Orthogonal range searching in moderate dimensions: K-D trees and range trees strike back

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We revisit the orthogonal range searching problem and the exact ℓ nearest neighbor searching problem for a static set of n points when the dimension d is moderately large. We give the first data structure with near linear space that achieves truly sublinear query time when the dimension is any constant multiple of log n. Specifically, the preprocessing time and space are O(n1+δ) for any constant δ > 0, and the expected query time is n1-1/O(c log c) for d = c log n. The data structure is simple and is based on a new "augmented, randomized, lopsided" variant of k-d trees. It matches (in fact, slightly improves) the performance of previous combinatorial algorithms that work only in the case of offline queries [Impagliazzo, Lovett, Paturi, and Schneider (2014) and Chan (SODA'15)]. It leads to slightly faster combinatorial algorithms for all-pairs shortest paths in general real-weighted graphs and rectangular Boolean matrix multiplication. In the offline case, we show that the problem can be reduced to the Boolean orthogonal vectors problem and thus admits an n2-1/O(log c)-time non-combinatorial algorithm [Abboud, Williams, and Yu (SODA'15)]. This reduction is also simple and is based on range trees. Finally, we use a similar approach to obtain a small improvement to Indyk's data structure [FOCS'98] for approximate ℓ nearest neighbor search when d = c log n.

Original languageEnglish (US)
Title of host publication33rd International Symposium on Computational Geometry, SoCG 2017
EditorsMatthew J. Katz, Boris Aronov
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages271-2715
Number of pages2445
ISBN (Electronic)9783959770385
DOIs
StatePublished - Jun 1 2017
Event33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia
Duration: Jul 4 2017Jul 7 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume77
ISSN (Print)1868-8969

Other

Other33rd International Symposium on Computational Geometry, SoCG 2017
Country/TerritoryAustralia
CityBrisbane
Period7/4/177/7/17

Keywords

  • Computational geometry
  • Data structures
  • Nearest neighbor searching
  • Range searching

ASJC Scopus subject areas

  • Software

Fingerprint

Dive into the research topics of 'Orthogonal range searching in moderate dimensions: K-D trees and range trees strike back'. Together they form a unique fingerprint.

Cite this