On two dimensional packing

  • Yossi Azar
  • Leah Epstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1097)


The paper considers packing of rectangles into an infinite bin. Similar to the Tetris game, the rectangles arrive from the top and, once placed, cannot be moved again. The rectangles are moved inside the bin to reach their place. For the case in which rotations are allowed, we design an algorithm whose performance ratio is constant. In contrast, if rotations are not allowed, we show that no algorithm of constant ratio exists. For this case we design an algorithm with performance ratio of O(log 1/ɛ), where ɛ is the minimum width of any rectangle. We also show that no algorithm can achieve a better ratio than Ω(√log 1/ɛ) for this case.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B.S. Baker, E.G. Coffman, Jr, and R.L. Rivest. Orthogonal packings in two dimensions. In The SIAM Journal of Computing, pages 846–855, 1980.Google Scholar
  2. 2.
    Y. Bartal, A. Fiat, H. Karloff, and R. Vohra. New algorithms for an ancient scheduling problem. In Proc. 24th ACM Symp. on Theory of Computing, pages 51–58, 1992.Google Scholar
  3. 3.
    Y. Bartal, H. Karloff, Y. Rabani. A better lower bound for on-line scheduling. In Information Processing Letters 50 pages 113–116, 1994.Google Scholar
  4. 4.
    B. Chen, and A. van Vliet. On the on-line scheduling algorithm RLS Report 9325/A, Econometric Institute, Erasmus University, Rotterdam. 1993.Google Scholar
  5. 5.
    B. Chen, A. van Vliet, and G.J. Woeginger. New lower and upper bounds for on-line scheduling. In Operations Research Letters, pages 222–230, 1994.Google Scholar
  6. 6.
    J. Csirik, J.B.G. Frenk, and M. Labbe. Two-dimensional rectangle packing: on-line methods and results. In Discrete applied Mathematics 45, pages 197–204, 1993.Google Scholar
  7. 7.
    E.G. Coffman, Jr, M.R. Garey, D.S. Johnson, and R.E. Tarjan. Performance bounds for level oriented two-dimensional packing Algorithms. In The SIAM Journal of Computing, pages 808–826, 1980.Google Scholar
  8. 8.
    D. Coppersmith, and P. Raghavan. Multidimensional on-line bin packing: algorithms and worst-case analysis. In Operation Research Letters, pages 17–20, 1989.Google Scholar
  9. 9.
    G. Galambos. A 1.6 lower bound for the two-dimensional on-line rectangle bin packing In Acta Cybernetica, pages 21–24, 1991.Google Scholar
  10. 10.
    G. Galambos, A. van Vliet. Lower bounds for 1-,2-and 3-dimensional on-line bin packing algorithms. In Computing, pages 281–297, 1994.Google Scholar
  11. 11.
    G. Galambos, and G.J. Woeginger. An on-line scheduling heuristic with better worst case than Graham's list scheduling. In SIAM J. Comput. 22, pages 349–355, 1993.Google Scholar
  12. 12.
    R.L. Graham. Bounds for certain multiprocessing anomalies. In Bell System Tech J. 45, pages 1563–1581, 1966.Google Scholar
  13. 13.
    D.S. Johnson, A. Demers, J.D. Ullman, M.R. Garey, and R.L. Graham. Worst-case performance bounds for simple one-dimensional packing algorithms. In The SIAM Journal of Computing, pages 299–325, 1974.Google Scholar
  14. 14.
    H. Karloff. Personal communication.Google Scholar
  15. 15.
    D.R. Karger, S.J. Phillips, and E. Torng. A better algorithm for an ancient scheduling problem. In Proc. 5 Ann. ACM-SIAM symposium on Discrete Algorithms, pages 132–140, 1994.Google Scholar
  16. 16.
    F. M. Liang. A lower bound for on-line bin packing. In Information Processing letters, pages 76–79, 1980.Google Scholar
  17. 17.
    A. van Vliet. Lower and upper bounds for on-line bin packing and scheduling heuristics. Ph.D. thesis. Thesis publishers, Amsterdam, 1995.Google Scholar
  18. 18.
    A.C. Yao. New algorithms for bin packing. In Journal of the ACM, pages 207–227, 1980.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Yossi Azar
    • 1
  • Leah Epstein
    • 2
  1. 1.Dept. of Computer ScienceTel-Aviv UniversityUSA
  2. 2.Dept. of Computer ScienceTel-Aviv UniversityUSA

Personalised recommendations