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/.
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].
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is δ = 8 /5π ≈ 0.509. This implies that any set of (not necessarily equal) squares of total area A ≤ 8 /5 can always be packed into a disk with radius 1; in contrast, for any ε > 0 there are sets of squares of total area 8 /5 + ε that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square ( 1 /2), circles in a square ( π /(3+2 √ 2) ≈ 0.539) and circles in a circle ( 1 /2) have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.