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Discrete & Computational Geometry

, Volume 8, Issue 3, pp 315–334 | Cite as

Efficient partition trees

  • Jiří Matoušek
Article

Abstract

We prove a theorem on partitioning point sets inEd (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space,O(n logn) deterministic preprocessing time, andO(n1−1/d(logn)O(1)) query time. WithO(nlogn) preprocessing time, where δ is an arbitrary positive constant, a more complicated data structure yields query timeO(n1−1/d(log logn)O(1)). This attains the lower bounds due to Chazelle [C1] up to polylogarithmic factors, improving and simplifying previous results of Chazelleet al. [CSW].

The partition result implies that, forrdn1−δ, a (1/r)-approximation of sizeO(rd) with respect to simplices for ann-point set inEd can be computed inO(n logr) deterministic time. A (1/r)-cutting of sizeO(rd) for a collection ofn hyperplanes inEd can be computed inO(n logr) deterministic time, provided thatrn1/(2d−1).

Keywords

Computational Geometry Query Time Partition Scheme Partition Tree Query Answering 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Jiří Matoušek
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czechoslovakia

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