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Algorithmica

, Volume 8, Issue 1–6, pp 407–429 | Cite as

Quasi-optimal upper bounds for simplex range searching and new zone theorems

  • Bernard Chazelle
  • Micha Sharir
  • Emo Welzl
Article

Abstract

This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in ℜd so that, given any query simplexq, the points inPq can be counted or reported efficiently. Ifm units of storage are available (n <m <n d ), then we show that it is possible to answer any query inO(n 1+ɛ/m 1/d ) query time afterO(m 1+ɛ) preprocessing. This bound, which holds on a RAM or a pointer machine, is almost tight. We also show how to achieveO(logn) query time at the expense ofO(n d) storage for any fixed ɛ > 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.

Key words

Computational geometry Range searching Space-time tradeoff 

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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Bernard Chazelle
    • 1
  • Micha Sharir
    • 2
    • 3
  • Emo Welzl
    • 4
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael
  4. 4.Fachbereich MathematikFreie Universität BerlinBerlin 33Federal Republic of Germany

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