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

In the circle packing problem for triangular containers, one asks whether a given set of circles can be packed into a given triangle. Packing problems like this have been shown to be NP-hard. In this paper, we present a new sufficient condition for packing circles into any right or obtuse triangle using only the circles’ combined area: It is possible to pack any circle instance whose combined area does not exceed the triangle’s incircle. This area condition is tight, in the sense that for any larger area, there are instances which cannot be packed. A similar result for square containers has been established earlier this year, using the versatile, divide-and-conquer-based Split Packing algorithm. In this paper, we present a generalized, weighted version of this approach, allowing us to construct packings of circles into asymmetric triangles. It seems crucial to the success of these results that Split Packing does not depend on an orthogonal subdivision structure. Beside realizing all packings below the critical density bound, our algorithm can also be used as a constant-factor approximation algorithm when looking for the smallest non-acute triangle of a given side ratio in which a given set of circles can be packed. An interactive visualization of the Split Packing approach and other related material can be found at https://morr.cc/split-packing/.