Optimal Ordered Binary Decision Diagrams for Tree-like Circuits

Many Boolean functions have short representations by OBDDs (ordered binary decision diagrams) if appropriate variable orderings are used. For tree-like circuits, which may contain EXOR-gates, it is proved that some depth rst traversal leads to an optimal variable ordering. Moreover, an optimal variable ordering and the resulting OBDD can be computed in time linear in the number of variables and the size of the OBDD, respectively. Upper and lower bounds on the OBDD size of the functions representable by tree-like circuits are derived. For, e. g., 1024 inputs, we show that all tree-like circuits have OBDDs of size at 5349 and we give an example of a tree-like circuit requiring an OBDD of size 5152. Acknowledgements The rst and second author have been supported by DFG grant We 1066/7-3.