Further results on colored range searching

Timothy M. Chan, Qizheng He, Yakov Nekrich

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

Abstract

We present a number of new results about range searching for colored (or “categorical”) data: 1. For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1,..., n}. Previous data structures require O(logloglognn + k) query time. Our result also implies improvements in higher constant dimensions. 2. Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log2 n) query time. 3. For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored “type-2” range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O(logloglognn + k log log n), where k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k logloglognn ) time. Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.

Original languageEnglish (US)
Title of host publication36th International Symposium on Computational Geometry, SoCG 2020
EditorsSergio Cabello, Danny Z. Chen
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771436
DOIs
StatePublished - Jun 1 2020
Event36th International Symposium on Computational Geometry, SoCG 2020 - Zurich, Switzerland
Duration: Jun 23 2020Jun 26 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume164
ISSN (Print)1868-8969

Conference

Conference36th International Symposium on Computational Geometry, SoCG 2020
CountrySwitzerland
CityZurich
Period6/23/206/26/20

Keywords

  • Geometric data structures
  • Random sampling
  • Randomized incremental construction
  • Range searching
  • Word RAM

ASJC Scopus subject areas

  • Software

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