Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Chan, Timothy M.; Har-Peled, Sariel; Jones, Mitchell http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-101140
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On Locality-Sensitive Orderings and Their Applications

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Abstract

For any constant d and parameter epsilon > 0, we show the existence of (roughly) 1/epsilon^d orderings on the unit cube [0,1)^d, such that any two points p, q in [0,1)^d that are close together under the Euclidean metric are "close together" in one of these linear orderings in the following sense: the only points that could lie between p and q in the ordering are points with Euclidean distance at most epsilon | p - q | from p or q. These orderings are extensions of the Z-order, and they can be efficiently computed. Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.

BibTeX - Entry

@InProceedings{chan_et_al:LIPIcs:2018:10114,
  author =	{Timothy M. Chan and Sariel Har-Peled and Mitchell Jones},
  title =	{{On Locality-Sensitive Orderings and Their Applications}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10114},
  URN =		{urn:nbn:de:0030-drops-101140},
  doi =		{10.4230/LIPIcs.ITCS.2019.21},
  annote =	{Keywords: Approximation algorithms, Data structures, Computational geometry}
}

Keywords: Approximation algorithms, Data structures, Computational geometry
Seminar: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue date: 2018
Date of publication: 2018


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