On the average path length in decision diagrams of multiple-valued functions

We consider the path length in decision diagrams for multiple-valued functions. This is an important measure of a decision diagram, since this models the time needed to evaluate the function. We focus on the average path length (APL), which is the sum of the path lengths over all assignments of values to the variables divided by the number of assignments. First, we show a multiple-valued function in which the APL is markedly affected by the order of variables. We show upper and lower bounds on the longest path length in a decision diagram of a multiple-valued function. Next, we derive the APL for individual functions, the MAX, ALL-MAX, and MODSUM functions. We show that the latter two functions achieve the lower and upper bound on the APL overall n-variable r-valued functions. Finally, we derive the average of the APL for two sets of functions, symmetric functions and all functions.