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A Simple Proof of Optimal Epsilon Nets

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References

  1. N. Alon: A non-linear lower bound for planar epsilon-nets, Discrete & Computational Geometry 47 2012, 235–244.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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. B. Chazelle: The Discrepancy Method: Randomness and Complexity, Cambridge University Press, 2000.

    Book  MATH  Google Scholar 

  7. B. Chazelle and J. Friedman: A deterministic view of random sampling and its use in geometry, Combinatorica 10 1990, 229–249.

    Article  MathSciNet  MATH  Google Scholar 

  8. K. L. Clarkson and P. W. Shor: Application of random sampling in computational geometry, II, Discrete & Computational Geometry 4 1989, 387–421.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. L. Clarkson and K. R. Varadarajan: Improved approximation algorithms for geometric set cover, Discrete & Computational Geometry 37 2007, 43–58.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Chapter  Google Scholar 

  12. S. Har-Peled, H. Kaplan, M. Sharir and S. Smorodinsky: Epsilon-nets for halfspaces revisited, CoRR, abs/1410.3154, 2014.

    Google Scholar 

  13. S. Har-Peled and M. Sharir: Relative (p, ϵ)-approximations in geometry, Discrete & Computational Geometry 45 2011, 462–496.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Haussler and E. Welzl: Epsilon-nets and simplex range queries, Discrete & Computational Geometry 2 1987, 127–151.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Komlós, J. Pach and G. Woeginger: Almost tight bounds for ϵ-nets, Discrete & Computational Geometry 7 1992, 163–173.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  18. J. Matoušek: On constants for cuttings in the plane, Discrete & Computational Geometry 20 1998, 427–448.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Matoušek: Geometric Discrepancy: An Illustrated Guide, Springer, 1999.

    Book  MATH  Google Scholar 

  20. J. Matoušek: Lectures in Discrete Geometry, Springer, 2002.

    Book  MATH  Google Scholar 

  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. J. Matoušek: Reporting points in halfspaces, Computational Geometry: Theory and Applications 2 1992, 169–186.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Matoušek: Derandomization in Computational Geometry, Journal of Algorithms 20 1996, 545–580.

    Article  MathSciNet  MATH  Google Scholar 

  24. N. H. Mustafa: A simple proof of the shallow packing lemma, Discrete & Computational Geometry 55 2016, 739–743.

    Article  MathSciNet  MATH  Google Scholar 

  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. J. Pach and P. K. Agarwal: Combinatorial Geometry, John Wiley & Sons, 1995.

    Book  MATH  Google Scholar 

  27. J. Pach and G. Tardos: Tight lower bounds for the size of epsilon-nets, Journal of the AMS 26 2013, 645–658.

    MathSciNet  MATH  Google Scholar 

  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. N. Sauer: On the density of families of sets, Journal of Combinatorial Theory, Series A 13 1972, 145–147.

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Shelah: A combinatorial problem, stability and order for models and theories in infinitary languages, Pacific Journal of Mathematics 41 1972, 247–261.

    MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  32. K. R. Varadarajan: Epsilon nets and union complexity, in: Proceedings of the 25th Annual ACM Symposium on Computational Geometry (SoCG), 11–16, 2009.

    Chapter  Google Scholar 

  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 

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Authors and Affiliations

  1. Université Paris-Est Laboratoire d’Informatique Gaspard-Monge, ESIEE Paris, Noisy-le-Grand, France

    Nabil H. Mustafa

  2. DataShape, INRIA Sophia Antipolis - Méditerranée Sophia Antipolis, Valbonne, France

    Kunal Dutta

  3. ACM Unit, Indian Statistical Institute, Kolkata, India

    Arijit Ghosh

Authors
  1. Nabil H. Mustafa
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  2. Kunal Dutta
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  3. Arijit Ghosh
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Corresponding author

Correspondence to Nabil H. Mustafa.

Additional information

The work of Nabil H. Mustafa in this paper has been supported by the grant ANR SAGA (JCJC-14-CE25-0016-01).

Kunal Dutta and Arijit Ghosh are supported by the European Research Council under the Advanced Grant 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions) and the Ramanujan Fellowship (No. SB/S2/RJN-064/2015) respectively. Part of this work was also done when Kunal Dutta and Arijit Ghosh were researchers in D1: Algorithms & Complexity, Max-Planck-Institute for Informatics, Germany, supported by the Indo-German Max Planck Center for Computer Science (IMPECS).

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Mustafa, N.H., Dutta, K. & Ghosh, A. A Simple Proof of Optimal Epsilon Nets. Combinatorica 38, 1269–1277 (2018). https://doi.org/10.1007/s00493-017-3564-5

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  • DOI: https://doi.org/10.1007/s00493-017-3564-5

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