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,
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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.
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Communicated by David Dobkin
D. Haussler gratefully acknowledges the support of ONR grant N00014-86-K-0454.
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Haussler, D., Welzl, E. ɛ-nets and simplex range queries. Discrete Comput Geom 2, 127–151 (1987). https://doi.org/10.1007/BF02187876
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DOI: https://doi.org/10.1007/BF02187876