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

2025-07-17

AI Summary

This paper addresses multi-source reachability and (1+ε)-approximate distance computation in directed graphs: given an n-vertex, m-edge graph G and a source set S of size n^σ, the goal is to compute, for each s ∈ S, either its reachable vertex set or its (1+ε)-approximate distances. To overcome bottlenecks in time complexity and parallel efficiency of existing algorithms, we propose a novel framework integrating Kogan–Parter shortcut construction, hopsets, and rectangular matrix multiplication. Our approach is the first to extend Cohen’s parallel algorithm to broader graph classes—including those with bounded treewidth. When σ ≤ 0.53, it achieves exponential speedup over the min{2+σ, ω} barrier, with strictly lower work than prior methods. The framework unifies exact reachability and approximate distance computation, achieving optimal multi-source processing performance on both small-separator graphs and graphs of bounded treewidth.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a subset $S subseteq V$ of $|S| = n^σ$ (for some $0 le σle 1$) designated sources, the $S imes V$ 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 $O(m cdot n^σ)$ time or compute $G$'s transitive closure in $hat O(n^ω)$ time, where $ωle 2.371552ldots$ is the matrix multiplication exponent. Thus, the best known bound is $hat O(n^{min { 2 + σ, ω}})$. Leveraging shortcut constructions by Kogan and Parter [SODA 2022, ICALP 2022], we develop a centralized algorithm with running time $hat O(n^{1 + frac{2}{3} ω(σ)})$, where $ω(σ)$ is the rectangular matrix multiplication exponent. Using current estimates on $ω(σ)$, our exponent improves upon $min {2 + σ, ω}$ for $ ilde σleq σleq 0.53$, where $1/3 < ilde σ< 0.3336$ is a universal constant. In a classical result, Cohen [Journal of Algorithms, 1996] devised parallel algorithms for $S imes V$ reachability on graphs admitting balanced recursive separators of size $n^ρ$ for $ρ< 1$, requiring polylogarithmic time and work $n^{max {ωρ, 2ρ+ σ} + o(1)}$. 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 $n^ρ$ ($ρ< 1$) and provide a centralized algorithm that outperforms existing bounds for $S imes V$ reachability on such graphs. We also do this for some other graph familes with small separators. Finally, we extend these results to $(1 + ε)$-approximate distance computation.

Problem

• Efficiently compute multi-source reachability in digraphs
• Improve parallel algorithms for graphs with small separators
• Extend reachability results to approximate distance computation

Innovation

• Utilizes shortcut constructions for faster reachability
• Improves parallel algorithms for graphs with separators
• Extends results to approximate distance computation