Maximizing the Parallelism Degree for Optimal Computing of Matrix Chain Products

Abstract

We are interested in a specific combinatorial optimization problem solved by an exact dynamic programming algorithm (DPA), namely the matrix chain product problem (MCPP) with two of its variants i.e. the standard (ST) one where the matrices are rectangular and the variant were the matrices are square and either dense or triangular (DT). Solving the MCPP consists in first determining an optimal parenthesization (OP) minimizing the number of required operations then computing the product matrix (PM) according to the OP which may be represented by a binary in-tree (BiT) where each node corresponds to a matrix product and is weighted by the number of required operations. Our aim is to design an efficient parallel computing of the PM. It turns out that, given an OP and the associated BiT, the cost of a critical path (CCP) is equal to the minimal cost of a parallel computing of the PM and the width of the BiT corresponds to the parallelism degree i.e. the number of required processors. It is within this scope that we propose a modified DPA based on a heuristic approach aiming to deter-mining a particular OP by maximizing the width of the associated BiT which leads to minimize its height hence the CCP. Our contribution is validated by achieving a series of experiments following a preliminary phase were we check for each input chain the existence of several OPs. Each obtained OP exhibiting the maximal parallelism degree is compared with a set of other OPs obtained in the ST variant of the MCPP by a particular approach determining several OPs and in the DT variant by specific exact greedy algorithms.