AN EFFICIENT PARALLEL ALGORITHM FOR THE MINIMAL ELIMINATION ORDERING (MEO) OF AN ARBITRARY GRAPH

We design the first efficient parallel algorithm for computing the min imal elimination ordering (MEO) of an arbitrary graph. The algorithm w orks in O(log(3) n) parallel time and O(nm) processors on CREW PRAM, f or an n-vertex, in-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 comb inatorial 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 stro ngly resisting sublinear parallel time (sublinear sequential storage) algorithms. As an application, this paper gives the first efficient pa rallel solutions to the problem of minimal fill-in for arbitrary graph s and connected combinatorial optimization problems (see e.g. Rose et al., 1976; Tarjan, 1985), and to the problem of the Gaussian eliminati on of sparse symmetric matrices (Rose, 1970; 1973) (The problem of com puting 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 so me new divide-and-conquer methods.