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.SoCG.2016.29
URN: urn:nbn:de:0030-drops-59218
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5921/
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Chang, Hsien-Chih ; Erickson, Jeff

Untangling Planar Curves

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LIPIcs-SoCG-2016-29.pdf (5 MB)

Abstract

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Omega(n^{3/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve.

BibTeX - Entry

@InProceedings{chang_et_al:LIPIcs:2016:5921,
  author =	{Hsien-Chih Chang and Jeff Erickson},
  title =	{{Untangling Planar Curves}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5921},
  URN =		{urn:nbn:de:0030-drops-59218},
  doi =		{10.4230/LIPIcs.SoCG.2016.29},
  annote =	{Keywords: computational topology, homotopy, planar graphs, Delta-Y transformations, defect, Reidemeister moves, tangles}
}

Keywords: computational topology, homotopy, planar graphs, Delta-Y transformations, defect, Reidemeister moves, tangles
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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