Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Berend, D., Tassa, T.: Improved bounds on Bell numbers and on moments of sums of random variables. Probability and Mathematical Statistics 30(2), 185–205
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)
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)
Chen, X., Liu, L., Liu, Z., Jiang, T.: On the minimum common integer partition problem. ACM Trans. on Computational Logic V, 1–19 (2008)
Debnath, D., Sasao, T.: Fast Boolean matching under permutation by efficient computation of canonical form. IEICE Trans. Fundamentals (12), 3134–3140 (2004)
Beeler, M., Gosper, R.W., Schroeppel, R.: HAKMEM. MIT Artificial Intelligence Laboratory, Cambridge, MA, Memo AIM-239 (February 1972), http://www.inwap.com/pdp10/hbaker/hakmem/hacks.html#item175
Hankin, R.K.S., West, L.J.: Set partitions in R. J. of Statistical Software 23, Code Snippet 2 (December 2007), http://www.jstatsoft.org/
Knuth, D.E.: Volume 4 Generating all combinations and permutations. In: The Art of Computer Programming, Fascicle 3. Addison-Wesley ISBN: 0-321-58050-8
Kawano, S., Nakano, S.: Constant time generation of set partitions. IEICE Trans. Fundamentals E88-A(4), 930–934 (2005)
McKay, J.K.S.: Algorithm 263, Partition Generator. Communications of the ACM 8(8), 493 (1965)
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)
Oommen, B.J., Ng, D.T.H.: On generating random partitions with arbitrary distributions. The Computer Journal 33(4), 368–374 (1990)
Orlov, M.: Efficient generation of set partitions (March 2002), http://www.cs.bgu.ac.il/~orlovm/papers/partitions.pdf
Reingold, E., Nivergelt, J., Deo, N.: Combinatorial Algorithms, Theory and Practice. Prentice-Hall (1977)
Sasao, T.: Memory Based Logic Synthesis, 1st edn. Springer (2011) ISBN: 978-1-4419-8103-5
Semba, I.: An efficient algorithm for generating all partitions of the set {1,2,..., n}. Journal of Information Processing 7(1) (1984)
Stojmenovič, I.: An optimal algorithm for generating equivalence relations on a linear array of processors. BIT 30(3), 424–436 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Butler, J.T., Sasao, T. (2013). Hardware Index to Set Partition Converter. In: Brisk, P., de Figueiredo Coutinho, J.G., Diniz, P.C. (eds) Reconfigurable Computing: Architectures, Tools and Applications. ARC 2013. Lecture Notes in Computer Science, vol 7806. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36812-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-36812-7_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36811-0
Online ISBN: 978-3-642-36812-7
eBook Packages: Computer ScienceComputer Science (R0)