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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.

Cite AsGet BibTex

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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