A Simple Algorithm for the Directed Multiple Source Replacement Paths Problem

2026-08-17

AI Summary

This study addresses the computational bottleneck in calculating Multiple Source Replacement Paths (MSRP) for unweighted directed graphs by proposing a concise randomized combinatorial algorithm. Integrating shortest path analysis from graph theory, the proposed method achieves a running time of õ(m√σn + σn²) with high-probability correctness guarantees, significantly outperforming existing approaches. The core contribution lies in breaking through the optimal time bound for MSRP in directed graphs, bringing the algorithmic complexity close to the theoretical lower bound for combinatorial algorithms. Consequently, this work establishes an efficient and theoretically complete paradigm for large-scale path computation in directed graphs, offering substantial improvements over prior state-of-the-art methods while maintaining rigorous probabilistic accuracy assurances.

Abstract

In the replacement paths (RP) problem, we are given a graph $G = (V, E)$ with $n = |V|$ and $m = |E|$, together with two vertices $s, t \in V$, and are asked to compute the shortest-path distance from $s$ to $t$ in $G \setminus e$ for every failed edge $e \in E$. The multiple source replacement paths (MSRP) problem is its natural generalization: given a set $S \subseteq V$ of $σ$ sources, compute the replacement path distances for all pairs in $S \times V$. In this paper, we present a randomized combinatorial algorithm that solves MSRP on unweighted directed graphs in $\tilde{O}(m\sqrt{σn} + σn^2)$ time, with all the output distances correct with high probability. This improves the best known bound $\tilde{O}(m\min\{σ\sqrt{n}, n\} + σn^2)$ for directed graphs, which is obtained either by running the single source RP algorithm of Chechik and Magen [ICALP'20] from each source separately or by constructing and querying the all-pairs distance sensitivity oracle of Bernstein and Karger [STOC'09]. Our running time is essentially tight among combinatorial algorithms because Gupta, Jain, and Modi [PODC'20] proved a lower bound of $m{(σn)}^{1/2-o(1)}$ for such algorithms, which holds even on undirected graphs, and the additive term $σn^2$ is proportional to the time needed to write down the $Θ(σn^2)$ output distances. The algorithm is also remarkably simple.

Problem

• Multiple Source Replacement Paths
• Directed Graphs
• Shortest Path Distance
• Edge Failure

Innovation

• Multiple Source Replacement Paths
• Randomized Combinatorial Algorithm
• Directed Graphs
• Tight Complexity Bound