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Some new bounds for Epsilon-nets

Authors:

János Pach,

Gerhard WoegingerAuthors Info & Claims

SCG '90: Proceedings of the sixth annual symposium on Computational geometry

Pages 10 - 15

https://doi.org/10.1145/98524.98529

Published: 01 May 1990 Publication History

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Abstract

Given any natural number d, 0 < ε < 1, let ƒ_d(ε) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension d has an ε-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if d ≥ 2, then ƒ_d(ε) > 1/48 d/ε log 1/ ε which is not far from being optimal, if d is fixed and ε → 0. Further, we prove that ƒ₁(ε) = max(2,⌈1/ε⌉ - 1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.

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Elbassioni K(2023)A Bicriteria Approximation Algorithm for the Minimum Hitting Set Problem in Measurable Range SpacesOperations Research Letters10.1016/j.orl.2023.07.005Online publication date: Aug-2023
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Bertschinger DEl Maalouly NKleist LMiltzow TWeber S(2023)The Complexity of Recognizing Geometric HypergraphsGraph Drawing and Network Visualization10.1007/978-3-031-49272-3_12(163-179)Online publication date: 20-Sep-2023
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Index Terms

Some new bounds for Epsilon-nets
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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SCG '90: Proceedings of the sixth annual symposium on Computational geometry

May 1990

371 pages

ISBN:0897913620

DOI:10.1145/98524

Chairman:
Raimund Seidel

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 May 1990

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June 7 - 9, 1990

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Elbassioni K(2023)A Bicriteria Approximation Algorithm for the Minimum Hitting Set Problem in Measurable Range SpacesOperations Research Letters10.1016/j.orl.2023.07.005Online publication date: Aug-2023
https://doi.org/10.1016/j.orl.2023.07.005
Bertschinger DEl Maalouly NKleist LMiltzow TWeber S(2023)The Complexity of Recognizing Geometric HypergraphsGraph Drawing and Network Visualization10.1007/978-3-031-49272-3_12(163-179)Online publication date: 20-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-49272-3_12
Dumitrescu AGhosh ATóth C(2021)Sparse hop spanners for unit disk graphsComputational Geometry10.1016/j.comgeo.2021.101808(101808)Online publication date: Jul-2021
https://doi.org/10.1016/j.comgeo.2021.101808
Bus NGarg SMustafa NRay S(2016)Tighter estimates for Ε-nets for disksComputational Geometry: Theory and Applications10.1016/j.comgeo.2015.12.00253:C(27-35)Online publication date: 1-Feb-2016
https://dl.acm.org/doi/10.1016/j.comgeo.2015.12.002
Alon N(2012)A Non-linear Lower Bound for Planar Epsilon-netsDiscrete & Computational Geometry10.5555/3115956.311602847:2(235-244)Online publication date: 1-Mar-2012
https://dl.acm.org/doi/10.5555/3115956.3116028
Pach JTardos GHurtado Fvan Kreveld M(2011)Tight lower bounds for the size of epsilon-netsProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998271(458-463)Online publication date: 13-Jun-2011
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Alon N(2011)A Non-linear Lower Bound for Planar Epsilon-netsDiscrete & Computational Geometry10.1007/s00454-010-9323-747:2(235-244)Online publication date: 5-Jan-2011
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Pach JAgarwal P(2011)BibliographyCombinatorial Geometry10.1002/9781118033203.biblio(319-341)Online publication date: 28-Oct-2011
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Mustafa NRay S(2010)Improved Results on Geometric Hitting Set ProblemsDiscrete & Computational Geometry10.5555/3115964.311610144:4(883-895)Online publication date: 1-Dec-2010
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Alon N(2010)A Non-linear Lower Bound for Planar Epsilon-NetsProceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science10.1109/FOCS.2010.39(341-346)Online publication date: 23-Oct-2010
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Epsilon nets and union complexity

Tight lower bounds for the size of epsilon-nets

Almost tight bounds forź-Nets

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Epsilon nets and union complexity

Tight lower bounds for the size of epsilon-nets

Almost tight bounds forź-Nets

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