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An 𝒪(3.82^k) Time FPT Algorithm for Convex Flip Distance

Authors Haohong Li, Ge Xia

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Haohong Li
  • Department of Computer Science, Lafayette College, Easton, PA, USA
Ge Xia
  • Department of Computer Science, Lafayette College, Easton, PA, USA

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Haohong Li and Ge Xia. An 𝒪(3.82^k) Time FPT Algorithm for Convex Flip Distance. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.STACS.2023.44

Abstract

Let P be a convex polygon in the plane, and let T be a triangulation of P. An edge e in T is called a diagonal if it is shared by two triangles in T. A flip of a diagonal e is the operation of removing e and adding the opposite diagonal of the resulting quadrilateral to obtain a new triangulation of P from T. The flip distance between two triangulations of P is the minimum number of flips needed to transform one triangulation into the other. The Convex Flip Distance problem asks if the flip distance between two given triangulations of P is at most k, for some given parameter k ∈ ℕ. 
We present an FPT algorithm for the Convex Flip Distance problem that runs in time 𝒪(3.82^k) and uses polynomial space, where k is the number of flips. This algorithm significantly improves the previous best FPT algorithms for the problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Flip distance
  • Rotation distance
  • Triangulations
  • Exact algorithms
  • Parameterized complexity

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