Binary Decision Diagram - Variable Ordering

Variable Ordering

The size of the BDD is determined both by the function being represented and the chosen ordering of the variables. There exist Boolean functions for which depending upon the ordering of the variables we would end up getting a graph whose number of nodes would be linear (in n) at the best and exponential at the worst case (e.g., a ripple carry adder). Let us consider the Boolean function Using the variable ordering, the BDD needs 2n+1 nodes to represent the function. Using the ordering, the BDD consists of 2n+2 nodes.

It is of crucial importance to care about variable ordering when applying this data structure in practice. The problem of finding the best variable ordering is NP-hard. For any constant c > 1 it is even NP-hard to compute a variable ordering resulting in an OBDD with a size that is at most c times larger than an optimal one. However there exist efficient heuristics to tackle the problem.

There are functions for which the graph size is always exponential — independent of variable ordering. This holds e. g. for the multiplication function (an indication as to the apparent complexity of factorization ).

Researchers have of late suggested refinements on the BDD data structure giving way to a number of related graphs, such as BMD (Binary Moment Diagrams), ZDD (Zero Suppressed Decision Diagram), FDD (Free Binary Decision Diagrams), PDD (Parity decision Diagrams), and MTBDDs (Multiple terminal BDDs).