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.ITCS.2019.21
URN: urn:nbn:de:0030-drops-101140
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/10114/
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Chan, Timothy M. ; Har-Peled, Sariel ; Jones, Mitchell

On Locality-Sensitive Orderings and Their Applications

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LIPIcs-ITCS-2019-21.pdf (0.6 MB)

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
Collection: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue Date: 2018
Date of publication: 08.01.2019

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