An improved lower bound on the minimum number of triangulations

Oswin Aichholzer, Victor Alvarez, Thomas Hackl, Alexander Pilz, Bettina Speckmann, Birgit Vogtenhuber

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Abstract

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Ω(2.631n) geometric triangulations. Our result improves the previously best general lower bound of Ω(2.43n) and also covers the previously best lower bound of Ω(2.63n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: 1. Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. 2. Generalized configurations of points that minimize the number of triangulations have at most ⌊n/2⌋ points on their convex hull. 3. We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.

Original languageEnglish
Title of host publication32nd International Symposium on Computational Geometry, SoCG 2016
PublisherSchloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH
Pages7.1-7.16
Volume51
ISBN (Electronic)9783959770095
DOIs
Publication statusPublished - 1 Jun 2016
Event32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States
Duration: 14 Jun 201617 Jun 2016

Conference

Conference32nd International Symposium on Computational Geometry, SoCG 2016
CountryUnited States
CityBoston
Period14/06/1617/06/16

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Triangulation

Keywords

  • Combinatorial geometry
  • Order types
  • Triangulations

ASJC Scopus subject areas

  • Software

Cite this

Aichholzer, O., Alvarez, V., Hackl, T., Pilz, A., Speckmann, B., & Vogtenhuber, B. (2016). An improved lower bound on the minimum number of triangulations. In 32nd International Symposium on Computational Geometry, SoCG 2016 (Vol. 51, pp. 7.1-7.16). Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. https://doi.org/10.4230/LIPIcs.SoCG.2016.7

An improved lower bound on the minimum number of triangulations. / Aichholzer, Oswin; Alvarez, Victor; Hackl, Thomas; Pilz, Alexander; Speckmann, Bettina; Vogtenhuber, Birgit.

32nd International Symposium on Computational Geometry, SoCG 2016. Vol. 51 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2016. p. 7.1-7.16.

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Aichholzer, O, Alvarez, V, Hackl, T, Pilz, A, Speckmann, B & Vogtenhuber, B 2016, An improved lower bound on the minimum number of triangulations. in 32nd International Symposium on Computational Geometry, SoCG 2016. vol. 51, Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, pp. 7.1-7.16, 32nd International Symposium on Computational Geometry, SoCG 2016, Boston, United States, 14/06/16. https://doi.org/10.4230/LIPIcs.SoCG.2016.7
Aichholzer O, Alvarez V, Hackl T, Pilz A, Speckmann B, Vogtenhuber B. An improved lower bound on the minimum number of triangulations. In 32nd International Symposium on Computational Geometry, SoCG 2016. Vol. 51. Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. 2016. p. 7.1-7.16 https://doi.org/10.4230/LIPIcs.SoCG.2016.7
Aichholzer, Oswin ; Alvarez, Victor ; Hackl, Thomas ; Pilz, Alexander ; Speckmann, Bettina ; Vogtenhuber, Birgit. / An improved lower bound on the minimum number of triangulations. 32nd International Symposium on Computational Geometry, SoCG 2016. Vol. 51 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2016. pp. 7.1-7.16
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