Jivri Matouaek
ABSTRACT
A finite set ? ? Rd is a weak e-net for an n -point set X ? R d (with respect to convex sets) if N intersects every convex set K with | K n X |= en. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al. [7], that every point set X in R d admits a weak e-net of cardinality O (e -d polylog(1/e)). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak eps-nets in time O ( n ln(1e)). We also prove, by a different method, a near-linear upper bound for points uniformly distributed on the (d--1)-dimensional sphere.
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- Noga Alon, Imre Barany, Zoltan Füredi, and Daniel Kleitman. Point selections and weak e-nets for convex hulls. Combin. Probab. Comput., 1(3):103--112, 1992.Google ScholarCross Ref
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- Bernard Chazelle, Herbert Edelsbrunner, David Eppstein, Michelangelo Grigni, Leonidas Guibas, Micha Sharir, and Emo Welzl. Algorithms for weak epsilon-nets. Manuscript, 1995, available at http://citeseer.nj.nec.com/24190.html.Google Scholar
- Bernard Chazelle, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas J. Guibas, Micha Sharir, and Emo Welzl. Improved bounds on weak e-nets for convex sets. Discrete Comput. Geom., 13:1--15, 1995.Google ScholarDigital Library
- Bernard Chazelle, Herbert Edelsbrunner, Leonidas Guibas, and Micha Sharir. A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. In Proc. 16th Internat. Colloq. Automata Lang. Program., volume 372 of Lecture Notes Comput. Sci., pages 179--192. Springer-Verlag, Berlin etc., 1989. Google ScholarDigital Library
- David Haussler and Emo Welzl. e-Nets and Simplex Range Queries. Discrete Comput. Geom., 2:127--151, 1987.Google ScholarDigital Library
- Vladlen Koltun. Almost tight upper bounds for vertical decompositions in four dimensions. In Proc. 42nd IEEE Symposium on Foundations of Computer Science, pages 56--65, 2001. Google ScholarDigital Library
- Jivri Matouaek. Efficient partition trees. Discrete Comput. Geom., 8(3):315--334, 1992.Google ScholarDigital Library
- Jivri Matouaek. Lectures on Discrete Geometry. Springer-Verlag, New York, 2002. Google ScholarDigital Library
- Jivri Matouaek. A lower bound for weak e-nets in high dimension. Discrete Comput. Geom., 28:45--48, 2002.Google ScholarCross Ref
Index Terms
- New constructions of weak epsilon-nets
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