Blockwise Variable Orderings for Shared BDDs

In this paper we consider the problem of characterizing optimal variable orderings for shared OBDDs of two Boolean functions fi = gi ⊗i hi, i = 1, 2, where ⊗i is an operator from the Base B∗ 2 and gi (resp. hi) depends only on x-variables (resp. y-variables). Tree-like circuits provide an example for such functions. In the special case f1 = f2, Sauerhoff, Wegener and Werchner [6] proved that there is some optimal ordering where all x-variables are tested before all y-variables or vice versa. We show that this is also true for arbitrary f1, f2 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.