Improved Bound for Online Square-into-Square Packing

Brubach, Brian

doi:10.1007/978-3-319-18263-6_5

Cited by 15 publications

(20 citation statements)

References 7 publications

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“…Another natural extension is the online version of the problem. The current best algorithm that packs squares into a square in an online fashion by Brubach [1], based on the work by Fekete and Hoffmann [4,5], gives a density guarantee of 2 5 . It is possible to directly use this algorithm to pack circles into a square in an online situation with a density of π 10 ≈ 0.3142.…”

Section: Resultsmentioning

confidence: 99%

Split Packing: Algorithms for Packing Circles with Optimal Worst-Case Density

Fekete

Morr

Scheffer

2018

Discrete Comput Geom

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In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be NP-hard. In this paper, we present new sufficient conditions for packing circles into square and triangular containers, using only the sum of the circles' areas: For square containers, it is possible to pack any set of circles with a combined area of up to ≈ 53.90% of the square's area. And when the container is a right or obtuse triangle, any set of circles whose combined area does not exceed the triangle's incircle can be packed. These area conditions are tight, in the sense that for any larger areas, there are sets of circles which cannot be packed. Similar results have long been known for squares, but to the best of our knowledge, we give the first results of this type for circular objects. Our proofs are constructive: We describe a versatile, divide-and-conquerbased algorithm for packing circles into various container shapes with optimal worst-case density. It employs an elegant subdivision scheme that recursively splits the circles into two groups and then packs these into subcontainers. We call this algorithm Split Packing. It can be used as a constant-factor approximation algorithm when looking for the smallest container in which a given set of circles can be packed, due to its polynomial runtime. A browser-based, interactive visualization of the Split Packing approach and other related material can be found at https://morr.cc/split-packing/.Extended abstracts presenting parts of this paper appeared in the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA 2017) [15] and the 15th Algorithms and Data Structures Symposium (WADS 2017) [6].

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Section: Resultsmentioning

confidence: 99%

Split Packing: Algorithms for Packing Circles with Optimal Worst-Case Density

Fekete

Morr

Scheffer

2018

Discrete Comput Geom

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“…The current best algorithm that packs squares into a square in an online fashion by Brubach [1], based on the work by Fekete and Hoffmann [4,5], gives a density guarantee of A related problem asks for the smallest area so that we can always cover the container with circles of that combined area. For example, we conjecture that for an isosceles right triangle, any circle instance with a total area of at least its excircle's area is sufficient to cover it.…”

Section: Resultsmentioning

confidence: 99%

Split Packing: An Algorithm for Packing Circles with Optimal Worst-Case Density

Morr¹

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the classic circle packing problem, one asks whether a given set of circles can be packed into the unit square. This problem is known to be NP-hard. In this paper, we present a new sufficient condition using only the circles' combined area: It is possible to pack any circle instance with a combined area of up to ≈ 0.5390. This bound is tight, in the sense that for any larger combined area, there are instances which cannot be packed, which is why we call this number the problem's critical density. Similar results have long been known for squares, but to the best of our knowledge, this paper gives the first results of this type for circular objects.Our proof is constructive: We describe a subdivision scheme which recursively splits the circles into groups and then packs these into subcontainers. We call this algorithm Split Packing. Beside realizing all packings up to the critical density bound, Split Packing also serves as a constant-factor approximation algorithm when looking for the smallest square in which a given set of circles can be packed.We believe that the ideas behind Split Packing are interesting and elegant on their own, and we see many opportunities to apply this technique in the context of other packing and covering problems.A browser-based, interactive visualization of the Split Packing approach and other related material can be found at https://morr.cc/split-packing/.

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“…Even the problem of handling a sequence of insertions of total volume at most one, without considering dynamic deletions and reallocation, requires underallocation. This problem is known as online square packing, see Fekete and Hoffmann [32,33]; currently, the best known approach results in 5/2-underallocation, see Brubach [52]. Rectangles of bounded aspect ratio k are dealt with in the same way.…”

Section: General Squares and Rectanglesmentioning

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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Improved Bound for Online Square-into-Square Packing

Cited by 15 publications

References 7 publications

Split Packing: Algorithms for Packing Circles with Optimal Worst-Case Density

Split Packing: Algorithms for Packing Circles with Optimal Worst-Case Density

Split Packing: An Algorithm for Packing Circles with Optimal Worst-Case Density

An efficient data structure for dynamic two-dimensional reconfiguration

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