Adaptive and approximate orthogonal range counting

Timothy M. Chan, Bryan T. Wilkinson

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

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 worst-case optimal query time O(logw n). We give an O(n log log n)-space adaptive data structure that improves the query time to O(log log n + logw, k), 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, Larsen, and Pǎtraş cu, SoCG 2011]. • We give an O(n log log n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1 + δ)-factor approximation to the count in O(log log n) time for any fixed constant δ > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem. • Lastly, 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 [SODA 2011] presented a linear-space adaptive data structure with query time O(log log n + logw k). We give a new linear-space structure that improves the query time to O(1 + logw k), exactly matching the lower bound proved by Jørgensen and Larsen.

Original languageEnglish (US)
Title of host publicationProceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
Pages241-251
Number of pages11
StatePublished - Apr 16 2013
Externally publishedYes
Event24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States
Duration: Jan 6 2013Jan 8 2013

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
CountryUnited States
CityNew Orleans, LA
Period1/6/131/8/13

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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