A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension

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Let (X, S) be a set system on an n-point set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S ∈ S and two-colorings x on X. We consider the scenario where, for any subset X' ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S to X' of size at most k is only O(md1kd-d1) for fixed integers d > 0 and 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case we show that there exists a coloring x with discrepancy bound O∗(|S|1/2-d1/(2d)n(d1-1)/(2d)), for each S ∈ S, where O∗(·) hides a polylogarithmic factor in n. This bound is tight up to a polylogarithmic factor [J. Matoušek, Discrete Comput. Geom., 13 (1995), pp. 593-601, Geometric Discrepancy, Algorithms Combin. 18, Springer-Verlag, Heidelberg, 1999], and the corresponding coloring ? can be computed in expected polynomial time using the very recent machinery of Lovett and Meka [Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, 2012, pp. 61-67] for constructive discrepancy minimization. Our bound improves and generalizes the bounds obtained from the machinery of Har-Peled and Sharir [Discrete Comput. Geom, 45 (2011), pp. 462-496] (and the follow-up work in [M. Sharir and S. Zaban, Output-Sensitive Tools for Range Searching in Higher Dimensions, unpublished manuscript, 2011; available online from www.cs.tau.ac.il/thesis/thesis/zaban.pdf]) for points and halfspaces in d-space for d ≥ 3. Last but not least, we show that our bound yields improved bounds for the size of relative (ε, δ)-approximations for set systems of the above kind.

Original languageEnglish
Pages (from-to)84-101
Number of pages18
JournalSIAM Journal on Computing
Issue number1
StatePublished - 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016 Society for Industrial and Applied Mathematics.


  • Entropy
  • Geometric discrepancy
  • Partial coloring
  • Relative approximations
  • Set systems of bounded primal shatter dimension
  • δ-packing


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