A Non-linear Lower Bound for Planar Epsilon-Nets

Alon, Noga

doi:10.1109/focs.2010.39

Cited by 25 publications

(44 citation statements)

References 36 publications

(24 reference statements)

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“…This consequence of Alon's theorem had been proved earlier, using the Hales-Jewett theorem [PaTT09]. Alon [Al10] proved that the same example also disproves that all range spaces consisting of straight-line ranges in the plane admit ε-nets of size O(1/ε).…”

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confidence: 84%

“…To choose the parameters d and r for a given ε we set d = ⌊log Once we have one example of a range space Σ = (X, R) that admits no small ε-net for a given value of ε, we can create arbitrarily large examples with the same property, by replacing each point p ∈ X with t new points, contained in the same ranges of R. (The same trick was applied in [Al10] and in the proof of Theorem 1.) This completes the proof of Theorem 4.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For the last two decades, "the prevailing conjecture" was that in "geometric scenarios," this bound is essentially tight: there always exists an ε-net of size O(1/ε) (see, e.g., [MaSW90,ArES10]). This conjecture had to be revised after Alon [Al10] discovered some geometric range spaces of small VC-dimension, in which the ranges are straight lines, rectangles or infinite strips in the plane, and which do not admit ε-nets of size O(1/ε). Alon's construction is based on the density version of the Hales-Jewett theorem [HaJ63], due to Furstenberg and Katznelson [FuK89,FuK91], and recently improved by participants of the Polymath blog project [Po09].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Alon [Al10] proved that there are n-element point sets X ⊂ R 2 and straight-line ranges that do not admit ε-nets of size O(1/ε). The standard duality between points and lines preserves incidences.…”

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confidence: 99%

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Tight lower bounds for the size of epsilon-nets

Pach¹,

Tardos²

2012

J. Amer. Math. Soc.

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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size O 1 ε log 1 ε . Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VCdimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1 ε , were found by Alon (2010). In his examples, the size of the smallest ε-nets is Ω 1 ε g( 1 ε ) , where g is an extremely slowly growing function, closely related to the inverse Ackermann function.We show that there exist geometrically defined range spaces, already of VCdimension 2, in which the size of the smallest ε-nets is Ω 1 ε log 1 ε . We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is Ω 1 ε log log 1 ε . By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

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mentioning

confidence: 84%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Tight lower bounds for the size of epsilon-nets

Pach¹,

Tardos²

2012

J. Amer. Math. Soc.

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“…This problem received a considerable amount of attention over the years, until it has finally been answered negatively in [5] and in [63], by constructions that have essential probabilistic ingredients. The following, however, is still open.…”

Section: Combinatorial Geometrymentioning

confidence: 99%

Paul Erdős and Probabilistic Reasoning

Alon

2013

Bolyai Society Mathematical Studies

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One of the major contributions of Paul Erdős is the development of the Probabilistic Method and its applications in Combinatorics, Graph Theory, Additive Number Theory and Combinatorial Geometry. This short paper describes some of the beautiful applications of the method, focusing on the long-term impact of the work, questions and results of Erdős. This is mostly a survey, but it contains a few novel results as well. The Probabilistic MethodThe Probabilistic Method is one of the most significant contributions of Paul Erdős, and part of his greatness is the fact that applications of the probabilistic method and of random graphs have become so common that it is now possible to use those without explicitly mentioning him. The method is a powerful tool with numerous applications in Combinatorics, Graph theory, Additive Number Theory and Geometry and had an immense impact on the development of theoretical Computer Science as well. The results and tools are far too numerous to cover in a short survey, even if the focus is only on those influenced directly by the work and problems of Erdős, and thus this paper is mainly a selection of topics that illustrate the method, and is not meant to be a comprehensive treatment of the whole area. Several books that contain more material on the subject are [13], [18], [54], [61].It is convenient to classify the applications of probabilistic techniques in Discrete Mathematics into three groups. The first one deals with the study of random combinatorial objects, like random graphs or random matrices. The results here are essentially results in Probability Theory, although many of them are motivated by problems in Combinatorics. The second group consists of probabilistic constructions. These are applications of probabilistic arguments in order to prove the existence of combinatorial structures which satisfy a list of prescribed properties. Existence proofs of this type often supply extremal examples to various questions in Discrete Mathematics. The third group, which contains some of the most striking examples, focuses on the application of probabilistic reasoning in the proofs of deterministic statements whose formulation does not give any indication that randomness may be helpful in their study.

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