# Two Proofs for Shallow Packings

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## Abstract

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let \(\mathcal {V}\) be a finite set system defined over an *n*-point set *X*; we view \(\mathcal {V}\) as a set of indicator vectors over the *n*-dimensional unit cube. A \(\delta \)-separated set of \(\mathcal {V}\) is a subcollection \(\mathcal {W}\), s.t. the Hamming distance between each pair \(\mathbf{u}, \mathbf{v}\in \mathcal {W}\) is greater than \(\delta \), where \(\delta > 0\) is an integer parameter. The \(\delta \)-packing number is then defined as the cardinality of a largest \(\delta \)-separated subcollection of \(\mathcal {V}\). Haussler showed an asymptotically tight bound of \(\Theta ((n/\delta )^d)\) on the \(\delta \)-packing number if \(\mathcal {V}\) has VC-dimension (or *primal shatter dimension*) *d*. We refine this bound for the scenario where, for any subset, \(X' \subseteq X\) of size \(m \le n\) and for any parameter \(1 \le k \le m\), the number of vectors of length at most *k* in the restriction of \(\mathcal {V}\) to \(X'\) is only \(O(m^{d_1} k^{d-d_1})\), for a fixed integer \(d > 0\) and a real parameter \(1 \le d_1 \le d\) (this generalizes the standard notion of *bounded primal shatter dimension* when \(d_1 = d\)). In this case when \(\mathcal {V}\) is “*k*-shallow” (all vector lengths are at most *k*), we show that its \(\delta \)-packing number is \(O(n^{d_1} k^{d-d_1}/\delta ^d)\), matching Haussler’s bound for the special cases where \(d_1=d\) or \(k=n\). We present two proofs, the first is an extension of Haussler’s approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler’s proof.

## Keywords

Packing lemma and shallow packing lemma Set systems of finite VC–dimension Primal shatter function Clarkson–Shor property## Mathematics Subject Classification

05B40 52C15 52C17 52C45 68R05## Notes

### Acknowledgments

We authors would like to thank two anonymous referees for their useful comments. The second author wishes to thank Boris Aronov, Sariel Har-Peled, Aryeh Kontorovich, and Wolfgang Mulzer for useful discussions and suggestions. Last but not least, the second author thanks Ramon Van Handel, for various discussions and for spotting an error in an earlier version of this paper. Work on this paper by Kunal Dutta and Arijit Ghosh has been supported by the Indo-German Max-Planck Center for Computer Science (IMPECS). Work on this paper by Esther Ezra has been supported by NSF Grants CCF-11-17336, CCF-12-16689, and NSF CAREER CCF-15-53354. A preliminary version of this paper appeared in *Proc. Sympos. Computational Geometry*, 2015, pp. 96–110 [8]

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