Optimal Distributed Replacement Paths

TL;DR

This paper determines the exact distributed round complexity for the replacement paths problem in unweighted directed graphs, establishing tight bounds and extending results to approximate weighted graphs, thus advancing understanding in distributed shortest path computations.

Contribution

It provides the first tight randomized complexity bounds for the replacement paths problem in the CONGEST model, improving previous bounds and extending to approximate solutions for weighted graphs.

Findings

Established tight randomized bounds of rac{n^{2/3}}{D} for unweighted directed graphs.

Extended the bounds to (1+b5)-approximation for weighted directed graphs.

Improved upon recent work by matching upper and lower bounds for the problem.

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 $P$.