Polylog-time and near-linear work approximation scheme for undirected shortest paths

Abstract

Shortest paths computations constitute one of the most fundamental network problems. Nonetheless, known parallel shortest-paths algorithms are generally inefficient: they perform significantly more work (product of time and processors) than their sequential counterparts. This gap, known in the literature as the "transitive closure bottleneck," poses a long-standing open problem. Our main result is an O(mnε0 + s(m + n1+ε0)) work polylog-time randomized algorithm that computes paths within (1 + O(1/polylog n)) of shortest from s source nodes to all other nodes in weighted undirected networks with n nodes and m edges (for any fixed ε0 > 0). This work bound nearly matches the Õ(sm) sequential time. In contrast, previous polylog-time algorithms required min{Õ(n3), Õ(m2)} work (even when s = 1), and previous near-linear work algorithms required near-O(n) time. We also present faster sequential algorithms that provide good approximate distances only between "distant" vertices: We obtain an O((m + sn)nε0) time algorithm that computes paths of weight (1 + O(1/polylog n)) dist + O(wmax polylog n), where dist is the corresponding distance and wmax is the maximum edge weight. Our chief instrument, which is of independent interest, are efficient constructions of sparse hop sets. A (d, ε)-hop set of a network G = (V, E) is a set E* of new weighted edges such that minimum-weight d-edge paths in (V, E ∪ E*) have weight within (1 + ε) of the respective distances in G. We construct hop sets of size O(n1+ε0) where ε = O(1/polylog n) and d = O(polylog n). General Terms: Algorithms, Theory © 2000 ACM.