Blockwise variable orderings for shared BDDs
A state-of-the-art data structure for the representation of Boolean functions are ordered binary decision diagrams (OBDDs). The size of an OBDD representing a Boolean function depends on the variable ordering. Finding a variable ordering with optimal (i.e. minimum) OBDD size is a central, but NP-hard problem. Thus it is of great interest to characterize optimal variable orderings from the structure of the given function. In this paper we investigate the problem of characterizing optimal variable orderings for shared OBDDs of two Boolean functions f i = g i ⊗ i h i , i = 1,2, where ⊗i is an operator from the base B 2 * , (the full binary basis consisting of all ten binary operations depending essentially on both inputs) and g i (resp. h i) depends only on x-variables (resp. y-variables). Tree-like circuits provide an example for such functions. In the special case f 1 = ¯f 2, Sauerhoff, Wegener and Werchner  proved that there is some optimal ordering where all x-variables are tested before all y-variables or vice versa (blockwise variable ordering). We show that this is also true for arbitrary f 1, f 2 provided that ⊗1 = ∧ and ⊗2 = ∀, and for shared OBDDs with complemented edges and arbitrary f1, f2 provided that ⊗1 = ∧ and ⊗2 = ⊕. For all other combinations of ⊗1 und ⊗2 we give counterexamples.
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- 2.R. E. Bryant: Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers, vol. C-35(8), pages 677–691, 1986.Google Scholar
- 5.S. Minato, N. Ishiura, S. Yajima: Shared binary decision diagrams with attributed edges for efficient Boolean function manipulation. Design Automation Conference, pages 52–57, 1990.Google Scholar
- 6.M. Sauerhoff, I. Wegener, R. Werchner: Optimal ordered binary decision diagrams for fanout-free circuits. Proc. of SASIMI 1996.Google Scholar
- 7.M. Sauerhoff, I. Wegener, R. Werchner: Optimal ordered binary decision diagrams for tree-like circuits. Forschungsbericht Nr. 613, Universität Dortmund 1996.Google Scholar