A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners

TL;DR

This paper presents an efficient algorithm for finding the best swap edges in tree spanners of graphs, improving previous methods and potentially enabling faster fault-tolerance solutions in network design.

Contribution

The authors develop an $O(n^2)$ time and space algorithm for computing best swap edges in tree spanners, surpassing the previous $O(n^2 ext{log}^4 n)$ complexity.

Findings

Algorithm runs in $O(n^2)$ time and space.

Significant improvement over previous algorithms.

Potential for breakthroughs in fault-tolerant network design.

Abstract

Given a 2-edge connected, unweighted, and undirected graph $G$ with $n$ vertices and $m$ edges, a $\sigma$-tree spanner is a spanning tree $T$ of $G$ in which the ratio between the distance in $T$ of any pair of vertices and the corresponding distance in $G$ is upper bounded by $\sigma$. The minimum value of $\sigma$ for which $T$ is a $\sigma$-tree spanner of $G$ is also called the {\em stretch factor} of $T$. We address the fault-tolerant scenario in which each edge $e$ of a given tree spanner may temporarily fail and has to be replaced by a {\em best swap edge}, i.e. an edge that reconnects $T-e$ at a minimum stretch factor. More precisely, we design an $O(n^2)$ time and space algorithm that computes a best swap edge of every tree edge. Previously, an $O(n^2 \log^4 n)$ time and $O(n^2+m\log^2n)$ space algorithm was known for edge-weighted graphs [Bil\`o et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a $o(n^2)$ time and space algorithm would be considered a breakthrough.