License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2015.500
URN: urn:nbn:de:0030-drops-49371
URL: https://drops.dagstuhl.de/opus/volltexte/2015/4937/
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Kanj, Iyad ; Xia, Ge

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

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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.

BibTeX - Entry

@InProceedings{kanj_et_al:LIPIcs:2015:4937,
  author =	{Iyad Kanj and Ge Xia},
  title =	{{Flip Distance Is in FPT Time O(n+ k * c^k)}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{500--512},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Ernst W. Mayr and Nicolas Ollinger},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/4937},
  URN =		{urn:nbn:de:0030-drops-49371},
  doi =		{10.4230/LIPIcs.STACS.2015.500},
  annote =	{Keywords: triangulations, flip distance, parameterized algorithms}
}

Keywords: triangulations, flip distance, parameterized algorithms
Collection: 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)
Issue Date: 2015
Date of publication: 26.02.2015


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