Faster Multi-Source Reachability and Approximate Distances via Shortcuts, Hopsets and Matrix Multiplication

TL;DR

This paper introduces faster algorithms for multi-source reachability and approximate distances in graphs, leveraging shortcuts, hopsets, and matrix multiplication to improve over previous bounds.

Contribution

It develops a centralized algorithm with improved running time using shortcut constructions and extends parallel algorithms to graphs with small separators and bounded treewidth.

Findings

Improved running time for multi-source reachability using shortcut-based methods.

Extended parallel algorithms to graphs with small separators and bounded treewidth.

Achieved better bounds for approximate distance computations in certain graph families.

Abstract

Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a subset $S \subseteq V$ of $|S| = n^{\sigma}$ (for some $0 \le \sigma \le 1$) designated sources, the $S \times V$ reachability problem is to compute the sets $\mathcal V_s$ of vertices reachable from $s$, for every $s \in S$. Naive centralized algorithms run BFS/DFS from each source in $O(m \cdot n^{\sigma})$ time or compute $G$'s transitive closure in $\hat O(n^{\omega})$ time, where $\omega \le 2.371552\ldots$ is the matrix multiplication exponent. Thus, the best known bound is $\hat O(n^{\min \{ 2 + \sigma, \omega\}})$. Leveraging shortcut constructions by Kogan and Parter [SODA 2022, ICALP 2022], we develop a centralized algorithm with running time $\hat O(n^{1 + \frac{2}{3} \omega(\sigma)})$, where $\omega(\sigma)$ is the rectangular matrix multiplication exponent. Using current estimates on $\omega(\sigma)$, our exponent improves upon $\min \{2 + \sigma, \omega \}$ for $\tilde \sigma \leq \sigma \leq 0.53$, where $1/3 < \tilde \sigma < 0.3336$ is a universal constant. In a classical result, Cohen [Journal of Algorithms, 1996] devised parallel algorithms for $S \times V$ reachability on graphs admitting balanced recursive separators of size $n^{\rho}$ for $\rho < 1$, requiring polylogarithmic time and work $n^{\max \{\omega \rho, 2\rho + \sigma \} + o(1)}$. We significantly improve, extend, and generalize Cohen's result. First, our parallel algorithm for graphs with small recursive separators has lower work complexity than Cohen's in boraod paramater ranges. Second, we generalize our algorithm to graphs of treewidth at most $n^{\rho}$ ($\rho < 1$) and provide a centralized algorithm that outperforms existing bounds for $S \times V$ reachability on such graphs. We also do this for some other graph familes with small separators. Finally, we extend these results to $(1 + \epsilon)$-approximate distance computation.