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DOI: 10.4230/LIPIcs.SoCG.2020.34

The Stretch Factor of Hexagon-Delaunay Triangulations

Authors: Michael Dennis, Ljubomir Perković, and Duru Türkoğlu

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract

The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of "easier" types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations.

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Michael Dennis, Ljubomir Perković, and Duru Türkoğlu. The Stretch Factor of Hexagon-Delaunay Triangulations. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dennis_et_al:LIPIcs.SoCG.2020.34,
  author =	{Dennis, Michael and Perkovi\'{c}, Ljubomir and T\"{u}rko\u{g}lu, Duru},
  title =	{{The Stretch Factor of Hexagon-Delaunay Triangulations}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.34},
  URN =		{urn:nbn:de:0030-drops-121920},
  doi =		{10.4230/LIPIcs.SoCG.2020.34},
  annote =	{Keywords: Delaunay triangulation, geometric spanner, plane spanner, stretch factor, spanning ratio}
}

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The Stretch Factor of Hexagon-Delaunay Triangulations

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