Efficient Construction of Directed Hopsets and Parallel Single-source Shortest Paths (Abstract).

The single-source shortest-path problem is as follows: given a graph with nonnegative edge weights and a designated source vertex s, return the distances from~s to each other vertex such. This paper presents a randomized parallel single-source shortest paths (SSSP) algorithm for directed graphs with non-negative integer edge weights that solves the exact SSSP problem in O (m) work and n^1/2+o(1) span, with high probability. All previous exact SSSP algorithms with nearly linear work have linear span, even for undirected unweighted graphs. To solve exact SSSP problem, we first show a deterministic reduction from exact SSSP to directed hopsets using the iterative gradual rounding technique. An (β, ε)-hopset is a set of weighted edges, also known as shortcuts, that when added to the graph, admit β-hop paths with weights no more than (1 + ε) times the true shortest path distances. We show that (β, ε)-hopsets can be used to solve the exact SSSP problem in O (m) work and O (β) span. Furthermore, we present the first nearly linear-work algorithm for constructing hopsets on directed graphs. Our sequential algorithm runs in O (m) time and constructs a hopset with O (n) edges and β = n^1/2+o(1) . We also provide a parallel version of the algorithm with O (m) work and n^1/2+o(1) span. The directed hopsets can be used to solve approximate SSSP problems efficiently, where the objective is to return estimates of the distances from the source vertex to every other vertex such that the estimate falls between the true distance and (1+ε) times the distance. Specifically, for constant ε and graphs with polynomially-bounded real edge weights, there is an algorithm solving approximate SSSP problem with O (m) work and n^1/2+o(1) span.