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

Abstract

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/.

Split Packing: Packing Circles into Triangles with Optimal Worst-Case Density

“…For the scenario with circular objects, Demaine, Fekete, and Lang [9] showed in 2010 that deciding whether a given set of disks can be packed into a unit square is NP-hard. Using a recursive procedure for partitioning the container into triangular pieces, Morr, Fekete and Scheffer [13,24] proved that the critical packing density of disks in a square is π /(3+2 √ 2) ≈ 0.539. More recently, Fekete, Keldenich and Scheffer [12] established the critical packing density of disks into a disk.…”

“…Figure 1 Worst-case optimal approaches and matching worst-case instances for packing: (a) Squares into a square with Shelf Packing by Moon and Moser [23]. (b) Disks into a square by Morr et al [24,13]. (c) Disks into a disk by Fekete et al [12].…”

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

“…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]. ?…”

A Reallocation Algorithm for Online Split Packing of Circles

“…We look at online circle packing, where we try to dynamically pack a set of circles of unequal areas into a unit square while allowing reallocations. Our work builds on insights from the Split Packing papers by Fekete, Morr, and Scheffer [14,25,24]. The Split Packing papers derive the critical density a of squares and obtuse triangles (triangles with an internal angle of at least 90 • ).…”

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The Split Packing algorithm [14,25, 24] is an offline algorithm that packs a set of circles into triangles and squares up to critical density. In this paper, we develop an online alternative to Split Packing to handle an online sequence of insertions and deletions, where the algorithm is allowed to reallocate circles into new positions at a cost proportional to their areas.

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