starting point are also derived from the information gleaned during the initializing phase. Numerical results on a large set of problems show the impact the initial trust region radius may have on trust region methods behavior and the usefulness of the proposed strategy.

Introduction . Trust region methods for unconstrained optimization were first introduced by Powell in [14]. Since then, these methods have enjoyed a good reputation on the basis of their remarkable numerical reliability in conjunction with a sound and complete convergence theory. They have been intensively studied and applied to unconstrained problems (see for instance [11], [14], and [15]), and also to problems including bound constraints (see [4], [7], [12]), convex constraints (see [2], [6], [18]), and non-convex ones (see [3], [5], and [19], for instance). At each iteration of a trust region method, the nonlinear objective function is replaced by a simple model centered on the current iterate. This model is built using first and possibly second order information available at this iterate and is therefore usually suitable only in a certain limited region surrounding this point. A trust region is thus defined where the model is supposed to agree adequately with the true objective function. Trust region approaches then consist of solving a sequence of subproblems in which the model is approximately minimized within the trust region, yielding a candidate for the next iterate. When a candidate is determined that guarantees a sufficient decrease on the model inside the trust region, the objective function is then evaluated at this candidate. If the objective value has decreased sufficiently, the candidate is accepted as next iterate and the trust region is possibly enlarged. Otherwise, the new point is rejected and the trust region is reduced. The updating of the trust region is directly dependant on a certain measure of agreement between the model and the objective function. A good choice for the trust region radius as the algorithm proceeds is crucial. Indeed, if the trust region is too large compared with the agreement between the model and the objective function, the approximate minimizer of the model is likely to be a poor indicator of an improved iterate for the true objective function. On the other hand, too small a trust region may lead to very slow improvement in the estimate of the solution. When implementing a trust region method, the question then arises of an appropriate choice for the initial trust region radius (ITRR). This should clearly reflect the region around the starting point, in which the model and objective function approximately agree. However, all the algorithms the author is aware of use a rather ad hoc value for this ITRR. In many algorithms, users are expected to provide their Department of Mathematics, Facult'es Universitaires N. D. de la Paix, 61 rue de Bruxelles, B- 5000, Namur, Belgium ([email protected]). This work was supported by the Belgian National Fund for Scientific Research. own choice based on their knowledge of the problem (see [8], and [9]). In other cases, the algorithm initializes the trust region radius to the distance to the Cauchy point (see [13]), or to a multiple or a fraction of the gradient norm at the starting point (see [8], and [9]). In each of these cases, the ITRR may not be adequate, and, even if the updating strategies used thereafter generally allow to recover in practice from a bad initial choice, there is usually some undesired additional cost in the number of iterations performed. Therefore, the ITRR selection may be considered as important, especially when the linear-algebra required per iteration is costly. In this paper, we propose a simple but efficient way of determining the ITRR, which consists of monitoring the agreement between the model and the objective function along the steepest descent direction computed at the starting point. Further improvements for the starting point will also be derived from the information gleaned during this initializing phase. Numerical experiments, using a modified version of the nonlinear optimization package LANCELOT (see [8]), on a set of relatively large test examples from the CUTE test suite (see [1]), show the merits of the proposed strategy. Section 2 of the paper develops the proposed automatic determination of a suitable ITRR. The detailed algorithm is described in x3. Computational results are presented and discussed in x4. We finally conclude in x5. 2. Automatic determination of an initial trust region. 2.1. Classical trust region update. We consider the solution of the unconstrained minimization problem The function f is assumed to be twice-continuously differentiable and a trust region method is used, whose iterations are indexed by k, to solve this problem. At iteration k, the quadratic model of f(x) around the current iterate x (k) is denoted by is a symmetric approximation of the Hessian (Subsequently, we will use the notation f (k) and g (k) for f(x (k) ) and g(x (k) ), respectively.) The trust region is defined as the region where Here \Delta (k) denotes the trust region radius and k \Delta k is a given norm. When a candidate for the next iterate, x say, is computed that approximately minimizes (2.2) subject to the constraint (2.3), a classical framework for the trust region radius update is to set for some selected fi (k) satisfying In (2.5), the quantity represents the ratio of the achieved to the predicted reduction of the objective function. The reader is referred to [8], [9], and [10] for instances of trust region updates using (2.4)-(2.5). 2.2. Initial trust region determination. The problem in determining an ITRR \Delta (0) is to find a cheap way to test agreement between the model (2.2) and the objective function at the starting point, x (0) . The method presented here is based on the use of information generally available at this point, namely the function value and the gradient. With the extra cost of some function evaluations, a reliable ITRR will be determined, whose use will hopefully reduce the number of iterations required to find the solution. As shown in x4, the possible saving produced in most cases largely warrants the extra cost needed to improve the ITRR. The basic idea is to determine a maximal radius that guarantees a sufficient agreement between the model and the objective function in the direction \Gammag (0) , using an iterative search along this direction. At each iteration i of the search, given a radius estimate \Delta (0) i , the model and the objective function values are computed at the point x Writing the ratio ae (0) is also calculated, and the algorithm then stores the maximal value among the estimates whose associated ae (0) ' is "close enough to one" (following some given criterion). It finally updates the current estimate \Delta (0) The updating phase for \Delta (0) i follows the framework presented in (2.4)-(2.5), but includes a more general test on ae (0) i because the predicted change in (2.8) (unlike that in (2.6)) is not guaranteed to be positive. That is, we set \Delta (0) for some Note that update (2.9) only takes the adequacy between the objective function and its model into consideration, without taking care of the minimization of the objective function f . That is, it may happen that the radius estimate is decreased (fi (0) i is not close enough to one (jae (0) even though a big reduction is made in the objective function (if f On the other hand, the radius estimate could be augmented (fi (0) i is close enough to one (jae (0) when actually the objective function has increased (if f for instance). This is not contradictory, as far as we forget temporarily about the minimization of f and concentrate exclusively on the adequacy between the objective function and its model to find a good ITRR. In the next section, we shall consider an extra feature that will take account of a possible decrease in f during the process. In order to select a suitable value for fi (0) satisfying (2.9), a careful strategy detailed below is applied, which takes advantage of the current available information. This strategy uses quadratic interpolation (as already done in some existing framework for trust region updates, see [9]), and has been inspired by the trust region updating rules developed in [8]. The univariate function f(x first modeled by the quadratic i (fi) that fits f (0) , f (0) , and the directional derivative \Gamma\Delta (0) where d (0) . Assuming that this quadratic does not coincide with the univariate quadratic m (0) used to provide candidates for fi (0) which the ratio ae (0) i would be close to one (slightly smaller and slightly larger than one, respectively) if f were the quadratic q (0) (fi). That is, equations are solved (where ' ? 0 is a small positive constant), yielding candidates and respectively. These two candidates will be considered as possible choices for a suitable satisfying (2.9), provided a careful analysis based on two principles is first performed. The first principle is to select and exploit, as much as possible, the relevant information that may be drawn from fi (0) i;1 and/or fi (0) i;2 . For instance, if fi (0) i;1 is greater than one and the radius estimate must be decreased (because jae (0) it should be ignored. The second principle consists in favouring the maximal value for fi (0) among the relevant ones. Based on the observation that the linear-algebraic costs during a trust region iteration are generally less when the trust region has been contracted (because part of the computation may be reused after a contraction but not after an expansion), this corresponds to favour an ITRR choice on the large rather than small side. As in (2.9), we distinguish three mutually exclusive cases. The first case, for which fi (0) occurs when jae (0) possibilities are considered in this first case, that produce choice (2.14). ffl Both fi (0) i;1 and fi (0) are irrelevant, that is, they recommend an increase of the radius estimate while in this case, in reality it should be decreased. These values are then ignored, and fi (0) i is set to a fixed constant ffl All the available relevant information provides a smaller value than the lower bound fl 1 . Set fi (0) ffl Either fi (0) i;1 (or fi (0) belongs to the appropriate interval, while fi (0) i;2 (or fi (0) respectively) is irrelevant or too small. The relevant one is selected. ffl Both fi (0) i;1 and fi (0) i;2 are within the acceptable bounds. The maximum is then chosen. These possibilities yield: if min(fi (0) In the second case (i.e. when jae (0) choice (2.15) is performed to select a suitable fi (0) based on the following reasoning. ffl Both fi (0) i;1 and fi (0) are irrelevant because they recommend a decrease of the radius estimate. fi (0) i is set to a fixed constant ffl At least one available piece of relevant information provides a larger value than the upper bound fl 2 . Since any maximal pertinent information is favoured, i is set to this bound. ffl Either fi (0) i;1 or fi (0) i;2 belongs to the appropriate interval, while the other is irrelevant. fi (0) i is set to the relevant one. ffl Both fi (0) i;1 and fi (0) i;2 are within the acceptable bounds. The maximum is then selected. This gives the following: Finally, three situations are considered in the third case for selecting fi (0) . Note that, since it is not clear from the value of ae (0) i that the radius estimate should be decreased or increased, fi (0) i;1 and fi (0) are trusted and indicate if a decrease or an increase is to be performed. ffl Both fi (0) i;1 and fi (0) i;2 advise a decrease of the radius estimate, but smaller than the lower bound allowed. This lower bound, fl 3 , is then adopted. ffl At least one among fi (0) i;1 and fi (0) i;2 recommends an increase of the radius es- timate, but larger than the upper bound allowed, fl 4 . This upper bound is used. ffl The maximal value, max(fi (0) belongs to the appropriate interval and defines fi (0) . The radius estimate is thus either increased or decreased, depending on this value. That is: 3. The algorithm. We are now in position to define our algorithm in full detail. as used in (2.5), (2.9), (2.12) and (2.13), the ITRR Algorithm depends on the constant - 0 ? 0. This one determines the lowest acceptable level of agreement between the model and the objective function that must be reached at a radius estimate to become a candidate for the ITRR. The iterations of Algorithm ITRR will be denoted by the index i. While the algorithm proceeds, \Delta max will record the current maximal radius estimate which guarantees a sufficient agreement between the model and the objective function. Fi- nally, the imposed limit on the number of iterations will be denoted by imax and fixes the degree of refinement used to determine the ITRR. ITRR Algorithm. Step 0. Initialization. Let the starting point x (0) be given. Compute and B (0) . Choose or compute an ITRR estimate \Delta (0) 0 and set Step 1. Maximal radius estimate update. Compute i as defined in (2.7) and (2.8). If set Step 2. Radius estimate update. If i - imax, go to Step 3. Otherwise, compute i;1 and fi (0) using (2.12) and (2.13), respectively, compute using using using (2.16) otherwise, and set Increment i by one and go to Step 1. Step 3. Final radius update. If Otherwise, set Stop ITRR Algorithm. The trust region algorithm may then begin, with \Delta (0) as ITRR. We end this section by introducing an extra feature in the above scheme, which takes advantage of the computations of f (0) i , the function values at the trial points (0) (see Step 1). That is, during the search of an improved radius estimate, we simply monitor a possible decrease in the objective function at each trial point. Doing so, at the end of Algorithm ITRR, rather than updating the final radius, we move to the trial point that produced the best decrease in the objective function (if at least one decrease has been observed). This point then becomes a new starting point, at which Algorithm ITRR is repeated to compute a good ITRR. Of course, a limit is needed on the number of times the starting point is allowed to change. Denoting by this limit and by j the corresponding counter (initialized to one in Step 0), this extra feature may be incorporated in Algorithm ITRR using two further instructions. The first one, added at the end of Step 1, is Here ffi denotes the current best decrease observed in the objective function and oe stores the associated radius. (These two quantities should be initialized to zero in Step 0). The second instruction, which comes at the beginning of Step 3, is increment j by one and go to Step 0. When starting a trust region algorithm with a rather crude approximation of the solution, this kind of improvement, which exploits the steepest descent direction, may be very useful. It is particularly beneficial when the cost of evaluating the function is reasonable. A similar concept is used in truncated Newton methods (see [16], and [17]). Note that a change in the starting point requires the computation of a new gradient and a new model, while the cost for determining the ITRR is estimated in terms of function evaluations. Suitable choices for the limits imax and jmax and for the constants used in Algorithm ITRR may depend on the problem type and will be discussed in x4. 4. Numerical results. For a good understanding of the results, it is necessary to give a rapid overview of the framework in which Algorithm ITRR has been embed- ded, namely the large-scale nonlinear optimization package LANCELOT/SBMIN (see [8]), designed for solving the bound-constrained minimization problem, minimize x2R n f(x) subject to the simple bound constraint l - x - u; where any of the bounds in (4.2) may be infinite. SBMIN is an iterative trust region method whose version used for our testing has the following characteristics: ffl Exact first and second derivatives are used. ffl The trust region is defined using the infinity norm in (2.3) for each k. ffl The trust region update strategy follows the framework (2.4)-(2.5), and implements a mechanism for contracting the trust region which is more sophisticated than that for expanding it (see [8], p. 116). ffl The solution of the trust region subproblem at each iteration is accomplished in two stages. In the first, the exact Cauchy point is obtained to ensure a sufficient decrease in the quadratic model. This point is defined as the first local minimizer of m (k) (x (k) +d (k) (t)), the quadratic model along the Cauchy arc d (k) (t) defined as d where l (k) , u (k) and the projection operator P (x; l are defined component-wise by l (k) l (k) The Cauchy arc (4.3) is continuous and piecewise linear, and the exact Cauchy point is found by investigating the model behaviour between successive pairs of breakpoints (points at which a trust region bound or a true bound is encountered along the Cauchy arc), until the model starts to increase. The variables which lie on their bounds at the Cauchy point (either a trust region bound or a true bound) are then fixed. ffl The second stage applies a truncated conjugate gradient method (in which an 11-band preconditioner is used), to further reduce the quadratic model by changing the values of the remaining free variables. The reader is referred to [8], Chapter 3, for a complete description of SBMIN. We selected our 77 test examples as the majority of large and/or difficult nonlinear unconstrained or bound-constrained test examples in the CUTE (see [1]) collection. Only problems which took excessive cpu time (more than 5 hours), or excessive number of iterations (more than 1500), were excluded, since it was not clear that they would have added much to the results. All experiments were made in double pre- cision, on a DEC 5000/200 workstation, using optimized (-O) Fortran 77 code and DEC-supplied BLAS. The values for the constants of Algorithm ITRR used in our tests are 0:25. The values for have been inspired from the trust region update strategy used in [8]. Suitable values for the other constants have been determined after extensive testing. (Note that, fortunately, slight variations for these constants have no significant impact on the behaviour of Algorithm ITRR). We set meaning that at most one move is allowed in the starting point, and 4, such that 5 radius estimates (including the first one) are examined per starting point. These values result from a compromise between the minimum number of radius estimates that should be sampled to produce a reasonable ITRR, and the maximum number of extra function evaluations which may amount to (imax 4.1. The quadratic case. Before introducing our results for the general non-linear case, a preliminary study of LANCELOT's behaviour on quadratic problems is presented in this section, that is intended to enlighten some of the characteristics of the specific trust region method implemented there. This should be helpful to set up a more adequate framework, in which a reliable interpretation of our testing for the general nonlinear case will become possible. When the objective function f in problem (4.1)-(4.2) is a quadratic function, model (2.2) is identical to f (since exact second derivatives are used in (2.2)). The region where this model should be trusted is therefore infinite at any stage of a trust region algorithm. Hence, a logical choice for the ITRR in that case is \Delta ever, when no particular choice is specified by the user for the ITRR, LANCELOT does not make any distinction when solving a quadratic problem and sets \Delta On the other hand, equations in (2.11) have no solution for (which is the case if f is a quadratic). There- fore, in order to circumvent this possibility, the next instruction has been added in Algorithm ITRR (before (3.1) in Step 1): If ae (0) and go to Step 3. Note that this test does not ensure that f is a quadratic. If needed, a careful strategy should rather be developed to properly detect this special situation. In order to compare both issues, we have tested quadratic problems from the collection, using LANCELOT with \Delta and with Algorithm ITRR in which (4.4) has been added (see LAN and ITRR, respectively, in the tables below). Results are presented in Tables 1 and 2 for a representative sample of quadratic problems (6 unconstrained and 6 bound-constrained). In these tables and the following ones, n denotes the number of variables in the problem, "#its" is the number of major iterations needed to solve the problem, "#cg" reports the number of conjugate gradient iterations performed beyond the Cauchy point, and the last column gives the cpu times in seconds. Note that, for all the tests reported in this section, only one additional function evaluation has been needed by Algorithm ITRR to set \Delta Table A comparison for the unconstrained quadratic problems. LAN ITRR LAN ITRR LAN ITRR LAN ITRR TESTQUAD 1000 Table 1 shows that, as expected, an infinite choice is the best when f is a quadratic function, and the problem is unconstrained. On the other hand, a substantial increase in the number of conjugate gradient iterations is observed in Table 2 (except for problem TORSIONF) when bound constraints are imposed, while the number of major iterations decreases. At first glance, these results may be quite surprising, but they closely depend on the LANCELOT package itself. This package includes a branch, after the conjugate gradient procedure, that allows re-entry of this conjugate gradient procedure when the convergence criterion (based on the relative residual) has been satisfied, but the step computed is small relative to the trust region radius and the model's gradient norm. This is intended to save major iterations, when possible. In Table A comparison for the bound-constrained quadratic problems. LAN ITRR LAN ITRR LAN ITRR LAN ITRR the absence of bound constraints, this avoids an early termination of the conjugate gradient process, allowing attainment of the solution in a single major iteration (see Table 1). In the presence of bounds however, these (possibly numerous) re-entries may not be justified as long as the correct set of active bounds has not yet been identified. This behaviour is detailed in Table 3 for a sequence of increasing initial radii, and exhibits, in particular, a high sensitivity to a variation of the ITRR, which is a rather undesirable feature. Table A comparison for a sequence of increasing initial trust region radii with LANCELOT. Problemn Initial radius \Delta (0) #its time #its OBSTCLAL 1024 #cg 48 55 70 76 93 117 time 14.64 15.73 18.52 18.28 20.97 24.75 #its 4 3 3 3 3 3 time 100.63 5.12 5.27 5.39 5.37 5.51 For comparison purposes, Tables 4 and 5 present the results when removing the aforementioned branch in LANCELOT. This time, an infinitely large value for the ITRR is justified. The conjugate gradient and timing results for Algorithm ITRR are much closer to those of LANCELOT in Table 4 than in Table 2, with a slightly better performance for problem OBSTCLAE and a slightly worse performance for problem JNLBRNG1 (even though a clear improvement occurred due to the branch removal). For problem JNLBRNG1 (as for others in our test set), a limited trust region acts as an extra safeguard to stop the conjugate gradient when the correct active set is not yet detected. This effect of the trust region may be considered as an advantage of trust region methods. In order to complete the above analysis, we now consider problem TORSIONF in Table 2. This problem is characterized by a very large number of active bounds at the optimal solution, while most of the variables are free at the starting point. Because of the very small ITRR, the identification process of the correct active set during the Cauchy point determination is hindered. That is, during the early major iterations, the trust region bounds are all activated at the Cauchy point, without any freedom left for the conjugate gradient procedure. When the trust region has been slightly enlarged, besides trust region bounds, some of the true bounds are also identified by the Cauchy point, but much fewer than the number that would be the case if the trust region was large enough. That is, the conjugate gradient procedure in LANCELOT is restarted each time a true bound is encountered (which occurs at almost every conjugate gradient iteration), in order to maintain the conjugacy between the directions, and the iteration is stopped only when a trust region bound is reached. All this produces extra linear-algebraic costs that greatly deteriorates the algorithm's performance. On the other hand, when starting with a large ITRR, a good approximation of the correct active set is immediately detected by the Cauchy point, and very little work has to be performed during the conjugate gradient process. This observation strengthens the priority given to a large choice for the ITRR, when possible. Table A comparison for the bound-constrained quadratic problems (modified version). Problemn #its #cg time LAN ITRR LAN ITRR LAN ITRR Table A comparison for a sequence of increasing initial trust region radii with LANCELOT (modified ver- sion). Problemn Initial radius \Delta (0) #its 9 6 6 6 time 33.43 30.38 30.57 30.33 #its 8 8 8 8 time 14.21 14.24 14.28 14.19 #its 5 4 4 4 time 102.02 5.66 5.61 5.55 In the light of the above analysis, we tested the 77 nonlinear problems with the original version of LANCELOT versus a modified version, where the extra feature to improve a too small step on output of the conjugate gradient process has been ignored. Slight differences in the results have generally been observed, that were more often in favour of the modified version. For this reason and in order to avoid an excessive sensitivity of the method to the trust region size as well as preventing a large choice for the ITRR (especially when this choice reflects a real adequacy between f and its model), we decided to use the modified version for the testing of the nonlinear case presented in the next section. 4.2. The general case. In order to test Algorithm ITRR, we ran LANCELOT successively ffl Algorithm ITRR, starting with \Delta (0) computed by LANCELOT when no other choice is made by the user); (the distance to the unconstrained Cauchy point, as suggested by Powell in [13]), except when the quadratic model is indefinite, in wich case we omitted the test. The detailed results are summarized in Tables 6 and 7 for the 64 unconstrained problems (possibly including some fixed variables), and in Table 8 for the 13 bound- constrained problems (see ITRR, LAN and CAU, respectively). For each case, the number of major iterations ("#its") and the cpu times in seconds ("time") are re- ported. Note that the number of function evaluations may then be easily deduced : for LANCELOT without Algorithm ITRR, it is the number of major iterations plus 1, while for LANCELOT with Algorithm ITRR, it is the number of major iterations plus 12 if the starting point is refined once (what is pointed out by an asterisk in the first column), and plus 6 otherwise. The tables also present the relative performances for the number of function evaluations, the number of major iterations and the cpu times (see "%f", "%its", and "%time", respectively), computed as \Theta 100 and \Theta 100; where "?" is, in turn, the number of function evaluations, "#its", and "time". In these tables, a "+" indicates when the performance is in favour of Algorithm ITRR and a "\Gamma" when not. Note that a difference of less than five percent in the cpu times is regarded as insignificant. The results first show that, all in all, Algorithm ITRR improves the overall cpu time performance of LANCELOT for a large number of problems: improvements against 13 deteriorations and ties when comparing with improvements against 19 deteriorations and 12 ties when comparing with CAU. More importantly, when they exist, these improvements may be quite significant (19 of them are greater than 30% when comparing with LAN, while 21 of them are greater than 30% when comparing with CAU), and confirm the impact the ITRR choice may have on the method behaviour. On the other hand, the damage is rather limited when it occurs (except for a few cases). Note that the larger number of improvements observed when comparing with LAN does not mean that the ITRR computed by LANCELOT is worse than the distance to the unconstrained Cauchy point. Actually, the improvements when comparing Algorithm ITRR with CAU are generally more significant, and the results show that, on average, LAN and CAU may be considered as equivalent (when compared together). As pointed out by the number of asterisks in the first column of Tables 6 to 8, a change in the starting point occurs very often and makes a significant contribution to the good performance observed. Columns 4 and 7 detail the relative extra cost in terms of function evaluations produced by Algorithm ITRR. Note that, in the current case where the starting point is refined once, the (fixed) extra cost incurs up to 11 extra function evaluations, which is quite high on average, compared with the total number of function evaluations. However, considering the relative performance in the cpu times, the extra cost is generally covered, sometimes handsomely, by the saving produced in the number of major iterations (that is, when %its is positive, %time is generally positive too). Only few cases produce a saving that just balances the extra A comparison for the unconstrained problems. Problemn ITRR LAN CAU ITRR LAN CAU #its #its %f %its #its %f %its time time %time time %time BROYDN7D 92 74 \Gamma39 \Gamma24 73 \Gamma41 \Gamma26 87.8 72.5 \Gamma21 71.1 \Gamma23 BRYBND 14 DQRTIC 1000 28 28 \Gamma17 0 28 \Gamma17 0 18.3 18.2 1 18.2 1 ERRINROS 59 67 \Gamma4 +12 68 \Gamma3 +13 2.9 3.2 +9 3.1 +6 GENROSE 1194 1290 +7 +7 1100 \Gamma10 \Gamma9 1023.8 1115.6 +8 920.3 \Gamma11 LIARWHD 14 LMINSURF 306 272 \Gamma16 \Gamma12 157 \Gamma101 \Gamma95 412.4 380.8 \Gamma8 279.7 \Gamma47 MSQRTALS MSQRTBLS 31 34 \Gamma23 +9 6573.9 6925.5 +5 A comparison for the unconstrained problems (end). Problemn ITRR LAN CAU ITRR LAN CAU #its #its %f %its #its %f %its time time %time time %time NONDIA 1000 POWER 1000 28 28 \Gamma17 0 28 \Gamma17 0 18.3 18.2 1 RAYBENDS 70 52 \Gamma55 \Gamma35 Table A comparison for the bound\Gammaconstrained problems. Problemn ITRR LAN CAU ITRR LAN CAU #its #its %f %its #its %f %its time time %time time %time 4.3 5.6 +23 9.0 +52 QRTQUAD 118 173 +25 +32 315 +59 +63 11.9 16.2 +27 28.4 +58 function evaluations (see the problems for which %its ? 0 and 0 - %time ! 5), while never a saving occurs which does not counterbalance the additional work. On the other hand, when a deterioration occurs in the cpu times (%time ! 0), it is rarely due to the extra cost of Algorithm ITRR exclusively (%its = 0). As a consequence, excepting when functions are very expensive, the use of Algorithm ITRR may be considered efficient and relatively cheap compared with the overall cost of the problem solution. We have observed that only 4 problem tests terminated Algorithm ITRR using update (3.3), while a successful maximal radius satisfying condition (3.1) was selected in the 73 other cases. We also experimented with a simpler choice for fi (0) when ae (0) i is close enough to one and fi (0) that resulted in a markedly performance compared with that of Algorithm ITRR. This proves the necessity of a sophisticated selection procedure for fi (0) i , that allows a swift recovery from a bad initial value for the ratio ae (0) We conclude this analysis by commenting on the negative results of Algorithm ITRR on problem TQUARTIC (see Table 7), when comparing with CAU, and on problem LINVERSE (see Table 8), especially when comparing with LAN. For problem TQUARTIC (a quartic), the ITRR computed by both LANCELOT and Algorithm ITRR is quite small and prevents from doing rapid progress to the solution. The trust region hence needs to be enlarged several times during the minimization algorithm. On the other hand, the distance to the Cauchy point is sufficiently large to allow solving the problem in one major iteration. For problem LINVERSE, the ITRR selected by Algorithm ITRR corresponds to an excellent agreement between the function and the model in the steepest descent direction. However, the starting point produced by Algorithm ITRR, although reducing significantly the objective function value, requires more work from the trust region method to find the solution. This is due to a higher nonlinearity of the objective function in the region where this new point is located and is, in a sense, just bad luck! When testing Algorithm ITRR with this problem, the same ITRR as LANCELOT had been selected, hence producing the same performance. On the other hand, we also tested a series of slightly perturbed initial trust region radii, and observed a rapid deterioration of the performance of the method. Problem LINVERSE is thus very sensitive to the ITRR choice. Note that this sensitivity has been observed on other problems during our testing, and leads to the conclusion that a good ITRR sometimes may not be a sufficient condition to guarantee an improvement of the method. We finally would like to note that no modification has been made in Algorithm ITRR (nor a constrained Cauchy point for CAU has been considered), when solving the bound-constrained problems reported in the paper. The purpose here was simply to illustrate the proposed method on a larger sample than only unconstrained problems. Of course, the author is aware that in the presence of bound constraints, a more reliable version of Algorithm ITRR should include a projection of each trial point onto the bound constraints. We end this section by briefly commenting on the choice of the constants and upper bounds on the iteration counters of Algorithm ITRR. Although a reasonable choice has been used for the testing presented in this paper, a specific one could be adapted depending on the a priori knowledge of a given problem. If, for instance, function evaluations are costly, a lower value for the bounds imax and jmax could be selected. Note however that imax should not be chosen excessively small, in order to be fairly sure that condition (3.1) will be satisfied (unless this condition is suitably relaxed by choosing the value of - 0 ). On the other hand, if the starting point is known to be far away from the solution, it may be worthwhile to increase the value of jmax, provided the function is cheap to evaluate. Improved values for the remaining constants closely depend on a knowledge of the level of nonlinearity of the problem. 5. Conclusions and perspectives. In this paper, we propose an automatic strategy to determine a reliable ITRR for trust region type methods. This strategy mainly investigates the adequacy between the objective function and its model in the steepest descent direction available at the starting point. It further includes a specific method for refining the starting point by exploiting the extra function evaluations performed during the ITRR search. Numerical tests are reported and discussed, showing the efficiency of the proposed approach and giving additional insights to trust region methods for unconstrained and bound-constrained optimization. The encouraging results suggest some direction for future research, such as the use of a truncated Newton direction computed at the starting point rather than the steepest descent direction for the search of an ITRR. An adaptation of the algorithm for methods designed to solve general constrained problems is presently being studied. Acknowledgement . The author wishes to thank an anonymous referee for suggesting a comparison of Algorithm ITRR with the choice of setting the ITRR to the distance to the Cauchy point (as in [13]). Thanks are also due to Andy Conn, Nick Gould, Philippe Toint and Michel Bierlaire who contributed to improve the present manuscript. --R CUTE: Constrained and Unconstrained Testing Environment A trust region algorithm for nonlinearly constrained optimization A trust region strategy for nonlinear equality constrained optimization Global convergence of a class of trust region algorithms for optimization using inexact projections on convex constraints Global convergence of a class of trust region algorithms for optimization with simple bounds Numerical methods for unconstrained optimization and nonlinear equations Practical Methods of Optimization: Unconstrained Optimization A Fortran subroutine for solving systems of nonlinear algebraic equations The conjugate gradient method and trust regions in large scale optimization Towards an efficient sparsity exploiting Newton method for minimization A trust region algorithm for equality constrained minimization: convergence properties and implementation --TR --CTR Wenling Zhao , Changyu Wang, Value functions and error bounds of trust region methods, Journal of Applied Mathematics and Computing, v.24 n.1, p.245-259, May 2007 Stefania Bellavia , Maria Macconi , Benedetta Morini, An affine scaling trust-region approach to bound-constrained nonlinear systems, Applied Numerical Mathematics, v.44 n.3, p.257-280, February Nicholas I. M. Gould , Dominique Orban , Philippe L. Toint, CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software (TOMS), v.29 n.4, p.373-394, December

initial trust region;trust region methods;numerical results;nonlinear optimization

273573

On the Stability of Null-Space Methods for KKT Systems.

This paper considers the numerical stability of null-space methods for Karush--Kuhn--Tucker (KKT) systems, particularly in the context of quadratic programming. The methods we consider are based on the direct elimination of variables, which is attractive for solving large sparse systems. Ill-conditioning in a certain submatrix A in the system is shown to adversely affect the method insofar as it is commonly implemented. In particular, it can cause growth in the residual error of the solution, which would not normally occur if Gaussian elimination or related methods were used. The mechanism of this error growth is studied and is not due to growth in the null-space basis matrix Z, as might have been expected, but to the indeterminacy of this matrix. When LU factors of A are available it is shown that an alternative form of the method is available which avoids this residual error growth. These conclusions are supported by error analysis and Matlab experiments on some extremely ill-conditioned test problems. These indicate that the alternative method is very robust in regard to residual error growth and is unlikely to be significantly inferior to the methods based on an orthogonal basis matrix. The paper concludes with some discussion of what needs to be done when LU factors are not available.

Introduction A Karush-Kuhn-Tucker (KKT) system is a linear system y version of this paper was presented at the Dundee Biennial Conference in Numerical Analysis, June, 1995 and the Manchester IMA Conference on Linear Algebra, July 1995. R. Fletcher and T. Johnson involving a symmetric matrix of the form Such systems are characteristic of the optimization problem subject to A T in which there are linear equality constraints, and the objective is a quadratic func- tion. The KKT system (1.1) represents the first order necessary conditions for a locally minimum solution of this problem, and y is a vector of Lagrange multipliers (see [3] for example). Problems like (1.3) arise in many fields of study, such as in Newton's method for nonlinear programming, and in the solution of partial differential equations involving incompressible fluid flows, incompressible solids, and the analysis of plates and shells. Also problems with inequality constraints are often solved by solving a sequence of equality constrained problems, most particularly in the active set method for quadratic programming. In (1.2) and (1.3), G is the symmetric n \Theta n Hessian matrix of the objective function, A is the n \Theta m Jacobian matrix of the linear constraints, and m n. We assume that A has full rank, for otherwise K would be singular. In some applications, A does not immediately have full rank, but can readily be reduced to a full rank matrix by a suitable transformation. There are various ways of solving KKT systems, most of which can be regarded as symmetry-preserving variants of Gaussian elimination with pivoting (see for example Forsgren and Murray [4]). This approach is suitable for a one-off solution of a large sparse KKT system, by incorporating a suitable data structure which permits fill-in in the resulting factors. Our interest in KKT systems arises in a Quadratic Programming (QP) context, where we are using the so-called null-space method to solve the sequence of equality constrained problems that arise. This method is described in Section 2. An important feature of QP is that the successive matrices K differ only in that one column is either added to or removed from A. The null-space method allows this feature to be used advantageously to update factors of the reduced Hessian matrix that arises when solving the KKT system. However in this paper we do not consider the updaing issue, but concentrate on the solution of a single problem like (1.3), but in a null-space context. In fact the null-space method is related to one of the above mentioned variants of Gaussian Elimination, and this point is discussed towards the end of Section 3. In this paper we study the numerical stability of the null-space method when the matrix K is ill-conditioned. This arises either when the matrix A is close to being rank deficient or when the reduced Hessian matrix is ill-conditioned. It is well known however that Gaussian elimination with pivoting usually enables ill-conditioned systems to be solved with small backward error (that is the computed solution is the exact solution of Stability of Null-Space Methods 3 a nearby problem). As Wilkinson [6] points out, the size of the backward error depends only on the growth in certain reduced matrices, and the amount of growth is usually negligible for an ill-conditioned matrix. Although it is possible for exponential growth to occur (we give an example for a KKT system), this is most unlikely in practice. A consequence of this is that if the computed solution is substituted into the system of equations, a very accurate residual is obtained. Thus variants of Gaussian elimination with pivoting usually provide a very stable method for solving ill-conditioned systems. However this argument does not carry over to the null-space method and we indicate at the end of Section 2 that there are serious concerns about numerical stability when A is nearly rank deficient. We describe some Matlab experiments in Section 6 which support these concerns. In particular the residual of the KKT system is seen to be proportional to the condition number of A. We present some error analysis in Section 4 which shows how this arises. When LU factors of A are available, we show in Section 3 that there is an alternative way of implementing a null-space method, which avoids the numerical instability. This is also supported by Matlab experiments. The reasons for this are described in Section 5, and we present some error analysis which illustrates the difference in the two approaches. In practice, when solving large sparse QP problems, LU factors are not usually available and it is more usual to use some sort of product form method. We conclude with some remarks about what can be done in this situation to avoid numerical instability. Null-Space Methods A null-space method (see e.g. [3]) is an important technique for solving quadratic programming problems with equality constraints. In this section we show how the method can be derived as a generalised form of constraint elimination. The key issue in this procedure is the formation of a basis for the null space of A. We determine the basis in such a way that we are able to solve large sparse problems efficiently. When A is ill-conditioned we argue that there is serious concern for the numerical stability of the method. The null space of A may be defined by and has dimension when A has full rank. Any matrix whose columns are a basis for N (A) will be referred to as a null-space matrix for A. Such a matrix satisfies A T and has linearly independent columns. A general specification for computing a null-space matrix is to choose an n \Theta (n \Gamma m) matrix V such that the matrix 4 R. Fletcher and T. Johnson is non-singular. Its inverse is then partitioned in the following way n\Gammam It follows from the properties of the inverse that A T . By construc- tion, the columns of Z are linearly independent, and it follows that these columns form a basis for N (A). The value of this construction is that it enables us to parametrize the solution set of the (usually) underdetermined system A T in (1.3) by Here Y b is one particular solution of A T any other solution x differs from Y b by a vector, Zv say, in N (A). Thus (2.2) provides a general way of eliminating the constraints, by expressing the problem in terms of the reduced variables v. Hence if (2.2) is substituted into the objective function of (1.3), we obtain the reduced problem We refer to Z T GZ as the reduced Hessian matrix and Z T (GY b\Gammac) as the reduced gradient vector (at the point sufficient condition for (2.3) to have a unique minimizer is that Z T GZ is positive definite. In this case there exist Choleski factors Z T and (2.3) can be solved by finding a stationary point, that is by solving the linear system Then substitution of v into (2.2) determines the solution x of (1.3). The vector Gx \Gamma c is the gradient of the objective function at the solution, so a vector y of Lagrange multipliers satisfying can then be obtained from by virtue of the property that Y T I. The vectors x and y also provide the solution to (1.1) as can readily be verified. In practice, when A is a large sparse matrix, the matrices Y and Z are usually substantially dense and it is impracticable to store them explicitly. Instead, products with Y and Z or their transposes are obtained by solving linear systems involving A. For example the vector could be computed by solving the linear system A by virtue of (2.1). Likewise solving the system A Stability of Null-Space Methods 5 provides the products u 1 t. These computations require an invertible representation of the matrix A to be available. Solving systems involving A is usually a major cost with the null-space method. To keep this cost as low as possible, it is preferable to choose the matrix V to be sparse. Other choices (for example based on the QR factors of A, see [3]) usually involve significantly more fill-in and computational expense. In particular it is attractive to choose the columns of V to be unit vectors, using some form of pivoting to keep A as well conditioned as possible. In this case, assuming that the row permutation has been incorporated into A, it is possible to write "I where A 1 is an m \Theta m nonsingular submatrix. Then (2.1) becomes and provides an explicit expression for Y and Z. In particular we see that We refer to this choice of V as direct elimination as it corresponds to directly using the first m variables to eliminate the constraints (see [3]). We shall adopt this choice of V throughout the rest of the paper. The reduced Hessian matrix Z T GZ is also needed for use in (2.3), and can be calculated in a similar way. The method is to compute the vectors Z T GZe k for of the unit matrix I n\Gammam . The computation is carried out from right to left by first computing the vector z by solving the system A T z Then the product Gz k is computed, followed by the solution of The partition u 2 is then column k of Z T GZ as required. The lower triangle of Z T GZ is then used to calculate the Choleski factor L. A similar approach is essentially used in an active set method for QP, in which the Choleski factor of Z T GZ is built up over a sequence of iterations. (If indefinite QP problems are solved, it may be required to solve KKT systems in which Z T GZ is indefinite. We note that such systems can also be solved in a numerically stable way which preserves symmetry, see Higham [5] in regard to the method of Bunch and Kaufmann [1]). 6 R. Fletcher and T. Johnson An advantage of the null-space approach is that we only need to have available a subroutine for the matrix product Gv. Thus we can take full advantage of sparsity or structure in G, without for example having to allow for fill-in as Gaussian elimination would require. The approach is most convenient when Z T GZ is sufficiently small to allow it to be stored as a dense matrix. In fact there is a close relationship between the null-space method and a variant of Gaussian elimination, as we shall see in the next section, and the matrix Z T GZ is the same submatrix in both methods. Thus it would be equally easy (or difficult) to represent Z T GZ in a sparse matrix format with either method. To summarize the content of this section we can enumerate the steps implied by (2.2) through (2.5) 1. Calculate Z T GZ as in (2.10) and (2.11). 2. Calculate by a solve with A T as in (2.6) with 3. Calculate requiring a product with G. 4. Calculate u 2 by a solve with A as in (2.7). 5. Solve Z T to determine v as in (2.4). 6. Calculate by a solve with A T as in (2.6). 7. Calculate requiring a product with G. 8. Calculate by a solve with A, which also provides z g. When direct elimination based on (2.9) is used, we shall refer to this as Method 1. Step 1 requires m) solves with either A or A T and products with G to set up the reduced Hessian matrix. The remaining steps require 4 solves and 2 products, plus a solve with Z T GZ. In some circumstances these counts can be reduced. If steps 2 and 3 are not required. If the multiplier part y of the solution is not of interest then steps 7 and 8 are not needed. We now turn to the concerns about the numerical stability of the null-space method when A (and hence A 1 and is ill-conditioned. In this case A is close to a rank deficient say, which has a null space of higher dimension. When we solve systems like (2.10) and (2.11), the matrix Z that we are implicitly using is badly determined. Therefore, because of round-off error, we effectively get a significantly different Z matrix each time we carry out a solve. Thus the computed reduced Hessian matrix Z T GZ does not correspond to any one particular Z matrix. As we shall see in the rest of the paper, this can lead to solutions with significant residual error. Stability of Null-Space Methods 7 3 Using LU factors of A In this section we consider the possibility that we can readily compute LU factors of A given by is unit lower triangular and U is upper triangular. We can assume that a row permutation has been made which enables us to bound the elements of L 1 and L 2 by 1. As we shall see, these factors permit us to circumvent the difficulties caused by ill-conditioning to a large extent. (Unfortunately, LU factors are not always available, and some indication is given in Section 7 as to what might be done in this situation.) We also describe how the steps in the null-space method are changed. Finally we explore some connections with Gaussian elimination and other methods, which provide some insight into the likelihood of growth in Z. A key observation is that if LU factors of A are available, then it is possible to express Z in the alternative form in which the UU \Gamma1 factors arising from (2.9) and (3.1) are cancelled out. A minor disadvantage, compared to (2.9), is that L 2 is needed, which is likely to be less sparse than A 2 and also requires additional storage. However if A is ill-conditioned, this is manifested in U (but not usually L) being ill-conditioned, so that (3.2) enables Z to be defined in a way which is well-conditioned. In calculating the reduced Hessian matrix it is convenient to define I and replace equations (2.10) and (2.11) by and I The steps of the resulting null-space method are as follows (using subscript 1 to denote the first m rows of a vector or matrix, and subscript 2 to denote the last 1. Calculate Z T GZ as in (3.4) and (3.5). 2. Calculate s 1 8 R. Fletcher and T. Johnson 3. Calculate requiring a product with G. 4. Calculate u 2 5. Solve Z T for v. 6. Calculate 7. Calculate 8. Calculate requiring a product with G. 9. Calculate 10. Calculate In the above, inverse operations involving L 1 and U are done by forward or backward substitution. The method is referred to as Method 2 in what follows. (For comparability with Method 1, we have also included the calculation of the reduced gradient z, although this would not normally be required.) Note that all solves with the n \Theta n matrix A are replaced by solves with smaller m \Theta m matrices. Also steps 1, 4, 6 and 10 use the alternative definition (3.2) of Z and so avoid a potentially ill-conditioned calculation with A (or A 1 We consider the numerical stability of both Method 1 and Method 2 in more detail in the next section. In the rest of this section, we explore some connections between this method and some variants of Gaussian elimination, and we examine the factored forms that are provided by these methods. It is readily observed (but not well known) that there are block factors of K corresponding to any null-space method in this general format. These are the factors I A T (using blanks to denote a zero matrix). This result is readily verified by using the equation I derived from (2.1). This expression makes it clear that inverse representations of the matrices A and Z T GZ will be required to solve (1.1). However these factors are not directly useful as a method of solution as they also involve the matrices Y T GY and Y T GZ whose computation we wish to avoid in a null-space method. Equation (3.6) also shows that K \Gamma1 will become large when either A or Z T GZ is ill-conditioned, and we would expect the spectral condition number to behave like A M where When using direct elimination (2.8) we may partition K in the form G 11 G 21 G 22 A Stability of Null-Space Methods 9 When A has LU factors (3.1) then it is readily verified that another way of factorizing K is given by6 4 G 11 G 21 G 22 A I Z U U T7 56 4 where Z is defined by (3.2) and G 1 ]. Note that the matrix U occurs on the reverse diagonal of the middle factor, but that no operations with U \Gamma1 are required in the calculation of the factors. Thus any ill-conditioning associated with U does not manifest itself until the factors are used in solving the KKT system (1.1). If there is no growth in Z then the backward error in (3.7) will be small, indicating the potential for a small residual solution of the KKT system. We show in Section 5 how this can come about. Another related observation is that if A is rank deficient, then the factors (3.6) do not exist (since the calculation of Y involves A \Gamma1and hence U be calculated without difficulty. The factorization (3.7) of K is closely related to some symmetry preserving variants of Gaussian elimination. Let us start by eliminating A 2 and the sub-diagonal elements of A 1 by row operations. (As before we can assume that row pivoting has been used.) The outcome of these row operations is that6 4 G 21 G 22 A I G 22 ]. Note that these row operations are exactly those used by Gaussian elimination to form (3.1). To restore symmetry in the factors, we repeat the above procedure in transposed form, that is we make column operations on A Tand A T, which gives rise to (3.7). We can also interleave these row and column operations without affecting the final result. If we pair up the first row and column operation, then the second row and column operation, and so on, then we get the method of 'ba' pivots described by Forsgren and Murray [4]. Thus these methods essentially share the same matrix factors. The difference is that in the null-space method, Z T GZ is calculated by matrix solves with A, as described in Section 2, whereas in these other methods it is obtained by row and column operations on the matrix K. This association with Gaussian elimination enables us to bound the growth in the R. Fletcher and T. Johnson factors of K. The bound is attained for the critical case typified by the matrix for which Row operations with pivots in the (1,7), (2,8), (3,9) and positions leads to the Then column operations with pivots in the (7,1), (8,2), (9,3) and (10,4) positions gives rise to 2 which corresponds to the middle factor in (3.7). In this case and the corresponding matrix Z is given by Stability of Null-Space Methods 11 In general it is readily shown that when m ! n, growth of 2 2m in the maximum modulus element of K can occur. For the null-space method based on (3.2), this example also illustrates the maximumpossible growth of 2 1. In practice however such growth is most unlikely and it is usual not to get any significant growth in Z. 4 Numerical Stability of Method 1 In this and the next section we consider the effect of ill-conditioning in the matrix K on the solutions obtained by null-space methods based on direct elimination. In particular we are interested to see whether or not we can establish results comparable to those for Gaussian elimination. We shall show that the forward error in x is not as severe as would be predicted by the condition number of K. We also look at the residual errors in the solution and show that Method 2 is very satisfactory in this respect, whereas Method 1 is not. In order to prevent the details of the analysis from obscuring the insight that we are trying to provide, we shall adopt the following simple convention. We imagine that we are solving a sequence of problems in which either A or M (the spectral condition numbers of A and increasing without bound. We then use the notation O(h) to indicate a quantity that is bounded in norm by ckhk on this sequence, where there exists an implied constant c that is independent of A or M , but may contain a modest dependence on n. Also we shall assume that the system is well scaled so that 1. This enables us for example to deduce that multiplication of an error bound O(") by A \Gamma1 causes the bound to be increased to O(A "). We also choose to assume that the KKT system models a situation in which the exact solution vectors x and y exist and are not unreasonably large in norm, that is A similar assumption is needed in order to show that Gaussian elimination provides accurate residuals, so we cannot expect to dispense with this assumption. Sometimes it may be possible to argue that we are solving a physical problem which is known to have a well behaved solution. Another assumption that we make is that the choice of the matrix V in (2.8) (and hence the partitioning of A) is made using some form of pivoting. Now the exact solution for Z is given by from (3.3), using the factors of A defined in (3.1). It follows that where L is the spectral condition number of L. Assuming that partial pivoting is used, so that jl ij j 1, and that negligible growth occurs in L \Gamma1, it then follows that negligible growth occurs in Z and we can assert that 12 R. Fletcher and T. Johnson Another consequence of this assumption is that we are able to neglect terms of O(L ") relative to terms of O(A ") when assessing the propagation of errors for Method 2. We shall now sketch some properties (Wilkinson [6]) of floating point arithmetic of relative precision ". If a nonsingular system of n linear equations solved by Gaussian elimination, the computed solution b x is the exact solution of a perturbed system referred to as the backward error. E can be bounded by an expression of the form aeOE(n)" in which ae measures the growth in A during the elimination and OE(n) is a modest quadratic in n. For ill-conditioned systems, and assuming that partial pivoting is used, growth is rare and can be ignored. Also this bound usually overstates the dependence on n which is unlikely to be a dominant factor. Hence for the backward error We can measure the accuracy of the solution either by the forward error b x or by computing the residual x. Using where A is some condition number of A. Since assuming that A " 1, it follows that b Likewise we can deduce that These bounds are likely to be realistic and tell us that for Gaussian elimination, ill-conditioning affects the forward error in x but not the residual r, as long as b x is of reasonable magnitude. Wilkinson also gives expressions for the backward error in a scalar product and hence in the product +Ax. The computed product b s is the exact product of a system in which the relative perturbation in each element of b and A is no more than n" where n is the dimension of x. We can express this as if we make the assumption that b and A are O(1). The first stage in a null-space calculation is the determination of Z T GZ, which we denote by M . In Method 1, this is computed as in (2.10) and (2.11). In (2.10) a column z k of the matrix Z is computed which, by applying (4.4), satisfies Stability of Null-Space Methods 13 where A is the spectral condition number of A. The product with G introduces negligible error, and the solution of (2.11) together with (4.5) shows that Ab Multiplying by L \Gamma1 and extracting the b partition gives using (4.7) and then (4.2). Hence we have established that c The argument has been given in some detail as it is important to see why the error in M is O(A ") and not O( 2 A "). We also observe that hence that c We now turn to the solution of the KKT system using Method 1. We shall assume that systems involving A and M are solved in such a way that (4.5) applies. Using (4.6), and assuming that the computed quantities b s; b t; are O(1), the residual errors in the sequence of calculations are then Ab c A y z These results, together with (4.8), may be combined to get the forward errors in the solution vectors b x and b y. Multiplying through equations (4.9) and (4.13) by A \GammaT magnifies the errors by a factor A (since we are assuming that A = O(1)), giving We can get a rather better bound from (4.11) and (4.15) by first multiplying through by using to give 14 R. Fletcher and T. Johnson from the second partition of the solution. However the first partition of (4.15) gives Combining (4.8) and (4.12) gives We can now chain through the forward errors, noting that a product with Z or Z T does not magnify the error in a previously computed quantity (by virtue of (4.2). However the product M \Gamma1 b in (4.21) magnifies the error in b by a factor M and the product in (4.20) magnifies the error in b g by a factor A . The outcome is that and A M "): (4.23) As we would expect, the forward errors are affected by the condition numbers of A and M . However although the condition number of K is expected to be of the order 2 A M , we see that this factor only magnifies the error in the y part of the solution, with the x part being less badly affected. When K is ill-conditioned we must necessarily expect that the forward errors are adversely affected. A more important question is to ask whether the solution satisfies the equations of the problem accurately. There are three measures of interest, the residuals of the KKT system (1.1), and the reduced gradient Gx is the negative gradient vector at the solution. We note that the vector z is computed as a by-product of step 8 of Method 1. If we compute r we obtain b as in (4.6), and it follows from (4.13) and the definition of A that A T b When computing q we obtain b z from (4.14) and (4.15). Thus the accuracy of b q depends on that of b z. From (4.19) and it follows that Stability of Null-Space Methods 15 from (4.17). (Notice that it is important not to use (4.22) here which would give an unnecessary factor of M .) Then (4.8), (4.12), (4.11) and (4.10) can be used, giving Thus we are able to predict under our assumptions that the reduced gradient b z and the residual b q are adversely affected by ill-conditioning in A, but not by ill-conditioning in M . However the residual b r is unaffected by ill-conditioning either in A or M . Simulations are described in Section 6 which indicate that these error bounds reliably predict the actual effects of ill-conditioning. Method 1 is seen to be unsatisfactory in that an accurate residual q cannot be obtained when A is ill-conditioned. We shall show in the next section that Method 2 does not share this disadvantage. The main results of this section and the next are summarised and discussed in Section 7. 5 Numerical Stability of Method 2 In this section we assess the behaviour of Method 2 in the presence of ill-conditioning in K. Although we cannot expect any improvement for the forward errors, we are able to show that Method 2 is able to give accurate residuals that are not affected by ill- conditioning. The relationship between Method 2 and Gaussian elimination described towards the end of Section 3 gives some hope of proving this result. However this is not immediate because Method 2 does not make direct use of the factors (3.7) in the same way that Gaussian elimination does. A fundamental difficulty with the analysis of Method 2 is that we can deduce from (4.7) that and this result cannot be improved if LU factors are available. To see this, we know that the computed factors of any square matrix A satisfy when there is no growth in b U . If are the exact factors, it follows that U say, where Q is the strict lower triangular part of L U \Gamma1 and R is the upper triangular part. Because L L is unit lower triangular and U b U \Gamma1 is upper triangular we can deduce that involves an inverse operation with b U we can expect that b L and L differ by O(A "). This result has been confirmed by computing the LU factors of a Hilbert matrix in single and double precision Fortran. On applying the result to our matrix A, it follows that (5.1) holds. R. Fletcher and T. Johnson Fortunately all is not lost because we are still able to compute a null-space matrix which accurately satisfies the equation Z T Z denote the null-space matrix obtained from b L in exact arithmetic. It follows that b and hence from (5.2) that We also have b long as A " 1. Our analysis will express the errors that arise in Method 2 in terms of b Z rather than Z and this enables us to avoid the A factor in the residuals. The first step in Method 2 is to compute GZ as in (3.4) and (3.5). In this section we denote c Z as the value computed from b Z in exact arithmetic and use c c M to denote the computed value of c M . It readily follows, using results like (4.2), that c c We now consider the solution of the KKT system using Method 2. As in equations through (4.15) we assume that the computed quantities b s; b t; are O(1). Then the residual errors in the sequence of calculations are A Tb s 1 c c It is readily seen from these equations that the forward errors will propagate in a similar way to Method 1. Turning to the residual errors, the computed value of the residual r is from (5.10), (5.9), (5.5) and (5.3). When computing q we obtain b for Method 1, and it follows from (5.12) that b q 1 O("). From (5.3) we can deduce that so it follows that Stability of Null-Space Methods 17 Thus the accuracy of the residual b q depends on that of b z, as for Method 1. For b z we can use (5.13), (5.11), (5.10) and (5.9) to get Now we can invoke (5.4) and (5.8) giving from (5.7) and (5.6). Thus we have established under our assumptions that all three measures of accuracy for the KKT system are O(") for Method 2 and are not affected by ill-conditioning in either A or M . These results are again supported by the simulations in the next section. Figure 1. Condition numbers of K, A and L 6 Numerical Experiments In order to check the predictions of Sections 4 and 5, some experiments have been carried out on artifically generated KKT systems. These experiments have been carried out in Matlab for which the machine precision is They suggest that the upper bounds given by the error analysis accurately reflect the actual behaviour of an ill-conditioned system. Another phenomenon that occurs when the ill-conditioning is very extreme is also explained. The KKT systems have been constructed in the following way. To make A ill-conditioned we have chosen it as the first m columns of the n \Theta n Hilbert matrix, where provides a sequence of problems for which the condition number of A increases exponentially. Factors are calculated by the Matlab routine lu which uses Gaussian Elimination with partial pivoting, and A is replaced by PA. In the first instance the matrix G is generated by random numbers in the range [\Gamma1; 1]. However to make positive definite, a multiple of the unit matrix is added to the G 22 partition of G, chosen so that the smallest eigenvalue of M R. Fletcher and T. Johnson is changed to 10 1\Gammak for some positive integer k. The assumptions of the analysis require that the KKT system has a solution that is O(1). To achieve this, exact solutions x and y are generated by random numbers in [\Gamma1; 1], and the right hand sides c and b are calculated from (1.1). For each value of m, 10 runs are made with a different random number seed and the statistics are averaged over these 10 runs. First of all we examine the effect of increasing the condition number of A whilst keeping M well-conditioned. To do this we increase m from 2 up to 10, whilst fixing 1. The resulting condition numbers of K, A and L are plotted in Figure 1. It can be seen that the slope of the unbroken line (K ) is about twice that of the dashed line 1, this is consistent with the estimate K 2 A M that we deduced in Section 3. The condition number of L (dotted line) shows negligible increase, showing that there is no growth in L \Gamma1, thus enabling us to assert that O(1). The levelling out of the K graph for is due to round-off error corrupting the least eigenvalue of K. Figure 2. Error growth vs. A for Method 1 Figure 3. Error growth vs. A for Method 2 The effect of the conditioning of A on the different types of error is illustrated in Figures 2 and 3. The forward error is shown by the two unbroken lines, the upper line being the error in y and the lower line being the error in x. The upper line has a slope of about 2 on the log-log scale, and the lower line has a slope of about 1, and both have an intercept with the y-axis of about 10 \Gamma16 . This is precisely in accordance with (4.23) and (4.22). It can also be seen that both methods exhibit the same forward error. The computed value of the residual error shown by the dashed line and both methods show the O(") behaviour as predicted by (4.24) and (5.14), with the increasing condition number having no effect. The difference between Methods 1 and 2 is shown by the computed values of the residual (dotted line) and the reduced gradient line). As we would expect from (4.27), these graphs are superimposed, and they clearly show the influence of A on the error growth for Method 1, as predicted by (4.28). Negligible error growth is observed for Method 2 as predicted by (5.16), except for an Stability of Null-Space Methods 19 increase in q for A greater than about 10 9 . This feature is explained later in the section. Figure 4. Error growth vs. M for Method 1 Figure 5. Error growth vs. M for Method 2 We now turn to see the influence of ill-conditioning in M on the errors. To do this we fix carry out a sequence of calculations with causes M to increase exponentially. Each calculation is the average of ten runs as above. The results are illustrated in Figures 3 and 4, using the same key. The forward errors are again seen to be the same for both methods and they both have a slope of about 1 on the log-log scale, corresponding to the M factor in (4.22) and (4.23). The upper line for the forward error in y lies about 10 5 units above that for the forward error in x, as the extra factor of A in (4.23) would predict. The residual r is seen to be unaffected by the conditioning of M as above. The residual q and the reduced gradient z are also unaffected by M , but the graphs for Method 1 lie above those for Method 2, due to the A factor in (4.28). All these effects are in accordance with what the error analysis predicts. To examine the anomalous behaviour of q in Figure 3 in more detail, we turn to a sequence of more ill-conditioned test problems obtained by using the last m columns of the Hilbert matrix to define A. The results for Method 2 are illustrated in Figure 6 and the anomalous behaviour (dotted line) is now very evident. The reason for this becomes apparent when it is noticed that it sets in when the forward error in y, and hence the value of b y, becomes greater than unity. This possibility has been excluded in our error analysis by the assumption that b O(1). The anomalous behaviour sets in when 2 that is A ' (M ") \Gamma1=2 , or in this case A ' 10 8 , much as Figures 3 and 6 illustrate. For greater values of A there is a term O(b y") in the expression for b indicating that the error is of the form 2 . The fact that this part of the graph of q is parallel to the graph of the forward error in y supports this conclusion. The above calculations have also been carried out using a Vandermonde matrix in place of the Hilbert matrix and very similar results have been obtained. R. Fletcher and T. Johnson Figure 6. Error growth for Method 2 for a more ill-conditioned matrix 7 Summary and Discussion In this paper we have examined the effect of ill-conditioning on the solution of a KKT system by null-space methods based on direct elimination. Such methods are important because they are well suited to take advantage of sparsity in large systems. However they have often been criticised for a lack of numerical stability, particularly when compared to methods based on QR factors. We have studied two methods: Method 1 in which an invertible representation of A in (2.8) is used to solve systems, and Method 2 in which LU factors (3.1) of A are available. We have presented error analysis backed up by numerical simulations which, under certain assumptions on growth, provide the following conclusions ffl Both methods have the same forward error bounds, with b A M "). ffl Both methods give accurate residuals if A is well conditioned, even if M is ill-conditioned gives an accurate residual for Method 1. ffl Both methods give an accurate residual if A is ill-conditioned. These conclusions do indicate that Method 1 is adversely affected by ill-conditioning in A, even though the technique for solving systems involving A is able to provide accurate residuals. The reasons for this are particularly interesting. For example one might expect that when A is ill-conditioned, then A \Gamma1 would be large and we might therefore expect from (2.1) that Z would be large. In fact we have seen that as long as V is chosen suitably, then growth in Z is very unlikely (the argument is similar to that for Gaussian elimination). Of course if V is badly chosen then Z can be large and this will cause Stability of Null-Space Methods 21 significant error. One might also expect that because the forward error in computing Z is necessarily of order O(A "), it would follow that no null-space method could provide accurate residuals. The way forward, which is exploited in the analysis for Method 2, is that Method 2 determines a matrix b Z for which b O("). Thus, although the null-space is inevitably badly determined when A is ill-conditioned, Method 2 fixes on one particular basis matrix Z that is well behaved. This basis is an exact basis for an O(") perturbation to A. Method 2 is able to solve this perturbed problem accurately. On the other hand Method 1 essentially obtains a different approximation to Z for every solve with A. Thus the computed reduced Hessian matrix Z T GZ does not correspond accurately to any one particular b Z matrix. In passing, it is interesting to remark that computing the factors and defining , also provides a stable approach, not so much because it avoids the growth in Z (we have seen that this is rarely a problem), but because it also provides a fixed null-space reference basis, which is an exact basis for an O(") perturbation to A. In the context of quadratic programming, a common solution method for large sparse systems is to use some sort of product form method (Gauss-Jordan, Bartels-Golub-Reid, Forrest-Tomlin etc. It is not clear that such methods provide O(") solutions to the systems involving A that are solved in Method 1 (although B-G-R may be stable in this respect). However the main difficulty comes when the product form becomes too unweildy and is re-inverted. If A is ill-conditioned, the refactorization of A is likely to determine a basis matrix Z that differs by O(A ") from that defined by the old product form. Thus the old reduced Hessian matrix Z T GZ would not correspond accurately to that defined by the new Z matrix after re-inversion. The only recourse would be to re-evaluate Z T GZ on re-inversion, which might be very expensive. Thus we do not see a product form method on its own as being suitable. Our paper has shown that if a fixed reference basis is generated then accurate residuals are possible. It is hoped to show how this might be done in a subsequent paper by combining a product form method with another method such as LU factorization. --R Parlett B. Erisman A. Practical Methods of Optimization Newton methods for large-scale linear equality- constrained minimization Stability of the Diagonal Pivoting Method with Partial Pivot- ing The Algebraic Eigenproblem --TR

null-space method;KKT system;ill-conditioning

273575

Spectral Perturbation Bounds for Positive Definite Matrices.

Let H and H positive definite matrices. It was shown by Barlow and Demmel and Demmel and Veselic that if one takes a componentwise approach one can prove much stronger bounds on $\lambda_i(H)/\lambda_i(H and the components of the eigenvectors of H and H than by using the standard normwise perturbation theory. Here a unified approach is presented that improves on the results of Barlow, Demmel, and Veselic. It is also shown that the growth factor associated with the error bound on the components of the eigenvectors computed by Jacobi's method grows linearly (rather than exponentially) with the number of Jacobi iterations required for convergence.

Introduction . If the positive definite matrix H can be written as where D is diagonal and A is much better conditioned than H then the eigenvalues and eigenvectors of H are determined to a high relative accuracy if the entries of the matrix H are determined to a high relative accuracy. This was shown by Demmel and Veseli'c [2], building on work of Barlow and Demmel [1]. In this paper we strengthen some of the perturbation bounds in [2], and present a unified approach to proving these results. We also show that, just as conjectured in [2], the growth factor that arises in the bound on the accuracy of the components of the eigenvectors computed by Jacobi's method is linear rather than exponential. We now give an outline of the paper and the main ideas in it and then define the notation. In Section 2 we quickly reprove some of the eigenvalue and eigenvector perturbation bounds from [2] in a perhaps more unified way and derive bounds on the sensitivity of the eigenvalues to perturbations in any given entry of the matrix. The main idea in this section is that the analysis is reduced to standard perturbation theory if one can express additive perturbations as multiplicative perturbations. In this respect our approach is similar to that of Eisenstat and Ipsen in [4], except that they assume a multiplicative perturbation and then go on to derive bounds, whereas we assume an additive perturbation, which we rewrite as a multiplicative perturbation, before performing the analysis. Our results are the same as those in [4] for eigenvalues, but not for eigenvectors. We briefly compare our approach to relative perturbation bounds with those in [1, 2, 4] in Section 2.1. We also show that the relative gap associated with an eigenvalue is a very good measure of the distance (in the scaled norm) to the nearest matrix with a repeated eigenvalue. In Section 3 we consider the components of the eigenvectors of a graded positive definite matrix 1 . The key idea here is that if H is a graded positive definite matrix and U is orthogonal such that H then U has a "graded" Department of Mathematics, College of William & Mary, Williamsburg, VA 23187. e-mail: [email protected]. This research was supported in part by National Science Foundation grant DMS-9201586 and much of it was done while the author was visiting the Institute for Mathematics and its Applications at the University of Minnesota. We say that the positive definite matrix H is graded if diagonal and A is much better conditioned than H. roy mathias structure related to that of H and H 1 . 2 This fact can be systematically applied to obtain component-wise perturbation bounds for the eigenvectors of "graded" positive definite matrices and component-wise bounds on the accuracy of the eigenvectors computed by Jacobi's method. The fact that the matrix of eigenvectors is ''graded'' has been observed in [1] and [2], however the results there were weaker than ours, and these papers did not exploit this "graded" structure to any great extent. The basic results on gradedness of eigenvectors are in Section 3.1 and the applications are in Section 3.2. Let Mm;n denote the space of m \Theta n real matrices , and let Mn j M n;n . For a symmetric matrix H we let - 1 its eigenvalues, ordered in decreasing order. For its singular values. The only norm that we use is the spectral norm (or 2-norm) and we denote it by k \Delta k, i.e., (X). When we say that a matrix has unit columns we mean that its columns have unit Euclidean norm. For a matrix or vector X, jXj denotes its entry-wise absolute value. For two matrices or vectors X and Y of the same dimensions we use minfX; Y g to denote their entry-wise minimum, and we use X - Y to mean that each entry of X is smaller than the corresponding entry of Y . To differentiate between the component-wise and positive semidefinite orderings we use A - B to mean that A and B are A is positive semidefinite. We use E to denote a matrix of ones and e to denote a column vector of ones-the dimension will be apparent from the context. In studying the perturbation theory of eigenvectors we use the two notions of the relative gap between the eigenvalues that were introduced in [1], but we use different notation. Given a positive vector - we define and relgap (-; One similarity between the two relative gaps is that it is sufficient to take the minimum either case. However, it is easy to see that relgap (-; i) is at most 1, while relgap(-; i) can be arbitrarily large and that then as we show at the end of the section relgap (- Unfortunately the result for the perturbation to relgap is more complicated, and this sometimes complicates analysis and results involving relgap. (See [2, proof of Proposition 2.6] for such an instance.) 2 By this we mean that both kD are not much larger than 1, where D and D 1 are diagonal matrices such that the diagonal elements of D 1. We use quotes because this not the usual definition of gradedness, but, none-the-less it is related to the gradedness of H and H 1 . perturbation bounds for positive definite matrices 3 It is not clear which relative gap one should use, or whether one should use both, or perhaps the relative gap used in [4]. In [2] it was suggested that relgap(-(H); i) is the appropriate measure of the relative gap between - i (H) and the rest of the eigenvalues of H, and that relgap(oe(G); i) is the appropriate measure of the relative gap between oe i (G) and the rest of the singular values of G. The eigenvector results in Theorems 3.5 and 2.9 and Corollary 2.10 and the singular vector results in in Theorem 2.8 suggest that this is not the case. Luckily, one is most interested in the relative gap when it is small and in this case it doesn't make much difference which definition one chooses. For example, if then one can check that One can also check that the left hand inequality is always valid by a simple application of the arithmetic-geometric mean inequality. Let us now prove (1.1). Define f on (0; 1) 2 by Then relgap (-; So in order to prove (1.1), it is sufficient to prove that for any - 1 which we must have Without loss of generality . The bound (1.3) implies that ~ - 1 - 2 . Since ~ Writing (1.6) as ~ ~ one sees that f( ~ of as a function of ff 1 , and ff 2 , is minimized then Substituting these values for ff 1 and ff 2 , substituting the expressions (1.5) and (1.6) in (1.4) we see that it is sufficient to prove 4 roy mathias or equivalently, which is equivalent to The left hand side of (1.9) is an increasing function of ffi and so in order to verify (1.9) it is sufficient to verify it when ffi is as large as possible - that is when Straight forward algebra shows that (1.9) holds with equality when one substitutes this value of ffi . Thus we have verified (1.1). The bound (1.1) is a slight improvement over [7, Proposition 3.3 equation (3.8)] in the case 2. A unified approach. In this section we give a unified approach to some of the inequalities in [2] and [1]. This approach also allows one to bound the relative perturbation in the eigenvalues of a positive definite matrix caused by a perturbation in a particular entry. The key idea in this section is to express the additive perturbation H + \DeltaH as a multiplicative perturbation of H. Given a multiplicative perturbation of a matrix it is quite natural that the perturbation of the eigenvalues and eigenvectors is also multiplicative. It is then a small step from this multiplicative perturbation to the component-wise perturbation bounds that we desire. There are two ways to write \DeltaH as a multiplicative perturbation and possible choice of Y is H 1 .) If one wants to prove eigenvalue inequalities it seems that both representations give the same bounds. If one uses the representation (2.1) then Ostrowski's Theorem [6, Theorem 4.5.9] yields the relation between the eigenvalues of H and H this is the route taken in [4]. We shall use (2.2) and the monotonicity principle (Theorem 2.1) because the proofs are slightly quicker. Demmel and Veseli'c [2] and Barlow and Demmel [1] used the Courant-Fisher min-max representation of the eigenvalues of a Hermitian matrix to derive similar results. In Jacobi's method one encounters positive definite matrices diagonal and A can be much better conditioned than H. For this reason Demmel and Veseli'c assumed the matrices D(\DeltaA)D with D diagonal to be the data in their work [2]. We consider a slightly more general situation and just assume that H and H + \DeltaH are positive definite. We consider this more general setting firstly to show that one can prove relative perturbation bounds for positive definite matrices without assuming that the matrices are graded and secondly because the results are slightly cleaner in the general case. perturbation bounds for positive definite matrices 5 (For example, the statement of Theorem 2.9, which deals with the general case, is cleaned than the statement of Corollary 2.10 which deals with the special case where D is diagonal.) Lemma 2.2 allows us to derive their results as corollaries of ours. Theorem 2.1. Monotonicity Principle [6, Corollary 4.3.3] Let A; B 2 Mn . If The following lemma will be useful in applying our general results in special situations. Lemma 2.2. Let H be positive definite and let \DeltaH be arbitrary. Let Y 2 Mn be such that Furthermore, if Proof. Since Y Y there must be an orthogonal matrix Q such that Q. Thus For the second part of the lemma take 2 and apply the first part. Then we have as required. 2 Note that if D is diagonal, as it will be in applications, then using the notation of Lemma 2.2 we have Our bounds are in terms of j while those of Demmel and Veseli'c in [2] are in terms of the larger quantity kA They assumed that the diagonal elements of A are all 1. This is not always necessary, though it is a good choice of A in that it approximately minimizes kA We only assume that the diagonal elements of A are 1 when it is necessary. 2.1. Eigenvalues and Singular Values. Our main eigenvalue perturbation theorem is Theorem 2.3. Let H; H +\DeltaH 2 Mn be positive definite and let k. Then Proof. Write 2 . Since 6 roy mathias we have The monotonicity principle (Theorem 2.1) now gives the required bounds. 2 Using the second part of Lemma 2.2 we obtain a result that is essentially the same as [2, Theorem 2.3]: Theorem 2.4. Let positive definite, assume that D diagonal, and let k. Then As another corollary of the monotonicity principle we have a useful relation between the diagonal elements of a positive definite matrix and its eigenvalues [2, Proposition 2.10]: Corollary 2.5. Let Mn be a positive definite matrix and assume that D is diagonal and that the main diagonal entries of A are all 1 while the main diagonal entries of H are ordered in decreasing order. Then Proof. Since -n (A)I - A - 1 (A)I it follows that -n (A)D 2 - DAD - 1 (A)D 2 . The matrix D 2 is diagonal so its eigenvalues are its diagonal elements and these are n. The result now follows from the monotonicity principle. 2 One would expect that the eigenvalues of H are more sensitive to perturbations in some entries of H and less sensitive to perturbations in others. Stating the bound in terms of allows one to derive stronger bounds on the sensitivity of the eigenvalues of H to a perturbation in any one of the entries (or two corresponding off-diagonal entries) of H than if we had replaced j by us assume the notation of the theorem. is the unit n-vector with ith component equal to 1. Suppose that that is a relative perturbation of ffl in the jth main diagonal entry, then and so In fact, we can say more. and so, from the monotonicity principle, we know than - i so the lower bound in (2.5) can be taken as 1, and vice versa if ffl ! 0. If kA as is quite possible for some values of j, then the bound (2.5) is much better than (2.3) with j replaced by perturbation in entries ij and ji, then for any perturbation bounds for positive definite matrices 7 Now taking and so for One may hope to prove a bound with j(A instead of of 2 . To see that such a bound is not possible consider the case A = I. Then the off diagonal elements of A \Gamma1 are 0, but clearly perturbing an off-diagonal element of A does change the eigenvalues of DAD. One can obtain similar bounds on the perturbation of the eigenvectors, singular values and singular vectors caused by a perturbation in one of the elements of the matrix. In the case of eigenvectors and singular vectors one can obtain norm-wise and component-wise bounds. The bounds for singular values and singular vectors involve a row of B is of full rank) rather than just one element of the inverse (or pseudo-inverse). 2.2. Eigenvectors and singular vectors. Now let us see how this approach gives norm-wise perturbation bounds for the eigenvectors of a graded positive definite matrix in terms of the relative gap between the eigenvalues. Let H be positive definite. Let U be an orthogonal matrix with jth column an eigenvector of H corresponding to - j (H) and let be a diagonal matrix with ii element - i (H). Then H, the first part of Lemma 2.2 implies that u be an eigenvector of 1 2 . Then ~ is an eigenvector of H+ \DeltaH . The vector is an eigenvector of H and so the norm-wise difference between u and ~ u is So to show that ~ u can be chosen such that ku \Gamma ~ uk is small we must show that - u can be chosen to be close to e j . We do this in Lemma 2.6, which follows easily from the standard perturbation theory given in [5, pp. 345-6]. We have used the fact that U is orthogonal in (2.8), and hence has norm 1, to obtain a norm-wise bound on u \Gamma ~ u. In Section 3.2 we use the component-wise bounds on U to derive a component-wise bound on u \Gamma ~ u. Lemma 2.6. Let diagonal elements ordered in decreasing order and assume that - j+1 be a symmetric matrix and let 8 roy mathias fflX. Then for ffl sufficiently small - is distinct, and one can choose - u(ffl) to be an eigenvector of H(ffl) such that and so If we take 2 in Lemma 2.6 then one can see that the coefficient of ffl on the right hand side of (2.11) is bounded by@ X element of \Delta. Substituting j for k\Deltak from (2.7) we get relgap(-; ?From (2.8) it follows that we have the same bound on ku \Gamma ~ uk. We summarize the argument in the following theorem: Theorem 2.7. Let H 2 Mn be positive definite and let k. Let Assume that - j (0) is a simple eigenvalue of H. Let u be a corresponding unit eigenvector of H. Then, for sufficiently small ffl, there is an eigenvector u(ffl) of H(ffl) corresponding to - j (ffl) such that relgap(-; As mentioned earlier, we may replace j by kA \Gamma1 kk\DeltaAk. The resulting bound improves that in [2, Theorem 2.5] by a factor of Eisenstat and Ipsen also give a bound on the perturbation of eigenvectors which involves a relative gap [4, Theorem 2.2]. Their bound relates the eigenvectors of H and those of KHK T , where K 2 Mn is non-singular. It is an absolute bound - not a first order bound. To obtain a bound of the form (2.12) from [4, Theorem 2.2] one must find a bound on k(H of the form It is shown in [9] that if (2.13) is to hold for all n \Theta n H and \DeltaH with H positive definite then the constant c must depend on n and must grow like log n. That is, a direct application of [4, Theorem 2.2] to the present situation does not yield (2.12). However one can derive (2.12) using the idea behind the proof of [4, Theorem 2.2] and a more careful argument [3]. perturbation bounds for positive definite matrices 9 Veseli'c and the author have used ideas similar to those in this section to prove a non-asymptotic relative perturbation bound on the eigenvectors of a positive definite matrix [13]. One can apply Lemma 2.6 to GG T and G T G and thereby remove the factor from the bound on the perturbation of the right and left singular vectors given in [2, Theorem 2.16]. Note that one must apply Lemma 2.6 directly in order to obtain the strongest result. If one applies Theorem 2.7 to G T G the resulting bound contains an extra factor Notice that the bound on the right and left singular vectors is not the same - the bound on the right singular vectors is potentially much smaller since relgap can be much larger than relgap . Theorem 2.8. Let G; G+ \DeltaG 2 M m;n and let G y be the pseudo-inverse of G. Assume that G is of rank ng and and that \DeltaG = \DeltaGG y G. and assume that oe j (G) is simple. Let u and v be left and right singular vectors of G corresponding to oe j (G). Then for sufficiently small ffl, there are left and right singular vectors of G(ffl), u(ffl) and v(ffl) corresponding to oe j (G(ffl)) such that relgap (oe 2 (G); Proof. Let U \SigmaV T be a singular value decomposition of G - here u and V are square and \Sigma is rectangular. First let us consider the right singular vectors, which are the eigenvectors of G T G. and hence has norm at most 2k\DeltaGG y Now from (2.11) one can choose ~ u a jth eigenvector of \Sigma T that differs in norm from e j by at most to first order in ffl. Hence, we have the same bound on ku \Gamma V ~ uk to first order in ffl. The vector V ~ u is an eigenvector (corresponding to jth eigenvalue) of which is equal to G T (ffl)G(ffl) up to O(ffl 2 ) terms. Since the jth singular value of G(ffl) is simple, it follows that V ~ u is a right singular vector of of G(ffl) up to O(ffl 2 ). Now let us consider the left singular vectors. As above we can show that roy mathias U and has norm at most j. So by (2.11) there is an eigenvector of that differs from e j in norm by at most to first order in ffl. In the same way as before we can now deduce that there is a vector v(ffl) with this distance of v. 2 2.3. Distance to nearest ill-posed problem. It was shown in [1, Proposition 9] that relgap(-(H); i) is approximately the distance from H to the nearest matrix with a multiple ith eigenvalue in the case that H is a scaled diagonally dominant symmetric matrix and distances are measured with respect to the grading of H. We show that there is a similar result for positive definite matrices. In Theorem 2.9 we show that relgap(-(H); i) is exactly the distance to the nearest matrix with a repeated ith eigenvalue when we use the norm k. We strengthen [1, Propostion 9] in Corollary 2.10 - our upper and lower bounds on the distance differ by a factor of -(A) while those in [1, Proposition 9] differ by a factor of about - 4 (A), a potentially large difference. Our bound is considerably simpler than that in [1], it doesn't involve factors of n (although one could replace by n) and it's validity doesn't depend on the value of the relative gap (the bound in [1] has the requirement diagonal examples show that not every eigenvalue of H will have the maximum sensitivity - \Gamma1 and so this difference in the upper and lower bounds is to be expected. That is to say that one cannot hope to improve the bound (2.17) by more than a factor of n. Our bound involves relgap while the bound in [1] involves relgap. All these reasons suggest that relgap \Gamma1 , and not relgap \Gamma1 , is the right measure of the distance to the nearest problem with a repeated ith eigenvalue. Theorem 2.9. Let H be positive definite. Let - i (H) be a simple eigenvalue of H, so that relgap (- \DeltaH ) is a multiple eigenvalue of H Then Proof. First we show that ffi - relgap (-(H); i). Let \DeltaH be a perturbation that attains the minimum in the definition of ffi . Then k. Let n. By Theorem 2.3 we know that Since \DeltaH has a multiple ith eigenvalue there is an index j 6= i for which - 0 . By (2.16) we must have perturbation bounds for positive definite matrices 11 which implies relgap (-(H); i) - Now we show that Choose a value j such that (One can easily show that this is possible.) Set where x i and x j are unit eigenvectors of H corresponding to - i and - j . One can check that \DeltaH). Because x i and x j are eigenvectors of H Because x i and x j are orthogonal it follows that as required. 2 Corollary 2.10. Let positive definite and assume that D is diagonal and that the main diagonal entries of A are 1. Let - i (H) be a simple eigenvalue of H, so that relgap (- \DeltaH) is a multiple eigenvalue of H+ \DeltaHg: Then Proof. Because and because, by Lemma 2.2, we have it follows that The result now follows from Theorem 2.9. 2 3. Eigenvector Components. It was shown in [1] that the eigenvectors of a scaled diagonally dominant matrix are scaled in the same way as the matrix. Essentially the same proof yields [2, Proposition 2.8]. We strengthen these by a factor -(A) in Corollaries 3.2 and 3.3. In Section 3.2 we strengthen many of the results in [2] by using the stronger results in Section 3.1, and show that the growth factor in the error bound on the eigenvectors computed by Jacobi's method is linear rather than exponential (Theorem 3.8). We also give improved component-wise bounds for the perturbation of singular vectors (Theorems 3.6 and 3.7). It is essential that the D i be diagonal in this section as we are considering the components of the eigenvectors. roy mathias 3.1. Gradedness of eigenvectors. Here we give some simple results on the "graded" structure of an orthogonal matrix that transforms one graded positive definite matrix into another, and use this to derive results on the eigenvectors of a graded positive definite matrix. Lemma 3.1. Let H where the main diagonal entries of the A i are 1 and the D i are diagonal. Assume that H 0 2 Mn and H 1 2 Mm are positive definite. Then Proof. It is easy to check that Now, using the fact that the main diagonal entries of A 1 2 Mm are all 1 for the first inequality, and the monotonicity principle (Theorem 2.1) applied to -n 1 for the second, we have Taking square roots and dividing by - 1n the asserted bound. 2 If the matrix C is orthogonal then H so we have a companion bound stated in the next result. Corollary 3.2. Let H where the main diagonal entries of the A i are 1 and the D i are diagonal. Assume that U is orthogonal. Then This says that if an orthogonal matrix U transforms H 0 into H 1 and -n (D \Gamma1 are not too small then U has a "graded" structure. In the special case that U is the matrix of eigenvectors of and we obtain n: It is useful to have bounds on the individual entries of U and we state a variety of such bounds in (3.5-3.7), but note that they are actually weaker than the norm-wise bounds in (3.4). The bounds (3.5-3.7) are stronger than those in [2, Proposition 2.8] and [1, Proposition 6] which have a factor - 3=2 (A) rather than - 1 2 (A) on the right hand side. The result in [1] is however applicable to scaled diagonally dominant symmetric matrices while our result is only for positive definite matrices. Corollary 3.3. Let positive definite and assume that D is diagonal and that the main diagonal entries of A are 1. Let U be an orthogonal matrix such that diagonal with diagonal entries - i . Then r s perturbation bounds for positive definite matrices 13 s r s r and the first inequality is stronger than the second and third. Proof. The fact that ju ij j is no larger than the first (second) quantity on the right hand side of (3.5) follows from the first (second) inequality in (3.4). The remaining inequalities can be derived from (3.5) using the relations between the eigenvalues of H and its main diagonal entries in Corollary 2.5. This also shows that they are weaker than (3.5). 2 Another way to state the bound in (3.5) is where the minimum is taken component-wise. Recall that E is the matrix of ones. 3.2. Applications of "graded" eigenvectors. Now we use the results in Section 3.1 to give another proof of the fact that components of the eigenvectors of a graded positive definite matrix are determined to a high relative accuracy, then show that relgap (-(H); i) is a good measure of the distance of a graded matrix from the nearest matrix with a multiple ith eigenvalue, where the distance is measured in a norm that respects that grading, and finally that Jacobi's method does indeed compute the eigenvectors to this accuracy (improving on [2, Theorem 3.4]). We now combine lemma 2.6 with the general technique used in Section 2 to obtain a lemma that will be useful in proving component-wise bounds for eigenvectors and singular vectors. Lemma 3.4. Let diagonal elements ordered in decreasing order and assume that - j+1 be a symmetric matrix and let U be an orthogonal matrix. be an eigenvector of H j H(0) associated with - j . Let - u be the upper bound on the jth eigenvector, that is, r s g: Then, for ffl sufficiently small, - is simple, and one can choose u(ffl) to be a unit eigenvector of H(ffl) corresponding to - j (ffl) such that relgap (-; Proof. Since U is the matrix of eigenvectors of H, the bound (3.5) gives r r g: Note that the vector - u defined in the statement of the theorem is just the jth column of the matrix - U just defined in (3.10). ?From Lemma 2.6 it follows that there is an eigenvector - u(ffl) such that 14 roy mathias where r is the vector given by is the element of X. Let u(ffl). So and we must now bound - Ur. The ith element of - Ur is min r r min r s r s min r s For the final equality note that the quantity min r s is now independent of k and is - defined in the statement of this lemma. 2 This result gives component-wise perturbation bounds for eigenvectors and singular vectors as simple corollaries. Theorem 3.5. Let positive definite and let and assume that it is a simple eigenvalue of H. Let u be a corresponding unit eigenvector of H. Let - u be the upper bound on the jth unit eigenvector, that is , r s g: Then, for sufficiently small ffl, - j (ffl) is simple and there is a unit eigenvector u(ffl) of H(ffl) corresponding to - j (ffl) such that relgap (-; perturbation bounds for positive definite matrices 15 Proof. Write where U is the matrix of eigenvectors of H and implies that j. The asserted bound now follows from Lemma 3.4. 2 Lemma 3.4 also yields a component-wise bound on singular vectors. Theorem 3.6. Let m;n be of rank n and let Assume that oe j is simple and that v is a corresponding unit right singular vector. Let - v be the upper bound on the jth right unit singular vector, that is g: Then, for sufficiently small ffl, oe j (ffl) is simple and there is a unit right singular vector v(ffl) of G(ffl) corresponding to oe j (ffl) such that relgap (oe Proof. Let has orthonormal columns, \Sigma 2 Mn is positive diagonal and V 2 Mn is orthogonal. We may write is the pseudo-inverse of B. Note that kFk - 2j. Since the jth singular value of G is simple the corresponding singular vector is differentiable and so in particular, v(ffl), the jth singular vector of G(ffl) (and eigenvector of G(ffl) T G(ffl)) and - v, the jth eigenvector of V \Sigma T differ by at most O(ffl 2 ). According to Lemma 3.4, we know that relgap (oe and hence the bound on v(ffl). 2 This improves [2, Proposition 2.20] in two ways. Firstly, our upper bound - v j is smaller than that in [2] by a factor of about oe \Gamma1 n (B), and secondly, we have a factor in the denominator while in [2] there is a factor oe 2 our bound is smaller by a factor of about oe \Gamma2 (B). The latter difference arises because in [2] the authors used the equivalent of Theorem 3.5 applied to G T G, whereas we use Lemma 3.4. The quantity relgap (oe can be hard to deal with when one perturbs G, and hence also its singular values. It would be more convenient to have relgap (oe; the bound. It is easy to check that relgap (oe; It is worth stating the stronger form of the inequality (3.11) as this is more natural when G is the Cholesky factor of a positive definite H (as is the case in [12]) . In this case oe 2 roy mathias Because we have no component-wise bound on the left singular vectors of we cannot get a component-wise bound on the difference between the left singular vectors of BD and (B \DeltaB)D. We now give a result on component-wise perturbations of singular vectors. Our bound is stronger than [2, Propositions 2.19 and 2.20] by a factor of about oe \Gamma3 upper bound - v is smaller than than in [2] by a factor of oe \Gamma2 n (B), and the denominator here contains a factor oe n (B) while that in [2] contained oe 2 (B)). We could give an improved bound for eigenvectors also, but we restrict ourselves to the case of singular vectors because that is what is important when one uses one sided Jacobi to compute eigenvectors of a positive definite matrix to high component-wise relative accuracy. Theorem 3.7. Let have rank n and assume that are positive and that B has unit columns. Choose ng and let v be a unit right singular vector corresponding to oe i (G). Let \DeltaBD and set Then and there is a vector - v that is a right singular vector of G+ \DeltaG such that Proof. The statement that - v is an upper bound on v follows from (3.5). Let t\DeltaG. The condition ensures that oe i (G(t)) is simple so there is a differentiable v(t) that is a right singular vector of G(t) such that and, from (3.12) we have the component-wise bound dt v(t) and B(t) has unit columns and D(t) is positive diagonal. So for a bound on jv \Gamma - need only bound each of the quantities that depend on t and then integrate the bound. Using the fact one can can show that for t 2 [0; 1] relgap (oe(G(t)); One can check that The condition necessarily less than 1. We use this in the final inequality in the display below. perturbation bounds for positive definite matrices 17 Using (3.5) for the first inequality and bounds on oe j (G(t)); oe n (B(t)) and d i (t) for the subsequent inequalities we have Substituting these bounds into (3.14) gives dt which when integrated yields the asserted inequality. 2 In right handed Jacobi one computes the singular values of G 0 2 Mn by generating a sequence G is an orthogonal matrix chosen to orthogonalize two columns of G i . One stops when One can obtain the right singular vectors of G by accumulating the J i . Demmel and Veseli'c show in [2, Theorem 3.4] that when implemented in finite precision arithmetic this algorithm gives the individual components of the eigenvectors to a high accuracy relative to their upper bounds (actually this is for two sided Jacobi, but the proof is essentially the same for 1 sided Jacobi). However, their bound contains a factor for which they say "linear growth is far more likely than exponential growth". In the next result we show that the growth is indeed linear. One can prove an analogous result for two sided Jacobi applied to a positive definite matrix. Let us denote the product J i J by J i:k . The key idea that allows us to derive a growth factor that is linear in M rather than exponential in M is that we bound J i:k directly, rather than bound it by jJ i:k j - jJ i jjJ bounding each of the terms on the right hand side. This idea has been used profitably in [11] also. Theorem 3.8. Let G has unit columns and D i is diagonal. Assume that where J i is orthogonal and Assume further that the columns of GM are almost orthogonal in the sense that GM satisfies (3.15) with tolerance tol. Let and assume that be the computed column of J 0:M \Gamma1 corresponding to oe j (G). Then there is a unit right singular vector u T of G roy mathias corresponding to oe j (G) such that, to first order in j; ffl; and tol, oe \Gamma2 min relgap (oe(G); where is the upper bound on u from (3.5). The bound (3.16) is a first order bound. The proof below would also yield a bound that takes into account all the higher order terms, but the resulting inequality would be much more complicated and probably not any more useful. If the G i and J i arise from right handed Jacobi applied to G in finite precision arithmetic with precision ffl then one can take Theorem 4.2 and the ensuing discussion]. Let us compare our bound with relgap (oe(G); min which is essentially the bound on the computed right singular values from [2, Theorem 4.4] stated in our notation. Our bound is stronger in several respects. The term q(M; n) is a growth factor that is exponential in M , while our bound is linear in M . As we have mentioned earlier, the upper bound vector - u in (3.17) is larger than - u by a factor of about oe \Gamma2 n (B), which could be quite significant. Also, we have two terms, one in oe \Gamma2 min and the other in (oe min \Delta relgap (oe(G); these quantities are less than (oe 2 min relgap (oe(G); which occurs in (3.17). A weakness of both bounds is that they contain the factor oe \Gamma1 min (defined in the statement of the theorem) rather than oe n (B). It has been observed experimentally [2, Section 7.4] that oe min =oe n (B) is generally 1 or close to 1, but no rigorous proof of this fact is known. Mascarenhas has shown that the ratio can be as large as n=4 [8]. One can also see that for a given ffl we can take tol, the stopping tolerance, as large as ffloe \Gamma1 significantly increasing the right hand side of (3.16). Typically, it is suggested that one take tol to be a modest multiple of ffl when one wants to compute the eigenvectors or eigenvalues to high relative accuracy [2]. Thus this larger value of tol may be useful in practice to save little computation through earlier termination. The rest of the paper is devoted to the rather lengthy proof of this theorem. Proof. The outline of the proof is as follows. First we will bound ju \Gamma - uj where u is the value of the jth column of J 0:M \Gamma1 computed in exact arithmetic. Next we will is for true) is an exact singular vector of G associated with oe j (G). The inequality (3.16) follows by combining these two bounds. Through out we will use the facts that oe j (GM min and Now consider uj. This bound depends only on the scaling of the J i:k and is independent of relgap (oe(G); j). If X;Y 2 Mn are multiplied in floating point arithmetic with precision ffl the result is XY Using this fact one can show by induction that (3. perturbation bounds for positive definite matrices 19 is the error in multiplying J 0:i\Gamma1 and J i and is the first order effect of this error in the computed value of J 0:M . Taking absolute values in (3.18) gives the component-wise error bound Now (D 0 jJ noe noe noe noe min E: Recall that E denotes the matrix of ones. We have used the first term in (3.8) and the fact that, up to first order, J i+1:M \Gamma1 diagonalizes G T for the first inequality, (3.2) twice for the third. Since GM has orthogonal columns, up to O(tol), and the singular values of GM are the same as those of G to O(j) it follows that DM = \Sigma, at least to first order. So, multiplying by D 0 and \Sigma \Gamma1 , we have, to first order min fflE: In the same way, we obtain the first order bound min fflE: These two bounds can be combined to give min where the minimum is taken component-wise. (Note that D D.) The jth column of this is the inequality we desire: u: This completes the first step. Now let us bound the error between u and a singular vector of G. If the columns of GM were orthogonal then in particular e j would be a right singular vector associated with oe j (GM ). If in addition all the \DeltaG i were 0 then would be a right singular vector associated with oe j (G). Neither of these hypotheses is true, though in each case they are 'almost true' and so u is close to being a singular vector of G 0 . We now bound he difference. First we will consider the fact that the columns of GM are not exactly orthogonal. Write Then each entry of A is at most tol in absolute value and so tol. The equation (3.20) implies that there is an orthogonal matrix Q such that One can check that roy mathias we will use this bound later. Now consider the effect of the \DeltaG i . It is easy to check, by induction for example, that where Now, using the assumption k\DeltaG for the first inequality and (3.2) for the second we have Together with (3.21) this yields Now we will combine these two results to show that GM + \DeltaG has a right singular vector close to e j and hence that G 0:M \Gamma1 has a right singular vector close to . The right singular vectors of GM + \DeltaG are the same as those of Q(GM is the orthogonal matrix introduced after equation (3.20). Also, The jth right singular vector of DM is e j and So by Theorem 3.7 there is a right singular vector v of G+ \DeltaG corresponding to its jth singular value such that where Now let us drop the second order terms in (3.22) and - v. The term - is O(tol)oe 1 (BM ) so we may drop all first order terms in - v and in the ratio in (3.22). In particular we may replace oe n (BM all by 1. We do not perturbation bounds for positive definite matrices 21 assume that relgap (oe(G M ); j) is large with respect to j and - so we cannot replace relgap (oe(G M ); However, as was shown at the end of the introduction, we have relgap (oe(G M ); min and hence relgap (oe(G M ); these substitutions we obtain the bound that is equivalent to (3.22) up to first order relgap (oe(G M ); where For convenience, let the coefficient of - v in (3.23) be denoted by c. construction it is a right singular vector of G 0 corresponding to oe j (G 0 ). Now we can complete the proof by bounding We have used a slight generalization of (3.5) for the penultimate inequality and have dropped second order terms in the last inequality. Similarly, and so Now combine the bound on ju \Gamma u Acknowledgment I thank the referee whose detailed comments, including the observation that grading is not necessary for relative perturbation bounds, have greatly improved the presentation of the results in this paper. --R Computing accurate eigensystems of scaled diagonally dominant Jacobi's method is more accurate than QR. personal communication. Relative perturbation techniques for singular value problems. Matrix Computations. Matrix Analysis. Relative perturbation theory: (I) eigenvalue and singular value variations. A note on Jacobi being more accurate then QR. A bound for the matrix square root with application to eigenvector perturbation. Matrix Anal. Accurate eigensystem computations by Jacobi methods. Instability of parallel prefix matrix multiplication. Fast accurate eigenvalue computations for graded positive definite matrices. A relative perturbation bound for positive definite matrices. --TR

jacobi's method;symmetric eigenvalue problem;error analysis;perturbation theory;positive definite matrix;graded matrix