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The Expanding Space Method in Sphere Packing Problem

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Lecture Notes in Computational Intelligence and Decision Making (ISDMCI 2020)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1246))

Abstract

A problem of packing unequal spheres, which radii are known, into containers of various shape are considered. An equivalent mathematical model to a standard sphere packing model is formed based on the assumption that the radii may be variable. In the new model formulated in a higher dimension space, a combinatorial structure is derived, and additional constraints are formed involving variable radii and providing its equivalence to the original packing problem. This approach of treating radii as variables allows improving local solutions of the original problem. Some practical tasks are indicated, which are reduced to packing unequal spheres.

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References

  1. Akeb, H.: A two-stage look-ahead heuristic for packing spheres into a three-dimensional bin of minimum length. In: Fidanova, S. (ed.) Recent Advances in Computational Optimization: Results of the Workshop on Computational Optimization, WCO 2014, pp. 127–144. Studies in Computational Intelligence. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-21133-6_8

  2. Amirgaliyeva, Z., Mladenović, N., Todosijević, R., Urošević, D.: Solving the maximum min-sum dispersion by alternating formulations of two different problems. Eur. J. Oper. Res. 260(2), 444–459 (2017). https://doi.org/10.1016/j.ejor.2016.12.039

    Article  MathSciNet  MATH  Google Scholar 

  3. Birgin, E.G., Sobral, F.N.C.: Minimizing the object dimensions in circle and sphere packing problems. Comput. Oper. Res. 35(7), 2357–2375 (2008). https://doi.org/10.1016/j.cor.2006.11.002

    Article  MathSciNet  MATH  Google Scholar 

  4. Burtseva, L., Salas, B.V., Werner, F., Petranovskii, V.: Packing of monosized spheres in a cylindrical container: models and approaches. Rev. Mex. Fis. 61(1), 20–27 (2015)

    MathSciNet  Google Scholar 

  5. Castillo, I., Kampas, F.J., Pintér, J.D.: Solving circle packing problems by global optimization: numerical results and industrial applications. Eur. J. Oper. Res. 191(3), 786–802 (2008). https://doi.org/10.1016/j.ejor.2007.01.054

  6. Chen, D.Z.: Sphere packing problem. In: Science, M.Y.K.P.O.C. (ed.) Encyclopedia of Algorithms, pp. 871–874. Springer, Heidelberg (2008). https://doi.org/10.1007/978-0-387-30162-4_391

  7. Conway, J., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York (1999)

    Google Scholar 

  8. Corwin, E.I., Clusel, M., Siemens, A.O.N., Brujic, J.: Model for random packing of polydisperse frictionless spheres. Soft Matter 6(13), 2949–2959 (2010). https://doi.org/10.1039/C000984A

    Article  Google Scholar 

  9. He, K., Huang, M., Yang, C.: An action-space-based global optimization algorithm for packing circles into a square container. Comput. Oper. Res. 58, 67–74 (2015). https://doi.org/10.1016/j.cor.2014.12.010

    Article  MathSciNet  MATH  Google Scholar 

  10. Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. Adv. Oper. Res. 7, 1–22 (2009). https://doi.org/10.1155/2009/150624

    Article  MATH  Google Scholar 

  11. Hifi, M., Yousef, L.: Handling lower bound and hill-climbing strategies for sphere packing problems. In: Fidanova, S. (ed.) Recent Advances in Computational Optimization: Results of the Workshop on Computational Optimization, WCO 2014. Studies in Computational Intelligence, pp. 145–164. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-21133-6_9

  12. Hifi, M., Yousef, L.: A global dichotomous search-based heuristic for the three-dimensional sphere packing problem. Int. J. Oper. Res. 33(2), 139–160 (2018). https://doi.org/10.1504/IJOR.2018.095195

    Article  MathSciNet  Google Scholar 

  13. Hifi, M., Yousef, L.: A local search-based method for sphere packing problems. Eur. J. Oper. Res. 274(2), 482–500 (2019). https://doi.org/10.1016/j.ejor.2018.10.016

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, W.Q., Li, Y., Akeb, H., Li, C.M.: Greedy algorithms for packing unequal circles into a rectangular container. J. Oper. Res. Soc. 56(5), 539–548 (2005). https://doi.org/10.1057/palgrave.jors.2601836

    Article  MATH  Google Scholar 

  15. Ke, C. (ed.): Recent Advances in Nanotechnology, 1 edn. Apple Academic Press (2011)

    Google Scholar 

  16. Kubach, T., Bortfeldt, A., Tilli, T., Gehring, H.: Greedy algorithms for packing unequal spheres into a cuboidal strip or a cuboid. Asia-Pacific J. Oper. Res. (APJOR) 28(06), 739–753 (2011)

    Article  MathSciNet  Google Scholar 

  17. Liu, J., Yao, Y., Zheng, Y., Geng, H., Zhou, G.: An effective hybrid algorithm for the circles and spheres packing problems. In: Du, D.Z., Hu, X., Pardalos, P.M. (eds.) Combinatorial Optimization and Applications. Lecture Notes in Computer Science, pp. 135–144. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02026-1_12

  18. López, C.O., Beasley, J.E.: Packing unequal circles using formulation space search. Comput. Oper. Res. 40(5), 1276–1288 (2013). https://doi.org/10.1016/j.cor.2012.11.022

  19. Pichugina, O., Kartashov, O.: Signed permutation polytope packing in VLSI design. In: 2019 IEEE 15th International Conference on the Experience of Designing and Application of CAD Systems (CADSM) Conference Proceedings, pp. 4/50–4/55. Lviv (2019). https://doi.org/10.1109/CADSM.2019.8779353

  20. Pichugina, O.S., Yakovlev, S.V.: Continuous representations and functional extensions in combinatorial optimization. Cybern. Syst. Anal. 52(6), 921–930 (2016). https://doi.org/10.1007/s10559-016-9894-2

    Article  MathSciNet  MATH  Google Scholar 

  21. Pichugina, O.S., Yakovlev, S.V.: Functional and analytic representations of the general permutation. Eastern Eur. J. Enterprise Technol. 79(1(4)), 27–38 (2016). https://doi.org/10.15587/1729-4061.2016.58550

  22. Shekhovtsov, S.B., Yakovlev, S.V.: Formalization and solution of a class of covering problems in the design of monitoring and control systems. Akademiya Nauk SSSR. Avtomatika i Telemekhanika 5, 160–168 (1989)

    MathSciNet  MATH  Google Scholar 

  23. Shirokanev, A., Kirsh, D., Ilyasova, N., Kupriyanov, A.: Investigation of algorithms for coagulate arrangement in fundus images. Comput. Opt. 42(4), 712–721 (2018). https://doi.org/10.18287/2412-6179-2018-42-4-712-721

  24. Stoyan, Y.G., Yakovlev, S.V.: Configuration space of geometric objects. Cybern. Syst. Anal. 716–726 (2018). https://doi.org/10.1007/s10559-018-0073-5

  25. Stoyan, Y., Yaskov, G.: Packing unequal circles into a strip of minimal length with a jump algorithm. Optim. Lett. 8(3), 949–970 (2014). https://doi.org/10.1007/s11590-013-0646-1

    Article  MathSciNet  MATH  Google Scholar 

  26. Stoyan, Y.G., Chugay, A.M.: Mathematical modeling of the interaction of non-oriented convex polytopes. Cybern. Syst. Anal. 48(6), 837–845 (2012). https://doi.org/10.1007/s10559-012-9463-2

    Article  MathSciNet  MATH  Google Scholar 

  27. Stoyan, Y.G., Scheithauer, G., Yaskov, G.N.: Packing unequal spheres into various containers. Cybern. Syst. Anal. 52(3), 419–426 (2016). https://doi.org/10.1007/s10559-016-9842-1

    Article  MATH  Google Scholar 

  28. Stoyan, Y.G., Yakovlev, S.V.: Theory and methods of Euclidean combinatorial optimization: current state and promising. Cybern. Syst. Anal. 56(3), 513–525 (2020)

    Article  Google Scholar 

  29. Stoyan, Y.G., Yakovlev, S.V., Emets, O.A., Valuĭskaya, O.A.: Construction of convex continuations for functions defined on a hypersphere. Cybern. Syst. Anal. 34(2), 27–36 (1998). https://doi.org/10.1007/BF02742066

  30. Stoyan, Y.Y., Yaskov, G., Scheithauer, G.: Packing of various radii solid spheres into a parallelepiped. Central Eur. J. Oper. Res. (CEJOR) 11(4), 389–407 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Sutou, A., Dai, Y.: Global optimization approach to unequal global optimization approach to unequal sphere packing problems in 3D. J. Optim. Theory Appl. 114(3), 671–694 (2002). https://doi.org/10.1023/A:1016083231326

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, J.: Packing of unequal spheres and automated radiosurgical treatment planning. J. Comb. Optim. 3(4), 453–463 (1999). https://doi.org/10.1023/A:1009831621621

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, Y., Nagarajan, M., Uhler, C., Shivashankar, G.V.: Orientation and repositioning of chromosomes correlate with cell geometry-dependent gene expression. Mol. Biol. Cell 28(14), 1997–2009 (2017). https://doi.org/10.1091/mbc.E16-12-0825

    Article  Google Scholar 

  34. Wäscher, G., Haußner, H., Schumann, H.: An improved typology of cutting and packing problems. Eur. J. Oper. Res. 183(3), 1109–1130 (2007). https://doi.org/10.1016/j.ejor.2005.12.047

  35. Yakovlev, S., Kartashov, O., Korobchynskyi, K., Skripka, B.: Numerical results of variable radii method in the unequal circles packing problem. In: 2019 IEEE 15th International Conference on the Experience of Designing and Application of CAD Systems (CADSM), pp. 1–4 (2019). https://doi.org/10.1109/CADSM.2019.8779288

  36. Yakovlev, S.V.: The method of artificial space dilation in problems of optimal packing of geometric objects. Cybern. Syst. Anal. 53(5), 725–731 (2017). https://doi.org/10.1007/s10559-017-9974-y

    Article  MATH  Google Scholar 

  37. Yakovlev, S.V.: Formalizing spatial configuration optimization problems with the use of a special function class. Cybern. Syst. Anal. 55(4), 581–589 (2019). https://doi.org/10.1007/s10559-019-00167-y

    Article  MATH  Google Scholar 

  38. Yakovlev, S.V., Pichugina, O.S.: Properties of combinatorial optimization problems over polyhedral-spherical sets. Cybern. Syst. Anal. 54(1), 99–109 (2018)

    Article  Google Scholar 

  39. Yakovlev, S.V., Valuiskaya, O.A.: Optimization of linear functions at the vertices of a permutation polyhedron with additional linear constraints. Ukrainian Math. J. 53(9), 1535–1545 (2001). https://doi.org/10.1023/A:1014374926840

    Article  MathSciNet  Google Scholar 

  40. Yakovlev, S.: Convex extensions in combinatorial optimization and their applications. In: Optimization Methods and Applications. Springer Optimization and Its Applications, pp. 567–584. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68640-0_27

  41. Yakovlev, S.V.: On some classes of spatial configurations of geometric objects and their formalization. J. Autom. Inf. Sci. 50(9), 38–50 (2018). https://doi.org/10.1615/JAutomatInfScien.v50.i9.30

    Article  Google Scholar 

  42. Yakovlev, S.V., Pichugina, O.S., Yarovaya, O.V.: Polyhedral-spherical configurations in discrete optimization problems. J. Autom. Inf. Sci. 51(1), 26–40 (2019). https://doi.org/10.1615/JAutomatInfScien.v51.i1.30

    Article  MathSciNet  Google Scholar 

  43. Yaskov, G.: Methodology to solve multi-dimentional sphere packing problems. J. Mech. Eng. 22(1), 67–75 (2019). https://doi.org/10.15407/pmach2019.01.067

  44. Ying, S.: A novel method for packing unequal circles into a rectangular container. Appl. Mech. Mater. 513, 3942–3945 (2014). https://doi.org/10.4028/www.scientific.net/AMM.513-517.3942

    Article  Google Scholar 

  45. Zong, C.: Sphere Packings. Springer, New York (1999)

    Google Scholar 

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Author information

Authors and Affiliations

  1. National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine

    Sergiy Yakovlev

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  1. Sergiy Yakovlev
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Correspondence to Sergiy Yakovlev .

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Editors and Affiliations

  1. Department of Informatics, Jan Evangelista Purkyně University in Ústí nad Labem, Ústí nad Labem, Czech Republic

    Sergii Babichev

  2. Department of Informatics and Computer Science, Kherson National Technical University, Kherson, Ukraine

    Volodymyr Lytvynenko

  3. Institute of Electronics and Information, Lublin University of Technology, Lublin, Poland

    Waldemar Wójcik

  4. Department of Informatics and Computer Science, Kherson National Technical University, Kherson, Ukraine

    Svetlana Vyshemyrskaya

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Yakovlev, S. (2021). The Expanding Space Method in Sphere Packing Problem. In: Babichev, S., Lytvynenko, V., Wójcik, W., Vyshemyrskaya, S. (eds) Lecture Notes in Computational Intelligence and Decision Making. ISDMCI 2020. Advances in Intelligent Systems and Computing, vol 1246. Springer, Cham. https://doi.org/10.1007/978-3-030-54215-3_10

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