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Halfspace range search: an algorithmic application of K-sets

Authors:

Bernard Chazelle,

Franco P. PreparataAuthors Info & Claims

SCG '85: Proceedings of the first annual symposium on Computational geometry

Pages 107 - 115

https://doi.org/10.1145/323233.323248

Published: 01 June 1985 Publication History

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Abstract

Given a fixed set S of n points in E³ and a query plane π, the halfspace range search problem asks for the retrieval of all points of S on a chosen side of π. We prove that with Ο(n(log n)³(log log n)⁴) storage it is possible to solve this problem in Ο(κ + log n) time, where κ is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the maximum number of κ-sets realized by a set of n points in E³ is Ogr;(nk^c) for a small positive constant c; a κ-set is any subset of S of size κ which can be separated from the rest of S by a plane. Incidentally, this result constitutes the only nontrivial upper bound, as a function of n and κ, known to date.

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Index Terms

Halfspace range search: an algorithmic application of K-sets
1. Mathematics of computing
  1. Discrete mathematics
2. Theory of computation
  1. Design and analysis of algorithms
    1. Data structures design and analysis
      1. Sorting and searching
  2. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

SCG '85: Proceedings of the first annual symposium on Computational geometry

June 1985

322 pages

ISBN:0897911636

DOI:10.1145/323233

Chairman:
Joseph O'Rourke
Johns Hopkins Univ., Baltimore, MD

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Association for Computing Machinery

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Publication History

Published: 01 June 1985

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SCG85

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SCG85: The 1st Annual ACM SIGACT/SIGGRAPH Symposium on Computational Geometry 1985

June 5 - 7, 1985

Maryland, Baltimore, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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