Branching Program Size Is Almost Linear in Formula Size

Abstract

Circuit size, formula size, and branching program size are the most important complexity measures for boolean functions. Although the corresponding computation models and the relations between these complexity measures have been investigated quite well in the past, we are still lacking a tight upper bound for the branching program size in terms of formula size. The best previously known simulation of B2-formulas of size ℓ by branching programs achieves a branching program size of O(ℓ1.195) (1999, M. Sauerhoff, I. Wegener, and R. Werchner, Lecture Notes in Computer Science, Vol. 1563, pp. 57–67, Springer-Verlag, Berlin/New York). This work presents a proof showing that every B2-formula of size ℓ can be transformed into a branching program of size O(ℓ1+ε) for an arbitrary ε>0. The proof is based on three major steps. First, a given formula is balanced appropriately. Then, a technique due to R. Cleve (1991, Comput. Complexity1, 91–105) to simulate balanced algebraic formulas of size s by algebraic straight-line programs that employ a constant number of registers and have length O(s1+ε) is applied. Finally, it is shown how branching programs of the stated size can be obtained from the constructed straight-line programs.