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Long Non-crossing Configurations in the Plane

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Abstract

We revisit several maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of $n$ points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching (b) a Hamiltonian path (c) a spanning tree. Here we obtain new results for (b) and (c), as well as for the Hamiltonian cycle problem: (i) For the longest non-crossing Hamiltonian path problem, we give an approximation algorithm with ratio $\frac{2}{\pi+1} \approx 0.4829$. The previous best ratio, due to Alon et al., was $1/\pi \approx 0.3183$. Moreover, the ratio of our algorithm is close to $2/\pi$ on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time. (ii) For the longest non-crossing spanning tree problem, we give an approximation algorithm with ratio $0.502$ which runs in $O(n \log{n})$ time. The previous ratio, $1/2$, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster $O(n \log{n})$-time algorithm and a very simple analysis. (iii) For the longest non-crossing Hamiltonian cycle problem, we give an approximation algorithm whose ratio is close to $2/\pi$ on a relatively broad class of instances: for point sets with the product $\bf{\langle}$~diameter~$\times$ ~convex hull size $\bf{\rangle}$ much smaller than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time. No previous approximation results were known for this problem.

BibTeX - Entry

@InProceedings{dumitrescu_et_al:LIPIcs:2010:2465,
  author =	{Adrian Dumitrescu and Csaba D. T{\'o}th},
  title =	{{Long Non-crossing Configurations in the Plane}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{311--322},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Jean-Yves Marion and Thomas Schwentick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2465},
  URN =		{urn:nbn:de:0030-drops-24655},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2010.2465},
  annote =	{Keywords: Longest non-crossing Hamiltonian path, longest non-crossing Hamiltonian cycle, longest non-crossing spanning tree, approximation algorithm.}
}

Keywords: Longest non-crossing Hamiltonian path, longest non-crossing Hamiltonian cycle, longest non-crossing spanning tree, approximation algorithm.
Seminar: 27th International Symposium on Theoretical Aspects of Computer Science
Issue date: 2010
Date of publication: 2010


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