Efficient Matrix Chain Ordering in Polylog Time

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(lg3 n)-time and n/lgn-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(lg2 n) time with n processors on the EREW PRAM and to O(lg2 n lg lgn) time with n/lg lgn 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 lg1.5 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].