A Simple Proof of the Shallow Packing Lemma

Mustafa, Nabil H.

doi:10.1007/s00454-016-9767-5

Cited by 12 publications

(20 citation statements)

References 6 publications

(8 reference statements)

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“…In case the reader is wondering if perhaps all the 'hard work' is hidden in the proof of shallow packings, we reproduce an elementary proof of a more general statement due to Mustafa [21], which shows that shallow packings follow directly from packings (Theorem D).…”

Section: Our Resultsmentioning

confidence: 99%

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A Simple Proof of Optimal Epsilon Nets

2017

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Showing the existence of small-sized-nets has been the subject of investigation for almost 30 years, starting from the breakthrough of Haussler and Welzl (1987). Following a long line of successive improvements, recent results have settled the question of the size of the smallest-nets for set systems as a function of their so-called shallow cell complexity. In this paper we give a short proof of this theorem in the space of a few elementary paragraphs. This immediately implies all known cases of results on unweighted-nets studied for the past 28 years, starting from the result of Matousek, Seidel and Welzl (SoCG 1990) to that of Clarkson and Varadajan (DCG 2007) to that of Varadarajan (STOC 2010) and Chan et al. (SODA 2012) for the unweighted case, as well as the technical and intricate paper of Aronov et al. (SIAM Journal on Computing, 2010). We find it quite surprising that such a simple elementary approach was missed in all earlier work, as all ingredients were already known in 1991.

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Section: Our Resultsmentioning

confidence: 99%

“…Such a P, where |∆(R, S)| ≥ δ for every R, S ∈ P, is called a δ-packing. In fact, a careful examination of the proof yields a stronger statement (this was later realized and formulated in [21]):…”

Section: Packing Lemmamentioning

confidence: 93%

A Simple Proof of Optimal Epsilon Nets

2017

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“…We note that after the submission of this paper we found a third proof by Mustafa [Mus16], based on the simplification of Chazelle's proof [Cha92] and combining Markov's inequality. While we find this proof simpler than ours, we emphasize the fact that the problem of shallow packings has initially been posed by the second author in [Ezr16], and the preliminary version of this paper [DEG15] is the first to resolve it (in the sense of Theorem 4).…”

Section: Our Resultsmentioning

confidence: 99%

Two Proofs for Shallow Packings

Dutta

Ezra

Ghosh

2016

Discrete Comput Geom

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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 δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v ∈ W is greater than δ, where δ > 0 is an integer parameter. The δ-packing number is then defined as the cardinality of a largest δ-separated subcollection of V. Haussler showed an asymptotically tight bound of Θ((n/δ) d) on the δ-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X ⊆ 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 d1 k d−d1), 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 δ-packing number is O(n d1 k d−d1 /δ 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.

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“…A set system (X, R) is an l-wise k-shallow δ-packing if it is an l-wise δ-packing and furthermore, |S| ≤ k, ∀S ∈ R. Building on the proof in [Mat99] and [Mus16], we prove the following, which simultaneously generalizes three theorems: that of Haussler [Hau95] Theorem 5 (l-Wise k-Shallow δ-Packing Lemma). Let (X, R) be a set system with |X| = n. Let d, k, l, δ > 0 be four integers such that VC-dim(R) ≤ d, and R is an l-wise k-shallow δ-packing.…”

Section: L-wise K-shallow δ-Packings (Proof In Section 5)mentioning

confidence: 88%

“…Recent efforts have been devoted to finding generalizations of the packing lemma to these finer classifications of set systems. For integers k and δ, call (X, R) a k-shallow δ-packing if R is a δ-packing, and |S| ≤ k for all S ∈ R. After an earlier bound [Ezr14], the following shallow packing lemma has been recently established in [DEG15,Mus16].…”

Section: Shallow Packing Lemmamentioning

confidence: 99%

Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

Dutta

Ghosh

Jartoux

et al. 2019

Discrete Comput Geom

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The packing lemma of Haussler states that given a set system (X, R) with bounded VC dimension, if every pair of sets in R are 'far apart' (i.e., have large symmetric difference), then R cannot contain too many sets. This has turned out to be the technical foundation for many results in geometric discrepancy using the entropy method (see [Mat99] for a detailed background) as well as recent work on set systems with bounded VC dimension [FPS + ar]. Recently it was generalized to the shallow packing lemma [DEG15, Mus16], applying to set systems as a function of their shallow cell complexity. In this paper we present several new results and applications related to packings: 1. an optimal lower bound for shallow packings, thus settling the open question in Ezra (SODA 2016) and Dutta et al. (SoCG 2015). 2. improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry (Annals of Mathematics, 1952). 3. simplifying and generalizing the main technical tool in Fox et al. (J. of the EMS, 2016).Besides using the packing lemma and a combinatorial construction, our proofs combine tools from polynomial partitioning and the probabilistic method.

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A Simple Proof of the Shallow Packing Lemma

Abstract: International audienceWe show that the shallow packing lemma follows from a simple modification of the standard proof, due to Haussler and simplified by Chazelle, of the packing lemma

Cited by 12 publications

References 6 publications

A Simple Proof of Optimal Epsilon Nets

A Simple Proof of Optimal Epsilon Nets

Two Proofs for Shallow Packings

Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

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