$\tilde{O}(1)$-Depth Parallel Reachability Faster than Transitive Closure

2026-08-13

AI Summary

This work addresses the challenge of achieving parallel directed graph reachability with depth $\tilde{O}(1)$ without computing the full transitive closure. To this end, the authors propose a randomized $d$-shortcut construction that effectively reduces graph diameter and circumvents the transitive closure bottleneck for the first time. For $d = 3$ and any even $d \geq 4$, the method surpasses existing conditional lower bounds: under the assumption that the matrix multiplication exponent $\omega = 2$, it achieves $\tilde{O}(T)$ work, improving upon the $T^{4/3 - o(1)}$ lower bound; using the current best-known value of $\omega$, it attains $\tilde{O}(T^{1.186})$ work, significantly outperforming the best sequential algorithm with $T^{1.3459 + o(1)}$ complexity.

Abstract

A $d$-shortcut of a directed graph $G=(V,E)$ is a subset of edges drawn from the transitive closure $TC(G)$ whose addition reduces the graph diameter to at most $d$. In the special case $d=1$, computing a $1$-shortcut is \emph{equivalent} to computing the transitive closure. For larger values of $d$, a lower bound of [Hesse, SODA 2003] shows that $n^δ$-shortcuts, for small constants $δ>0$, may still contain a large fraction of the edges of $TC(G)$, suggesting that shortcut construction may remain as hard as transitive closure even in this regime. Consequently, since $\widetilde{O}(d)$-depth parallel reachability algorithms rely on computing $d$-shortcuts, achieving $\widetilde{O}(1)$ depth by this approach has so far required computing the full transitive closure. Assuming $ω=2$, the PS-AE-Triangle hypothesis of [Abboud, Bringmann, Fischer, and Künnemann, SODA 2024] yields a conditional $T^{4/3-o(1)}$ time barrier for computing transitive closure when $T\leq n^{3/2}$, where $T=|TC(G)|$. In this work, we bypass the transitive-closure barrier for $\widetilde{O}(1)$-depth parallel reachability. We introduce randomized $d$-shortcut constructions that already circumvent this barrier for $d=3$ and, more generally, for every even $d\geq4$ up to $O(\log n)$. Our approach yields a randomized $\widetilde{O}(1)$-depth parallel reachability algorithm with total work $\widetilde{O}(T^{ω/2})$, which becomes $\widetilde{O}(T)$ when $ω=2$, falling below this conditional $T^{4/3-o(1)}$ barrier throughout that regime. Under the current bound of $ω$, this gives $\widetilde{O}(T^{1.186})$ work, improving on the current $T^{1.3459+o(1)}$ sequential-time bound for transitive closure due to Abboud et al. Thus, although $\widetilde{O}(1)$-shortcuts might be almost as dense as the full transitive closure, they can nevertheless be computed substantially faster.

Problem

• parallel reachability
• transitive closure
• graph shortcuts
• computational complexity
• diameter reduction

Innovation

• parallel reachability
• d-shortcut
• transitive closure
• randomized algorithm
• matrix multiplication exponent