Optimal Distributed Replacement Paths

2025-02-21

AI Summary

We study the edge-fault-tolerant replacement path problem for $s$-$t$ shortest paths in the distributed CONGEST model: for each edge $e$ on a shortest $s$-$t$ path $P$, compute the shortest $s$-$t$ distance avoiding $e$. We establish the first tight randomized round complexity bound of $widetilde{Theta}(n^{2/3} + D)$ for this problem, and extend it to a $(1+varepsilon)$-approximation for weighted graphs; our lower bound also applies to the second-shortest path problem. Technically, we combine hierarchical sampling, distance estimation, and randomized message compression to achieve the optimal upper bound and provide a matching lower bound proof. Compared to prior work (SIROCCO’24), our results eliminate dependence on $h_{st}$ (the length of $P$), significantly improving both the precision and generality of the bounds.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Replanning and Plan Repair

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
• Systems and Infrastructure for Web, Mobile and WoT: Cloud, edge and content delivery systems for the Web

Abstract

We study the replacement paths problem in the $mathsf{CONGEST}$ model of distributed computing. Given an $s$-$t$ shortest path $P$, the goal is to compute, for every edge $e$ in $P$, the shortest-path distance from $s$ to $t$ avoiding $e$. For unweighted directed graphs, we establish the tight randomized round complexity bound for this problem as $widetilde{Theta}(n^{2/3} + D)$ by showing matching upper and lower bounds. Our upper bound extends to $(1+epsilon)$-approximation for weighted directed graphs. Our lower bound applies even to the second simple shortest path problem, which asks only for the smallest replacement path length. These results improve upon the very recent work of Manoharan and Ramachandran (SIROCCO 2024), who showed a lower bound of $widetilde{Omega}(n^{1/2} + D)$ and an upper bound of $widetilde{O}(n^{2/3} + sqrt{n h_{st}} + D)$, where $h_{st}$ is the number of hops in the given $s$-$t$ shortest path $P$.

Problem

• Optimal distributed replacement paths
• Shortest-path avoiding edges
• Tight round complexity bounds

Innovation

• Distributed CONGEST model
• Optimal replacement paths
• Randomized round complexity