Discrete & Computational Geometry

, Volume 2, Issue 2, pp 127–151

ɛ-nets and simplex range queries

  • David Haussler
  • Emo Welzl

DOI: 10.1007/BF02187876

Cite this article as:
Haussler, D. & Welzl, E. Discrete Comput Geom (1987) 2: 127. doi:10.1007/BF02187876


We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,d≥2, with linear storage andO(nα) query time,
$$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$

These bounds are better than those previously published for alld≥2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an ɛ-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an ɛ-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • David Haussler
    • 1
  • Emo Welzl
    • 2
  1. 1.Department of Computer and Information SciencesUniversity of California at Santa CruzSanta CruzUSA
  2. 2.Institutes for Information ProcessingTechnical University of GrazGrazAustria