Empty Axis-Parallel Boxes

Bukh, Boris; Chao, Ting-Wei

doi:10.1093/imrn/rnab123

Cited by 6 publications

(2 citation statements)

References 15 publications

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“…Depending on the relation between d and ε −1 , (1.1) can sometimes be improved. We refer to [7,26] for recent results in this direction as well as for further references. Let us stress, that we are mainly interested in the regime, where d is (much) larger than ε −1 and where the logarithmic dependence on d is crucial.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Deterministic Constructions of High-Dimensional Sets with Small Dispersion

Ullrich

Vybíral

2022

Algorithmica

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The dispersion of a point set $$P\subset [0,1]^d$$ P ⊂ [ 0 , 1 ] d is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect P. It was observed only recently that, for any $$\varepsilon >0$$ ε > 0 , certain randomized constructions provide point sets with dispersion smaller than $$\varepsilon $$ ε and number of elements growing only logarithmically in d. Based on deep results from coding theory, we present explicit, deterministic algorithms to construct such point sets in time that is only polynomial in d. Note that, however, the running-time will be super-exponential in $$\varepsilon ^{-1}$$ ε - 1 . Our construction is based on the apparently new insight that low-dispersion point sets can be deduced from solutions of certain k-restriction problems, which are well-known in coding theory.

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Section: Introduction and Resultsmentioning

confidence: 99%

Deterministic Constructions of High-Dimensional Sets with Small Dispersion

Ullrich

Vybíral

2022

Algorithmica

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“…It is known that 1.504 󰃑 lim n→∞ nA(n) 󰃑 1.895; see also [1,30,34,37]. The lower bound is a recent result of Bukh and Chao [6] and the upper bound is another recent result of Kritzinger and Wiart [31]. The upper bound ϕ 4 /(ϕ 2 + 1) = 1.8945 .…”

Section: Related Workmentioning

confidence: 93%

Piercing All Translates of a Set of Axis-Parallel Rectangles

Dumitrescu,

Tkadlec

2024

Electron. J. Combin.

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For a given shape $S$ in the plane, one can ask what is the lowest possible density of a point set $P$ that pierces ("intersects", "hits") all translates of $S$. This is equivalent to determining the covering density of $S$ and as such is well studied. Here we study the analogous question for families of shapes where the connection to covering is altered. That is, we require that a single point set $P$ simultaneously pierces each translate of each shape from some family $\mathcal{F}$. We denote the lowest possible density of such an $\mathcal{F}$-piercing point set by $\pi_T(\mathcal{F})$. Specifically, we focus on families $\mathcal{F}$ consisting of axis-parallel rectangles. When $|\mathcal{F}|=2$ we exactly solve the case when one rectangle is more squarish than $2\times 1$, and give bounds (within $10\,\%$ of each other) for the remaining case when one rectangle is wide and the other one is tall. When $|\mathcal{F}|\ge 2$ we present a linear-time constant-factor approximation algorithm for computing $\pi_T(\mathcal{F})$ (with ratio $1.895$).

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Piercing All Translates of a Set of Axis-Parallel Rectangles

Dumitrescu

Tkadlec

2021

Lecture Notes in Computer Science

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Empty Axis-Parallel Boxes

Cited by 6 publications

References 15 publications

Deterministic Constructions of High-Dimensional Sets with Small Dispersion

Deterministic Constructions of High-Dimensional Sets with Small Dispersion

Piercing All Translates of a Set of Axis-Parallel Rectangles

Piercing All Translates of a Set of Axis-Parallel Rectangles

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