Computation of Matrix Chain Products on Parallel Machines

Abstract

The Matrix Chain Ordering Problem is a well studied optimization problem, aiming at finding optimal parentheses assignment for minimizing the number of arithmetic operations required when computing a chain of matrix multiplications. Existing algorithms include the O(N^3) dynamic programming of Godbole (1973) and the faster O(NlogN) algorithm of Hu and Shing (1982). We show that both may result in sub-optimal parentheses assignment for modern machines as they do not take into account inter-processor communication costs that often dominate the running time. Further, the optimal solution may change when using fast matrix multiplication algorithms. We adapt the O(N^3) dynamic programming algorithm to provide optimal solutions for modern machines and modern matrix multiplication algorithms, and obtain an adaption of the O(NlogN) algorithm that guarantees a constant approximation.