Discrete & Computational Geometry

, Volume 1, Issue 1, pp 83–93

Halfspace range search: An algorithmic application ofk-sets

  • B. Chazelle
  • F. P. Preparata

DOI: 10.1007/BF02187685

Cite this article as:
Chazelle, B. & Preparata, F.P. Discrete Comput Geom (1986) 1: 83. doi:10.1007/BF02187685


Given a fixed setS ofn points inE3 and a query planeπ, the halfspace range search problem asks for the retrieval of all points ofS on a chosen side ofπ. We prove that withO(n(logn)8 (loglogn)4) storage it is possible to solve this problem inO(k+logn) time, wherek is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the total number ofj-sets (j=1, ...,k) realized by a set ofn points inE3 isO(nk5); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • B. Chazelle
    • 1
  • F. P. Preparata
    • 2
  1. 1.Department of Computer ScienceBrown UniversityProvidence
  2. 2.Coordinated Science LaboratoryUniversity of IllinoisUrbana-Champaign