Consider any known sequential algorithm for matrix multiplication over an arbitrary ring with time complexity O(N/sup /spl alpha//), where 2/spl alpha//spl les/3. We show that such an algorithm can be parallelized on a distributed memory parallel computer (DMPC) in O (log N) time by using N/sup /spl alpha///log N processors. Such a parallel computation is cost optimal and matches the performance of PRAM. Furthermore, our parallelization on a DMPC can be made fully scalable, that is, for all 1/spl les/p/spl les/N/spl alpha//sup /spl alpha///log N, multiplying two N/spl times/N matrices can be performed by a DMPC with p processors in O(N/sup /spl alpha///p) rime, i.e., linear speedup and cost optimality can be achieved in the range [1..N/sup /spl alpha///log N]. This unifies all known algorithms for matrix multiplication on DMPC, standard or non-standard, sequential or parallel. Extensions of our methods and results to other parallel systems are also presented. The above claims result in significant progress in scalable parallel matrix multiplication (as well as solving many other important problems) on distributed memory systems, both theoretically and practically.
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