Abstract
The multistage shuffle-exchange network is generalized, by replacing the perfect shuffle with an arbitrary permutation π, in order to pass all permutations using aminimal number of stages. The universality of such a network is shown to be equivalent to theprimitivity of a related regular digraph, thus showing that universality is decidable in polynomial time. Extensive computational results are presented that characterizeall nonisomorphic minimal networks with up to 12 inputs. In addition, strong evidence is presented relating the minimal number of network stages needed to achieve universality totwice the index of primitivity of the corresponding digraph.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Anand, V. C. Dumir, and H. Gupta, A combinatorial distribution problem,Duke Math. J. 33, 757–769 (1966).
W. Brown, ed.,Reviews in Graph Theory, Vols. 1 and 2, American Mathematical Society, Providence, RI, 1980.
R. G. Busacker and T. L. Saaty,Finite Graphs and Networks, McGraw-Hill, New York, 1965.
H. Cam and J. Fortes, Rearrangeability of shuffle-exchange networks,Proceedings of the Third Symposium on Frontiers of Massively Parallel Computation, 1990.
R. F. Chamberlain and C. M. Fiduccia, Universality of iterated networks,Proceedings of the Fourth Annual ACM Symposium on Parallel Algorithms and Architectures, July 1992.
S. W. Golomb,Shift Register Sequences, Aegean Park Press, Laguna Hills, CA, 1982.
Hall,Combinatorial Theory, 2nd edn., Wiley, New York, 1986.
Harary and Palmer,Graphical Enumeration, Academic Press, New York, 1973.
D. H. Lawrie, Access and alignment of data in an array processor,IEEE Trans. Comput. 25, 1145–1155 (1976).
H. Mine,Nonnegative Matrices, Wiley, New York, 1987.
D. S. Parker, Notes on shuffle/exchange type networks,IEEE Trans. Comput. 29, 213–222 (1980).
M. C. Pease, The indirect binaryn-cube microprocessor array,IEEE Trans. Comput. 26(5), 458–473 (1977).
G. PóPolya,Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (translated by R. C. Read), Springer-Verlag, New York, 1987.
C. S. Raghavendra and A. Varma, Rearrangeability of the 5-stage shuffle/exchange network forN = 8,Proceedings of the 1986 International Conference on Parallel Processing, pp. 119–122.
R. C. Read, The enumeration of locally restricted graphs, I and H,J. London Math. Soc. 34, 417–436 (1959);35, 344–351 (1960).
R. C. Read, The use ofS-functions in combinatorial analysis,Canad. J. Math. 20, 808–841 (1968).
J. H. Redfield, The theory of group-reduced distributions,Amer. J. Math. 49, 433–455 (1927).
H. J. Siegel, The universality of various types of SIMD machine interconnection networks,Proceedings of the Fourth Annual Symposium on Computer Architecture, 1977.
S. Skiena,Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematics, Addison-Wesley, Reading, MA, 1990.
D. Steinberg, Invariant properties of the shuffle-exchange and a simplified cost-effective version of the omega network,IEEE Trans. Comput. 32(5), 444–450 (1983).
H. S. Stone, Parallel processing with the perfect shuffle,IEEE Trans. Comput. 20, 153–161 (1971).
A. A. Waksman, A permutation network,J. Assoc. Comput. Mach. 15, 159–163 (1968).
H. S. Wilf,Generatingfunctionology, Academic Press, San Diego, CA, 1990.
S. Wolfram,Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, Reading, MA, 1988.
C. L. Wu and T. Y. Feng, The universality of the shuffle-exchange network,IEEE Trans. Comput. 30(5), 324–332 (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chamberlain, R.F., Fiduccia, C.M. Universality of iterated networks. Math. Systems Theory 27, 381–430 (1994). https://doi.org/10.1007/BF01184932
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01184932