Discrepancy and approximations for bounded VC-dimension

Abstract

Let (X, ℛ) be a set system on ann-point setX. For a two-coloring onX, itsdiscrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in ℛ. We show that if for anym-point subset\(Y \subseteq X\) the number of distinct subsets induced by ℛ onY is bounded byO(m d) for a fixed integerd, then there is a coloring with discrepancy bounded byO(n 1/2−1/2d(logn)1+1/2d). Also if any subcollection ofm sets of ℛ partitions the points into at mostO(m d) classes, then there is a coloring with discrepancy at mostO(n 1/2−1/2dlogn). These bounds imply improved upper bounds on the size of ε-approximations for (X, ℛ). All the bounds are tight up to polylogarithmic factors in the worst case. Our results allow to generalize several results of Beck bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    R. Alexander: Geometric methods in the theory of uniform distribution,Combinatorica 10 (1990), 115–136.

    Google Scholar 

  2. [2]

    J. Beck: Some upper bounds in the theory of irregularities of distribution,Acta Arith. 43 (1984), 115–130.

    Google Scholar 

  3. [3]

    J. Beck, andW. Chen:Irregularities of distribution, Cambridge University Press, 1987.

  4. [4]

    J. Beck: Quasi-random 2-colorings of point sets,Random Structures and Algorithms, to appear.

  5. [5]

    A. Blumer, A. Ehrenfeucht, D. Haussler, andM. Warmuth: Classifying learnable geometric concepts with the Vapnik-Chervonenkis dimension,Journal of the ACM 36 (1989), 929–965.

    Google Scholar 

  6. [6]

    K. L. Clarkson: Applications of random sampling in computational geometry,Discrete & Computational Geometry 2 (1987), 195–222.

    Google Scholar 

  7. [7]

    K. L. Clarkson: Applications of random sampling in computational geometry II, in:Proc. 4. ACM Symposium on Computational Geometry, 1988 1–11.

  8. [8]

    B. Chazelle, andE. Welzl: Quasi-optimal range searching in spaces of finite VC-dimension,Discrete & Computational Geometry 4 (1989), 467–490.

    Google Scholar 

  9. [9]

    E. Hlawka:The theory of uniform distribution. A B Academic, 1984.

  10. [10]

    D. Haussler, andE. Welzl: ε-nets and simplex range queries,Discrete & Computational Geometry 2 (1987), 127–151.

    Google Scholar 

  11. [11]

    J. Komlós, J. Pach, andG. Wöginger: Almost tight bounds for epsilon-nets.Discrete & Computational Geometry 1991, to appear.

  12. [12]

    J. Matoušek: Approximations and optimal geometric divide-and-conquer, in:Proc. 23. ACM Symposium on Theory of Computing, 1991, to appear.

  13. [13]

    J. Matoušek: Efficient partition trees, in:Proc. 7th ACM Symposium on Computational Geometry, 1991, to appear.

  14. [14]

    J. Pach, andG. Wöginger: Some new bounds for epsilon-nets, in:Proc. 6. ACM Symposium on Computational Geometry, 1990, 10–15.

  15. [15]

    N. Sauer: On the density of families of sets,Journal of Combin. Theory Ser. A,13, (1972) 145–147.

    Google Scholar 

  16. [16]

    R. Spencer:Ten lectures on the probabilistic method, CBMS-NSF, SIAM, 1987.

  17. [17]

    V. N. Vapnik, andA. Ya. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities,Theory Probab. Appl. 16 (1971), 264–280.

    Google Scholar 

  18. [18]

    E. Welzl: Partition trees for triangle counting and other range searching problems, in:Proc. 4. ACM Symposium on Computational Geometry, 1988, 23–33.

References

  1. [1]

    D. Haussler: Sphere packing numbers for subsets of the booleann-cube with bounded Vapnik-Chervonenkis dimension. Technical Report UCSU-CRL-91-41, University of California at Santa Cruz, 1991, to appear inJ. of Comb. Theory A.

  2. [2]

    L. Wernisch: Note on stabbing numbers and sphere packing numbers, manuscript, 1992.

Download references

Author information

Affiliations

Authors

Additional information

Work of J.M. and E.W. was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract no. 3075 (project ALCOM). L.W. acknowledges support from the Deutsche Forschungsgemeinschaft under grant We 1265/1-3, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Matoušek, J., Welzl, E. & Wernisch, L. Discrepancy and approximations for bounded VC-dimension. Combinatorica 13, 455–466 (1993). https://doi.org/10.1007/BF01303517

Download citation

AMS subject classification codes (1991)

  • 05 C 65
  • 52 C 99
  • 05 A 05
  • 52 C 10