License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.29
URN: urn:nbn:de:0030-drops-178790
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17879/
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Eppstein, David

Non-Crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

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LIPIcs-SoCG-2023-29.pdf (0.8 MB)

Abstract

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.

BibTeX - Entry

@InProceedings{eppstein:LIPIcs.SoCG.2023.29,
  author =	{Eppstein, David},
  title =	{{Non-Crossing Hamiltonian Paths and Cycles in Output-Polynomial Time}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17879},
  URN =		{urn:nbn:de:0030-drops-178790},
  doi =		{10.4230/LIPIcs.SoCG.2023.29},
  annote =	{Keywords: polygonalization, non-crossing structures, output-sensitive algorithms}
}

Keywords: polygonalization, non-crossing structures, output-sensitive algorithms
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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