Document Open Access Logo

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

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

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2019.63.pdf
  • Filesize: 1.49 MB
  • 6 pages

Document Identifiers

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.

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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).

Cite As Get 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

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.

References

  1. Ignacio Castillo, Frank J. Kampas, and János D. Pintér. Solving circle packing problems by global optimization: numerical results and industrial applications. European Journal of Operational Research, 191(3):786-802, 2008. Google Scholar
  2. Erik D. Demaine, Sándor P. Fekete, and Robert J. Lang. Circle Packing for Origami Design is Hard. In Origami⁵: 5th International Conference on Origami in Science, Mathematics and Education, AK Peters/CRC Press, pages 609-626, 2011. URL: http://arxiv.org/abs/1105.0791.
  3. Sándor P. Fekete, Phillip Keldenich, and Christian. Scheffer. Packing Disks into Disks with Optimal Worst-Case Density. In Proceedings of the 35th Annual Symposium on Computational Geometry (SoCG), LIPIcs, pages 35:1-35:19, 2019. These proceedings; full version at URL: https://arxiv.org/abs/1903.07908.
  4. Sándor P. Fekete, Sebastian Morr, and Christian Scheffer. Split Packing: Algorithms for Packing Circles with Optimal Worst-Case Density. Discrete & Computational Geometry, page 562–594, 2018. URL: http://dx.doi.org/10.1007/s00454-018-0020-2.
  5. F. Fodor. The Densest Packing of 19 Congruent Circles in a Circle. Geometriae Dedicata, 74:139–145, 1999. Google Scholar
  6. F. Fodor. The Densest Packing of 12 Congruent Circles in a Circle. Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry), 41:401–409, 2000. Google Scholar
  7. F. Fodor. The Densest Packing of 13 Congruent Circles in a Circle. Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry), 44:431–440, 2003. Google Scholar
  8. Hamish J. Fraser and John A. George. Integrated container loading software for pulp and paper industry. European Journal of Operational Research, 77(3):466-474, 1994. Google Scholar
  9. M. Goldberg. Packing of 14, 16, 17 and 20 circles in a circle. Mathematics Magazine, 44:134–139, 1971. Google Scholar
  10. R.L. Graham, B.D. Lubachevsky, K.J. Nurmela, and P.R.J. Östergøard. Dense Packings of Congruent Circles in a Circle. Discrete Mathematics, 181:139–154, 1998. Google Scholar
  11. Mhand Hifi and Rym M'Hallah. A literature review on circle and sphere packing problems: Models and methodologies. Advances in Operations Research, 2009. Article ID 150624. Google Scholar
  12. S. Kravitz. Packing cylinders into cylindrical containers. Mathematics Magazine, 40:65–71, 1967. Google Scholar
  13. Robert J. Lang. A computational algorithm for origami design. Proceedings of the Twelfth Annual Symposium on Computational Geometry (SoCG), pages 98-105, 1996. Google Scholar
  14. Joseph Y. T. Leung, Tommy W. Tam, Chin S. Wong, Gilbert H. Young, and Francis Y. L. Chin. Packing squares into a square. Journal of Parallel and Distributed Computing, 10(3):271–275, 1990. Google Scholar
  15. B.D. Lubachevsky and R.L. Graham. Curved Hexagonal Packings of Equal Disks in a Circle. Discrete & Computational Geometry, 18:179–194, 1997. Google Scholar
  16. H. Melissen. Densest Packing of Eleven Congruent Circles in a Circle. Geometriae Dedicata, 50:15–25, 1994. Google Scholar
  17. John W. Moon and Leo Moser. Some packing and covering theorems. In Colloquium Mathematicae, volume 17, pages 103-110. Institute of Mathematics, Polish Academy of Sciences, 1967. Google Scholar
  18. Sebastian Morr. Split Packing: An Algorithm for Packing Circles with Optimal Worst-Case Density. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 99-109, 2017. Google Scholar
  19. Norman Oler. A finite packing problem. Canadian Mathematical Bulletin, 4:153–155, 1961. Google Scholar
  20. G.E. Reis. Dense Packing of Equal Circles within a Circle. Mathematics Magazine, issue 48:33–37, 1975. Google Scholar
  21. Eckard Specht. Packomania, 2015. URL: http://www.packomania.com/.
  22. Kokichi Sugihara, Masayoshi Sawai, Hiroaki Sano, Deok-Soo Kim, and Donguk Kim. Disk packing for the estimation of the size of a wire bundle. Japan Journal of Industrial and Applied Mathematics, 21(3):259-278, 2004. Google Scholar
  23. Péter Gábor Szabó, Mihaly Csaba Markót, Tibor Csendes, Eckard Specht, Leocadio G. Casado, and Inmaculada García. New Approaches to Circle Packing in a Square. Springer US, 2007. Google Scholar
  24. Huaiqing Wang, Wenqi Huang, Quan Zhang, and Dongming Xu. An improved algorithm for the packing of unequal circles within a larger containing circle. European Journal of Operational Research, 141(2):440–453, September 2002. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact