# Online Circle Packing

## TL;DR

This paper introduces an online algorithm for packing circles into a square with an improved area utilization of approximately 35%, surpassing previous bounds, and also addresses packing into rectangles with optimality conditions.

## Contribution

The paper presents a novel online circle packing algorithm that achieves a higher area utilization than previous methods and extends to packing into rectangles with proven optimality.

## Key findings

- Achieves an area utilization of 0.350389 of the square's area.
- Improves upon the previous best of 0.31416 for online circle packing.
- Provides an online algorithm for packing into rectangles with optimality for certain aspect ratios.

## 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 0.350389 of the square's area, improving the previous best value of {\pi}/10 \approx 0.31416; even in an offline setting, there is an upper bound of {\pi}/(3 + 2 \sqrt{2}) \approx 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\times b rectangle with b \geq 1. This algorithm is worst case-optimal for b \geq 2.36.