Separating Points in a Rectangle

Schwartz, Benjamin L.

doi:10.2307/2689032

Cited by 8 publications

(6 citation statements)

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“…It is often equivalent to a circle packing problem. For a circular disc, results can be found in Coxeter et al (1968), Melissen (1994b) and Pirl (1969), for a square in Kirchner and Wengerodt (1987), Melissen (1994a), Nurmela &Östergård (1998), Peikert et al (1991), Schaer & Meir (1965), Schaer (1965), Schwartz (1970) and Wengerodt (1983Wengerodt ( , 1987a, and for an equilateral triangle, Melissen (1993Melissen ( , 1994b, Melissen & Schuur (1995) and Oler (1961). There is an enormous amount of literature dealing with the problem on a two-sphere in three-space where it is known as Tammes's problem (Tammes 1930), but exact solutions are known only in a few special cases.…”

Section: Introductionmentioning

confidence: 93%

How different can colours be? Maximum separation of points on a spherical octant

Melisseny

1998

Proc. R. Soc. Lond. A

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Section: Introductionmentioning

confidence: 93%

How different can colours be? Maximum separation of points on a spherical octant

Melisseny

1998

Proc. R. Soc. Lond. A

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“…The dispersal packing problem tries to maximize the radius of a given number of circles packed in a square. A lot of effort has been made in finding the optimal radius and the corresponding packing when the number of circles is a small constant; see [38,37,39,32]. Heuristic methods have also been used in finding approximations when the number of circles gets large [25,41].…”

Section: Related Workmentioning

confidence: 99%

Efficient Approximations for the Online Dispersion Problem

Chen

2019

SIAM J. Comput.

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The dispersion problem has been widely studied in computational geometry and facility location, and is closely related to the packing problem. The goal is to locate n points (e.g., facilities or persons) in a k-dimensional polytope, so that they are far away from each other and from the boundary of the polytope. In many real-world scenarios however, the points arrive and depart at different times, and decisions must be made without knowing future events. Therefore we study, for the first time in the literature, the online dispersion problem in Euclidean space.There are two natural objectives when time is involved: the all-time worst-case (ATWC) problem tries to maximize the minimum distance that ever appears at any time; and the cumulative distance (CD) problem tries to maximize the integral of the minimum distance throughout the whole time interval. Interestingly, the online problems are highly non-trivial even on a segment. For cumulative distance, this remains the case even when the problem is time-dependent but offline, with all the arriving and departure times given in advance.For the online ATWC problem on a segment, we construct a deterministic polynomialtime algorithm which is (2 ln 2 + ǫ)-competitive, where ǫ > 0 can be arbitrarily small and the algorithm's running time is polynomial in 1 ǫ . We show this algorithm is actually optimal. For the same problem in a square, we provide a 1.591-competitive algorithm and a 1.183 lower-bound. Furthermore, for arbitrary k-dimensional polytopes with k ≥ 2, we provide a 2 1−ǫ -competitive algorithm and a 7 6 lower-bound. All our lower-bounds come from the structure of the online problems and hold even when computational complexity is not a concern. Interestingly, for the offline CD problem in arbitrary k-dimensional polytopes, we provide a polynomial-time blackbox reduction to the online ATWC problem, and the resulting competitive ratio increases by a factor of at most 2. Our techniques also apply to online dispersion problems with different boundary conditions.

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“…Two are the main directions which research in this field has followed: the first deals with finding packings with proven optimality. This way proofs have been found for optimal packings of up to 9 disks , , Schwartz (1970)] and for some isolated larger n values (n = 14 ], n = 16 [Wengerodt (1983)], n = 25 ], n = 36 ]). A different research direction within this stream has aimed towards finding computer-aided proofs.…”

Section: Introductionmentioning

confidence: 98%

Disk Packing in a Square: A New Global Optimization Approach

Addis

Locatelli

Schoen

2008

INFORMS Journal on Computing

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We present a new computational approach to the problem of placing n identical non overlapping disks in the unit square in such a way that their radius is maximized. The problem has been studied in a large number of papers, both from a theoretical and from a computational point of view. In this paper we conjecture that the problem possesses a so-called funneling landscape, a feature which is commonly found in molecular conformation problems. Based upon this conjecture we develop a stochastic search algorithm which displays excellent numerical performance. Thanks to this algorithm we could improve over previously known putative optima in the range n ≤ 130 in as many as 32 instances, the smallest of which is n = 53. First experiments in the related problem of packing equal spheres in the unit cube led us to an improvement for n = 28 spheres.History:

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Separating Points in a Rectangle

Cited by 8 publications

References 0 publications

How different can colours be? Maximum separation of points on a spherical octant

How different can colours be? Maximum separation of points on a spherical octant

Efficient Approximations for the Online Dispersion Problem

Disk Packing in a Square: A New Global Optimization Approach

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