Efficient matrix chain ordering in polylog time

Bradford, Phillip G.; Rawlins, Gregory J. E.; Shannon, Gregory E.

doi:10.1109/ipps.1994.288295

Cited by 6 publications

(8 citation statements)

References 25 publications

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“…In only 4% of the test cases, other solutions are more than 1.1 times faster than the generated code. In at least 10% of the test cases (from a minimum of 10% for Armadillo recommended to 25% for Eigen naive), the other implementations are more than 10 times slower than the GMC 8 solutions. In the worst case, the naive Eigen and Matlab solutions are about 200 times slower than the best solution.…”

Section: Resultsmentioning

confidence: 99%

“…e naive implementation uses inv(), while the recommended one uses the / and \ operators. 8 Development version of Julia 0.7 from September 4, 2017. 9 Version 2017a.…”

Section: Resultsmentioning

confidence: 99%

“…One such example is the reduction of a symmetric matrix to tridiagonal form [22]. As a second example, consider the matrix chain ABCDE with matrix sizes (from le to right) 130, 700, 383, 1340, 193 and 900. e parenthesization that results in the smallest number of FLOPs is (((AB)C)D)E with 3.16 × 10 8 FLOPs. e parenthesization that results in the shortest execution time is ((AB)(CD))E, even though the number of FLOPs is slightly higher with 3.32 × 10 8 .…”

Section: Cost Functionsmentioning

confidence: 99%

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The generalized matrix chain algorithm

Barthels

Copik

Bientinesi

2018

Proceedings of the 2018 International Symposium on Code Generation and Optimization

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In this paper, we present a generalized version of the matrix chain algorithm to generate efficient code for linear algebra problems, a task for which human experts o en invest days or even weeks of works. e standard matrix chain problem consists in finding the parenthesization of a matrix product M := A 1 A 2 · · · A n that minimizes the number of scalar operations. In practical applications, however, one frequently encounters more complicated expressions, involving transposition, inversion, and matrix properties. Indeed, the computation of such expressions relies on a set of computational kernels that offer functionality well beyond the simple matrix product. e challenge then shi s from finding an optimal parenthesization to finding an optimal mapping of the input expression to the available kernels. Furthermore, it is o en the case that a solution based on the minimization of scalar operations does not result in the optimal solution in terms of execution time. In our experiments, the generated code outperforms other libraries and languages on average by a factor of about 9. e motivation for this work comes from the fact that-despite great advances in the development of compilers-the task of mapping linear algebra problems to optimized kernels is still to be done manually. In order to relieve the user from this complex task, new techniques for the compilation of linear algebra expressions have to be developed.

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Section: Resultsmentioning

confidence: 99%

“…e naive implementation uses inv(), while the recommended one uses the / and \ operators. 8 Development version of Julia 0.7 from September 4, 2017. 9 Version 2017a.…”

Section: Resultsmentioning

confidence: 99%

Section: Cost Functionsmentioning

confidence: 99%

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The generalized matrix chain algorithm

Barthels

Copik

Bientinesi

2018

Proceedings of the 2018 International Symposium on Code Generation and Optimization

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“…Many parallel algorithms aimed at reducing the ordering time have been studied using the dynamic programming method [13], [14], [15], [16] and the convex polygon triangulation method [17], [18], [19]. Bradford et al [16] proposed a parallel algorithm based on dynamic programming, which runs in Oðlog 3 ðnÞÞ time with n= logðnÞ processors on the CRCW PRAM model. Czumaj [17] proposed an Oðlog 3 ðnÞÞ time algorithm based on the triangulation of a convex polygon, which runs with n 2 = log 3 ðnÞ processors on the CREW PRAM.…”

Section: Parallel Evaluation Timementioning

confidence: 99%

“…The optimal product sequence can be found in Oðn logðnÞÞ time using a sequential algorithm [10], [11]. In addition, many parallel algorithms that run in polylog time have been studied [16], [17], [19]. Thus, by using these parallel algorithms, it is possible to find the optimal product sequence within polylog time on P processor systems.…”

Section: Optimal Product Sequence By the Mcopmentioning

confidence: 99%

Processor allocation and task scheduling of matrix chain products on parallel systems

Lee¹,

Kim

Hong

et al. 2003

IEEE Trans. Parallel Distrib. Syst.

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The problem of finding an optimal product sequence for sequential multiplication of a chain of matrices (the matrix chain ordering problem, MCOP) is well-known and has been studied for a long time. In this paper, we consider the problem of finding an optimal product schedule for evaluating a chain of matrix products on a parallel computer (the matrix chain scheduling problem, MCSP). The difference between the MCSP and the MCOP is that the MCOP pertains to a product sequence for single processor systems and the MCSP pertains to a sequence of concurrent matrix products for parallel systems. The approach of parallelizing each matrix product after finding an optimal product sequence for single processor systems does not always guarantee the minimum evaluation time on parallel systems since each parallelized matrix product may use processors inefficiently. We introduce a new processor scheduling algorithm for the MCSP which reduces the evaluation time of a chain of matrix products on a parallel computer, even at the expense of a slight increase in the total number of operations. Given a chain of n matrices and a matrix product utilizing at most P =k processors in a P -processor system, the proposed algorithm approaches kðn À 1Þ=ðn þ k logðkÞ À kÞ times the performance of parallel evaluation using the optimal sequence found for the MCOP. Also, experiments performed on a Fujitsu AP1000 multicomputer show that the proposed algorithm significantly decreases the time required to evaluate a chain of matrix products in parallel systems.Index Terms-Matrix chain product, parallel matrix multiplication, matrix chain scheduling problem, processor allocation, task scheduling. ae 394 Fig. 7. Comparison of experimental results with analysis. (a) ÉðP Þ as a function of m with P ¼ 512 and n ¼ 100 or 200. (b) ÉðP Þ as a function of n with P ¼ 512 and m ¼ 50 or 100. (c) Expected behavior of ÉðP Þ as a function of P and n.

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Dynamic Programming for Energy Minimization

Jeong¹

2014

Architectures for Computer Vision

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Efficient matrix chain ordering in polylog time

Cited by 6 publications

References 25 publications

The generalized matrix chain algorithm

The generalized matrix chain algorithm

Processor allocation and task scheduling of matrix chain products on parallel systems

Dynamic Programming for Energy Minimization

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