Randomized vs. deterministic decision tree complexity for read-once Boolean functions

The authors consider the deterministic and the randomized decision-tree complexities for Boolean functions, denoted DC( f) and RC(f), respectively. It is well known that RC(f)⩾DC(f)^0.5 for every Boolean function f (called 0.5-exponent), but no better lower bound is known for all Boolean functions, whereas the best known upper bound is RC(f)=Θ(DC(f )⁰_.^{753 . .}) (or 0.753 . . .-exponent) for some Boolean function f. The present result is a 0.51 lower bound on the exponent for all read-once functions representable by formulae in which each input variable appears exactly once. To obtain it the authors generalize an existing lower bound technique and combine it with restrictions arguments. This result provides a lower bound of n^0.51 on the number of positions that have to be evaluated by any randomized α-β pruning algorithm computing the value of any two-person zero-sum game tree with n final positions