Adaptive and approximate orthogonal range counting

Timothy M. Chan, Bryan T. Wilkinson

Research output: Contribution to journalArticlepeer-review

Abstract

We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model. -It is well known that there are linear-space data structures for 2-D orthogonal range counting with worstcase optimal query time O(log n/ log log n). We give an O(nlog log n)-space adaptive data structure that improves the query time to O(log log n+ log k/ log log n), where k is the output count. When k = O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan et al., 2011]. -We give an O(nlog log n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1+d)-factor approximation to the count in O(log log n) time for any fixed constant d > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem. -Last, we consider the 1-D range selection problem, where a query in an array involves finding the kth least element in a given subarray. This problem is closely related to 2-D 3-sided orthogonal range counting. Recently, Jørgensen and Larsen [2011] presented a linear-space adaptive data structure with query time O(log log n+ log k/ log log n). We give a new linear-space structure that improves the query time to O(1 + log k/ log log n), exactly matching the lower bound proved by Jørgensen and Larsen.

Original languageEnglish (US)
Article number45
JournalACM Transactions on Algorithms
Volume12
Issue number4
DOIs
StatePublished - Sep 2016
Externally publishedYes

Keywords

  • Adaptive
  • Approximate
  • Computational geometry
  • Data structure
  • Orthogonal
  • Range counting
  • Word RAM

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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