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

2026-05-05

AI Summary

This work proposes the first parallel algorithm for single-source reachability and shortest paths on non-sparse directed graphs that achieves near-linear work (Õ(m)) and depth strictly below √n. By integrating graph decomposition, recursive contraction, and parallel search—augmented with refined load balancing and depth-optimized scheduling—the algorithm attains o(√n) depth on graphs with m ≥ n^{1+o(1)} edges. In dense regimes, such as when m = Ω(n²), the depth improves further to n^{0.136} for reachability and n^{0.25+o(1)} for shortest paths. These results substantially advance the state of the art by overcoming longstanding bottlenecks in the depth–work trade-off for these fundamental graph 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 $m=Ω(n^{2})$, 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 $Ω(\sqrt{n})$ depth in all density regimes.

Problem

• parallel algorithms
• reachability
• shortest paths
• non-sparse digraphs
• computational depth

Innovation

• parallel algorithms
• reachability
• shortest paths
• near-linear work
• sub-square-root depth