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

TL;DR

The results imply that the original restoration lemma is exactly tight in the case $k=1, but can be significantly improved for larger $k, and an asymptotically matching lower bound is shown.

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.

Figures (6)

• Figure 1: (Left) Suppose that two blue dashed edges in the graph fail. (Right) The restoration lemma guarantees that any shortest $s \leadsto t$ path (indicated with wavy edges) in the post-failure network can be partitioned into three subpaths, which are each a shortest path in the pre-failure network.
• Figure 2: Under the equal subpath and first fault assumptions, we can reach contradiction if we assume that there are three different subpaths that all have shortcuts that use $e$ as their first fault.
• Figure 3: In order to relax the first fault assumption, instead of counting $(e_1, \pi_i)$ and $(e_2, \pi_i)$ as pairs, we can map these to distinct FS pairs $(e^*_1, \pi_i), (e^*_2, \pi_i)$. Our main technical step is to show that this distinct mapping is always possible.
• Figure 4: If the blue and yellow edges fail (i.e. all straight-line edges on the inside of the outer semicircle), then we can't partition the remaining shortest path (black edges along the outer semicircle) into two subpaths that are both $(f-2)$-fault replacement paths. For clarity, only power of two vertices are drawn here, but the outer cycle contains $2^{g+1}-1$ vertices.
• Figure 5: The top figure depicts one copy of $G_{f/k}$, and the bottom depicts all the copies combined together.

Theorems & Definitions (35)

• Definition 1: Replacement Paths
• Theorem 2: Original Restoration Lemma restoration
• Corollary 3: restoration
• proof : Proof of Corollary \ref{['cor:baserestoration']}, given Theorem \ref{['thm:baserestoration']}
• Theorem 4: Main Result
• Theorem 5: Weighted Restoration Lemma restoration
• Theorem 6: Main Result, Weighted Setting
• Theorem 7: Unweighted Algorithmic Restoration Lemma
• Theorem 8: Weighted Algorithmic Restoration Lemma
• Lemma 9: Hall's Theorem