Using Selective Path-Doubling for Parallel Shortest-Path Computations

We11A preliminary version appeared in the “Proceedings of the 2nd Israeli Symposium on the Theory of Computing and Systems, 1993.”consider parallel shortest-paths computations in weighted undirected graphsG=(V,E), wheren=|V| andm=|E|. The standardO(n3) work path-doubling (Floyd-Warshall) algorithm consists ofO(logn) phases, where in each phase, for every triplet of vertices (u1,u2,u3)?V3, the distance betweenu1andu3is updated to be no more than the sum of the previous-phase distances between {u1,u2} and {u2,u3}. We introduce a new NC algorithm that for ?=o(n), considers onlyO(n?2) triplets. Our algorithm performsO(n?2) work and augmentsEwithO(n?) new weighted edges such that between every pair of vertices, there exists a minimum weight path of size (number of edges)O(n/?) (whereO(f)?O(fpolylogn)). To compute shortest-paths, we apply to the augmented graph algorithms that are efficient for small-size shortest paths. We obtain anO(t) timeO(|S|n2+n3/t2) work deterministic PRAM algorithm for computing shortest-paths from |S| sources to all other vertices, wheret?nis a parameter. When the ratio of the largest edge weight and the smallest edge weight isnO(polylogn), the algorithm computes shortest paths. When weights are arbitrary, it computes paths within a factor of 1+n??(polylogn)of shortest.

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