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Lexicographic Optimal Homologous Chains and Applications to Point Cloud Triangulations

Authors David Cohen-Steiner, André Lieutier, Julien Vuillamy

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David Cohen-Steiner
  • Université Côte d'Azur, Sophia Antipolis, France
  • Inria Sophia Antipolis - Mediterranée, France
André Lieutier
  • Dassault Systèmes Provence, Aix-en-Provence, France
Julien Vuillamy
  • Université Côte d'Azur, Sophia Antipolis, France
  • Inria Sophia Antipolis - Mediterranée, France
  • Dassault Systèmes Provence, Aix-en-Provence, France

Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments and suggestions on the submitted version of this paper.

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David Cohen-Steiner, André Lieutier, and Julien Vuillamy. Lexicographic Optimal Homologous Chains and Applications to Point Cloud Triangulations. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.32

Abstract

This paper considers a particular case of the Optimal Homologous Chain Problem (OHCP) for integer modulo 2 coefficients, where optimality is meant as a minimal lexicographic order on chains induced by a total order on simplices. The matrix reduction algorithm used for persistent homology is used to derive polynomial algorithms solving this problem instance, whereas OHCP is NP-hard in the general case. The complexity is further improved to a quasilinear algorithm by leveraging a dual graph minimum cut formulation when the simplicial complex is a pseudomanifold. We then show how this particular instance of the problem is relevant, by providing an application in the context of point cloud triangulation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • OHCP
  • simplicial homology
  • triangulation
  • Delaunay

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