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Fast and scalable parallel algorithms for matrix chain product and matrix powers on distributed memory systems

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1 Author(s)
Keqin Li ; Dept. of Comput. Sci., State Univ. of New York, New Paltz, NY, USA

Given N matrices A1, A2,…, AN of size N×N, the matrix chain product problem is to compute A1×A2×···AN . Given an N×N matrix A, the matrix powers problem is to calculate the first N powers of A, i.e., A, A2A3,…, AN. We consider distributed memory systems (DMS) with p processors that can support one-to-one communications in O(T(p)) time. Assume that the time complexity of the best known sequential algorithm for matrix multiplication is O(Nα), where α<2.3755. Let p be arbitrarily chosen in the range 1⩽p⩽Nα+1/log N. We show that the two problems can be solved on a p-processor DMS in T chain(N,p)=O(Nα+1/p+T(p)(N 2(1+1/α)/p2/ α(log p/N)1-2α/+log(p log N/Nα) log N)) and Tpower(N,p)=0(Nα+1/p+T(p)(N 2(1+1/α)/p2/ α(log p/log N) 1-2α/+(log N)2)) times, respectively. We also give instantiation of the above results in distributed memory parallel computers and DMS with hypercubic networks, and show that our parallel algorithms are either fully scalable or highly scalable

Published in:

Parallel and Distributed Processing Symposium., Proceedings 15th International

Date of Conference:

Apr 2001