Online circle packing

Abstract

We consider the online problem of packing circles into a square container. A sequence of circles has to be packed one at a time, without knowledge of the following incoming circles and without moving previously packed circles. We present an algorithm that packs any online sequence of circles with a combined area not larger than 0.350389 of the square’s area, improving the previous best value of π/10 ≈ 0.31416 ; even in an offline setting, there is an upper bound of π/(3+22)≈0.5390. If only circles with radii of at least 0.026622 are considered, our algorithm achieves the higher value 0.375898. As a byproduct, we give an online algorithm for packing circles into a 1 × b rectangle with b≥ 1. This algorithm is worst case-optimal for b≥ 2.36.