Analysis of Parallel Algorithms for Matrix Chain Product and Matrix Powers on Distributed Memory Systems

Keqin Li

doi:10.1109/tpds.2007.1027

Abstract

Given N matrices A 1 , A 2, ...,A N of size NtimesN, the matrix chain product problem is to compute A 1 timesA 2 times...timesA N . Given an NtimesN matrix A, the matrix powers problem is to calculate the first N powers of A, that is, A, A 2 , A 3 ,..., A N . We solve the two problems on distributed memory systems (DMSs) with p processors that can support one-to-one communications in T(p) time. Assume that the fastest sequential matrix multiplication algorithm has time complexity O(N alpha ), where the currently best value of a is less than 2.3755. Let p be arbitrarily chosen in the range 1lesplesN alpha+1 /(log N) 2 . We show that the two problems can be solved by a DMS with p processors in T chain (N,p)=O((N alpha+1 /p)+T(p))((N 2(2+1/alpha /p 2/alpha )(log+p/N) 1-2/alpha +log+((p log N)/N alpha )) and T power (N,p)=O(N alpha+1 /p+T(p)((N 2(1+1/alpha) /p 2/alpha )(log+p/2 log N) 1-2/alpha +(log N) 2 ))) times, respectively, where the function log+ is defined as follows: log+ x=log x if xges1 and log+ x=1 if 0<x<1. We also give instantiations of the above results on several typical DMSs and show that computing matrix chain product and matrix powers are fully scalable on distributed memory parallel computers (DMPCs), highly scalable on DMSs with hypercubic networks, and not highly scalable on DMSs with mesh and torus networks.

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