On optimal k-iinear scheduling of tree-like task graphs for LogP-machines

  • Wolf Zimmermann
  • Martin Middendorf
  • Weif Löwe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1470)


A k-linear schedule may map up to k directed paths of a task graph onto one processor. We consider k-linear scheduling algorithms for the communication cost model of the LogP-machine, i.e. without assumption on processor bounds. The main result of this paper is that optimal k-linear schedules of trees and tree-like task graphs G with n tasks can be computed in time O(n k+2 log n) and O(n k+3 log n), respectively, if o ≥ g. These schedules satisfy a capacity constraint, i.e., there are at most ⌈L/g⌋ messages in transit from any or to any processor at any time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F.D. Anger, J. Hwang, and Y. Chow. Scheduling with sufficient loosely coupled processors. Journal of Parallel and Distributed Computing, 9:87–92, 1990.CrossRefGoogle Scholar
  2. 2.
    P. Chretienne. A polynomial algorithm to optimally schedule tasks over an ideal distributed system under tree-like precedence constraints. European Journal of Operations Research, 2:225–230, 1989.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Culler, R. Karp, D. Patterson, A. Sahay, K. E. Schauser, E. Santos, R. Subramonian, and T. von Eicken. LogP: Towards a realistic model of parallel computation. In 4th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (PPOPP 93), pp. 1–12, 1993. published in: SIGPLAN Notices (28) 7.Google Scholar
  4. 4.
    D. Culler, R. Karp, D. Patterson, A. Sahay, K. E. Schauser, E. Santos, R. Subramonian, and T. von Eicken. LogP: A practical model of parallel computation. Communications of the ACM, 39(11):78–85, 1996.CrossRefGoogle Scholar
  5. 5.
    S. Darbha and D.P. Agrawal. SBDS: A task duplication based optimal algorithm. In Scalable High Performance Conference, 1994.Google Scholar
  6. 6.
    B. Di Martino and G. Ianello. Parallelization of non-simultaneous iterative methods for systems of linear equations. In Parallel Processing: CONPAR 94 — VAPP VI, volume 854 of Lecture Notes in Computer Science, pp. 253–264. Springer, 1994.Google Scholar
  7. 7.
    J. Eisenbiegler, W. Löwe, and W. Zimmermann. Optimizing parallel programs on machines with expensive communication. In Europar’ 96 Parallel Processing Vol. 2, volume 1124 of Lecture Notes in Computer Science, pp. 602–610. Springer, 1996.Google Scholar
  8. 8.
    Jrn Eisenbiegler, Welf Löwe, and Andreas Wehrenpfennig. On the optimization by redundancy using an extended LogP model. In International Conference on Advances in Parallel and Distributed Computing (APDC’97), pp. 149–155. IEEE Computer Society Press, 1997.Google Scholar
  9. 9.
    A. Gerasoulis and T. Yang. On the granularity and clustering of directed acyclic task graphs. IEEE Transactions on Parallel and Distributed Systems, 4:686–701, June 1993.CrossRefGoogle Scholar
  10. 10.
    J.A. Hoogreven, J.K. Lenstra, and B. Veltmann. Three, four, five, six or the complexity of scheduling with communication delays. Operations Research Letters, 16:129–137, 1994.MathSciNetCrossRefGoogle Scholar
  11. 11.
    H. Jung, L. M. Kirousis, and P. Spirakis. Lower bounds and efficient algorithms for multiprocessor scheduling of directed acyclic graphs with communication delays. Information and Computation, 105:94–104, 1993.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    I. Kort and D. Trystram. Scheduling fork graphs under LogP with an unbounded number of processors. Submitted to Europar ’98, 1998.Google Scholar
  13. 13.
    E.L. Lawler. Optimal sequencing of a single machine subject to precedence constraints. Management Science, 19:544–546, 1973.MATHCrossRefGoogle Scholar
  14. 14.
    J.K. Lenstra, M. Veldhorst, and B. Veltmann. The complexity of scheduling trees with communication delays. Journal of Algorithms, 20:157–173, 1996.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    W. Löwe and W. Zimmermann. On finding optimal clusterings in task graphs. In N. Mirenkov, editor, Parallel Algorithms/Architecture Synthesis pAs’95, pp. 241–247. IEEE, 1995.Google Scholar
  16. 16.
    W. Löwe and W. Zimmermann. Programming data-parallel — executing process parallel. In P. Fritzson and L. Finmo, editors, Parallel Programming and Applications, pp. 50–64. IOS Press, 1995.Google Scholar
  17. 17.
    W. Lwe and W. Zimmermann. Upper time bounds for executing PRAM-programs on the LogP-machine. In M. Wolfe, editor, Proceedings of the 9th ACM International Conference on Supercomputing, pp. 41–50. ACM, 1995.Google Scholar
  18. 18.
    Welf Löwe, Wolf Zimmermann, and Jörn Eisenbiegler. On linear schedules for task graphs for generalized LogP-machines. In Europar’97: Parallel Processing, volume 1300 of Lecture Notes in Computer Science, pp. 895–904, 1997.MATHGoogle Scholar
  19. 19.
    W. Lwe, J. Eisenbiegler, and W. Zimmermann. Optimizing parallel programs on machines with fast communication. In 9. International Conference on Parallel and Distributed Computing Systems, pp. 100–103, 1996.Google Scholar
  20. 20.
    M. Middendorf, W. Löwe, and W. Zimmermann. Scheduling inverse trees under the communication model of the LogP-machine. Accepted for publication in Theoretical Computer Science.Google Scholar
  21. 21.
    C.H. Papadimitriou and M. Yannakakis. Towards an architecture-independent analysis of parallel algorithms. SIAM Journal on Computing, 19(2):322–328, 1990.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    C. Picouleau. New complexity results on the uet-uct scheduling algorithm. In Proc. Summer School on Scheduling Theory and its Applications, pp. 187–201, 1992.Google Scholar
  23. 23.
    J. Siddhiwala and L.-F. Cha. Path-based task replication for scheduling with communication cost. In Proceedings of the International Conference on Parallel Processing, volume II, pp. 186–190, 1995.Google Scholar
  24. 24.
    J. Verriet. Scheduling tree-structured programs in the LogP-model. Technical Report UU-CS-1997-18, Dept. of Computer Science, Utrecht University, 1997.Google Scholar
  25. 25.
    T. Yang and A. Gerasoulis. DSC: Scheduling parallel tasks on an unbounded number of processors. IEEE Transactions on Parallel and Distributed Systems, 5(9):951–967, 1994.CrossRefGoogle Scholar
  26. 26.
    W. Zimmermann and W. Löwe. An approach to machine-independent parallel programming. In Parallel Processing: CONPAR 94 — VAPP VI, volume 854 of Lecture Notes in Computer Science, pp. 277–288. Springer, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Wolf Zimmermann
    • 1
  • Martin Middendorf
    • 2
  • Weif Löwe
    • 1
  1. 1.Institut für Programmstrukturen und DatenorganisationUniversität KarlsruheKarlsruheGermany
  2. 2.Institut für Angewandte Informatik und Formale BeschreibungsverfahrenUniversität KarlsruheKarlsruheGermany

Personalised recommendations