License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.52
URN: urn:nbn:de:0030-drops-179024
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17902/
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Papadopoulou, Evanthia

Abstract Voronoi-Like Graphs: Extending Delaunay’s Theorem and Applications

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Abstract

Any system of bisectors (in the sense of abstract Voronoi diagrams) defines an arrangement of simple curves in the plane. We define Voronoi-like graphs on such an arrangement, which are graphs whose vertices are locally Voronoi. A vertex v is called locally Voronoi, if v and its incident edges appear in the Voronoi diagram of three sites. In a so-called admissible bisector system, where Voronoi regions are connected and cover the plane, we prove that any Voronoi-like graph is indeed an abstract Voronoi diagram. The result can be seen as an abstract dual version of Delaunay’s theorem on (locally) empty circles.
Further, we define Voronoi-like cycles in an admissible bisector system, and show that the Voronoi-like graph induced by such a cycle C is a unique tree (or a forest, if C is unbounded). In the special case where C is the boundary of an abstract Voronoi region, the induced Voronoi-like graph can be computed in expected linear time following the technique of [Junginger and Papadopoulou SOCG'18]. Otherwise, within the same time, the algorithm constructs the Voronoi-like graph of a cycle C′ on the same set (or subset) of sites, which may equal C or be enclosed by C. Overall, the technique computes abstract Voronoi (or Voronoi-like) trees and forests in linear expected time, given the order of their leaves along a Voronoi-like cycle. We show a direct application in updating a constraint Delaunay triangulation in linear expected time, after the insertion of a new segment constraint, simplifying upon the result of [Shewchuk and Brown CGTA 2015].

BibTeX - Entry

@InProceedings{papadopoulou:LIPIcs.SoCG.2023.52,
  author =	{Papadopoulou, Evanthia},
  title =	{{Abstract Voronoi-Like Graphs: Extending Delaunay’s Theorem and Applications}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{52:1--52:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17902},
  URN =		{urn:nbn:de:0030-drops-179024},
  doi =		{10.4230/LIPIcs.SoCG.2023.52},
  annote =	{Keywords: Voronoi-like graph, abstract Voronoi diagram, Delaunay’s theorem, Voronoi tree, linear-time randomized algorithm, constraint Delaunay triangulation}
}

Keywords: Voronoi-like graph, abstract Voronoi diagram, Delaunay’s theorem, Voronoi tree, linear-time randomized algorithm, constraint Delaunay triangulation
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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