An efficient parallel algorithm for the minimal elimination ordering (MEO) of an arbitrary graph

Abstract

We design the first efficient parallel algorithm for computing the minimal elimination ordering (MEO) of an arbitrary graph. The algorithm works in O(log 3 n) parallel time and O( nm) processors on CREW PRAM, for an n-vertex, m-edge graph, and is optimal up to a polylogarithmic factor with respect to the best sequential algorithm of Rose, et al. (1976). The MEO problem for arbitrary graphs arises in a number of combinatorial optimization problems, as well as in database applications, scheduling problems, and the sparse Gaussian elimination on symmetric matrices. It was believed before to be inherently sequential, and strongly resisting sublinear parallel time (sublinear sequential storage) algorithms. As an application, this paper gives the first efficient parallel solutions to the problem of minimal fill-in for arbitrary graphs and connected combinatorial optimization problems (see e.g. Rose et al., 1976; Tarjan, 1985), and to the problem of the Gaussian elimination of sparse symmetric matrices (Rose, 1970; 1973) (The problem of computing a minimum fill-in is known to be NP-complete (Yannakakis, 1981). The method of solution involves a development of new techniques for solving connected minimal set system problem, and combining it with some new divide-and-conquer methods.