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$$\epsilon$$-nets and simplex range queries. (English) Zbl 0619.68056
The main problem may be described as follows: given a set of n points in d-dimensional Euclidean space, find a data structure that uses linear storage such that the number of points in any query half space can be determined in sublinear time $$O(n^{\alpha})$$. A data structure with $$\alpha =d(d-1)/(d(d-1)+1)+\gamma$$ for any $$\gamma >0$$ is exhibited. These bounds for $$\alpha$$ are better than those previously published for all $$d\geq 2$$ by A. Yao and F. Yao [A general approach to d- dimensional geometric queries. Proc. 17th Symp. Theory of Computing, 163- 169 (1985)].
Let X be a set and R be a set of subsets of X, which have a finite dimension in Vapnik-Chervonenkis sense [V. N. Vapnik and A. Ya. Chervonenkis: The theory of pattern recognition (Russian) (1974; Zbl 0284.68070)], A be a finite subset of X and $$0\leq \epsilon \leq 1$$. A subset N of A is an $$\epsilon$$-net of A (for R) if N contains a point in each $$r\in R$$ such that $$| A\cap r| /| A| >\epsilon$$. The authors prove that for $$0<\epsilon$$, $$\delta <1$$, if N is a subset of A obtained by $$m\geq \max (4/\epsilon \log 2/\delta,8d/\epsilon \log 8d/\epsilon)$$ random independent draws, then N is an $$\epsilon$$-net of A with probability at least 1-$$\delta$$. Using this result, a partition tree structure that achieves the above query time is constructed.
Reviewer: I.Molchanov

##### MSC:
 68P10 Searching and sorting 52A22 Random convex sets and integral geometry (aspects of convex geometry) 60C05 Combinatorial probability 05B99 Designs and configurations
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