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

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 (n involution r/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 (n squared/6) nodes, for large n, compared to (N sqaured/2), for general binary symmetric functions.