Online Circle Packing
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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 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 + 2 √(2)) ≈ 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.
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