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<p><b>Abstract</b>—We derive the average and worst case number of nodes in decision diagrams of <it>r</it>-valued symmetric functions of <it>n</it> variables. We show that, for large <it>n</it>, both numbers approach <tmath>${\textstyle{{{n^r} \over {r\,!}}}}.$</tmath> For binary decision diagrams (<it>r</it> = 2), we compute the distribution of the number of functions on <it>n</it> 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 <tmath>${\textstyle{n^2} \over 6}$</tmath> nodes, for large <it>n</it>, compared to <tmath>${\textstyle{{{n^2} \over 2}}},$</tmath> for general binary symmetric functions.</p>
Decision diagrams, BDD, symmetric functions, multiple-valued functions, complexity, asymptotic approximation, average case.
Tsutomu Sasao, Robert J. Barton III, David S. Herscovici, Jon T. Butler, "Average and Worst Case Number of Nodes in Decision Diagrams of Symmetric Multiple-Valued Functions", IEEE Transactions on Computers, vol. 46, no. , pp. 491-494, April 1997, doi:10.1109/12.588065
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