Probabilistic Analysis of Online Stacking Algorithms

  • Martin OlsenEmail author
  • Allan Gross
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9335)


Consider the situation where some items arrive to a storage location where they are temporarily stored in bounded capacity LIFO stacks until their departure. We consider the problem of deciding where to put an arriving item with the objective of using as few stacks as possible. The decision has to be made as soon as an item arrives and we assume that we only have information on the departure times for the arriving item and the items currently at the storage area. We are only allowed to put an item on top of another item if the item below departs at a later time. We assume that the numbers defining the storage time intervals are picked independently and uniformly at random from the interval [0, 1]. We present a simple polynomial time online algorithm for the problem and prove the following: For any positive real numbers \(\epsilon _1, \epsilon _2 > 0\) there exists an \(N > 0\) such that the algorithm uses no more than \((1+\epsilon _1)OPT\) stacks with probability at least \(1-\epsilon _2\) if the number of items is at least N where OPT denotes the optimal number of stacks. The result even holds if the stack capacity is \(o(\sqrt{n})\) where n is the number of items.


Probabilistic Analysis Competitive Ratio Online Algorithm Storage Location Interval Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldous, D., Diaconis, P.: Longest increasing subsequences: From patience sorting to the baik-deift-johansson theorem. Bull. Amer. Math. Soc. 36, 413–432 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avriel, M., Penn, M., Shpirer, N.: Container ship stowage problem: complexity and connection to the coloring of circle graphs. Discrete Applied Mathematics 103(1–3), 271–279 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borgman, B., van Asperen, E., Dekker, R.: Online rules for container stacking. OR Spectrum 32(3), 687–716 (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cornelsen, S., Di Stefano, G.: Track assignment. Journal of Discrete Algorithms 5(2), 250–261 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Demange, M., Di Stefano, G., Leroy-Beaulieu, B.: On the online track assignment problem. Discrete Applied Mathematics 160(7–8), 1072–1093 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jansen, K.: The mutual exclusion scheduling problem for permutation and comparability graphs. Information and Computation 180(2), 71–81 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kobayashi, H., Mark, B.L., Turin, W.: Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance. Cambridge University Press (2012)Google Scholar
  8. 8.
    König, F.G., Lübbecke, M., Möhring, R.H., Schäfer, G., Spenke, I.: Solutions to real-world instances of PSPACE-complete stacking. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 729–740. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  9. 9.
    Pacino, D., Jensen, R.M.: Fast Generation of Container Vessel Stowage Plans: using mixed integer programming for optimal master planning and constraint based local search for slot planning. PhD thesis, IT University of Copenhagen (2012)Google Scholar
  10. 10.
    Pilpel, S.: Descending subsequences of random permutations. Journal of Combinatorial Theory, Series A 53(1), 96–116 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rei, R.J., Pedroso, J.P.: Tree search for the stacking problem. Annals OR 203(1), 371–388 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Scheinerman, E.R.: Random interval graphs. Combinatorica 8(4), 357–371 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Tierney, K., Pacino, D., Jensen, R.M.: On the complexity of container stowage planning problems. Discrete Applied Mathematics 169, 225–230 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wang, N., Zhang, Z., Lim, A.: The stowage stack minimization problem with zero rehandle constraint. In: Ali, M., Pan, J.-S., Chen, S.-M., Horng, M.-F. (eds.) IEA/AIE 2014, Part II. LNCS, vol. 8482, pp. 456–465. Springer, Heidelberg (2014) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Business Development and TechnologyAarhus UniversityAarhusDenmark

Personalised recommendations