Recursively divisible problems

  • Rolf Niedermeier
Session 5b: Invited Presentation
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1178)


We introduce the concept of a (p, d)-divisible problem, where p reflects the number of processors and d the number of communication phases of a parallel algorithm solving the problem. We call problems that are recursively (n, O(1))-divisible in a work-optimal way with 0 <ε < 1ideally divisible and give motivation drawn from parallel computing for the relevance of that concept. We show that several important problems are ideally divisible. For example, sorting is recursively (n1/3, 4)-divisible in a work-optimal way. On the other hand, we also provide some results of lower bound type. For example, ideally divisible problems appear to be a proper subclass of the functional complexity class FP of sequentially feasible problems. Finally, we also give some extensions and variations of the concept of (p, d)-divisibility.


Parallel Algorithm Communication Phase Sorting Problem Proper Subclass Parallel Random Access Machine 
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  1. 1.
    A. Aggarwal, A. K. Chandra, and M. Snir. Communication Complexity of PRAMs. Theoretical Comput. Sci., 71:3–28, 1990.CrossRefGoogle Scholar
  2. 2.
    D. A. Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. J. Comput. Syst. Sci., 38:150–164, 1989.CrossRefGoogle 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, pages 1–12, May 1993.Google Scholar
  4. 4.
    R. Cypher and J. L. Sanz. Cubesort: A parallel algorithm for sorting N data items with S-sorters. Journal of Algorithms, 13:211–234, 1992.CrossRefMathSciNetGoogle Scholar
  5. 5.
    P. de la Torre and C. P. Kruskal. Towards a single model of efficient computation in real parallel machines. Future Generation Computer Systems, 8:395–408, 1992.CrossRefGoogle Scholar
  6. 6.
    P. de la Torre and C. P. Kruskal. A structural theory of recursively decomposable parallel processor-networks. In IEEE Symp. on Parallel and Distributed Processing, 1995.Google Scholar
  7. 7.
    R. Greenlaw, H. J. Hoover, and W. L. Ruzzo. Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, 1995.Google Scholar
  8. 8.
    T. Heywood and C. Leopold. Models of parallelism. In J. R. Davy and P. M. Dew, editors, Abstract Machine Models for Highly Parallel Computers, chapter 1, pages 1–16. Oxford University Press, 1995.Google Scholar
  9. 9.
    T. Heywood and S. Ranka. A practical hierarchical model of parallel computation: I and II. Journal of Parallel and Distributed Computing, 16:212–249, November 1992.CrossRefGoogle Scholar
  10. 10.
    C. P. Kruskal, L. Rudolph, and M. Snir. A complexity theory of efficient parallel algorithms. Theoretical Comput. Sci., 71:95–132, 1990.CrossRefGoogle Scholar
  11. 11.
    T. Leighton. Tight bounds on the complexity of parallel sorting. IEEE Transactions on Computers, C-34(4):344–354, April 1985.Google Scholar
  12. 12.
    W. L. Ruzzo. On uniform circuit complexity. J. Comput. Syst. Sci., 22:365–383, 1981.CrossRefGoogle Scholar
  13. 13.
    L. G. Valiant. A bridging model for parallel computation. Commun. ACM, 33(8):103–111, 1990.CrossRefGoogle Scholar
  14. 14.
    J. S. Vitter and R. A. Simons. New classes for parallel complexity: A study of unification and other complete problems for P. IEEE Trans. Comp., C-35(5):403–418, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Rolf Niedermeier
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany

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