Discrete & Computational Geometry

, Volume 2, Issue 2, pp 127-151

First online:

ɛ-nets and simplex range queries

  • David HausslerAffiliated withDepartment of Computer and Information Sciences, University of California at Santa Cruz
  • , Emo WelzlAffiliated withInstitutes for Information Processing, Technical University of Graz

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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.