Minimal Covering of the Space by Domino Tiles

Hoffmann, Rolf; Désérable, Dominique; Seredyński, Franciszek

doi:10.1007/978-3-030-86359-3_35

Cited by 4 publications

(7 citation statements)

References 11 publications

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“…In the most regular 9-point pattern the distance between points is 2, and the cover level distribution is 1 9+0 2 18 3 0 4 9 . The last two 9-point patterns are similar under rotation and shift, and their cover level distribution both is 1 9+0 2 12 17a. The number of points range from 7 to 10.…”

Section: × 6 Patternsmentioning

confidence: 92%

“…Rule A works also fine for other tile shapes, like dominoes [9,11,12]. Depending on the shape, Rule A may produce patterns with uncovered cells (gaps).…”

Section: Additional Rule B: Deleting Gapsmentioning

confidence: 99%

“…This new approach was successfully applied to place a maximal number of dominoes in a 2D field [9], to find a sensor point pattern (to cover an area by active sensors) [10], to place a maximal number of dominoes in the diamond [11], and to cover a space with a minimal number of dominoes [12].…”

Section: Introductionmentioning

confidence: 99%

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Forming Point Patterns by a Probabilistic Cellular Automata Rule

Hoffmann¹

2022

Preprint

Self Cite

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The objective is to find a Cellular Automata rule that can form a 2D point pattern with a maximum number of points (1-cells). Points are not allowed to touch each other, they have to be separated by 0-cells, and every 0-cell can find at least one point in its Moore-neighborhood. Probabilistic rules are designed that can solve this task with asynchronous updating and cyclic boundary condition. The task is considered as a tiling problem, where point tiles are used to cover the space with overlaps. A point tile consists of a center pixel (the kernel with value 1) and 8 surrounding pixels forming the hull with value 0. The term pixel is used to distinguish the cells of a tile from the cells of a cellular automaton. For each of the 9 tile pixels a so-called template is defined by a shift of the point tile. In the rule application, the 9 templates are tested at the actual cell position. If all template pixels (except the central reference pixel) of a template match with the corresponding neighbors of the actual cell under consideration, the cell's state is adjusted to the reference pixel's value. Otherwise the cell is set to the random value 0 or 1 with a certain probability. The hull pixels are allowed to overlap. In order to evolve a maximum of points, the overlap between tiles has to be maximized. To do that, the number of template hits is counted. Depending on the hit-number, additional noise is injected with certain probabilities. Thereby optimal patterns with the maximum number of points can be evolved. The behavior and performance of the designed rules is evaluated for different parameter settings.

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Section: × 6 Patternsmentioning

confidence: 92%

“…Rule A works also fine for other tile shapes, like dominoes [9,11,12]. Depending on the shape, Rule A may produce patterns with uncovered cells (gaps).…”

Section: Additional Rule B: Deleting Gapsmentioning

confidence: 99%

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Forming Point Patterns by a Probabilistic Cellular Automata Rule

Hoffmann¹

2022

Preprint

Self Cite

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“…Then, a probabilistic CA rule was used to generate point patterns [1], sensor point patterns [3], and domino patterns [2,5]. It was possible to aim at a maximal or a minimal number of dominos by injecting noise depending on the level of tile overlapping.…”

Section: Related Workmentioning

confidence: 99%

“…This paper is a good basis for further studies and extensions to the n-dimensional case. There are a lot of applications for overlapping tilings, like the dense parking of cars in a parking lot or constructing a sieve for maximal particle flow (using domino tiles) [5], optimizing task scheduling (overlapping communication and computation) [12], or building nanostructures [13].…”

Section: Related Workmentioning

confidence: 99%

Generating Loop Patterns with a Genetic Algorithm and a Probabilistic Cellular Automata Rule

Hoffmann

2023

Algorithms

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The objective is to find a Cellular Automata (CA) rule that can generate “loop patterns”. A loop pattern is given by ones on a zero background showing loops. In order to find out how loop patterns can be locally defined, tentative loop patterns are generated by a genetic algorithm in a preliminary stage. A set of local matching tiles is designed and checked whether they can produce the aimed loop patterns by the genetic algorithm. After having approved a certain set of tiles, a probabilistic CA rule is designed in a methodical way. Templates are derived from the tiles, which then are used in the CA rule for matching. In order to drive the evolution to the desired patterns, noise is injected if the templates do not match or other constraints are not fulfilled. Simulations illustrate that loops and connected loops can be evolved by the CA rule.

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Expanding the Cellular Automata Topologies Library for Parallel Implementation of Synchronous Cellular Automata

Medvedev

Kireev

2023

Lecture Notes in Computer Science

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Minimal Covering of the Space by Domino Tiles

Cited by 4 publications

References 11 publications

Forming Point Patterns by a Probabilistic Cellular Automata Rule

Forming Point Patterns by a Probabilistic Cellular Automata Rule

Generating Loop Patterns with a Genetic Algorithm and a Probabilistic Cellular Automata Rule

Expanding the Cellular Automata Topologies Library for Parallel Implementation of Synchronous Cellular Automata

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