Graph Spanners for Group Steiner Distances

Authors Davide Bilò , Luciano Gualà , Stefano Leucci , Alessandro Straziota



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Davide Bilò
  • Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Italy
Luciano Gualà
  • Department of Enterprise Engineering, University of Rome "Tor Vergata", Italy
Stefano Leucci
  • Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Italy
Alessandro Straziota
  • Department of Enterprise Engineering, University of Rome "Tor Vergata", Italy

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Davide Bilò, Luciano Gualà, Stefano Leucci, and Alessandro Straziota. Graph Spanners for Group Steiner Distances. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.25

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Mathematics of computing → Graph algorithms
Keywords
  • Network sparsification
  • Graph spanners
  • Group Steiner tree
  • Distance oracles

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