Discrete & Computational Geometry

, Volume 2, Issue 2, pp 127–151 | Cite as

ɛ-nets and simplex range queries

  • David Haussler
  • Emo Welzl


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(n α ) 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.


Internal Node Range Query Query Time Finite Dimension Range Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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

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