Optimal Task Scheduling of a Complete K-Ary Tree with Communication Delays

  • Noriyuki Fujimoto
  • Kenichi Hagihara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2328)


It is known that task scheduling problem of a complete k-ary intree with unit time tasks and general communication delays onto an unlimited number of processors is NP-complete. In this paper, we show that such a problem can be solved in linear time if we restrict communication delays within the range from (k − 1) to k unit times. We also show that naive scheduling is optimal if communication delays are constant and at most (k − 1) unit times.


Completion Time Optimal Schedule Task Schedule Task Graph Communication Delay 
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  1. 1.
    Ahmad, I. and Kwok, Y.: On Exploiting Task Duplication in Parallel Program Scheduling, IEEE Trans. on Parallel and Distributed Systems, Vol. 9, No. 9 (1998) 872–892CrossRefGoogle Scholar
  2. 2.
    Blazewicz, J., Guinand, F., Penz, B., and Trystram, D.: Scheduling Complete Trees on Two Uniform Processors with Integer Speed Ratios and Communication Delays, Parallel Processing Letters, Vol. 10, No. 4 (2000) 267–277CrossRefGoogle Scholar
  3. 3.
    Chrétienne, P.: A Polynomial Algorithm to Optimally Schedule Tasks on a Virtual Distributed System under Tree-like Precedence Constraints, European J. Oper. Res. Vol. 43 (1989) 225–230MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chrétienne, P. and Picouleau, C.: Scheduling with Communication Delays: A Survey, Scheduling Theory and Its Applications, Wiley (1995) 65–90Google Scholar
  5. 5.
    Colin, J. Y. and Chritienne, P.: C.P.M. Scheduling with Small Communication Delays and Task Duplication, Operations Research, Vol. 39, No. 4 (1991) 680–684zbMATHCrossRefGoogle Scholar
  6. 6.
    Darbha, S. and Agrawal, D. P.: Optimal Scheduling Algorithm for Distributed-Memory Machines, IEEE Trans. on Parallel and Distributed Systems, Vol. 9, No. 1 (1998) 87–95CrossRefGoogle Scholar
  7. 7.
    El-Rewini, H., Lewis, T.G., and Ali, H.H.: TASK SCHEDULING in PARALLEL and DISTRIBUTED SYSTEMS, PTR Prentice Hall (1994)Google Scholar
  8. 8.
    Gerasoulis, A. and Yang, T.: On the Granularity and Clustering of Directed Acyclic Task Graphs, IEEE Transactions on Parallel and Distributed Systems, Vol. 4, No. 6 (1993) 686–701CrossRefGoogle Scholar
  9. 9.
    Graham, R. L., Lawler, E. L., Lenstra, J. K., and Rinnooy Kan, A. H. G.: Optimization and Approximation in Deterministic Sequencing and Scheduling: A Survey, Ann. Discrete Math. Vol. 5 (1979) 287–326zbMATHCrossRefGoogle Scholar
  10. 10.
    Guinand, F. and Trystram, D.: Optimal Scheduling of UECT Trees on Two Processors, RAIRO Operations Research, Vol. 34, No. 2(2000) 131–144MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Jakoby, A. and Reischuk, R.: The Complexity of Scheduling Problems with Communication Delays for Trees, Lecture Notes in Computer Science, Vol. 621. Springer-Verlag (1992) 165–177Google Scholar
  12. 12.
    Jung, H., Kirousis, L., and Spirakis, P.: Lower bounds and Efficient Algorithms for Multiprocessor Scheduling of Dags with Communication Delays, the Information and Computation Journal, Vol. 105, No. 1 (1993) 94–104MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lawler, E. L.: Scheduling Trees on Multiprocessors with Unit Communication Delays, Presented at the First Workshop on Models and Algorithms for Planning and Scheduling Problems, Villa Vigoni, Lake Como, Italy, unpublished manuscript, June (1993)Google Scholar
  14. 14.
    Lenstra, J. K., Veldhorst, M., and Veltman, B.: The Complexity of Scheduling Trees with Communication Delays, Journal of Algorithms, Vol. 20 (1996) 157–173MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Picouleau, C.: Etude de Problèmes les Systèmes Distribués, Ph.D. thesis, Univ. Piere et Marie Curie, Paris, France (1992)Google Scholar
  16. 16.
    Thurimella, R. and Yesha, Y.: A scheduling principle for precedence graphs with communication delay, International Conference on Parallel Processing, 3 (1992) 229–236Google Scholar
  17. 17.
    Varvarigou, T. A., Roychowdhury, V. P., Kailath, T., and Lawler, E.: Scheduling In and Out Forests in the Presence of Communication Delays, IEEE Trans. on Parallel and Distributed Systems, Vol. 7, No. 10 (1996) 1065–1074CrossRefGoogle Scholar
  18. 18.
    Yang, T. and Gerasoulis, A.: DSC: Scheduling Parallel Tasks on an Unbounded Number of Processors, IEEE Trans. on Parallel and Distributed Systems, Vol. 5, No. 9 (1994) 951–967CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Noriyuki Fujimoto
    • 1
  • Kenichi Hagihara
    • 1
  1. 1.Graduate School of Engineering ScienceOsaka UniversityOsakaJapan

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