Average and Worst Case Number of Nodes in Decision Diagrams of Symmetric Multiple-Valued Functions

Authors:
Jon T. Butler;David S. Herscovici;Tsutomu Sasao;Robert J. Barton, III
Affiliations:
Naval Postgraduate School, Monterey, CA;St. Mary's College, Moraga, CA;Kyushu Institute of Technology, Iizuka, Japan;Fraunhofer Center for Research in Computer Graphics, Providence, RI
Venue:
IEEE Transactions on Computers
Year:
1997

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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.