Algorithms for the Circle-Packing Problem via Extended Sequence-Pair

Conference paper


The circle-packing problem is a problem of packing circles into a two dimensional area such that none of them overlap with each other. Each of the former methods has its own difficulty; some of them are only applicable to the case that the area the circles are to be packed into has a special shape; some of them require different search technique according as the shape of the area. Also, most of the former methods search in a restricted neighbor. In addition, there exist unsearchable location of circles. These facts mean former methods cannot assure global optimization. Hence, in the present paper, we propose sequence-pair for circle packing (SPC), a method of representing relative location of circle pairs, which is an extended version of sequence-pair for rectangles. We propose also a method of obtaining an approximate solution of the circle-packing problem, where all constraints are replaced by approximate linear inequalities.


Circle-packing Linear approximation Linear programming Non-linear programming Packing density Reducing redundancy Sequence-pair 


  1. 1.
    E.G. Birgin, J.M. Gentil, New and improved results for packing identical unitary radius circles within triangles, rectangles and strips. Comput. Oper. Res. 37, 1318–1327 (2010)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    H. Wang, W. Huang, Q. Zhang, D. Xu, An improved algorithm for the packing of unequal circles within a larger containing circle. Eur. J. Oper. Res. 141, 440–453 (2002)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    D. Zhang, A. Deng, An effective hybrid algorithm for the problem of packing circles into a larger containing circle. Comput. Oper. Res. 32, 1941–1951 (2005)CrossRefMATHGoogle Scholar
  4. 4.
    H. Akeb, M. Hifi, R. M’Hallah, A beam search algorithm for the circular packing problem. Comput. Oper. Res. 36, 1513–1528 (2009)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    C.O. Lo′pez, J.E. Beasley, A Packing unequal circles using formulation space search. Comput. Oper. Res. 40, 1276–1288 (2013)CrossRefMathSciNetGoogle Scholar
  6. 6.
    T. Sawa, A. Nagao, T. Kambe, I. Shirakawa, K. Chihara, A method for rectangle packing problem. Inst. Electr. Inf. Commun. Eng. 97(137), 159–166 (1997)Google Scholar
  7. 7.
    H. Murata, K. Fujiyoshi, S. Nakatake, Y. Kajitani, VLSI module placement based on rectangle-packing by the sequence-pair. IEEE Trans. CAD 15(12), 1518–1524 (1996)CrossRefGoogle Scholar
  8. 8.
    S. Morinaga, H. Ohta, M. Nakamori: A method for solving circle-packing problem by using extended seqence-pair, in The 26th Workshop on Circuits and Systems, pp. 489–494, 2013Google Scholar
  9. 9.
    S. Morinaga, H. Ohta, M. Nakamori: An algorithm for the circle-packing problem via extended sequence-pair with nonlinear optimization, lecture notes in engineering and computer science, in Proceedings of The World Congress on Engineering and Computer Science 2013, WCECS 2013, San Francisco, USA, pp. 1222–1227 23–25 Oct. 2013Google Scholar
  10. 10.
    E. Specht, packomania. Accessed 02 July 2013

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Shuhei Morinaga
    • 1
  • Hidenori Ohta
    • 1
  • Mario Nakamori
    • 1
  1. 1.Tokyo University of Agriculture and TechnologyKoganei-shiJapan

Personalised recommendations