Online Makespan Minimization with Parallel Schedules

  • Susanne Albers
  • Matthias Hellwig
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)


Online makespan minimization is a classical problem in which a sequence of jobs σ = J1, …, Jn has to be scheduled on m identical parallel machines so as to minimize the maximum completion time of any job. In this paper we investigate the problem in a model where extra power/resources are granted to an algorithm. More specifically, an online algorithm is allowed to build several schedules in parallel while processing σ. At the end of the scheduling process the best schedule is selected. This model can be viewed as providing an online algorithm with extra space, which is invested to maintain multiple solutions.

As a main result we develop a (4/3+ε)-competitive algorithm, for any 0 < ε ≤ 1, that uses a constant number of schedules. The constant is equal to 1/εO(log(1/ε)). We also give a (1 + ε)-competitive algorithm, for any 0 < ε ≤ 1, that builds a polynomial number of (m/ε)O(log(1/ε) / ε) schedules. This value depends on m but is independent of the input σ. The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4/3 must construct Ω(m) schedules. On the technical level, our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of σ to within a factor of 1 + ε and (2) guess the job processing times and their frequencies in σ. In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant.


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  1. 1.
    Albers, S.: Better bounds for online scheduling. SIAM J. Comput. 29, 459–473 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Albers, S., Hellwig, M.: On the value of job migration in online makespan minimization. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 84–95. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Angelelli, E., Nagy, A.B., Speranza, M.G., Tuza, Z.: The on-line multiprocessor scheduling problem with known sum of the tasks. J. Scheduling 7, 421–428 (2004)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Angelelli, E., Speranza, M.G., Tuza, Z.: Semi-on-line scheduling on two parallel processors with an upper bound on the items. Algorithmica 37, 243–262 (2003)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Angelelli, E., Speranza, M.G., Tuza, Z.: New bounds and algorithms for on-line scheduling: two identical processors, known sum and upper bound on the tasks. Discrete Mathematics & Theoretical Computer Science 8, 1–16 (2006)MATHMathSciNetGoogle Scholar
  6. 6.
    Azar, Y.: On-line load balancing. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms 1996. LNCS, vol. 1442, pp. 178–195. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Azar, Y., Regev, O.: On-line bin-stretching. Theor. Comput. Sci. 268, 17–41 (2001)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Azar, Y., Epstein, L., van Stee, R.: Resource augmentation in load balancing. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 189–199. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Bartal, Y., Karloff, H., Rabani, Y.: A better lower bound for on-line scheduling. Infomation Processing Letters 50, 113–116 (1994)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bartal, Y., Fiat, A., Karloff, H., Vohra, R.: New algorithms for an ancient scheduling problem. Journal of Computer and System Sciences 51, 359–366 (1995)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R.: On the advice complexity of the k-server problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 207–218. Springer, Heidelberg (2011)Google Scholar
  12. 12.
    Cheng, T.C.E., Kellerer, H., Kotov, V.: Semi-on-line multiprocessor scheduling with given total processing time. Theor. Comput. Sci. 337, 134–146 (2005)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theor. Comput. Sci. 2412(24), 2642–2656 (2011)CrossRefGoogle Scholar
  14. 14.
    Englert, M., Özmen, D., Westermann, M.: The power of reordering for online minimum makespan scheduling. In: Proc. 49th IEEE FOCS, pp. 603–612 (2008)Google Scholar
  15. 15.
    Faigle, U., Kern, W., Turan, G.: On the performance of on-line algorithms for partition problems. Acta Cybernetica 9, 107–119 (1989)MATHMathSciNetGoogle Scholar
  16. 16.
    Fleischer, R., Wahl, M.: Online scheduling revisited. J. Scheduling 3, 343–353 (2000)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Galambos, G., Woeginger, G.: An on-line scheduling heuristic with better worst case ratio than Graham’s list scheduling. SIAM J. Comput. 22, 349–355 (1993)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Graham, R.L.: Bounds for certain multi-processing anomalies. Bell System Technical Journal 45, 1563–1581 (1966)CrossRefGoogle Scholar
  19. 19.
    Gormley, T., Reingold, N., Torng, E., Westbrook, J.: Generating adversaries for request-answer games. In: Proc. 11th ACM-SIAM SODA, pp. 564–565 (2000)Google Scholar
  20. 20.
    Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: Theoretical and practical results. J. ACM 34, 144–162 (1987)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Karger, D.R., Phillips, S.J., Torng, E.: A better algorithm for an ancient scheduling problem. Journal of Algorithms 20, 400–430 (1996)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Kellerer, H., Kotov, V., Speranza, M.G., Tuza, Z.: Semi on-line algorithms for the partition problem. Operations Research Letters 21, 235–242 (1997)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Raghavan, P., Snir, M.: Memory versus randomization in on-line algorithms. IBM Journal of Research and Development 38, 683–708 (1994)CrossRefGoogle Scholar
  24. 24.
    Renault, M.P., Rosén, A., van Stee, R.: Online Algorithms with advice for bin packing and scheduling problems. CoRR abs/1311.7589 (2013)Google Scholar
  25. 25.
    Rudin III., J.F.: Improved bounds for the on-line scheduling problem. Ph.D. Thesis (2001)Google Scholar
  26. 26.
    Rudin III., J.F., Chandrasekaran, R.: Improved bounds for the online scheduling problem. SIAM J. Comput. 32, 717–735 (2003)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Sanders, P., Sivadasan, N., Skutella, M.: Online scheduling with bounded migration. Mathematics of Operations Reseach 34(2), 481–498 (2009)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Communications of the ACM 28, 202–208 (1985)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Susanne Albers
    • 1
  • Matthias Hellwig
    • 2
  1. 1.Technische Universität MünchenGermany
  2. 2.Humboldt-Universität zu BerlinGermany

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