Processor Allocation and Task Scheduling of Matrix Chain Products on Parallel Systems
IEEE Transactions on Parallel and Distributed Systems
Optimal solution to matrix parenthesization problem employing parallel processing approach
EC'07 Proceedings of the 8th Conference on 8th WSEAS International Conference on Evolutionary Computing - Volume 8
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The matrix chain ordering problem is to find the cheapest way to multiply a chain of n matrices, where the matrices are pairwise compatible but of varying dimensions. Here we give several new parallel algorithms including $O(\lg^3 n)$-time and $n/\!\lg n$-processor algorithms for solving the matrix chain ordering problem and for solving an optimal triangulation problem of convex polygons on the common CRCW PRAM model. Next, by using efficient algorithms for computing row minima of totally monotone matrices, this complexity is improved to $O(\lg^2 n)$ time with $n$ processors on the EREW PRAM and to $O(\lg^2 n \lg \lg n)$ time with $n/ \! \lg \lg n$ processors on a common CRCW PRAM\@. A new algorithm for computing the row minima of totally monotone matrices improves our parallel MCOP algorithm to $O(n \lg^{1.5} n)$ work and polylog time on a CREW PRAM\@. Optimal log-time algorithms for computing row minima of totally monotone matrices will improve our algorithm and enable it to have the same work as the sequential algorithm of Hu and Shing [SIAM J. Comput., 11 (1982), pp. 362--373; SIAM J. Comput., 13 (1984), pp. 228--251].