On range searching with semialgebraic sets II

Pankaj K. Agarwal, Jiří Matoušek, Micha Sharir

Research output: Contribution to journalConference articlepeer-review

Abstract

Let P be a set of n points in ℝd. We present a linear-size data structure for answering range queries on P with constant-complexity semi algebraic sets as ranges, in time close to O(n1-1/d). It essentially matches the performance of similar structures for simplex range searching, and, for d ≥ 5, significantly improves earlier solutions by the first two authors obtained in~1994. This almost settles a long-standing open problem in range searching. The data structure is based on the polynomial-partitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter r, 1 < r ≤ n, there exists a d-variate polynomial f of degree O(r1/d) such that each connected component of ℝd \ Z(f) contains at most n/r points of P, where Z(f) is the zero set of f. We present an ef?cient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications.

Original languageEnglish
Article number6375320
Pages (from-to)420-429
Number of pages10
JournalProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
DOIs
StatePublished - 2012
Event53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States
Duration: 20 Oct 201223 Oct 2012

Keywords

  • Range searching
  • ham-sandwich cuts
  • polynomial partition
  • semialgebraic sets

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