A simple proof of optimal epsilon nets

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Mathematics Subject Classification (2010)

52C45 05D15 

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References

  1. [1]
    N. Alon: A non-linear lower bound for planar epsilon-nets, Discrete & Computational Geometry 47 2012, 235–244.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    B. Aronov, E. Ezra and M. Sharir: Small-size ϵ-nets for axis-parallel rectangles and boxes, SIAM Journal on Computing 39 2010, 3248–3282.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    N. Bus, S. Garg, N. H. Mustafa and S. Ray: Tighter estimates for ϵ-nets for disks, Computational Geometry: Theory and Applications 53 2016, 27–35.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    T. M. Chan, E. Grant, J. Könemann and M. Sharpe: Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling, in: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 1576–1585, 2012.CrossRefGoogle Scholar
  5. [5]
    B. Chazelle: A note on Haussler's packing lemma, 1992, See Section 5.3 from Geometric Discrepancy: An Illustrated Guide by J. Matoušek.Google Scholar
  6. [6]
    B. Chazelle: The Discrepancy Method: Randomness and Complexity, Cambridge University Press, 2000.CrossRefMATHGoogle Scholar
  7. [7]
    B. Chazelle and J. Friedman: A deterministic view of random sampling and its use in geometry, Combinatorica 10 1990, 229–249.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    K. L. Clarkson and P. W. Shor: Application of random sampling in computational geometry, II, Discrete & Computational Geometry 4 1989, 387–421.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    K. L. Clarkson and K. R. Varadarajan: Improved approximation algorithms for geometric set cover, Discrete & Computational Geometry 37 2007, 43–58.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    K. Dutta, E. Ezra and A. Ghosh: Two proofs for shallow packings, in: Proceedings of the 31st Annual Symposium on Computational Geometry (SoCG), 96–110, 2015.Google Scholar
  11. [11]
    E. Ezra: A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension, in: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 1378–1388, 2014.CrossRefGoogle Scholar
  12. [12]
    S. Har-Peled, H. Kaplan, M. Sharir and S. Smorodinsky: Epsilon-nets for halfspaces revisited, CoRR, abs/1410.3154, 2014.Google Scholar
  13. [13]
    S. Har-Peled and M. Sharir: Relative (p, ϵ)-approximations in geometry, Discrete & Computational Geometry 45 2011, 462–496.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    D. Haussler: Sphere packing numbers for subsets of the boolean n-cube with bounded Vapnik-Chervonenkis dimension, Journal of Combinatorial Theory, Series A 69 1995, 217–232.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    D. Haussler and E. Welzl: Epsilon-nets and simplex range queries, Discrete & Computational Geometry 2 1987, 127–151.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J. Komlós, J. Pach and G. Woeginger: Almost tight bounds for ϵ-nets, Discrete & Computational Geometry 7 1992, 163–173.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    A. Kupavskii, N. H. Mustafa and J. Pach: New lower bounds for epsilon-nets, in: Proceedings of the 32nd International Symposium on Computational Geometry, SoCG 2016, 54:1–54:16, 2016.Google Scholar
  18. [18]
    J. Matoušek: On constants for cuttings in the plane, Discrete & Computational Geometry 20 1998, 427–448.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    J. Matoušek: Geometric Discrepancy: An Illustrated Guide, Springer, 1999.CrossRefMATHGoogle Scholar
  20. [20]
    J. Matoušek: Lectures in Discrete Geometry, Springer, 2002.CrossRefMATHGoogle Scholar
  21. [21]
    J. Matoušek, R. Seidel and E. Welzl: How to net a lot with little: Small epsilonnets for disks and halfspaces, in: Proceedings of the 6th Annual ACM Symposium on Computational Geometry (SoCG), 16–22, 1990.Google Scholar
  22. [22]
    J. Matoušek: Reporting points in halfspaces, Computational Geometry: Theory and Applications 2 1992, 169–186.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    J. Matoušek: Derandomization in Computational Geometry, Journal of Algorithms 20 1996, 545–580.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    N. H. Mustafa: A simple proof of the shallow packing lemma, Discrete & Computational Geometry 55 2016, 739–743.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    N. H. Mustafa and S. Ray: Near-optimal generalisations of a theorem of Macbeath, in: Proceedings of the 31st Symposium on Theoretical Aspects of Computer Science (STACS), 578–589, 2014.Google Scholar
  26. [26]
    J. Pach and P. K. Agarwal: Combinatorial Geometry, John Wiley & Sons, 1995.CrossRefMATHGoogle Scholar
  27. [27]
    J. Pach and G. Tardos: Tight lower bounds for the size of epsilon-nets, Journal of the AMS 26 2013, 645–658.MathSciNetMATHGoogle Scholar
  28. [28]
    E. Pyrga and S. Ray: New existence proofs for ϵ-nets, in: Proceedings of the 24th Annual ACM Symposium on Computational Geometry (SoCG), 199–207, 2008.Google Scholar
  29. [29]
    N. Sauer: On the density of families of sets, Journal of Combinatorial Theory, Series A 13 1972, 145–147.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    S. Shelah: A combinatorial problem, stability and order for models and theories in infinitary languages, Pacific Journal of Mathematics 41 1972, 247–261.MATHGoogle Scholar
  31. [31]
    V. N. Vapnik and A. Y. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and its Applications 16 1971, 264–280.CrossRefMATHGoogle Scholar
  32. [32]
    K. R. Varadarajan: Epsilon nets and union complexity, in: Proceedings of the 25th Annual ACM Symposium on Computational Geometry (SoCG), 11–16, 2009.CrossRefGoogle Scholar
  33. [33]
    K. R. Varadarajan: Weighted geometric set cover via quasi-uniform sampling, in: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC), 641–648, 2010.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Université Paris-Est Laboratoire d’Informatique Gaspard-MongeESIEEParisFrance
  2. 2.DataShapeINRIA Sophia Antipolis - Méditerranée SophiaAntipolisFrance
  3. 3.ACM UnitIndian Statistical InstituteKolkataIndia

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