On range searching with semialgebraic sets

  • Pankaj K. Agarwal
  • Jiří Matoušek
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 629)


Let P be a set of n points in ℝd (d a small fixed positive integer), and let Γ be a collection of subsets of ℝd, each of which is defined by a constant number of bounded degree polynomials. The Γ-range searching problem is defined as: Preprocess P into a data structure, so that all points of P lying in a given γ Γ can be counted (or reported) efficiently. Generalizing the simplex range searching techniques, we construct a data structure for Γ-range searching with nearly linear space and preprocessing time, which can answer a query in time O(n 1−1/b+δ ), where d≤b≤ 2d−3 and δ>0 is an arbitrarily small constant. The actual value of b is related to the problem of partitioning arrangements of algebraic surfaces into constant-complexity cells.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Jiří Matoušek
    • 2
    • 3
  1. 1.Computer Science DepartmentDuke UniversityDurham
  2. 2.Katedra aplikované matamatikyUniversita KarlovaPraha
  3. 3.Institut für InformatikFreie Universität BerlinGermany

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