The Split Packing algorithm is an offline algorithm that packs a set of circles into shapes (triangles and squares) at an optimal packing 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 achieve optimal packing density as defined in our paper, with an amortized reallocation cost of @math for squares, and @math for right angled triangles, where @math is the ratio of the lengths of the second shortest side to the shortest, when inserting a circle of area @math . When insertions and deletions are considered, we achieve a packing density of @math of the optimal, where @math can be made arbitrarily small, for an additional amortized reallocation cost of @math .
@cite_1: In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receive an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares, isosceles right triangles of leg length one, and unit cubes, respectively. For Online Circle Packing in Squares, we improve the previous best-known competitive ratio for the bounded space version, when at most a constant number of bins can be open at any given time, from 2.439 to 2.3536. For Online Circle Packing in Isosceles Right Triangles and Online Sphere Packing in Cubes we show bounded space algorithms of asymptotic competitive ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and 2.7707 on the competitive ratio of any online bounded space algorithm for these two problems. We also considered the online unbounded space variant of these three problems which admits a small reorganization of the items inside the bin after their packing, and we present algorithms of competitive ratios 2.3105, 2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes, respectively.
@cite_25: In this paper, we study 1-space bounded 2-dimensional bin packing and square packing. A sequence of rectangular items (square items, respectively) arrive over time, which must be packed into square bins of size 1×1. 90°-rotation of an item is allowed. When an item arrives, we must pack it into an active bin immediately without any knowledge of the future items. The objective is to minimize the total number of bins used for packing all the items in the sequence. In the 1-space bounded variant, there is only one active bin for packing the current item. If the active bin does not have enough space to pack the item, it must be closed and a new active bin is opened.
There has also been some recent work on the online circle packing. However, this work on online circle packing focuses on minimising the number of square bins used to pack an online sequence of circles. @cite_25 provides an asymptotic competitive ratio of @math , and gives a lower bound of @math for the problem. The upper bound was improved to @math in @cite_1 . We have not found existing work that takes into account the possibility of reallocations for online circle packing.