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Improved Mixing for the Convex Polygon Triangulation Flip Walk

Authors David Eppstein, Daniel Frishberg

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David Eppstein
  • Department of Computer Science, University of California, Irvine, CA, USA
Daniel Frishberg
  • Department of Computer Science, University of California, Irvine, CA, USA

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David Eppstein and Daniel Frishberg. Improved Mixing for the Convex Polygon Triangulation Flip Walk. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 56:1-56:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.56

Abstract

We prove that the well-studied triangulation flip walk on a convex point set mixes in time O(n³ log³ n), the first progress since McShine and Tetali’s O(n⁵ log n) bound in 1997. In the process we give lower and upper bounds of respectively Ω(1/(√n log n)) and O(1/√n) - asymptotically tight up to an O(log n) factor - for the expansion of the associahedron graph K_n. The upper bound recovers Molloy, Reed, and Steiger’s Ω(n^(3/2)) bound on the mixing time of the walk. To obtain these results, we introduce a framework consisting of a set of sufficient conditions under which a given Markov chain mixes rapidly. This framework is a purely combinatorial analogue that in some circumstances gives better results than the projection-restriction technique of Jerrum, Son, Tetali, and Vigoda. In particular, in addition to the result for triangulations, we show quasipolynomial mixing for the k-angulation flip walk on a convex point set, for fixed k ≥ 4.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • associahedron
  • mixing time
  • mcmc
  • Markov chains
  • triangulations
  • quadrangulations
  • k-angulations
  • multicommodity flow
  • projection-restriction

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