Improved Sourcewise Roundtrip Spanners with Constant Stretch

Eli Stafford, Chunjiang Zhu

Abstract: Graph spanners are a sparse subgraph of a graph such that shortest-path distances for all pairs of vertices are approximately preserved with a factor called stretch, and roundtrip-spanners are defined for directed graphs to preserve roundtrip distances instead of one-way distances. Sourcewise roundtrip-spanners can approximate roundtrip distances for only some pairs of vertices \(S\times V\) for source vertices \(S\subseteq V\) and are more generalized than traditional all-pairs roundtrip-spanners. While general roundtrip-spanners have made progress in the realm of constant stretch, it is unknown whether constant stretch (with small number of edges dependent on |S|) can be achieved in the sourcewise setting. In this paper, we provide an algorithm that, for a weighted, directed graph with n vertices, m edges G and a set of sources S of size s, constructs a sourcewise roundtrip-spanner with stretch 3 and \(\tilde{O}(n\sqrt{s})\) expected edges in \(\tilde{O}(ms)\) time. Moreover, we develop a faster \(\tilde{O}(m\sqrt{n}/\epsilon ^2)\)-time algorithm with stretch \((5+\epsilon )\) and \(\tilde{O}(n\sqrt{s}/\epsilon ^2)\) edges when S is randomly picked with size \(s=\varOmega (\sqrt{n})\). Our algorithms combine ideas from [RTZ08, RTZ05] and adapt the algorithm of [DW20] to the sourcewise case.