Online Circle and Sphere Packing
Carla Negri Lintzmayer, Fl\'avio Keidi Miyazawa, Eduardo Candido, Xavier

TL;DR
This paper advances online packing algorithms for circles and spheres in various geometric containers, improving competitive ratios and establishing bounds for both bounded and unbounded space scenarios.
Contribution
It introduces improved algorithms and bounds for online circle and sphere packing in squares, triangles, and cubes, covering both bounded and unbounded space variants.
Findings
Improved competitive ratio for circles in squares from 2.439 to 2.3536.
Established bounds for circles in triangles and spheres in cubes.
Presented algorithms with competitive ratios for unbounded space variants.
Abstract
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…
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