## Abstract

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(m^{d}_{1}k^{d-d}_{1}) for fixed integers d > 0 and 1 ≤ d_{1} ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d_{1} = 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(and the follow-up work in [M. Sharir and S. Zaban, Output-Sensitive Tools for Range Searching in Higher Dimensions, unpublished manuscript, 2011; available online from www.cs.tau.ac.il/thesis/thesis/zaban.pdf]) for points and halfspaces in d-space for d ≥ 3.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 language | English |
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Pages (from-to) | 84-101 |

Number of pages | 18 |

Journal | SIAM Journal on Computing |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

### Bibliographical note

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

## Keywords

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