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Average and Worst Case Number of Nodes in Decision Diagrams of Symmetric Multiple-Valued Functions
April 1997 (vol. 46 no. 4)
pp. 491-494

Abstract—We derive the average and worst case number of nodes in decision diagrams of r-valued symmetric functions of n variables. We show that, for large n, both numbers approach ${\textstyle{{{n^r} \over {r\,!}}}}.$ For binary decision diagrams (r = 2), we compute the distribution of the number of functions on n variables with a specified number of nodes. Subclasses of symmetric functions appear as features in this distribution. For example, voting functions are noted as having an average of ${\textstyle{n^2} \over 6}$ nodes, for large n, compared to ${\textstyle{{{n^2} \over 2}}},$ for general binary symmetric functions.

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Index Terms:
Decision diagrams, BDD, symmetric functions, multiple-valued functions, complexity, asymptotic approximation, average case.
Citation:
Jon T. Butler, David S. Herscovici, Tsutomu Sasao, Robert J. Barton III, "Average and Worst Case Number of Nodes in Decision Diagrams of Symmetric Multiple-Valued Functions," IEEE Transactions on Computers, vol. 46, no. 4, pp. 491-494, April 1997, doi:10.1109/12.588065
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