J. C. Wyllie. The Complexity of Parallel Computations. Ph.d. dissertation, Cornell University, Computer Science Department, Ithaca, NY, 1981. 29  Home/Search   Document Not in Database   Summary   Related Articles   Check

....value initially 1, and performing a prefix sum on those numbers. Conversely, if we could rank we could perform prefix computation, by using the ranks to move the data into an array and performing the array prefix computation algorithm described above. A simple algorithm for ranking, due to Wyllie [75], is as follows. 1) Give each list member of the list a data value of 1. 2) For each list member other than the first, add the value of the previous member in the list into the value at this list member. 3) Recursively call steps (2) and (3) on a new linked list formed by linking each list ....

....by replacing each tree edge with two directed arcs, one in each direction. Such a tour is a directed cycle such that each arc appears exactly once in the cycle. The vertices of the tree typically appear many times, so it is not a simple cycle. Euler tours for binary trees were first used by Wyllie [75]. The Euler tour technique for general trees was introduced by Tarjan and Vishkin [66] who used it as part of an algorithm for finding biconnected components of a graph. The examples we give of tree functions computable using Euler tours were also introduced in the same paper. For each vertex v, ....

J.C. Wyllie, The Complexity of Parallel Computation. Tech. Rep. TR 79-387, 1979, Dept. of Computer Science, Cornell University, Ithaca, New York.

....the inherent serial part in the application, the work imbalance between the executing threads of control, any redundant computation that may be performed, and additional work introduced by the parallelization. The achievable performance of an application even on an ideal machine such as the PRAM [7] would be lower than the available curve as depicted by the achievable curve in Figure 1 due to algorithmic overheads. The implementation of the application on an actual machine would result in further slowdown due to software and hardware interaction overheads. Software interaction overheads ....

....Such a feature can narrow down bottlenecks in the execution and help in application restructuring. For each executing processor, SPASM accumulates the following statistics for each mode of execution: ffl computation time: This is the time taken to execute on an ideal machine such as the PRAM [7]. ffl latency: Accesses to variables in a shared memory system may involve the network, and the physical limitations of the network tend to contribute to overheads in the execution. The latency overhead is defined as the total amount of time spent by a processor waiting for messages due to the ....

J. C. Wyllie, The Complexity of Parallel Computations, Ph.D. thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

....applications as a subroutine in other problems. The theoretical interest of list ranking is evident: it is one of the most basic problems, and in the theory of parallel computation (and thus by inheritance also in the theory of external computation) Therefore it has been considered extensively [25, 6, 7, 1, 2]. List ranking appears as a subroutine in many graph problems particularly because it is the key ingredient of the Euler tour technique [23] List ranking has linear sequential complexity, and can be solved efficiently by a trivial algorithm. This makes it very hard to achieve good speed ups on a ....

....extent, determined by the time for computing the Euler array, which boils down to solving a list ranking problem on an Euler tour of the tree. The depths can be computed by keeping track of some additional information. 1.2 Previous Results PRAMs. On PRAMs, the basic approach is pointer jumping [25]. This technique can be used in a list ranking algorithm which, for a list of length N , runs in O(log N) time with O(N log N) work. Using accelerated cascading, the work of this algorithm is reduced to the optimal O(N) while maintaining running time O(log N) 6] These improved algorithms ....

Wyllie, J.C., The Complexity of Parallel Computations, PhD Thesis, Computer Science Department, Cornell University, Ithaca, NY, 1979. 14

....the list compares itself with the element on its right and the element on its left. This indicates the elements that are the ends of the intervals. Then, every other member assigns itself to the corresponding interval. This can be done in O(log n) time using the path doubling technique of Wyllie [27]. function BUILD( h) begin L : a list of the edges with the colors of their endpoints listed in increasing order; sort L lexicographically by the colors of the endpoints; determine the set of intervals of L; A NEW ALGORITHM FOR MIS 5 for each interval L ij in parallel do ( the ....

J. C. Wyllie, The complexity of parallel computations, Ph. D. thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

....with the architecture. The algorithmic overhead arises from factors such as the serial part and work imbalance in the algorithm, and is captured by the ideal time metric provided by SPASM. Ideal time is the time taken by the parallel program to execute on an ideal machine such as the PRAM [31]. This metric includes the algorithmic overheads but does not include any overheads arising from architectural limitations. Of the interaction component, the latency and contention introduced by network limitations are the important overheads that are of relevance to this study. The time that a ....

J. C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

....data. Since this modified algorithm requires no more than approximately n noncontiguous memory accesses while running in O(n) computation time, it is optimal according to our model. 7 The first fast parallel algorithm for prefix computations was probably the list ranking algorithm of Wyllie [18], which requires at least n log n non contiguous accesses. Other parallel algorithms which improved upon this result include those of Vishkin [16] 5n non contiguous accesses) Anderson and Miller [5] 4n noncontiguous accesses) and Reid Miller and Blelloch [13] 2n non contiguous accesses see ....

J.C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Department of Computer Science, Cornell University, Ithica, NY, 1979.

....parallel systems it is tempting to investigate promising alternatives from the large body of research done on parallel sorting in the field of theoretical computer science. Such theoretical algorithms are mainly described for the PRAM (Parallel Random Access Machine) model of computation [13, 26]. This paper shows how this kind of investigation can be done on a simple but versatile environment for programming and measuring of PRAM algorithms [18, 19] The practical value of Cole s Parallel Merge Sort algorithm [10,11] have been investigated by comparing it with Batcher s bitonic sorting ....

....programmed, executed and measured on a CREW PRAM simulator. The Original CREW PRAM Unfortunately, there is a lot of different opinions on how the (CREW) PRAM should be programmed. My work is originally inspired by James Wyllie s well written Ph.D. thesis The Complexity of Parallel Computations [26]. The thesis gives a high level and succinct description of how a PRAM may be programmed. Background The P RAM (parallel random access machine) was first presented by Steven Fortune and James Wyllie [13] It was further elaborated in Wyllie s Ph.D. thesis [26] The P RAM is based on random access ....

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J. C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Dept. of Computer Science, Cornell University, 1979.

....of the algorithmic overhead would help in re structuring the algorithm to improve its performance. Algorithmic overhead is the difference between the linear curve and that which would be obtained (the ideal curve in Figure 1) by executing the algorithm on an ideal machine such as the PRAM [32]. Such a machine idealizes the parallel architecture by assumingan infinite number of processors, and unit costs for communication and synchronization. A real execution could deviate significantly from the ideal execution due to the overheads such as latency, contention, synchronization, ....

....(SPASM) to be presented in the next sub section, provides an elegant set of mechanisms for quantifying the different overheads we discussed earlier. The algorithmic overhead is quantified by computing the time taken for execution of a given parallel program on an ideal machine such as the PRAM [32] and measuring its deviation from a linear speedup curve. Further, we separate this overhead into that due to the serial part (serial overhead) and that due to work imbalance (work imbalance overhead) As we mentioned earlier, the interaction overhead should be separated into its component parts. ....

J. C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Department of Computer Science, Cornell University, 1979.

....k ERCW k CREW k CRCW k Figure 1: Relations between several types of PRAMs the end rather than from the beginning, i.e. s i = P n k=i x k . Let v(i) be the value of the ith node in the linked list and F (i) its successor, then the following simple algorithm solves the prefix sums problem [Wyl81]: do dlog ne times for i = 1; n pardo v(i) v(i) v(F (i) F (i) F (F (i) W.l.o.g. we have to assume F (l) 0, F (0) 0, and v(l) 0, where l is the last element in the list. This algorithm was intended to be EREW, but it is in fact EROW. However, it is in no way an owner read ....

J. C. Wyllie. The Complexity of Parallel Computations. Ph.d. dissertation, Cornell University, Computer Science Department, Ithaca, NY, 1981.

....A special case is that of determining for every element in the list its distance from the head (or tail) of the list. Once the position of an element in the list is known, the algorithm for prefix sum can be applied to the linked list prefix sums. Although an O(log n) time algorithm due to Wyllie [88] was known for a while, it used n processors and was not optimal. The following result was obtained independently by Anderson and Miller [10] Han [47] and Cole and Vishkin [22] Lemma 2.2 The list ranking problem can be solved in an EREW model optimally in O(log n) time. 2.2 Parallel Sorting ....

J. Wyllie. The Complexity of Parallel Computation. PhD thesis, Cornell University, 1979.

....See Figure 1. As output we require that every processor stores for each of its n=p nodes x 2 S the value dist(x) A trivial sequential algorithm solves the list ranking problem in optimal linear time by traversing the list. Several PRAM list ranking algorithms have been proposed [15] 20] Wyllie [25] proposed a non optimal O(log n) time algorithm with total work greater than O(n) The first optimal O(log n) EREW PRAM algorithm is due to Cole and Vishkin [7] Proc.1 Proc.2 Proc.3 Proc.4 Figure 1: A Linear Linked List Stored In A Distributed Memory Multiprocessor Another optimal deterministic ....

....is bounded, with high probability, by 118, and we conjecture that actual number of communication rounds required will not exceed 50. Our randomization technique is very different from the ones used in [1, 17, 18] In the above model, our algorithm uses considerably fewer communication rounds than [1, 2, 4, 5, 7, 15, 17, 18, 20, 21, 25]. The simple version of our algorithm is a generalization of the algorithm used in ReidMiller s [21] empirical study of parallel list ranking for the Cray C 90. She uses the Cray C 90 in shared memory mode. The analysis of our simple list ranking algorithm improves the estimates on the load ....

J. C. Wyllie, "The complexity of parallel computation", Technical Report TR 79-387, Department of Computer Science, Cornell University, 1979. ln (i+1) (n) (2p/3) 2i+1 i k

....order in which they are visited during execution of the greedy DFS algorithm on H (Lemma 5.1) The first two steps of REDUCE can be implemented in O(log 2 n) time using M(n) processors on an EREW PRAM. The third step of REDUCE only requires constant time. By the results of Cole [6] and Wyllie [30], it follows that steps 4 and 5 can be implemented within the same bounds as step 1 on the EREW PRAM. This shows REDUCE can be implemented using O(log 2 n) time and M(n) processors on an EREW PRAM. The proof of the theorem is complete contingent on the proof of the lemma. A layered dag is a dag ....

J.C. Wyllie. A Complexity of Parallel Computations. PhD thesis, Cornell University, 1981.

....small number of processors are available on a symmetric multiprocessor, any parallel algorithm must be competitive even on a single processor in order to be practically significant. The first fast parallel algorithm for prefix computations was probably the list ranking algorithm of Wyllie [16] (note that his algorithm was originally developed for the suffix computation problem, and our presentation here makes appropriate modifications for our prefix computation problem) After initially traversing the list to establish the predecessor links, the algorithm consists of iteratively ....

....expect that every step in the traversal would involve accessing a non contiguous location in memory. However, what distinguishes the [S] benchmark from the [R] benchmark is the the constant stride, which can take advantage of cache pre fetching. Number of Threads Benchmark Step: 1] 2] 4] 8] [16] [R] S] O] R] S] O] R] S] O] R] S] O] R] S] O] 1) 3) 0.59 0.87 0.66 0.34 0.40 0.34 0.18 0.21 0.18 0.10 0.12 0.10 0.08 0.08 0.08 (4) 6.69 1.86 2.33 3.40 1.08 1.17 1.75 0.57 0.59 0.96 0.31 0.30 0.74 0.22 0.18 (5) 0.01 0.12 0.01 0.01 0.04 0.01 0.01 0.05 0.01 0.01 0.06 0.01 0.01 ....

J.C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Department of Computer Science, Cornell University, Ithica, NY, 1979.

....cf(n)g 1 n g(c) where c0 is a fixed constant and g(c) is a polynomial in c with g(c) 1 for c 1 [19] A trivial sequential algorithm solves the list ranking problem in optimal linear time by traversing the list. Several PRAM list ranking algorithms have been proposed [15] 20] Wyllie [25] proposed a non optimal O(log n) time algorithm with total work greater than O(n) The first optimal O(log n) EREW PRAM algorithm is due to Cole and Vishkin [7] Another optimal deterministic algorithm is given by Anderson and Miller [2] Parallel list ranking algorithms using randomization were ....

....is bounded, with high probability, by 118, and we conjecture that actual number of communication rounds required will not exceed 50. Our randomization technique is very different from the ones used in [1, 17, 18] In the above model, our algorithm uses considerably fewer communication rounds than [1, 2, 4, 5, 7, 15, 17, 18, 20, 21, 25]. The simple version of our algorithm is a generalization of the algorithm used in ReidMiller s [21] empirical study of parallel list ranking for the Cray C 90 in shared memory mode. The analysis of our simple list ranking algorithm improves the estimates on the load imbalance provided in [21] ....

J. C. Wyllie, "The complexity of parallel computation", Technical Report TR 79-387, Department of Computer Science, Cornell University, 1979. Proc.1 Proc.2 Proc.3 Proc.4

....parallel time on n processors if for each vertex a pointer to its successor and its left sibling is given. If we dedicate each processor to log n tasks we can compute the Euler path in O(log n) time on n= log n processors. A technique similar to the Euler path technique was used by Wyllie [Wyl79] for tree traversals. As a general technique it was introduced by Tarjan and Vishkin [TV85] A prefix sum algorithm for the EREW PRAM running in O(log n) time using n= log n processors was developed by Ladner and Fischer [LF80] For optimal list ranking on the EREW PRAM we can either use the ....

J. C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Computer Science Department, Cornell University, Ithaca, NY, 1979.

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J. C. Wyllie. The Complexity of Parallel Computations. Ph.d. dissertation, Cornell University, Computer Science Department, Ithaca, NY, 1981. 29

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J. C. Wyllie, The complexity of parallel computation, TR 79-387, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

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J. C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Dept. of Computer Science, Cornell University, 1979. 245

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. J. C. Wyllie. The complexity of parallel computation. TR 79-387, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

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Wyllie, J. C. The complexity of parallel computation. Tech. Rep. 79-387, Department of Computer Science, Cornell University, Ithaca, NY,

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. J.C. Wyllie. The complexity of parallel computation, TR 79-387, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

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. J. C. Wyllie. The complexity of parallel computation, TR 79-387, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

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J. C. Wyllie, "The complexity of parallel computation," TR 79-387, Dep. Comput. Sci., Cornell

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. J. C. Wyllie. The complexity of parallel computation, TR 79-387, Department of Computer Science, Cornell University, Ithaca, NY, 1979.

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J. C. Wyllie. The Complexity of Parallel Computations. PhD thesis, Computer Science Department, Conell University, Ithaca, NY, 1979.

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