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https://doi.org/10.1007/BF02187876

ɛ-nets and simplex range queries - Discrete & Computational Geometry

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



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ɛ-nets and simplex range queries - Discrete & Computational Geometry

https://doi.org/10.1007/BF02187876

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



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https://doi.org/10.1007/BF02187876

ɛ-nets and simplex range queries - Discrete & Computational Geometry

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

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      ɛ-nets and simplex range queries - Discrete & Computational Geometry
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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, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey% ypa0ZaaSaaaeaacaWGKbGaaiikaiaadsgacqGHsislcaaIXaGaaiyk% aaqaaiaadsgacaGGOaGaamizaiabgkHiTiaaigdacaGGPaGaey4kaS% IaaGymaaaacqGHRaWkcqaHZoWzieaacaWFGaGaa8hiaiaa-bcacaWF% GaGaa8hiaiaa-bcacaWFGaGaa8Nzaiaa-9gacaWFYbGaa8hiaiaa-f% gacaWFSbGaa8hBaiaa-bcacaWFGaGaa8hiaiabeo7aNjaa-5dacaWF% Waaaaa!574F! $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ .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.

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