 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.96
URN: urn:nbn:de:0030-drops-51493
URL: https://drops.dagstuhl.de/opus/volltexte/2015/5149/
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### 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 V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A delta-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v in W is greater than delta, where delta > 0 is an integer parameter. The delta-packing number is then defined as the cardinality of the largest delta-separated subcollection of V. Haussler showed an asymptotically tight bound of Theta((n / delta)^d) on the delta-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X' of X of size m <= n and for any parameter 1 <= k <= m, the number of vectors of length at most k in the restriction of V to X' is only O(m^{d_1} k^{d-d_1}), for a fixed integer d > 0 and a real parameter 1 <= d_1 <= d (this generalizes the standard notion of bounded primal shatter dimension when d_1 = d). In this case when 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.

### BibTeX - Entry

```@InProceedings{dutta_et_al:LIPIcs:2015:5149,
author =	{Kunal Dutta and Esther Ezra and Arijit Ghosh},
title =	{{Two Proofs for Shallow Packings}},
booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
pages =	{96--110},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-83-5},
ISSN =	{1868-8969},
year =	{2015},
volume =	{34},
editor =	{Lars Arge and J{\'a}nos Pach},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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