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Packing Geometric Objects with Optimal Worst-Case Density (Multimedia Exposition)

Authors Aaron T. Becker , Sándor P. Fekete , Phillip Keldenich , Sebastian Morr , Christian Scheffer

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Author Details

Aaron T. Becker
  • Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005 USA
Sándor P. Fekete
  • Department of Computer Science, TU Braunschweig, Mühlenpfordtstr. 23, 38106 Braunschweig, Germany
Phillip Keldenich
  • Department of Computer Science, TU Braunschweig, Mühlenpfordtstr. 23, 38106 Braunschweig, Germany
Sebastian Morr
  • Department of Computer Science, TU Braunschweig, Mühlenpfordtstr. 23, 38106 Braunschweig, Germany
Christian Scheffer
  • Department of Computer Science, TU Braunschweig, Mühlenpfordtstr. 23, 38106 Braunschweig, Germany

Funding

  • Becker, Aaron T.: Supported by the National Science Foundation under Grant No. IIS-1553063.
  • Keldenich, Phillip: Supported by the German Research Foundation under Grant No. FE 407/17-2.

Cite AsGet BibTex

Aaron T. Becker, Sándor P. Fekete, Phillip Keldenich, Sebastian Morr, and Christian Scheffer. Packing Geometric Objects with Optimal Worst-Case Density (Multimedia Exposition). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 63:1-63:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.63

Abstract

We motivate and visualize problems and methods for packing a set of objects into a given container, in particular a set of {different-size} circles or squares into a square or circular container. Questions of this type have attracted a considerable amount of attention and are known to be notoriously hard. We focus on a particularly simple criterion for deciding whether a set can be packed: comparing the total area A of all objects to the area C of the container. The critical packing density delta^* is the largest value A/C for which any set of area A can be packed into a container of area C. We describe algorithms that establish the critical density of squares in a square (delta^*=0.5), of circles in a square (delta^*=0.5390 ...), regular octagons in a square (delta^*=0.5685 ...), and circles in a circle (delta^*=0.5).

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Computational geometry
Keywords
  • Packing
  • complexity
  • bounds
  • packing density

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Related Versions

  • Parts of this contribution describe the conference paper [Fekete et al., 2019], which is part of SoCG 2019; a full version of that paper can be found at https://arxiv.org/abs/1903.07908.

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