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# Packing Disks into Disks with Optimal Worst-Case Density

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LIPIcs.SoCG.2019.35.pdf
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• 19 pages

## Acknowledgements

We thank Sebastian Morr for joint previous work.

## Cite As

Sándor P. Fekete, Phillip Keldenich, and Christian Scheffer. Packing Disks into Disks with Optimal Worst-Case Density. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.35

## Abstract

We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area delta <= 1/2 can always be packed into a disk of area 1; on the other hand, for any epsilon>0 there are sets of disks of area 1/2+epsilon that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Packing and covering problems
• Theory of computation → Computational geometry
##### Keywords
• Disk packing
• packing density
• tight worst-case bound
• interval arithmetic
• approximation

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## References

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