An O(n) Algorithm for Determining a Near-Optimal Computation Order of Matrix Chain Products

Abstract

This paper discusses the computation of matrix chain products of the form M1 × M22 × ··· × Mn where Mi‘s are matrices. The order in which the matrices are computed affects the number of operations. A sufficient condition about the association of the matrices in the optimal order is presented. An O(n) algorithm to find an order of computation which takes less than 25 percent longer than the optimal time Topt is also presented. In most cases, the algorithm yields the optimal order or an order which takes only a few percent longer than Topt (less than 1 percent on the average). © 1978, ACM. All rights reserved.