Fast Exact Minimization of BDDsRolf

We present a new exact algorithm for nding the optimal variable ordering for reduced ordered Binary Decision Diagrams (BDDs). The algorithm makes use of a lower bound technique known from VLSI design. Up to now this technique has been used only for theoretical considerations and it is adapted here for our purpose. Furthermore, the algorithm supports symmetry aspects and makes use of a hash-ing based data structure. Experimental results are given to demonstrate the eeciency of our approach. We succeeded in minimizing adder functions with up to 64 variables, while all other previously presented approaches fail. 1 Introduction Recently, several design methods have been proposed that are based on ordered Binary Decision Diagrams (BDDs) 7]. The resulting circuits have very nice properties, like e.g. testability 2, 1] and low power 17]. For synthesis approaches based on Pass Transistor Logic (PTL) BDDs seem to be a good starting point. First promising results on how to transform a decision diagram to a circuit based on PTL have been reported in 24, 9, 3]. One drawback of BDDs is that they are very sensitive to the variable ordering, i.e. the size of the representation may vary from linear to exponential. Therefore in the last few years several methods have been presented to determine good orderings. However, nding the optimal variable ordering starting from a given BDD representation is known to be NP-hard 5]. Existing methods for the determination of good variable orderings can be classiied into three categories. The rst are initial heuristics starting from a circuit 13], the second are gradual improvement heuristics based on the exchange of variables in the BDD 15, 14, 21, 19, 20], and the third are exact methods to nd an optimal ordering 12, 15, 16]. Obviously, it is desirable to determine the exact result. In applications like the ones discussed above it is even more important to nd the best variable ordering since it turned out by experiments that the best greedy approaches are up to a factor of two worse than the optimal result. But ex