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Chains, Koch Chains, and Point Sets with Many Triangulations

Authors Daniel Rutschmann, Manuel Wettstein

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Daniel Rutschmann
  • Department of Computer Science, ETH Zürich, Switzerland
Manuel Wettstein
  • Department of Computer Science, ETH Zürich, Switzerland

Acknowledgements

The material presented in this paper originates from the first author’s Master’s thesis [Rutschmann, 2021] under the second author’s direct supervision. Both authors wish to express their gratitude to Emo Welzl, the official advisor in this endeavor.

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Daniel Rutschmann and Manuel Wettstein. Chains, Koch Chains, and Point Sets with Many Triangulations. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.59

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Planar Point Set
  • Chain
  • Koch Chain
  • Triangulation
  • Maximum Number of Triangulations
  • Lower Bound

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