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 logO(1) n black-box approximate distance computations. They apply equally to parallel, distributed, and semi-streaming settings. Strongly (1+ε)-approximate distances are equivalent to exact distances in a (1+ε)-perturbed graph and approximately satisfy the subtractive triangle inequality. In directed graphs, this is sufficient to reduce even exact distance computation to arbitrary (1+ε)-approximate ones.