The size of reduced OBDDs and optimal read-once branching programs for almost all Boolean functions

Abstract

Boolean functions are often represented by ordered binary decision diagrams (OBDDs) introduced by Bryant [2]. Liaw and Lin [7] have proved upper and lower bounds on the minimal OBDD size of almost all Boolean functions. Now tight bounds are proved for the minimal OBDD size for arbitrary or optimal variable orderings and for the minimal read-once branching program size of almost all functions. Almost all Boolean functions have a sensitivity of almost 1, i. e. the minimal OBDD size for an optimal variable ordering differs from the minimal OBDD size for a worst variable ordering by a factor of at most 1 + ɛ(n) where ɛ(n) converges exponentially fast to 0.