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Making Multicurves Cross Minimally on Surfaces

Author Loïc Dubois

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Author Details

Loïc Dubois
  • LIGM, Université Gustave Eiffel, Marne-la-Vallée, France
  • LORIA, INRIA Nancy, France

Funding

This work was partially supported by the grant ANR-19-CE40-0014 (Min-max) of the French National Research Agency.

Acknowledgements

The author thanks Éric Colin de Verdière, Vincent Despré, and Arnaud de Mesmay for their help and discussions.

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Loïc Dubois. Making Multicurves Cross Minimally on Surfaces. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.50

Abstract

On an orientable surface S, consider a collection Γ of closed curves. The (geometric) intersection number i_S(Γ) is the minimum number of self-intersections that a collection Γ' can have, where Γ' results from a continuous deformation (homotopy) of Γ. We provide algorithms that compute i_S(Γ) and such a Γ', assuming that Γ is given by a collection of closed walks of length n in a graph M cellularly embedded on S, in O(n log n) time when M and S are fixed.
The state of the art is a paper of Despré and Lazarus [SoCG 2017, J. ACM 2019], who compute i_S(Γ) in O(n²) time, and Γ' in O(n⁴) time if Γ is a single closed curve. Our result is more general since we can put an arbitrary number of closed curves in minimal position. Also, our algorithms are quasi-linear in n instead of quadratic and quartic. Most importantly, our proofs are simpler, shorter, and more structured.
We use techniques from two-dimensional topology and from the theory of hyperbolic surfaces. Most notably, we prove a new property of the reducing triangulations introduced by Colin de Verdière, Despré, and Dubois [SODA 2024], reducing our problem to the case of surfaces with boundary. As a key subroutine, we rely on an algorithm of Fulek and Tóth [JCO 2020].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Algorithms
  • Topology
  • Surfaces
  • Closed Curves
  • Geometric Intersection Number

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