Faster Multi-Source Directed Reachability via Shortcuts and Matrix Multiplication
Michael Elkin, Chhaya Trehan
TL;DR
The paper tackles multi-source reachability (S × V-direachability) in directed graphs, improving over naive BFS-based and transitive-closure approaches by exploiting directed shortcuts and rectangular matrix multiplication. It first presents an unconditional two-step method achieving $\tilde{O}(n^{1 + \frac{2}{3}\omega(\sigma)})$ time, with concrete gains for sparse-to-dense regimes (notably $\tilde{\sigma} \le \sigma \le 0.53$) and in dense graphs via density-dependent exponents. It then introduces conditional improvements via a recursive framework $\mathcal{A}_k$ under a Path Direachability Assumption, showing that for any $\tilde{\sigma} < \sigma < 1$ there exists a depth $k$ yielding better bounds in the $\mu=2$ case; crucially, these results connect to efficiently computing a rectangular max-min matrix product. The work relies on Kogan and Parter's directed-shortcut constructions and highlights a path to further improvements by achieving near-$\tilde{O}(n^{\omega(\sigma)})$ time for rectangular products. Overall, the paper advances multi-source reachability by weaving shortcut-based diameter reduction with fast rectangular matrix multiplication and a recursive, assumption-rich framework that broadens the regimes where sub-$n^{\omega}$ time is achievable.
Abstract
Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a set $S \subseteq V$, $|S| = n^σ$ (for some $0 < σ\le 1$) of designated sources, the $S \times V$-direachability problem is to compute for every $s \in S$, the set of all the vertices reachable from $s$ in $G$. Known naive algorithms for this problem either run a BFS/DFS separately from every source, and as a result require $O(m \cdot n^σ)$ time, or compute the transitive closure of $G$ in $\tilde O(n^ω)$ time, where $ω< 2.371552\ldots$ is the matrix multiplication exponent. Hence, the current state-of-the-art bound for the problem on graphs with $m = Θ(n^μ)$ edges in $\tilde O(n^{\min \{μ+ σ, ω\}})$. Our first contribution is an algorithm with running time $\tilde O(n^{1 + \tiny{\frac{2}{3}} ω(σ)})$ for this problem, where $ω(σ)$ is the rectangular matrix multiplication exponent. Using current state-of-the-art estimates on $ω(σ)$, our exponent is better than $\min \{2 + σ, ω\}$ for $\tilde σ\le σ\le 0.53$, where $1/3 < \tilde σ< 0.3336$ is a universal constant. Our second contribution is a sequence of algorithms $\mathcal A_0, \mathcal A_1, \mathcal A_2, \ldots$ for the $S \times V$-direachability problem. We argue that under a certain assumption that we introduce, for every $\tilde σ\le σ< 1$, there exists a sufficiently large index $k = k(σ)$ so that $\mathcal A_k$ improves upon the current state-of-the-art bounds for $S \times V$-direachability with $|S| = n^σ$, in the densest regime $μ=2$. We show that to prove this assumption, it is sufficient to devise an algorithm that computes a rectangular max-min matrix product roughly as efficiently as ordinary $(+, \cdot)$ matrix product. Our algorithms heavily exploit recent constructions of directed shortcuts by Kogan and Parter.