Hardware Index to Set Partition Converter

  • Jon T. Butler
  • Tsutomu Sasao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7806)


We demonstrate, for the first time, high-speed circuits that generate partitions on a set S of n objects. We offer two versions. In the first, partitions are produced in lexicographical order in response to successive clock pulses. In the second, an index input determines the set partition produced. Such circuits are needed in the hardware implementation of the optimum distribution of tasks to processors. Our circuits are combinational. For large n, they can have large delay. However, one can easily pipeline them to produce one set partition per clock period. We show 1) analytical and 2) experimental time/complexity results that quantify the efficiency of our designs. Our results show that a hardware partition generator running on a 100 MHz FPGA produces partitions at a rate that is approximately 10 times the rate of a software implementation on a processor running at 2.26 GHz.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berend, D., Tassa, T.: Improved bounds on Bell numbers and on moments of sums of random variables. Probability and Mathematical Statistics 30(2), 185–205Google Scholar
  2. 2.
    Butler, J.T., Sasao, T.: Index to Constant Weight Codeword Converter. In: Koch, A., Krishnamurthy, R., McAllister, J., Woods, R., El-Ghazawi, T. (eds.) ARC 2011. LNCS, vol. 6578, pp. 193–205. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Butler, J.T., Sasao, T.: Hardware index to permutation converter. In: 19th Reconfigurable Architectures Workshop (RAW 2012), Proc. of the 26th IEEE International Parallel and Distributed Processing Symposium, Shanghai, China, May 21-22, pp. 424–429 (2012)Google Scholar
  4. 4.
    Chen, X., Liu, L., Liu, Z., Jiang, T.: On the minimum common integer partition problem. ACM Trans. on Computational Logic V, 1–19 (2008)Google Scholar
  5. 5.
    Debnath, D., Sasao, T.: Fast Boolean matching under permutation by efficient computation of canonical form. IEICE Trans. Fundamentals (12), 3134–3140 (2004)Google Scholar
  6. 6.
    Beeler, M., Gosper, R.W., Schroeppel, R.: HAKMEM. MIT Artificial Intelligence Laboratory, Cambridge, MA, Memo AIM-239 (February 1972),
  7. 7.
    Hankin, R.K.S., West, L.J.: Set partitions in R. J. of Statistical Software 23, Code Snippet 2 (December 2007),
  8. 8.
    Knuth, D.E.: Volume 4 Generating all combinations and permutations. In: The Art of Computer Programming, Fascicle 3. Addison-Wesley ISBN: 0-321-58050-8Google Scholar
  9. 9.
    Kawano, S., Nakano, S.: Constant time generation of set partitions. IEICE Trans. Fundamentals E88-A(4), 930–934 (2005)Google Scholar
  10. 10.
    McKay, J.K.S.: Algorithm 263, Partition Generator. Communications of the ACM 8(8), 493 (1965)CrossRefGoogle Scholar
  11. 11.
    Nagayama, S., Sasao, T., Butler, J.T.: Analysis of multi-state systems with multi-state components using EVBDDs. In: Proc. 42nd International Symposium on Multiple-Valued Logic, Victoria, Canada, May 14-16, pp. 122–127 (2012)Google Scholar
  12. 12.
    Oommen, B.J., Ng, D.T.H.: On generating random partitions with arbitrary distributions. The Computer Journal 33(4), 368–374 (1990)CrossRefGoogle Scholar
  13. 13.
    Orlov, M.: Efficient generation of set partitions (March 2002),
  14. 14.
    Reingold, E., Nivergelt, J., Deo, N.: Combinatorial Algorithms, Theory and Practice. Prentice-Hall (1977)Google Scholar
  15. 15.
    Sasao, T.: Memory Based Logic Synthesis, 1st edn. Springer (2011) ISBN: 978-1-4419-8103-5Google Scholar
  16. 16.
    Semba, I.: An efficient algorithm for generating all partitions of the set {1,2,..., n}. Journal of Information Processing 7(1) (1984)Google Scholar
  17. 17.
    Stojmenovič, I.: An optimal algorithm for generating equivalence relations on a linear array of processors. BIT 30(3), 424–436 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jon T. Butler
    • 1
  • Tsutomu Sasao
    • 2
  1. 1.Naval Postgraduate SchoolMontereyUSA
  2. 2.Kyushu Institute of TechnologyIizukaJapan

Personalised recommendations