On Locality-Sensitive Orderings and Their Applications

Authors Timothy M. Chan, Sariel Har-Peled, Mitchell Jones

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Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA
Sariel Har-Peled
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA
Mitchell Jones
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA

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Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones. On Locality-Sensitive Orderings and Their Applications. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Approximation algorithms
  • Data structures
  • Computational geometry


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