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DOI: 10.4230/LIPIcs.SoCG.2017.35
URN: urn:nbn:de:0030-drops-71838
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7183/
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Despré, Vincent ; Lazarus, Francis

Computing the Geometric Intersection Number of Curves

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LIPIcs-SoCG-2017-35.pdf (0.6 MB)


Abstract

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most l on a combinatorial surface of complexity n we describe simple algorithms to (1) compute the geometric intersection number of c in O(n+ l^2) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O(n+l^4) time, (3) decide if the geometric intersection number of c is zero, i.e. if c is homotopic to a simple curve, in O(n+l log^2 l) time. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O(n+g^2l^2) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2). Interestingly, our solution to problem (3) is the first quasi-linear algorithm since the problem was raised by Poincare more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most l in O(n+ l^2) time.

BibTeX - Entry

@InProceedings{despr_et_al:LIPIcs:2017:7183,
  author =	{Vincent Despr{\'e} and Francis Lazarus},
  title =	{{Computing the Geometric Intersection Number of Curves}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{35:1--35:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Boris Aronov and Matthew J. Katz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7183},
  URN =		{urn:nbn:de:0030-drops-71838},
  doi =		{10.4230/LIPIcs.SoCG.2017.35},
  annote =	{Keywords: computational topology, curves on surfaces, combinatorial geodesic}
}

Keywords: computational topology, curves on surfaces, combinatorial geodesic
Seminar: 33rd International Symposium on Computational Geometry (SoCG 2017)
Issue Date: 2017
Date of publication: 08.06.2017


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