Randomized vs.Deterministic Decision Tree Complexity for Read-Once Boolean Functions

13 years 7 months ago

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We consider the deterministic and the randomized decision tree complexities for Boolean functions, denoted DC(f) and RC(f), respectively. A major open problem is how small RC(f) can be with respect to DC(f). It is well known that RC(f) ≥ DC(f)0.5 for every Boolean function f (called “0.5-exponent”). On the other hand, some Boolean function f is known to have RC(f) = Θ(DC(f)0.753...) (or “0.753...-exponent”). It is not known whether there is a Boolean function with exponent smaller than 0.753... Likewise, no lower bound for arbitrary Boolean functions with exponent greater than 0.5 is known. Our result is a 0.51 lower bound on the exponent for every read-once function. Read-once means that each input variable appears exactly once in the Boolean formula representing the function. To obtain this result we generalize an existing lower bound technique and combine it with restriction arguments. This result provides a lower bound of n0.51 on the number of positions that have to be ...

Rafi Heiman, Avi Wigderson

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