Improved processor bounds for parallel algorithms for weighted directed graphs

Abstract

We present a parallel algorithm that solves the single-source shortest path problem (SSSP) for a weighted digraph G=( V, E) in time O(log 2 n) using M( n) processors on an exclusive-read exclusive-write parallel random access machine (EREW PRAM), where n=\ | V\ |, edge weights are drawn from the set \\s{0, 1,…, k\\s} for some constant k, and M( n) is the number of processors necessary to multiply two n× n integer matrices over a ring in O(log n) time (currently, M( n)= n 2.376 ). This algorithm is a generalization of the O(log 2 n) time, M( n) processor EREW PRAM algorithm due to Gazit and Miller for the SSSP problem in an unweighted digraph. We also briefly explain how our solution of the SSSP problem for a weighted digraph can be used to reduce the previous known processor bounds for a number of digraph problems to M( n) from Θ( n 3) (within a polylog factor) without increasing the running time.