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.
@InProceedings{fekete_et_al:LIPIcs.SoCG.2019.35, author = {Fekete, S\'{a}ndor P. and Keldenich, Phillip and Scheffer, Christian}, title = {{Packing Disks into Disks with Optimal Worst-Case Density}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {35:1--35:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.35}, URN = {urn:nbn:de:0030-drops-104398}, doi = {10.4230/LIPIcs.SoCG.2019.35}, annote = {Keywords: Disk packing, packing density, tight worst-case bound, interval arithmetic, approximation} }
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