Parallel Matrix Multiplication on a Linear Array with a Reconfigurable Pipelined Bus System

The known fast sequential algorithms for multiplying two $N\times N$ matrices (over an arbitrary ring) have time complexity $O(N^\alpha)$, where $2 < \alpha < 3$. The current best value of $\alpha$ is less than 2.3755. We show that, for all $1 \le p \le N^{\alpha}$, multiplying two $N\times N$ matrices can be performed on a p-processor linear array with a reconfigurable pipelined bus system (LARPBS) in $ O({N^{\alpha}\over p}+({N^2\over p^{2/\alpha}})\log p)$ time. This is currently the fastest parallelization of the best known sequential matrix multiplication algorithm on a distributed memory parallel system. In particular, for all $1 \le p \le N^{2.3755}$, multiplying two $N\times N$ matrices can be performed on a p-processor LARPBS in $ O({N^{2.3755}\over p}+({N^2\over p^{0.8419}})\log p) $ time and linear speedup can be achieved for $p$ as large as $O(N^{2.3755}/(\log N)^{6.3262})$. Furthermore, multiplying two $N\times N$ matrices can be performed on an LARPBS with $O(N^\alpha)$ processors in $O(\log N)$ time. This compares favorably with the performance on a PRAM.