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 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 2 6 nodes, for large n, compared to n 2 2 , for general binary symmetric functions.