Abstract
Our main result is an optimal online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize makespan. The algorithm is deterministic, yet it is optimal even among all randomized algorithms. In addition, it is optimal for any fixed combination of speeds of the machines, and thus our results subsume all the previous work on various special cases. Together with a new lower bound it follows that the overall competitive ratio of this optimal algorithm is between 2.054 and e≈2.718. We also give a complete analysis of the competitive ratio for three machines.
Similar content being viewed by others
References
Albers, S.: On randomized online scheduling. In: Proc. 34th Symp. Theory of Computing (STOC), pp. 134–143. ACM, New York (2002)
Berman, P., Charikar, M., Karpinski, M.: On-line load balancing for related machines. J. Algorithms 35, 108–121 (2000)
Chen, B., van Vliet, A., Woeginger, G.J.: Lower bounds for randomized online scheduling. Inf. Process. Lett. 51, 219–222 (1994)
Chen, B., van Vliet, A., Woeginger, G.J.: An optimal algorithm for preemptive on-line scheduling. Oper. Res. Lett. 18, 127–131 (1995)
Ebenlendr, T., Sgall, J.: Optimal and online preemptive scheduling on uniformly related machines. In: Proc. 21st Symp. on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Comput. Sci., vol. 2996, pp. 199–210. Springer, Berlin (2004)
Ebenlendr, T., Sgall, J.: Online preemptive scheduling on four uniformly related machines. In preparation
Epstein, L.: Optimal preemptive scheduling on uniform processors with non-decreasing speed ratios. Oper. Res. Lett. 29, 93–98 (2001)
Epstein, L., Noga, J., Seiden, S.S., Sgall, J., Woeginger, G.J.: Randomized on-line scheduling for two uniform machines. J. Sched. 4, 71–92 (2001)
Epstein, L., Sgall, J.: A lower bound for on-line scheduling on uniformly related machines. Oper. Res. Lett. 26(1), 17–22 (2000)
Fleischer, R., Wahl, M.: On-line scheduling revisited. J. Sched. 3, 343–353 (2000)
Gonzales, T.F., Sahni, S.: Preemptive scheduling of uniform processor systems. J. ACM 25, 92–101 (1978)
Graham, R., Lawler, E., Lenstra, J., Kan, A.R.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Ann. Discrete Math. 5, 287–326 (1979)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)
Hochbaum, D.S., Shmoys, D.: A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach. SIAM J. Comput. 17, 539–551 (1988)
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. J. ACM 34, 144–162 (1987)
Horwath, E., Lam, E.C., Sethi, R.: A level algorithm for preemptive scheduling. J. ACM 24, 32–43 (1977)
McNaughton, R.: Scheduling with deadlines and loss functions. Manag. Sci. 6, 1–12 (1959)
Rudin, J.F. III: Improved bound for the online scheduling problem. PhD thesis, The University of Texas at Dallas (2001)
Sgall, J.: A lower bound for randomized on-line multiprocessor scheduling. Inf. Process. Lett. 63, 51–55 (1997)
Tichý, T.: Randomized on-line scheduling on 3 processors. Oper. Res. Lett. 32, 152–158 (2004)
Wen, J., Du, D.: Preemptive on-line scheduling for two uniform processors. Oper. Res. Lett. 23, 113–116 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
T. Ebenlendr and J. Sgall partially supported by Institutional Research Plan No. AV0Z10190503, by Inst. for Theor. Comp. Sci., Prague (project 1M0545 of MŠMT ČR), grant 201/05/0124 of GA ČR, and grant IAA1019401 of GA AV ČR.
W. Jawor supported by NSF grants CCF-0208856 and OISE-0340752.
Rights and permissions
About this article
Cite this article
Ebenlendr, T., Jawor, W. & Sgall, J. Preemptive Online Scheduling: Optimal Algorithms for All Speeds. Algorithmica 53, 504–522 (2009). https://doi.org/10.1007/s00453-008-9235-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-008-9235-6