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

Dutta, Kunal; Ghosh, Arijit; Jartoux, Bruno; Mustafa, Nabil H.

doi:10.1007/s00454-019-00075-0

Cited by 8 publications

(9 citation statements)

References 35 publications

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“…They constructed small -Mnets (i.e., such families with small ) for several classes of geometric hypergraphs. These results were extended by Dutta et al [19] using polynomial partitioning.…”

Section: Hypergraphs and Vc-dimensionmentioning

confidence: 67%

“…Hypergraphs with VC-dimension 1 admit O( 1 )-sized -nets [27] and -Mnets [19]. The latter fact yields the following result, albeit with worse constants.…”

Section: Hypergraphs Of Finite Vc-dimension Have Small -T-netsmentioning

confidence: 80%

“…In this case, the existence of an -net of size O( 1) was proved in [27]. The next theorem could be derived from results on Mnets [19], at the cost of poor multiplicative constants. Here we give a simpler proof for it.…”

Section: Small -Nets Small -2-netsmentioning

confidence: 95%

“…A specific case of the following result could also be derived from previous results on Mnets [19], if one adds the additional assumption that the regions have bounded "semi-algebraic description complexity". (The proof of [19] is involved and uses algebraic arguments).…”

Section: Small Union Complexitymentioning

confidence: 96%

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The $ε$-$t$-Net Problem

Alon,

Jartoux,

Keller

et al. 2020

Preprint

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We study a natural generalization of the classical -net problem (Haussler-Welzl 1987), which we call the -t-net problem: Given a hypergraph on n vertices and parameters t and ≥ t n , find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least n contains a set in S. When t = 1, this corresponds to the -net problem.We prove that any sufficiently large hypergraph with VC-dimension d admits an -t-net of size O( (1+log t)d log 1 ). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O( 1)-sized -t-nets.We also present an explicit construction of -t-nets (including -nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of -nets (i.e., for t = 1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.

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Section: Hypergraphs and Vc-dimensionmentioning

confidence: 67%

“…Hypergraphs with VC-dimension 1 admit O( 1 )-sized -nets [27] and -Mnets [19]. The latter fact yields the following result, albeit with worse constants.…”

Section: Hypergraphs Of Finite Vc-dimension Have Small -T-netsmentioning

confidence: 80%

Section: Small -Nets Small -2-netsmentioning

confidence: 95%

Section: Small Union Complexitymentioning

confidence: 96%

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The $ε$-$t$-Net Problem

Alon,

Jartoux,

Keller

et al. 2020

Preprint

Self Cite

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“…They have also been applied to several problems in the field of computational geometry. However, most previous results were either in the form of lower bounds [9,12,17] or focused on existential results [6,7,23,30].…”

Section: Techniquesmentioning

confidence: 99%

Near-Optimal $\varepsilon$-Kernel Construction and Related Problems

Arya,

da Fonseca,

Mount

2017

Preprint

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The computation of (i) ε-kernels, (ii) approximate diameter, and (iii) approximate bichromatic closest pair are fundamental problems in geometric approximation. In this paper, we describe new algorithms that offer significant improvements to their running times. In each case the input is a set of n points in R d for a constant dimension d ≥ 3 and an approximation parameter ε > 0. We reduce the respective running times), and (iii) from O(n/ε d/3 ) to O(n/ε d/4+α ), for an arbitrarily small constant α > 0. Result (i) is nearly optimal since the size of the output ε-kernel is Θ(1/ε (d−1)/2 ) in the worst case.These results are all based on an efficient decomposition of a convex body using a hierarchy of Macbeath regions, and contrast to previous solutions that decompose space using quadtrees and grids. By further application of these techniques, we also show that it is possible to obtain near-optimal preprocessing time for the most efficient data structures to approximately answer queries for (iv) nearest-neighbor searching, (v) directional width, and (vi) polytope membership.

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