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

Bernard Chazelle, Micha Sharir, Emo Welzl

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76 Scopus citations


This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a set P of n points in ℜd so that, given any query simplex q, the points in P ∩q can be counted or reported efficiently. If m units of storage are available (n <m <n d ), then we show that it is possible to answer any query in O(n 1+e{open}/m 1/d ) query time after O(m 1+e{open}) preprocessing. This bound, which holds on a RAM or a pointer machine, is almost tight. We also show how to achieve O(log n) query time at the expense of O(n d+e{open}) storage for any fixed e{open} > 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.

Original languageEnglish (US)
Pages (from-to)407-429
Number of pages23
Issue number1-6
StatePublished - Dec 1992
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Computer Science
  • Applied Mathematics
  • Computer Science Applications


  • Computational geometry
  • Range searching
  • Space-time tradeoff


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