Path-Fault-Tolerant Approximate Shortest-Path Trees

TL;DR

This paper introduces a sparse auxiliary structure that enhances shortest-path trees to efficiently tolerate path failures, providing approximate shortest paths with low stretch and size, and enabling quick rerouting in networks.

Contribution

It presents a novel sparse structure for fault-tolerant approximate shortest-path trees that effectively handles path failures with low stretch and size, improving upon previous solutions.

Findings

Achieves a stretch factor of (2k-1)(2|F|+1) with size O(kn·f^{1+1/k})

Refines to a stretch of 3 and size O(n log n) for f=2

Enables efficient approximate-distance sensitivity queries

Abstract

Let $G=(V,E)$ be an $n$-nodes non-negatively real-weighted undirected graph. In this paper we show how to enrich a {\em single-source shortest-path tree} (SPT) of $G$ with a \emph{sparse} set of \emph{auxiliary} edges selected from $E$, in order to create a structure which tolerates effectively a \emph{path failure} in the SPT. This consists of a simultaneous fault of a set $F$ of at most $f$ adjacent edges along a shortest path emanating from the source, and it is recognized as one of the most frequent disruption in an SPT. We show that, for any integer parameter $k \geq 1$, it is possible to provide a very sparse (i.e., of size $O(kn\cdot f^{1+1/k})$) auxiliary structure that carefully approximates (i.e., within a stretch factor of $(2k-1)(2|F|+1)$) the true shortest paths from the source during the lifetime of the failure. Moreover, we show that our construction can be further refined to get a stretch factor of $3$ and a size of $O(n \log n)$ for the special case $f=2$, and that it can be converted into a very efficient \emph{approximate-distance sensitivity oracle}, that allows to quickly (even in optimal time, if $k=1$) reconstruct the shortest paths (w.r.t. our structure) from the source after a path failure, thus permitting to perform promptly the needed rerouting operations. Our structure compares favorably with previous known solutions, as we discuss in the paper, and moreover it is also very effective in practice, as we assess through a large set of experiments.