An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner

TL;DR

This paper presents a highly efficient algorithm for computing all the best swap edges in a tree spanner, significantly improving the computational complexity over previous methods, which is crucial for network resilience and maintenance.

Contribution

The authors introduce an optimized algorithm that computes all best swap edges of a tree spanner in $O(n^2 ext{log}^4 n)$ time, a substantial improvement over prior approaches.

Findings

Algorithm runs in $O(n^2 ext{log}^4 n)$ time

Significant reduction in computational complexity for dense graphs

Enhances network robustness by efficient swap edge computation

Abstract

A tree $\sigma$-spanner of a positively real-weighted $n$-vertex and $m$-edge undirected graph $G$ is a spanning tree $T$ of $G$ which approximately preserves (i.e., up to a multiplicative stretch factor $\sigma$) distances in $G$. Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in $T$, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be rather prohibitive. To circumvent this drawback, an effective alternative is that of associating with each tree edge a best possible (in terms of resulting stretch) swap edge -- a well-established approach in the literature for several other tree topologies. Correspondingly, the problem of computing all the best swap edges of a tree spanner is a challenging algorithmic problem, since solving it efficiently means to exploit the structure of shortest paths not only in $G$, but also in all the scenarios in which an edge of $T$ has failed. For this problem we provide a very efficient solution, running in $O(n^2 \log^4 n)$ time, which drastically improves (almost by a quadratic factor in $n$ in dense graphs!) on the previous known best result.