Improved Shortest Path Restoration Lemmas for Multiple Edge Failures: Trade-offs Between Fault-tolerance and Subpaths

TL;DR

This paper introduces a new multiplicative tradeoff for shortest path restoration after multiple edge failures, improving upon the classic additive result, with implications for fault-tolerant network design.

Contribution

It presents a multiplicative tradeoff between fault tolerance and subpath count, extending the classic restoration lemma with tighter bounds and efficient algorithms.

Findings

Shows a multiplicative partitioning of replacement paths into subpaths with reduced failure tolerance

Provides asymptotically matching lower bounds for the new tradeoff

Extends results to weighted graphs with efficient computation algorithms

Abstract

The restoration lemma is a classic result by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [PODC '01], which relates the structure of shortest paths in a graph $G$ before and after some edges in the graph fail. Their work shows that, after one edge failure, any replacement shortest path avoiding this failing edge can be partitioned into two pre-failure shortest paths. More generally, this implies an additive tradeoff between fault tolerance and subpath count: for any $f, k$, we can partition any $f$-edge-failure replacement shortest path into $k+1$ subpaths which are each an $(f-k)$-edge-failure replacement shortest path. This generalized result has found applications in routing, graph algorithms, fault tolerant network design, and more. Our main result improves this to a multiplicative tradeoff between fault tolerance and subpath count. We show that for all $f, k$, any $f$-edge-failure replacement path can be partitioned into $O(k)$ subpaths that are each an $(f/k)$-edge-failure replacement path. We also show an asymptotically matching lower bound. In particular, our results imply that the original restoration lemma is exactly tight in the case $k=1$, but can be significantly improved for larger $k$. We also show an extension of this result to weighted input graphs, and we give efficient algorithms that compute path decompositions satisfying our improved restoration lemmas.