A Reallocation Algorithm for Online Split Packing of Circles

The Split Packing algorithm \cite{splitpacking_ws, splitpackingsoda, splitpacking} 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. The algorithm can be used to pack circles into squares and right angled triangles. If only insertions are considered, our algorithm is also able to pack to critical density, with an amortised reallocation cost of $O(c\log \frac{1}{c})$ for squares, and $O(c(1+s^2)\log_{1+s^2}\frac{1}{c})$ for right angled triangles, where $s$ is the ratio of the lengths of the second shortest side to the shortest side of the triangle, when inserting a circle of area $c$. When insertions and deletions are considered, we achieve a packing density of $(1-\epsilon)$ of the critical density, where $\epsilon>0$ can be made arbitrarily small, with an amortised reallocation cost of $O(c(1+s^2)\log_{1+s^2}\frac{1}{c} + c\frac{1}{\epsilon})$.