All-Pairs 2-Reachability in $\mathcal{O}(n^{\omega}\log n)$ Time

TL;DR

This paper introduces an efficient algorithm for all-pairs 2-reachability in directed graphs, running close to the time of transitive closure, and enabling constant-time connectivity queries after preprocessing.

Contribution

The paper presents a novel $ ilde{O}(n^{ ext{omega}})$ time algorithm for all-pairs 2-reachability, producing witnesses and enabling fast connectivity queries.

Findings

Runs in $ ilde{O}(n^{ ext{omega}})$ time, close to transitive closure.

Produces witnesses for non-2-reachable pairs.

Enables constant-time connectivity queries after preprocessing.

Abstract

In the $2$-reachability problem we are given a directed graph $G$ and we wish to determine if there are two (edge or vertex) disjoint paths from $u$ to $v$, for a given pair of vertices $u$ and $v$. In this paper, we present an algorithm that computes $2$-reachability information for all pairs of vertices in $\mathcal{O}(n^{\omega}\log n)$ time, where $n$ is the number of vertices and $\omega$ is the matrix multiplication exponent. Hence, we show that the running time of all-pairs $2$-reachability is only within a $\log$ factor of transitive closure. Moreover, our algorithm produces a witness (i.e., a separating edge or a separating vertex) for all pair of vertices where $2$-reachability does not hold. By processing these witnesses, we can compute all the edge- and vertex-dominator trees of $G$ in $\mathcal{O}(n^2)$ additional time, which in turn enables us to answer various connectivity queries in $\mathcal{O}(1)$ time. For instance, we can test in constant time if there is a path from $u$ to $v$ avoiding an edge $e$, for any pair of query vertices $u$ and $v$, and any query edge $e$, or if there is a path from $u$ to $v$ avoiding a vertex $w$, for any query vertices $u$, $v$, and $w$.