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Oriented Spanners

Authors Kevin Buchin , Joachim Gudmundsson , Antonia Kalb , Aleksandr Popov , Carolin Rehs , André van Renssen , Sampson Wong

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Author Details

Kevin Buchin
  • Technical University of Dortmund, Germany
Joachim Gudmundsson
  • University of Sydney, Australia
Antonia Kalb
  • Technical University of Dortmund, Germany
Aleksandr Popov
  • Technical University of Eindhoven, The Netherlands
Carolin Rehs
  • Technical University of Dortmund, Germany
André van Renssen
  • University of Sydney, Australia
Sampson Wong
  • BARC, University of Copenhagen, Denmark

Funding

  • Popov, Aleksandr: Supported by the Dutch Research Council (NWO) under the project number 612.001.801.

Cite AsGet BibTex

Kevin Buchin, Joachim Gudmundsson, Antonia Kalb, Aleksandr Popov, Carolin Rehs, André van Renssen, and Sampson Wong. Oriented Spanners. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.26

Abstract

Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in 𝒪(n⁸) time for n points, and a greedy algorithm that computes a 5-spanner in 𝒪(nlog n) time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented 𝒪(1)-spanner.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • computational geometry
  • spanner
  • oriented graph
  • greedy triangulation

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