Online Packing of Rectangular Items into Square Bins

  • Janusz Januszewski
  • Łukasz ZielonkaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10787)


Any list of rectangular items of total area not greater than 0.2837 can be packed online into the unit square (\(90^{\circ }\)-rotations are allowed). Furthermore, we describe a 4.84-competitive 1-space bounded 2-dimensional bin packing algorithm and present the lower bound of 3.246 for the competitive ratio.


Online Algorithms Competitive analysis Two dimensional bin packing 


  1. 1.
    Brubach, B.: Improved online square-into-square packing.
  2. 2.
    Brubach, B.: Improved bound for online square-into-square packing. In: Bampis, E., Svensson, O. (eds.) WAOA 2014. LNCS, vol. 8952, pp. 47–58. Springer, Cham (2015). Scholar
  3. 3.
    Zhang, Y., Chen, J., Chin, F.Y.L., Han, X., Ting, H.-F., Tsin, Y.H.: Improved online algorithms for 1-space bounded 2-dimensional bin packing. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6507, pp. 242–253. Springer, Heidelberg (2010). Scholar
  4. 4.
    Chin, F.Y.L., Han, X., Poon, C.K., Ting, H.-F., Tsin, Y.H., Ye, D., Zhang, Y.: Online algorithms for 1-space bounded 2-dimensional bin packing and square packing. Theor. Comput. Sci. 554, 135–149 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chin, F.Y.L., Ting, H.-F., Zhang, Y.: 1-space bounded algorithms for 2-dimensional bin packing. In: Proceedings of the 20th Annual International Symposium on Algorithms and Computation (ISAAC), pp. 321–330 (2009)Google Scholar
  6. 6.
    Chin, F.Y.L., Ting, H.-F., Zhang, Y.: One-space bounded algorithms for two-dimensional bin packing. Int. J. Found. Comput. Sci. 21(6), 875–891 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Epstein, L., van Stee, R.: Optimal online algorithms for multidimensional packing problems. SIAM J. Comput. 35(2), 431–448 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fekete, S.P., Hoffmann, H.-F.: Online square-into-square packing. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds.) APPROX/RANDOM -2013. LNCS, vol. 8096, pp. 126–141. Springer, Heidelberg (2013). Scholar
  9. 9.
    Grzegorek, P., Januszewski, J.: Online algorithms for \(3\)-space bounded \(2\)-dimensional bin packing and square packing. Rom. J. Inf. Sci. Technol. 17(2), 189–202 (2014)Google Scholar
  10. 10.
    Grzegorek, P., Januszewski, J.: A note on one-space bounded square packing. Inf. Process. Lett. 115, 872–876 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Han, X., Iwama, K., Zhang, G.: Online removable square packing. Theory Comput. Syst. 43(1), 38–55 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Januszewski, J.: Packing rectangles into the unit square. Geom. Dedicata 8, 13–18 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Januszewski, J., Lassak, M.: On-line packing sequences of cubes in the unit cube. Geom. Dedicata 67, 285–293 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Januszewski, J., Zielonka, Ł.: Improved online algorithms for 2-space bounded 2-dimensional bin packing. Int. J. Found. Comput. Sci. 27(4), 407–429 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Johnson, D.S.: Fast algorithms for bin packing. J. Comput. Syst. Sci. 8, 272–314 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lassak, M.: On-line potato-sack algorithm efficient for packing into small boxes. Periodica Math. Hung. 34, 105–110 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lee, C.C., Lee, D.T.: A simple on-line bin packing algorithm. J. ACM 32, 562–572 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Moser, L.: Poorly formulated unsolved problems of combinatorial geometry. Mimeographed (1966)Google Scholar
  19. 19.
    Moon, J., Moser, L.: Some packing and covering theorems. Colloq. Math. 17, 103–110 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhao, X., Shen, H.: On-line algorithms for 2-space bounded cube and hypercube packing. Tsinghua Sci. Technol. 20(3), 255–263 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and PhysicsUTP University of Science and TechnologyBydgoszczPoland

Personalised recommendations