Parallel Reachability and Shortest Paths on Non-sparse Digraphs: Near-linear Work and Sub-square-root Depth

TL;DR

This paper introduces parallel algorithms for reachability and shortest paths on dense directed graphs, achieving near-linear work and significantly reduced depth compared to previous methods.

Contribution

The paper develops parallel algorithms with near-linear work and sub-square-root depth for dense graphs, improving over the state-of-the-art in all density regimes.

Findings

Achieves depth of $n^{0.136}$ for reachability on dense graphs.

Achieves depth of $n^{0.25+o(1)}$ for shortest paths on dense graphs.

Requires near-linear work $ ilde{O}(m)$ for both problems.

Abstract

We present parallel algorithms for computing single-source reachability and shortest paths on directed $n$-vertex $m$-edge graphs using near-linear $\tilde{O}(m)$ work and $o(\sqrt{n})$ depth whenever $m\ge n^{1+o(1)}$. At the extreme of $m=\Omega(n^{2})$, our reachability and shortest path algorithms have depth only $n^{0.136}$ and $n^{0.25+o(1)}$, respectively. The state-of-the-art parallel algorithms with near-linear work for both problems require $\Omega(\sqrt{n})$ depth in all density regimes.