On Random Orderings of Variables for Parity OBDDs

Ordered binary decision diagrams (OBDDs) are a model for representing Boolean functions. There is also a more powerful variant called parity OBDDs. The size of the representation of a given function depends in both these models on an ordering of the variables. It is known that there are functions such that almost all orderings of its variables yield an OBDD of polynomial size, but there are also some exceptional orderings, for which the size is exponential. We prove that for parity OBDDs, the size for a random ordering and the size for the worst ordering are polynomially related. More exactly, for every " > 0 there is a number c > 0 such that the following holds. If a Boolean function f is such that a random ordering of the variables yields a parity OBDD for f of size at most s with probability at least ", then every ordering of the variables yields a parity OBDD for f of size at most s c .