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Lower bounds for on-line scheduling with precedence constraints on identical machines

  • Leah Epstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1444)

Abstract

We consider the on-line scheduling problem of jobs with precedence constraints on m parallel identical machines. Each job has a time processing requirement, and may depend on other jobs (has to be processed after them). A job arrives only after its predecessors have been completed. The cost of an algorithm is the time that the last job is completed. We show lower bounds on the competitive ratio of on-line algorithms for this problem in several versions. We prove a lower bound of 2 − 1/m on the competitive ratio of any deterministic algorithm (with or without preemption) and a lower bound of 2 − 2/(m + 1) on the competitive ratio of any randomized algorithm (with or without preemption). The lower bounds for the cases that preemption is allowed require arbitrarily long sequences. If we use only sequences of length O(m 2), we can show a lower bound of 2 − 2/(m + 1) on the competitive ratio of deterministic algorithms with preemption, and a lower bound of 2 − O(1/m) on the competitive ratio of any randomized algorithm with preemption. All the lower bounds hold even for sequences of unit jobs only. The best algorithm that is known for this problem is the well known List Scheduling algorithm of Graham. The algorithm is deterministic and does not use preemption. The competitive ratio of this algorithm is 2 − 1/m. Our randomized lower bounds are very close to this bound (a difference of 0(1/m)) and our deterministic lower bounds match this bound.

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References

  1. 1.
    B. Chen and A. Vestjens. Scheduling on identical machines: How good is lpt in an on-line setting? To appear in Oper. Res. Lett. Google Scholar
  2. 2.
    Fabian A. Chudak, David B. Shmoys. Approximation algorithms for precedence constrained scheduling problems on parallel machines that run at different speeds. Proc. of the 8th Ann. ACM-SIAM Symp. on Discrete Algorithms, 581–590, 1997.Google Scholar
  3. 3.
    E. Davis and J. M. Jaffe. Algorithms for scheduling tasks on unrelated processors. J. ACM. 28(4):721–736, 1981.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Feldmann, M.-Y. Kao, J. Sgall and S.-H. Teng. Optimal online scheduling of parallel jobs with dependencies. Proc. of the 25th Ann. ACM symp. on Theory of Computing, pages 642–651.Google Scholar
  5. 5.
    A. Feldmann, B. Maggs, J. Sgall, D. D. Sleator and A. Tomkins. Competitive analysis of call admission algorithms that allow delay. Technical Report CMU-CS-95-102, Carnegie-Mellon University, Pittsburgh, PA, U.S.A., 1995.Google Scholar
  6. 6.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness, W. H. Freeman, New-York, 1979.Google Scholar
  7. 7.
    T. Gonzalez and D. B. Johnson. A new algorithm for preemptive scheduling of trees. J. Astoc. Comput. Mach., 27:287–312, 1980.MATHMathSciNetGoogle Scholar
  8. 8.
    R.L. Graham. Bounds for certain multiprocessor anomalies. Bell System Technical Journal, 45:1563–1581, 1966.Google Scholar
  9. 9.
    R.L. Graham. Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math, 17:263–269, 1969.Google Scholar
  10. 10.
    K. S. Hong and J. Y.-T. Leung. On-line scheduling of real-time tasks. IEEE Transactions on Computers, 41(10):1326–1331, 1992.CrossRefGoogle Scholar
  11. 11.
    Jeffrey M. Jaffe Efficient scheduling of tasks without full use of processor resources. The. Computer Science, 12:1–17, 1980.MATHMathSciNetGoogle Scholar
  12. 12.
    J.W.S. Liu and C.L. Liu. Bounds on scheduling algorithms for heterogeneous computing systems. Information Processing 74, North Holland, 349–353,1974.Google Scholar
  13. 13.
    S. Sahni and Y. Cho. Nearly on line scheduling of a uniform processor system with release times. Siam J. Comput. 8(2):275–285, 1979.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Sgall. On-Line Scheduling — A Survey 1997Google Scholar
  15. 15.
    D. B. Shmoys, J. Wein and D. P. Williamson. Scheduling parallel machines on line. Siam J. of Computing, 24:1313–1331, 1995.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    A. P. A. Vestjens. Scheduling uniform machines on-line requires nondecreasing speed ratios. Technical Report Memorandum COSOR 94-35, Eindhoven University of Technology, 1994. To appear in Math. Programming.Google Scholar
  17. 17.
    A. P. A. Vestjens. On-line Machine Scheduling. Ph.D. thesis, Eindhoven University of Technology, The Netherlands, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Leah Epstein
    • 1
  1. 1.Dept. of Computer ScienceTel-Aviv UniversityIsrael

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