On Greedy Algorithms for Solving the Matrix Chain Product Problem with Square Dense and Triangular Matrices

We address a combinatorial optimization problem (COP), namely a particular variant of the matrix chain product problem (MCPP) where the chain involves square dense and triangular matrices. Besides the classical dynamic programming algorithm of cubic complexity, exact greedy algorithms (GA) of linear complexity are also known in the literature. We present in this paper new exact GA’s of linear complexity too and leading in general to different optimal chain parenthesizations (OCP). Our objective is to optimize, while building an OCP represented by a binary tree, its parallelism degree. Indeed, this criterion is quite useful for an efficient parallel computation of the chain product according to a given OCP. Our theoretical contribution is validated by an experimental study permitting to establish comparisons between several OCP’s through their representative binary trees, thus precising the practical interest of the approaches we developed. Keywords— Binary tree, complexity, dynamic programming, grain, greedy algorithm, matrix chain product, parallel algorithm, parenthesization.