Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Eppstein, David; Khodabandeh, Hadi https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-138328
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On the Edge Crossings of the Greedy Spanner

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Abstract

The greedy t-spanner of a set of points in the plane is an undirected graph constructed by considering pairs of points in order by distance, and connecting a pair by an edge when there does not already exist a path connecting that pair with length at most t times the Euclidean distance. We prove that, for any t > 1, these graphs have at most a linear number of crossings, and more strongly that the intersection graph of edges in a greedy t-spanner has bounded degeneracy. As a consequence, we prove a separator theorem for greedy spanners: any k-vertex subgraph of a greedy spanner can be partitioned into sub-subgraphs of size a constant fraction smaller, by the removal of O(√k) vertices. A recursive separator hierarchy for these graphs can be constructed from their planarizations in linear time, or in near-linear time if the planarization is unknown.

BibTeX - Entry

@InProceedings{eppstein_et_al:LIPIcs.SoCG.2021.33,
  author =	{Eppstein, David and Khodabandeh, Hadi},
  title =	{{On the Edge Crossings of the Greedy Spanner}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13832},
  URN =		{urn:nbn:de:0030-drops-138328},
  doi =		{10.4230/LIPIcs.SoCG.2021.33},
  annote =	{Keywords: Geometric Spanners, Greedy Spanners, Separators, Crossing Graph, Sparsity}
}

Keywords: Geometric Spanners, Greedy Spanners, Separators, Crossing Graph, Sparsity
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021


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