Graph Spanners for Group Steiner Distances

TL;DR

This paper introduces new graph spanners tailored for the group Steiner metric, enabling efficient approximation of shortest paths that traverse specified groups of vertices, with various size-stretch trade-offs and applications to distance oracles.

Contribution

It presents novel constructions of group Steiner spanners, explores their relationship with existing spanners, and provides near-optimal solutions for the singleton case and sourcewise spanners.

Findings

Provided spanners with optimal size-stretch trade-offs for singleton groups

Developed constructions for all-pairs and single-source cases

Achieved improved query times for group Steiner distance oracles

Abstract

A spanner is a sparse subgraph of a given graph $G$ which preserves distances, measured w.r.t.\ some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t.\ the group Steiner metric, which generalizes the recently introduced beer distance metric. In such a metric we are given a collection of groups of required vertices, and we measure the distance between two vertices as the length of the shortest path between them that traverses at least one required vertex from each group. We discuss the relation between group Steiner spanners and classic spanners and we show that they exhibit strong ties with sourcewise spanners w.r.t.\ the shortest path metric. Nevertheless, group Steiner spanners capture several interesting scenarios that are not encompassed by existing spanners. This happens, e.g., for the singleton case, in which each group consists of a single required vertex, thus modeling the setting in which routes need to traverse certain points of interests (in any order). We provide several constructions of group Steiner spanners for both the all-pairs and single-source case, which exhibit various size-stretch trade-offs. Notably, we provide spanners with almost-optimal trade-offs for the singleton case. Moreover, some of our spanners also yield novel trade-offs for classical sourcewise spanners. Finally, we also investigate the query times that can be achieved when our spanners are turned into group Steiner distance oracles with the same size, stretch, and building time.