Skip to main content
SpringerLink
Log in

On Online Algorithms for Bin, Strip, and Box Packing, and Their Worst-Case and Average-Case Analysis

  • Published:
Programming and Computer Software Aims and scope Submit manuscript

Abstract

In this survey, we consider online algorithms for bin packing and strip packing problems, as well as their generalizations (multidimensional bin packing, multiple strip packing, and packing into strips of different widths). For the latter problem, only the worst-case analysis is described; for the other problems, both the worst-case and average-case (probabilistic) asymptotic ratios are presented. The best lower and upper bounds are considered. Basic algorithms and methods for their analysis are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.

Similar content being viewed by others

REFERENCES

  1. Pastukhov, R.K., Korshunov, A.V., Turdakov, D.Y., and Kuznetsov, S.D., Improving quality of graph partitioning using multi-level optimization, Program. Comput. Software, 2015, vol. 41, no. 5, pp. 302–306. https://doi.org/10.1134/S0361768815050096

    Article  MathSciNet  Google Scholar 

  2. López, J., Kushik, N., and Yevtushenko, N., Source code optimization using equivalent mutants, Inf. Software Technol., 2018, pp. 138–141. https://doi.org/10.1016/j.infsof.2018.06.013

  3. Massobrio, R., Nesmachnow, S., Tchernykh, A., Avetisyan, A., and Radchenko, G., Towards a cloud computing paradigm for big data analysis in smart cities, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2016, vol. 28, no. 6, pp. 121–140. https://doi.org/10.15514/ISPRAS-2016-28(6)-9

  4. Anichkin, A.S. and Semenov, V.A., Mathematical formalization of project scheduling problems, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2017, vol. 29, no. 2, pp. 231–256. https://doi.org/10.15514/ISPRAS-2017-29(2)-9

  5. Zelenova, S.A. and Zelenov, S.V., Non-conflict scheduling criterion for strict periodic tasks, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2017, vol. 29, no. 6, pp. 183–202. https://doi.org/10.15514/ISPRAS-2017-29(6)-10

  6. Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., and Graham, R.L., Worst-case performance bounds for simple one-dimensional packing algorithms, SIAM J. Comput., 1974, vol. 3, no. 4, pp. 299–325.

    Article  MathSciNet  MATH  Google Scholar 

  7. Ghalam, L. and Grosu, D., A parallel approximation algorithm for scheduling identical machines, Proc. IEEE Int. Parallel and Distributed Processing Symposium Workshops, 2017, pp. 619–628.

  8. Sheikhalishahi, M., Wallace, R.M., Grandinetti, L., Vazquez-Poletti, J.L., and Guerriero, F., A multi-dimensional job scheduling, Future Gener. Comput. Syst., 2016, vol. 54, pp. 123–131.

    Article  Google Scholar 

  9. Feoktistov, A., Sidorov, I., Tchernykh, A., Edelev, A., Zorkalzev, V., Kostromin, R., Gorsky, S., Bychkov, I., and Avetisyan, A., Multi-agent approach for dynamic elasticity of virtual machines provisioning in heterogeneous distributed computing environment, Proc. Int. Conf. High Performance Computing and Simulation (HPCS), 2018, pp. 909–916. https://doi.org/10.1109/HPCS.2018.00145

  10. Tchernykh, A., Babenko, M., Chervyakov, N., Miranda-Lopez, V., Cortes-Mendoza, J.M., Du, Z., Navaux, P.O., and Avetisyan, A., Analysis of secured distributed cloud data storage based on multilevel RNS, Proc. IEEE Conf. Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), 2018, pp. 382–386. https://doi.org/10.1109/EIConRus.2018.8317112

  11. Tchernykh, A., Schwiegelshohn, U., Yahyapour, R., and Kuzjurin, N., On-line hierarchical job scheduling on grids with admissible allocation, J. Scheduling, 2010, vol. 13, no. 5, pp. 545–552.

    Article  MathSciNet  MATH  Google Scholar 

  12. Tshernykh, A., Ramirez, J.M., Avetisyan, A., Kuzjurin, N., Grushin, D., and Zhuk, S., Two-level job-scheduling strategies for a computational grid, Lect. Notes Comput. Sci., vol. 3911, pp. 774–781.

  13. Cohil, B., Shah, S., Goleshha, Y., and Patel, D., A Comparative analysis of virtual machine placement techniques in the cloud environment, Int. J. Comput. Appl., 2016, vol. 156, no. 14, pp. 12–18.

    Google Scholar 

  14. Garey, M.R., Graham, R.L., and Ullman, J.D., Worst-case analysis of memory allocation algorithms, Proc. 4th Annu. ACM Symp. Theory of Computing, 1972, pp. 143–150.

  15. Trushnikov, M.A., Probabilistic analysis of a new strip packing algorithm, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2013, vol. 24, pp. 457–468. https://doi.org/10.15514/ISPRAS-2013-24-21

  16. Varnovskiy, N.P., Martishin, S.A., Khrapchenko, M.V., and Shokurov, A.V., Secure cloud computing based on threshold homomorphic encryption, Program. Comput. Software, 2015, vol. 41, no. 4, pp. 215–218. https://doi.org/10.1134/S0361768815040088

    Article  MathSciNet  MATH  Google Scholar 

  17. Shor, P.W., The average-case analysis of some on-line algorithms for bin packing, Combinatorica, 1986, vol. 6, no. 4, pp. 179–200.

    Article  MathSciNet  MATH  Google Scholar 

  18. Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman: San Francisco, 1979.

    MATH  Google Scholar 

  19. Johnson, D.S., Near-optimal bin packing algorithms, PhD (Department of Mathematics) Thesis, Massachusetts Institute of Technology, Cambridge, 1973.

  20. Garey, M.R., Graham, R.L., Johnson, D.S., and Yao, A.C., Resource constrained scheduling as generalized bin packing, J. Comb. Theory, Ser. A, 1976, vol. 21, no. 3, pp. 257–298.

    Article  MathSciNet  MATH  Google Scholar 

  21. Gambosi, G., Postiglione, A., and Talamo, M.M., New algorithms for online bin packing, Proc. 1st Ital. Conf. Algorithms and Complexity, 1990, pp. 44–59.

  22. Ivcovi, Z. and Lloyd, E., Fully dynamic algorithms for bin packing: Being (mostly) myopic helps, Lect. Notes Comput. Sci., vol. 726, pp. 224–235.

  23. Yao, A.C., New algorithms for bin packing, J. ACM, 1981, vol. 27, no. 2, pp. 207–227.

    Article  MathSciNet  MATH  Google Scholar 

  24. Seiden, S.S., On the online bin packing problem, Lect. Notes Comput. Sci., 2002, vol. 2076, pp. 207–227.

    MATH  Google Scholar 

  25. Brown, J.D., A lower bound for on-line one dimensional bin packing algorithms, Technical report R-864, Coordinated Science laboratory, Univ. of Illinois, Urbana, 1979.

  26. Vliet, A., An improved lower bound for on-line bin packing algorithms, Inf. Process. Lett., 1992, vol. 43, no. 5, pp. 277–284.

    Article  MathSciNet  MATH  Google Scholar 

  27. Breitgand, D. and Epstein, A., Improving consolidations of virtual machines with risk-aware bandwidth oversubscription in compute clouds, Proc. IEEE INFCOM, 2012, pp. 2861–2865.

  28. Coffman, E.G. and Shor, P.W., A simple proof of the sqrt(n log3/4 n) upright matching bound, SIAM J. Discrete Math., 1991, vol. 4, no. 1, pp. 48–57.

    Article  MathSciNet  MATH  Google Scholar 

  29. Karp, R.M., Luby, M., and Marchetti-Spaccamela, A., A probabilistic analysis of multidimensional bin packing problem, Proc. 16th Annu. ACM Symp. Theory of Computing, 1984, pp. 289–298.

  30. Ajtai, M., Komlós, J., and Tusnadi, G., On optimal matchings, Combinatorica, 1984, vol. 4, no. 4, pp. 259–264.

    Article  MathSciNet  MATH  Google Scholar 

  31. Leighton, F.T. and Shor, P., Tight bonds for minimax grid matching with application to the average-case analysis of algorithms, Proc. 18th Annu. ACM Symp. Theory of Computing, 1986, pp. 91–103.

  32. Coffman, E.G., Courcoubetis, C., Garey, M.R., Johnson, D.S., McGeoch, L.A., Shor, P.W., Weber, R., and Yannakakis, M., Fundamental discrepancies between average-case analysis under discrete and continuous distributions: A bin packing study, Proc. 21st Annu. ACM Symp. Theory of Computing, 1991, pp. 230–240.

  33. Shor, P.W., How to pack better than Best Fit: Tight bounds for average-case online bin packing, Proc. 32nd Annu. Symp. Foundations of Computer Science, 1991, pp. 752–759.

  34. Galambos, G. and van Vliet, A., Lower bounds for 1-, 2-, and 3- dimensional on-line bin packing algorithms, Comput., 1994, vol. 52, no. 3, pp. 281–297.

    Article  MathSciNet  MATH  Google Scholar 

  35. Han, X., Chin, F.Y.L., Ting, H.-F., Zhang, G., and Zhang, Y., A new upper bound 2.5545 on 2D online bin packing, ACM Trans. Algorithms, 2011, vol. 7, no. 4.

  36. Csirik, J. and van Vliet, A., An on-line algorithm for multidimensional bin packing, Oper. Res. Lett., 1993, vol. 13, no. 3, pp. 149–158.

    Article  MathSciNet  MATH  Google Scholar 

  37. Epstein, L. and van Stee, R., Optimal online algorithms for multidimensional packing problems, Proc. 15th Annu. ACM-CIAM Symp. Discrete Algorithms, 2004, pp. 214–223.

  38. Chang, E.-C., Wang, W., and Kankanhalli, M.S., Multidimensional on-line bin-packing: An algorithm and its average-case analysis, Inf. Process. Lett., 1993, vol. 48, no. 3, pp. 121–125.

    Article  MathSciNet  MATH  Google Scholar 

  39. Baker, B.S., Coffman, E.G., and Rivest, R.L., Orthogonal packings in two dimensions, SIAM J. Comput., 1980, vol. 9, no. 4, pp. 846–855.

    Article  MathSciNet  MATH  Google Scholar 

  40. Baker, B.S. and Schwarz, J.S., Shelf algorithms for two-dimensional packing problems, SIAM J. Comput., 1983, vol. 12, no. 3, pp. 508–525.

    Article  MathSciNet  MATH  Google Scholar 

  41. Csirik, J. and Woeginger, G.J., Shelf algorithms for on-line strip packing, Inf. Process. Lett., 1997, vol. 63, no. 4, pp. 171–175.

    Article  MathSciNet  MATH  Google Scholar 

  42. Han, X., Iwama, K., Ye, D., and Zhang, G., Strip packing vs. bin packing, Lect. Notes Comp. Sci., 2007, vol. 4508, pp. 358–367.

    Article  MATH  Google Scholar 

  43. Coffman, E.G. and Shor, P.W., Packing in two dimensions: Asymptotic average-case analysis of algorithms, Algorithmica, 1993, vol. 9, no. 3, pp. 253–277.

    Article  MathSciNet  MATH  Google Scholar 

  44. Kuzjurin, N.N. and Pospelov, A.I., Probabilistic analysis of different shelf algorithms for packing rectangles into a strip, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2007, vol. 12, pp. 17–26.

  45. Kuzyurin, N.N. and Pospelov, A.I., Probabilistic analysis of a new class of strip packing algorithms, Comput. Math. Math. Phys., 2011, vol. 51, no. 10.

  46. Trushnikov, M.A., On one problem of Koffman–Shor connected to strip packing problem, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2012, vol. 22, pp. 456–462. https://doi.org/10.15514/ISPRAS-2012-22-24

  47. Lazarev, D.O. and Kuzyrin, N.N., An algorithm for multiple strip package and its average case evaluation, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2017, vol. 29, no. 6, pp. 221–228. https://doi.org/10.15514/ISPRAS-2017-29(6)-13

  48. Kuzjurin, N.N., Grushin, D.A., and Fomin, A., Two-dimensional packing problems and optimization in distributed computing systems, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2014, vol. 26, no. 1, pp. 483–502. https://doi.org/10.15514/ISPRAS-2014-26(1)-21

  49. Ye, D., Han, X., and Zhang, G., Online multiple-strip packing, Theor. Comput. Sci., 2011, vol. 412, no. 3, pp. 233–239.

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhuk, S.N., Approximation algorithms for packing rectangles into several strips, Discrete Math. Appl., 2006, vol. 16, no. 1, pp. 73–85.

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhuk, S.N., Analysis of some heuristics of packing rectangles into several strips, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2004, vol. 6, pp. 13–26.

  52. Zhuk, S.N., Online algorithm for packing rectangles into several strips with guaranteed accuracy estimates, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2007, vol. 12, pp. 7–16.

  53. Zhuk, S.N., On-line algorithms for packing rectangles into several strips, Discrete Math. Appl., 2007, vol. 17, no. 5, pp. 517–531.

    Article  MATH  Google Scholar 

  54. Zhuk, S.N., On-line algorithm for scheduling parallel tasks on a group of related clusters, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2012, vol. 23, pp. 447–454. https://doi.org/10.15514/ISPRAS-2012-23-27

Download references

Author information

Authors and Affiliations

  1. Ivannikov Institute for System Programming, Russian Academy of Sciences, ul. Solzhenitsyna 25, 109004, Moscow, Russia

    D. O. Lazarev & N. N. Kuzyurin

  2. Moscow Institute of Physics and Technology, Institutskii per. 9, 141700, Dolgoprudnyi, Moscow oblast, Russia

    D. O. Lazarev & N. N. Kuzyurin

Authors
  1. D. O. Lazarev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. N. Kuzyurin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to D. O. Lazarev or N. N. Kuzyurin.

Additional information

Translated by Yu. Kornienko

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lazarev, D.O., Kuzyurin, N.N. On Online Algorithms for Bin, Strip, and Box Packing, and Their Worst-Case and Average-Case Analysis. Program Comput Soft 45, 448–457 (2019). https://doi.org/10.1134/S0361768819080036

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768819080036

Search

Navigation