Average Number of Nodes in Binary Decision Diagrams of Fibonacci Functions*

A binary decision diagram (BDD) is a directed graph representation of a switching function f(xly x2,...,xn). Subfunctions of/ correspond to nodes in the BDD; /itself is represented by a source node, i.e., a node with no incoming arcs. Attached to this node are two outgoing arcs, labeled 0 and 1, that go to descendent nodes representing /(x1,x2,. . . ,0) and f(xt,x2,..., 1), respectively. Attached to each of these nodes are descendent nodes, where xn_x is replaced by 0 and 1, etc. This process is repeated until all variables are assigned values. The last assigned functions are a constant 0 and 1, which correspond to sink nodes, i.e., nodes with no outgoing arcs. If two nodes represent the same function, they are merged into one node, and if the descendents of one node 77 are the same, 77 is removed. If / = 1 (0) for some assignment of values to x1? x2,..., and xn, then there is a path in the BDD for/from the source node to the sink node 1 (0) for that assignment. Figure 1(a) shows the BDD of the OR function on four variables. As is usual, the arrows are omitted; all arcs are assumed to be directed down. As can be seen, there is a path from the source node to the node labeled 1 if and only if at least one variable is 1. Figure 1(b) shows the BDD of the AND function of four variables, which is the mirror image of the OR function BDD.