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.2015.240
URN: urn:nbn:de:0030-drops-51132
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5113/
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Bokal, Drago ; Cabello, Sergio ; Eppstein, David

Finding All Maximal Subsequences with Hereditary Properties

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Abstract

Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i) such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results:

- In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1.

- In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1.

- In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction.

The same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.

BibTeX - Entry

@InProceedings{bokal_et_al:LIPIcs:2015:5113,
  author =	{Drago Bokal and Sergio Cabello and David Eppstein},
  title =	{{Finding All Maximal Subsequences with Hereditary Properties}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{240--254},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5113},
  URN =		{urn:nbn:de:0030-drops-51132},
  doi =		{10.4230/LIPIcs.SOCG.2015.240},
  annote =	{Keywords: convex hull, diameter, monotone path, sequence of points, trajectory}
}

Keywords: convex hull, diameter, monotone path, sequence of points, trajectory
Collection: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 12.06.2015

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