Online Stochastic Reordering Buffer Scheduling

  • Hossein Esfandiari
  • MohammadTaghi Hajiaghayi
  • Mohammad Reza Khani
  • Vahid Liaghat
  • Hamid Mahini
  • Harald Räcke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


In this paper we consider online buffer scheduling problems in which an online stream of n items (jobs) with different colors (types) has to be processed by a machine with a buffer of size k. In the standard model initially introduced by Räcke, Sohler, and Westermann [31], the machine chooses an active color and processes items whose color matches that color until no item in the buffer has the active color (note that the buffer is refilled in each step). In the block-operation model, the machine chooses an active color and can–in each step–process all items of that color in the buffer. Motivated by practical applications in real-world, we assume we have prior stochastic information about the input. In particular, we assume that the colors of items are drawn i.i.d. from a possibly unknown distribution, or more generally, the items are coming in a random order. In the random order setting, an adversary determines the color of each item in advance, but then the items arrive in a random order in the input stream. To the best of our knowledge, this is the first work which considers the reordering buffer problem in stochastic settings.

Our main result is demonstrating constant competitive online algorithms for both the standard model and the block operation model in the unknown distribution setting and more generally in the random order setting. This provides a major improvement of the competitiveness of algorithms in stochastic settings; the best competitive ratio in the adversarial setting is Θ(loglogk) for both the standard and the block-operation models by Avigdor-Elgrabli and Rabani [8] and Adamaszek et al. [3]. Along the way, we also show that in the random order setting, designing competitive algorithms with the same competitive ratios (up to constant factors) in both the block operation model and the standard model are equivalent. To the best of our knowledge this is the first result of this type which relates an algorithm for the standard model to an algorithm for the block-operation model. Last but not least, we show in the uniform distribution setting, in which the probabilities of appearances of all colors are the same, a simple greedy algorithm is the best online algorithm in both models.


Input Sequence Competitive Ratio Online Algorithm Buffer Management Active Color 
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.
    Aboud, A.: Correlation clustering with penalties and approximating the reordering buffer management problem. Master’s thesis (2008)Google Scholar
  2. 2.
    Adamaszek, A., Czumaj, A., Englert, M., Räcke, H.: Almost tight bounds for reordering buffer management. In: STOC (2011)Google Scholar
  3. 3.
    Adamaszek, A., Czumaj, A., Englert, M., Räcke, H.: Optimal online buffer scheduling for block devices. In: STOC (2012)Google Scholar
  4. 4.
    Agrawal, S., Wang, Z., Ye, Y.: A dynamic near-optimal algorithm for online linear programming. CoRR (2009)Google Scholar
  5. 5.
    Asahiro, Y., Kawahara, K., Miyano, E.: NP-hardness of the sorting buffer problem on the uniform metric. Discrete Appl. Math. 160(10-11) (2012)Google Scholar
  6. 6.
    Avigdor-Elgrabli, N., Rabani, Y.: An improved competitive algorithm for reordering buffer management. In: SODA (2010)Google Scholar
  7. 7.
    Avigdor-Elgrabli, N., Rabani, Y.: A constant factor approximation algorithm for reordering buffer management. In: SODA (2013)Google Scholar
  8. 8.
    Avigdor-Elgrabli, N., Rabani, Y.: An optimal randomized online algorithm for reordering buffer management. In: FOCS (2013)Google Scholar
  9. 9.
    Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: SODA (2007)Google Scholar
  10. 10.
    Bahmani, B., Kapralov, M.: Improved Bounds for Online Stochastic Matching. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 170–181. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Bansal, N., Buchbinder, N., Naor, J(S.): A primal-dual randomized algorithm for weighted paging. In: FOCS (2007)Google Scholar
  12. 12.
    Bateni, M., Hajiaghayi, M., Zadimoghaddam, M.: Submodular secretary problem and extensions. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010. LNCS, vol. 6302, pp. 39–52. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Blandford, D., Blelloch, G.: Index compression through document reordering. In: DCC (2002)Google Scholar
  14. 14.
    Cai, Y., Daskalakis, C., Weinberg, S.M.: On optimal multidimensional mechanism design. SIGecom Exch. 10(2), 29–33 (2011)CrossRefGoogle Scholar
  15. 15.
    Chan, H.-L., Megow, N., van Stee, R., Sitters, R.: The sorting buffer problem is NP-hard. CoRR (2010)Google Scholar
  16. 16.
    Devanur, N.R., Jain, K., Sivan, B., Wilkens, C.A.: Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In: EC (2011)Google Scholar
  17. 17.
    Devenur, N.R., Hayes, T.P.: The adwords problem: online keyword matching with budgeted bidders under random permutations. In: EC (2009)Google Scholar
  18. 18.
    Dynkin, E.B.: The optimum choice of the instant for stopping a markov process. Soviet Math. Dokl 4 (1963)Google Scholar
  19. 19.
    Englert, M., Westermann, M.: Reordering buffer management for non-uniform cost models. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 627–638. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Feldman, J., Mehta, A., Mirrokni, V., Muthukrishnan, S.: Online stochastic matching: beating 1 − 1/e. In: FOCS (2009)Google Scholar
  21. 21.
    Freeman, P.R.: The secretary problem and its extensions: a review. Inter. Statistical Review 51(2), 189–206 (2011)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: SODA (2008)Google Scholar
  23. 23.
    Hajiaghayi, M.T., Kleinberg, R.D., Leighton, T., Räcke, H.: New lower bounds for oblivious routing in undirected graphs. In: SODA (2006)Google Scholar
  24. 24.
    Hajiaghayi, M.T., Kleinberg, R., Parkes, D.C.: Adaptive limited-supply online auctions. In: EC (2004)Google Scholar
  25. 25.
    Hajiaghayi, M., Kim, J.H., Leighton, T., Räcke, H.: Oblivious routing in directed graphs with random demands. In: STOC (2005)Google Scholar
  26. 26.
    Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching with unknown distributions. In: STOC (2011)Google Scholar
  27. 27.
    Krokowski, J., Räcke, H., Sohler, C., Westermann, M.: Reducing state changes with a pipeline buffer. In: VMV (2004)Google Scholar
  28. 28.
    Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In: STOC (2011)Google Scholar
  29. 29.
    Manshadi, V.H., Gharan, S.O., Saberi, A.: Online stochastic matching: online actions based on offline statistics. In: SODA (2011)Google Scholar
  30. 30.
    Panconesi, A., Srinivasan, A.: Randomized distributed edge coloring via an extension of the chernoff–hoeffding bounds. SIAM Journal on Computing 26(2), 350–368 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Räcke, H., Sohler, C., Westermann, M.: Online scheduling for sorting buffers. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, p. 820. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  32. 32.
    Spieckermann, S., Gutenschwager, K., VoB̈, S.: A sequential ordering problem in automotive paint shops. International journal of production research 42(9), 1865–1878 (2004)CrossRefzbMATHGoogle Scholar
  33. 33.
    Teorey, T.J., PinkertonA, T.B.: comparative analysis of disk scheduling policies. Communications of the ACM 15(3), 177–184 (1972)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hossein Esfandiari
    • 1
  • MohammadTaghi Hajiaghayi
    • 1
  • Mohammad Reza Khani
    • 1
  • Vahid Liaghat
    • 1
  • Hamid Mahini
    • 1
  • Harald Räcke
    • 2
  1. 1.Computer Science DepartmentUniversity of MarylandCollege ParkUSA
  2. 2.Institut für InformatikTechnische Universität MünchenMünchenGermany

Personalised recommendations