Online Square Packing with Gravity

Fekete, Sándor P.; Kamphans, Tom; Schweer, Nils

doi:10.1007/s00453-012-9713-8

Cited by 6 publications

(10 citation statements)

References 24 publications

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“…Proof. Our proof for the constant upper bound is very similar to a proof presented in [4] to show that the online BottomLeft algorithm has approximation ratio 3.5. The main idea is to prove that the holes in a bottom-left packing can be covered by surrounding squares.…”

Section: Bounds For the Worst-order Bottom-left Packingmentioning

confidence: 52%

“…The main idea is to prove that the holes in a bottom-left packing can be covered by surrounding squares. Contrary to [4] we do not aim to get the best possible constant. This makes our proof shorter than the one presented in [4] at the cost of getting a worse constant (16 instead of 3.5).…”

Section: Bounds For the Worst-order Bottom-left Packingmentioning

confidence: 99%

“…Contrary to [4] we do not aim to get the best possible constant. This makes our proof shorter than the one presented in [4] at the cost of getting a worse constant (16 instead of 3.5). We present the details of our proof for the upper bound in the appendix.…”

Section: Bounds For the Worst-order Bottom-left Packingmentioning

confidence: 99%

See 2 more Smart Citations

On the nearest neighbor rule for the metric traveling salesman problem

Hougardy

Mirko

2015

Discrete Applied Mathematics

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We present a very simple family of traveling salesman instances with n cities where the nearest neighbor rule may produce a tour that is Θ(log n) times longer than an optimum solution. Our family works for the graphic, the euclidean, and the rectilinear traveling salesman problem at the same time. It improves the so far best known lower bound in the euclidean case and proves for the first time a lower bound in the rectilinear case.keywords: traveling salesman problem; nearest neighbor rule; approximation algorithm

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Section: Bounds For the Worst-order Bottom-left Packingmentioning

confidence: 52%

Section: Bounds For the Worst-order Bottom-left Packingmentioning

confidence: 99%

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On the nearest neighbor rule for the metric traveling salesman problem

Hougardy

Mirko

2015

Discrete Applied Mathematics

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“…The online version of square packing has been studied by Januszewski and Lassak [30] and Han et al [31], with more recent progress due to Fekete and Hoff-mann [32,33]. A different kind of online square packing was considered by Fekete et al [34,35]. The container is an unbounded strip, into which objects enter from above in a Tetrislike fashion; any new object must come to rest on a previously placed object, and the path to its final destination must be collision-free.…”

Section: Related Workmentioning

confidence: 99%

An efficient data structure for dynamic two-dimensional reconfiguration

Fekete

Reinhardt

Scheffer

2017

Journal of Systems Architecture

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In the presence of dynamic insertions and deletions into a partially reconfigurable FPGA, fragmentation is unavoidable. This poses the challenge of developing efficient approaches to dynamic defragmentation and reallocation. One key aspect is to develop efficient algorithms and data structures that exploit the two-dimensional geometry of a chip, instead of just one. We propose a new method for this task, based on the fractal structure of a quadtree, which allows dynamic segmentation of the chip area, along with dynamically adjusting the necessary communication infrastructure. We describe a number of algorithmic aspects, and present different solutions. We also provide a number of basic simulations that indicate that the theoretical worst-case bound may be pessimistic.

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“…A different kind of online square packing was considered by Fekete et al [21,22]. The container is an unbounded strip, into which objects enter from above in a Tetris-like fashion; any new object must come to rest on a previously placed object, and the path to its final destination must be collision-free.…”

Section: Related Workmentioning

confidence: 99%

Online Square-into-Square Packing

Fekete

Hoffmann

2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects by decreasing size, and packing them greedily into shelves. Since then, the online version of the problem has remained open; the best upper bound is still 1/2, while the currently best lower bound is 1/3, due to Han et al. (2008). In this paper, we present a new lower bound of 11/32, based on a dynamic shelf allocation scheme, which may be interesting in itself.We also give results for the closely related problem in which the size of the square container is not fixed, but must be dynamically increased in order to accommodate online sequences of objects. For this variant, we establish an upper bound of 3/7 for the critical density, and a lower bound of 1/8. When aiming for accommodating an online sequence of squares, this corresponds to a 2.82 . . .competitive method for minimizing the required container size, and a lower bound of 1.33 . . . for the achievable factor.

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Online Square Packing with Gravity

Cited by 6 publications

References 24 publications

On the nearest neighbor rule for the metric traveling salesman problem

On the nearest neighbor rule for the metric traveling salesman problem

An efficient data structure for dynamic two-dimensional reconfiguration

Online Square-into-Square Packing

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