Parallel algorithm for the matrix chain product and the optimal triangulation problems

Abstract

This paper considers the problem of finding an optimal order of the multiplication chain of matrices and the problem of finding an optimal triangulation of a convex polygon. For both these problems the best sequential algorithms run in θ(n log n) time. All parallel algorithms known use the dynamic programming paradigm and run in a polylogarithmic time using, in the best case, O(n6/lagk n) processors for a constant k. We give a new algorithm which uses a different approach and reduces the problem to computing certain recurrence in a tree. We show that this recurrence can be optimally solved which enables us to improve the parallel bound by a few factors. Our algorithm runs in O(log3 n) time using n2/log3 n processors on a CREW PRAM. We also consider the problem of finding an optimal triangulation in a monotone polygon. An O(log2 n) time and n processors algorithm on a CREW PRAM is given.