Parallel Breadth-First Search and Exact Shortest Paths and Stronger Notions for Approximate Distances

This paper introduces stronger notions for approximate single-source shortest-path distances and gives simple reductions to compute them from weaker standard notions of approximate distances. Strongly-approximate distances isolate, capture, and address the well-known barriers for using approximate distances algorithmically and their reductions directly address these barriers in a clean and modular manner. The reductions are model-independent and require only log(O(1)) black-box approximate distance computations. They apply equally to parallel, distributed, and semi-streaming settings. Strongly (1 +epsilon)-approximate distances are equivalent to exact distances in a (1 + epsilon)-perturbed graph and approximately satisfy the subtractive triangle inequality. In directed graphs, this is sufficient to reduce even exact distance computation to arbitrary (1 + epsilon)-approximate ones. Overall, this paper simplifies, unifies, and cleans up problemspecific ad-hoc solutions developed in many prior works, e.g., for ball-growing routines in undirected graphs and for computing exact distances in directed graphs. Several algorithmic results for parallel and distributed algorithms - some known, some new - are directly implied by our reductions. Applications of particular interest include the first work-efficient sublinear-depth parallel algorithm for breadth-first search and computing exact single-source shortest paths - both major open problems in parallel computing. Given a source vertex in a directed graph with polynomiallybounded nonnegative integer lengths the algorithm computes an exact shortest path tree in.. log(O(1)) work and n(1/2+o(1)) depth. Previously, no parallel algorithm improving the trivial linear depths of Dijkstra's algorithm without significantly increasing the work was known, even for the case of undirected and unit-length graphs, i.e., for computing a breadth-first-search tree. This result was independently obtained by Fineman and Cao.