, Volume 78, Issue 2, pp 492–520 | Cite as

Online Makespan Minimization with Parallel Schedules

  • Susanne AlbersEmail author
  • Matthias Hellwig


Online makespan minimization is a classical problem in which a sequence of jobs \(\sigma = J_1, \ldots , J_n\) 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 with an essentially new model of resource augmentation. More specifically, an online algorithm is allowed to build several schedules in parallel while processing \(\sigma \). 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. The setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions. As a main result we develop a \((4/3+\varepsilon )\)-competitive algorithm, for any \(0<\varepsilon \le 1\), that uses a constant number of schedules. The constant is equal to \(1/\varepsilon ^{O(\log (1/\varepsilon ))}\). We also give a \((1+\varepsilon )\)-competitive algorithm, for any \(0<\varepsilon \le 1\), that builds a polynomial number of \((m/\varepsilon )^{O(\log (1/\varepsilon ) / \varepsilon )}\) schedules. This value depends on m but is independent of the input \(\sigma \). The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4 / 3 must construct \(\Omega (m)\) schedules. Our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of a job sequence \(\sigma \) to within a factor of \(1+\varepsilon \) and (2) guess the job processing times and their frequencies in \(\sigma \). In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant. We remark that the competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation. Furthermore they are at least as good and in most cases better than the ratios obtained with other means of resource augmentation for makespan minimization.


Scheduling Makespan minimization Online algorithms Competitive analysis 



We thank two anonymous referees for helpful comments improving the presentation of this paper.


  1. 1.
    Albers, S.: Better bounds for online scheduling. SIAM J. Comput. 29, 459–473 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albers, S., Hellwig, M.: On the value of job migration in online makespan minimization. In: Proceedings of the 20th Annual European Symposium on Algorithms, Springer LNCS 7501, pp. 84–95 (2012)Google 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. Sched. 7, 421–428 (2004)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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 Math. Theor. Comput. Sci. 8, 1–16 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Azar, Y.: On-line load balancing. In: Fiat, A., Woeginger, G. (eds.) Online Algorithms: The State of the Art, vol. 1441, pp. 178–195. Springer, New York (1998)CrossRefGoogle Scholar
  7. 7.
    Azar, Y., Regev, O.: On-line bin-stretching. Theor. Comput. Sci. 268, 17–41 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bartal, Y., Karloff, H., Rabani, Y.: A better lower bound for on-line scheduling. Inf. Process. Lett. 50, 113–116 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bartal, Y., Fiat, A., Karloff, H., Vohra, R.: New algorithms for an ancient scheduling problem. J. Comput. Syst. Sci. 51, 359–366 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bender, M.A., Slonim, D.K.: The Power of team exploration: two robots can learn unlabeled directed graphs. In: 35th Annual Symposium on Foundations of Computer Science, pp. 75–85 (1994)Google Scholar
  11. 11.
    Blum, A., Chalasani, P.: An online algorithm for improving performance in navigation. SIAM J. Comput. 29, 1907–1938 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Blum, A., Raghavan, P., Schieber, B.: Navigating in unfamiliar geometric terrain. SIAM J. Comput. 26, 110–137 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Böckenhauer, H.J., Komm, D., Královic, R., Královic, R.: On the advice complexity of the k-server problem. In: Proceedings of the 38th International Colloquium on Automata, Languages and Programming, Springer LNCS 6755, pp. 207–218 (2011)Google Scholar
  14. 14.
    Böhm, M., Sgall, J., van Stee, R., Veselý, P.: Better algorithms for online bin stretching. In: Proceedings of the 12th International Workshop on Approximation and Online Algorithms, Springer LNCS 8952, pp. 23–34 (2014)Google Scholar
  15. 15.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Deng, X., Kameda, T., Papadimitriou, C.H.: How to learn an unknown environment I: the rectilinear case. J. ACM 45, 215–245 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theor. Comput. Sci. 2412(24), 2642–2656 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Englert, M., Özmen, D., Westermann, M.: The power of reordering for online minimum makespan scheduling. SIAM J. Comput. 43(3), 1220–1237 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Faigle, U., Kern, W., Turan, G.: On the performance of on-line algorithms for partition problems. Acta Cybern. 9, 107–119 (1989)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Feller, W.: An Introduction to Probability Theory and its Applications. Wiley, New York (1968)zbMATHGoogle Scholar
  21. 21.
    Fleischer, R., Wahl, M.: Online scheduling revisited. J. Sched. 3, 343–353 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fraigniaud, P., Gasieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48, 166–177 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gabay, M., Kotov, V., Brauner, N.: Semi-online bin stretching with bunch tech- niques. HAL preprint hal-00869858 (2013)Google Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Graham, R.L.: Bounds for certain multi-processing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)CrossRefzbMATHGoogle Scholar
  26. 26.
    Gormley, T., Reingold, N., Torng, E., Westbrook, J.: Generating adversaries for request-answer games. In: Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 564–565 (2000)Google Scholar
  27. 27.
    Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. J. ACM 34, 144–162 (1987)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Irani, S.: Coloring inductive graphs on-line. Algorithmica 11, 53–72 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Karger, D.R., Phillips, S.J., Torng, E.: A better algorithm for an ancient scheduling problem. J. Algorithms 20, 400–430 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pp. 352–358 (1990)Google Scholar
  31. 31.
    Kellerer, H., Kotov, V.: An efficient algorithm for bin stretching. Oper. Res. Lett. 41(4), 343–346 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kellerer, H., Kotov, V., Speranza, M.G., Tuza, Z.: Semi on-line algorithms for the partition problem. Oper. Res. Lett. 21, 235–242 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    López-Ortiz, A., Schuierer, S.: On-line parallel heuristics, processor scheduling and robot searching under the competitive framework. Theor. Comput. Sci. 310, 527–537 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lovász, L., Saks, M.E., Trotter, W.A.: An on-line graph coloring algorithm with sublinear performance ratio. Discrete Math. 75, 319–325 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Raghavan, P., Snir, M.: Memory versus randomization in on-line algorithms. IBM J. Res. Dev. 38, 683–708 (1994)CrossRefGoogle Scholar
  36. 36.
    Renault, M.P., Rosén, A., van Stee, R.: Online Algorithms with advice for bin packing and scheduling problems. Theor. Comput. Sci. 600, 155–170 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Rudin III, J.F.: Improved bounds for the on-line scheduling problem. Ph.D. Thesis, The University of Texas at Dallas, May (2001)Google Scholar
  38. 38.
    Rudin III, J.F., Chandrasekaran, R.: Improved bounds for the online scheduling problem. SIAM J. Comput. 32, 717–735 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sanders, P., Sivadasan, N., Skutella, M.: Online scheduling with bounded migration. Math. Oper. Res. 34(2), 481–498 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28, 202–208 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnische Universität MünchenGarchingGermany
  2. 2.Department of Computer ScienceHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations