License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.SoCG.2020.57
URN: urn:nbn:de:0030-drops-122152
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Mulzer, Wolfgang ; Valtr, Pavel

Long Alternating Paths Exist

LIPIcs-SoCG-2020-57.pdf (0.6 MB)


Let P be a set of 2n points in convex position, such that n points are colored red and n points are colored blue. A non-crossing alternating path on P of length š“ is a sequence pā‚, ā€¦ , p_š“ of š“ points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors, for i ā‰  j.
We show that there is an absolute constant Īµ > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + Īµ)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + Īµ)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3+o(n).

BibTeX - Entry

  author =	{Wolfgang Mulzer and Pavel Valtr},
  title =	{{Long Alternating Paths Exist}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{57:1--57:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-122152},
  doi =		{10.4230/LIPIcs.SoCG.2020.57},
  annote =	{Keywords: Non-crossing path, bichromatic point sets}

Keywords: Non-crossing path, bichromatic point sets
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020

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