, Volume 2, Issue 1, pp 127-151

ɛ-nets and simplex range queries

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Abstract

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.