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Flip Distance Is in FPT Time O(n+ k * c^k)

Authors Iyad Kanj, Ge Xia



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LIPIcs.STACS.2015.500.pdf
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Iyad Kanj
Ge Xia

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Iyad Kanj and Ge Xia. Flip Distance Is in FPT Time O(n+ k * c^k). In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 500-512, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.STACS.2015.500

Abstract

Let T be a triangulation of a set P of n points in the plane, and let e be an edge shared by two triangles in T such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q 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 Flip Distance problem asks if the flip distance between two given triangulations of P is k, for some given k \in \mathbb{N}. It is a fundamental and a challenging problem. In this paper we present an algorithm for the Flip Distance problem that runs in time O(n + k \cdot c^{k}), for a constant c \leq 2 \cdot 14^11, which implies that the problem is fixed-parameter tractable. The NP-hardness reduction for the Flip Distance problem given by Lubiw and Pathak can be used to show that, unless the exponential-time hypothesis (ETH) fails, the Flip Distance problem cannot be solved in time O^*(2^o(k)). Therefore, one cannot expect an asymptotic improvement in the exponent of the running time of our algorithm.
Keywords
  • triangulations
  • flip distance
  • parameterized algorithms

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