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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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