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A New Algorithm for Generation of Exactly M–Block Set Partitions in Associative Model

  • Zbigniew Kokosiński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3911)

Abstract

In this paper a new parallel algorithm is presented for generation of all exactly m–block partitions of n–element set. The basic building blocks of the algorithm are an associative generator of combinations and a complex parallel counter. Consecutive objects are generated in lexicographic order, with O(1) time per object. The algorithm can be used for generation of all partitions within the given range of the parameter m, where 1 ≤ m 1mm 2n.

Keywords

Parallel Algorithm Choice Function Lexicographic Order Parallel Generation Combinatorial Object 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Akl, S.G., Gries, D., Stojmenović, I.: An optimal parallel algorithm for generating combinations. Information Processing Letters 33, 135–139 (1989/90)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Akl, S.G., Stojmenović, I.: Generating combinatorial objects on a linear array of processors. In: Zomaya, A.Y. (ed.) Parallel Computing. Paradigms and Applications, pp. 639–670. Int. Thompson Comp. Press (1996)Google Scholar
  3. 3.
    Djokić, B., et al.: A fast iterative algorithm for generating set partitions. The Computer Journal 32, 281–282 (1989)CrossRefGoogle Scholar
  4. 4.
    Djokić, B., et al.: Parallel algorithms for generating subset and set partitions. In: Asano, T., Imai, H., Ibaraki, T., Nishizeki, T. (eds.) SIGAL 1990. LNCS, vol. 450, Springer, Heidelberg (1990)Google Scholar
  5. 5.
    Er, M.C.: A fast algorithm for generating set partitions. The Computer Journal 31, 283–284 (1988)CrossRefMATHGoogle Scholar
  6. 6.
    Even, S.: Algorithmic Combinatorics. Macmillan, New York (1973)MATHGoogle Scholar
  7. 7.
    Hutchinson, G.: Partitioning algorithms for finite sets. Comm. ACM 6, 613–614 (1963)CrossRefGoogle Scholar
  8. 8.
    Kapralski, A.: New methods for generation permutations, combinations and other combinatorial objects in parallel. J. Parallel and Distrib. Computing 17, 315–326 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kapralski A.: Modeling arbitrary sets of combinatorial objects and their sequential and parallel generation. Studia Informatica 21(2)(40) (2000)Google Scholar
  10. 10.
    Kaye, R.: A Gray code for set partitions. Inform. Process. Letters 5, 171–173 (1976)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kokosiński, Z.: On generation of permutations through decomposition of symmetric groups into cosets. BIT 30, 583–591 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kokosiński, Z.: Circuits generating combinatorial configurations for sequential and parallel computer systems. Monografia 160, Politechnika Krakowska, Kraków, Poland (in Polish) (1993)Google Scholar
  13. 13.
    Kokosiński, Z.: Mask and pattern generation for associative supercomputing. In: Proc. IASTED Int. Conference AI 1994, Annecy, France, pp. 324–326 (1994)Google Scholar
  14. 14.
    Kokosiński, Z.: On parallel generation of set partitions in associative processor architectures. In: Proc. Int. Conf. PDPTA 1999, Las Vegas, USA, pp. 1257–1262 (1999)Google Scholar
  15. 15.
    Kokosiński, Z.: On parallel generation of combinations in associative processor architectures. In: Proc. IASTED Int. Conf. Euro–PDS 1997, Barcelona, Spain, pp. 283–289 (1997)Google Scholar
  16. 16.
    Lee, W.-T., Tsay, J.-C., Chen, H.-S., Tseng, T.-J.: An optimal systolic algorithm for the set partitioning problem. Parallel Algorithm and Applications 10, 301–314 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lehmer, D.H.: Teaching combinatorial tricks to a computer. In: Proc. of Symposium Appl. Math., Combinatorial Analysis 10, pp. 179–193. Amer. Math. Society, Providence (1960)Google Scholar
  18. 18.
    Lehmer, D.H.: The machine tools of combinatorics. In: Beckenbach, E.F. (ed.) Applied combinatorial mathematics, pp. 5–31. John Wiley, N.Y. (1964)Google Scholar
  19. 19.
    Lin, C.J., Tsay, J.C.: A systolic generation of combinations. BIT 29, 23–36 (1989)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mirsky, L.: Transversal theory. Academic Press, N.Y. (1971)MATHGoogle Scholar
  21. 21.
    Semba, I.: An efficient algorithm for generating all partitions of the set {1,2,.,n}. Journal of Information Processing 7, 41–42 (1984)MathSciNetMATHGoogle Scholar
  22. 22.
    Stojmenović, I.: An optimal algorithm for generating equivalence relations on a linear array of processors. BIT 30, 424–436 (1990)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    von zur Gathen, J.: Parallel linear algebra. In: Reiff, J.H. (ed.) Synthesis of parallel algorithms, pp. 573–617. Morgan Kaufman, San Francisco (1993)Google Scholar
  24. 24.
    Williamson, S.G.: Ranking algorithms for lists of partitions. SIAM Journal of Computing 5, 602–617 (1976)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zbigniew Kokosiński
    • 1
  1. 1.Faculty of Electrical & Computer Eng.Cracow University of TechnologyKrakówPoland

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