Parallel shortest paths using radius stepping

Abstract

The single-source shortest path problem (SSSP) with nonnegative edge weights is notoriously difficult to solve efficiently in parallel-it is one of the graph problems said to suffer from the transitive-closure bottleneck. Yet, in practice, the Δ-stepping algorithm of Meyer and Sanders (J. Algorithms, 2003) often works efficiently but has no known theoretical bounds on general graphs. The algorithm takes a sequence of steps, each increasing the radius by a user-specified value Δ. Each step settles the vertices in its annulus but can take θ(n) substeps, each requiring θ(m) work (n vertices and m edges). Building on the success of Δ-stepping, this paper describes Radius-Stepping, an algorithm with one of the best-known tradeoffs between work and depth bounds for SSSP with nearly-linear (Õ(m)) work. The algorithm is a Δ-stepping-like algorithm but uses a variable instead of a fixed-size increase in radii, allowing us to prove a bound on the number of steps. In particular, by using what we define as a vertex k-radius, each step takes at most k + 2 substeps. Furthermore, we define a pk, ρ)-graph property and show that if an undirected graph has this property, then the number of steps can be bounded by Opn/ρ · log ρL), for a total of Opkn/ρ · logρL) substeps, each parallel. We describe how to preprocess a graph to have this property. Altogether, for an arbitrary input graph with n vertices and m edges, Radius-Stepping, after preprocessing, takes Oppm+nρ) logn) work and On/ρ-logn log p ρL)) depth per source. The preprocessing step takes p mlogn + nρ2) work and O pρ log ρ) depth, adding no more than Opnρ) edges.