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.2022.59
URN: urn:nbn:de:0030-drops-160678
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16067/
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Rutschmann, Daniel ; Wettstein, Manuel

Chains, Koch Chains, and Point Sets with Many Triangulations

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LIPIcs-SoCG-2022-59.pdf (0.7 MB)

Abstract

We introduce the abstract notion of a chain, which is a sequence of n points in the plane, ordered by x-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations.
We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω(9.08ⁿ) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω(8.65ⁿ) for the maximum number of triangulations of planar point sets.

BibTeX - Entry

@InProceedings{rutschmann_et_al:LIPIcs.SoCG.2022.59,
  author =	{Rutschmann, Daniel and Wettstein, Manuel},
  title =	{{Chains, Koch Chains, and Point Sets with Many Triangulations}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{59:1--59:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16067},
  URN =		{urn:nbn:de:0030-drops-160678},
  doi =		{10.4230/LIPIcs.SoCG.2022.59},
  annote =	{Keywords: Planar Point Set, Chain, Koch Chain, Triangulation, Maximum Number of Triangulations, Lower Bound}
}

Keywords: Planar Point Set, Chain, Koch Chain, Triangulation, Maximum Number of Triangulations, Lower Bound
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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