Towards Better Separation between Deterministic and Randomized Query Complexity

We show that there exists a Boolean function F which gives the following separations among deterministic query complexity (D(F )), randomized zero error query complexity (R0(F )) and randomized one-sided error query complexity (R1(F )): R1(F ) = Õ( √ D(F )) and R0(F ) = Õ(D(F ))3/4. This refutes the conjecture made by Saks and Wigderson that for any Boolean function f , R0(f) = Ω(D(f))0.753... This also shows widest separation between R1(f) and D(f) for any Boolean function. The function F was defined by Göös, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous nondeterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function F certify optimal (quadratic) separation between D(f) and R0(f), and polynomial separation between R0(f) and R1(f). Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considered in the work of Ambainis et al are different variants of F , in this work we show that the original function F itself is sufficient to refute the Saks-Wigderson conjecture and obtain widest possible separation between the deterministic and one-sided error randomized query complexity. 1998 ACM Subject Classification F.1.1 [Computation by Abstract Devices]: Models of Computation – Relations between models, F.1.2 [Computation by Abstract Devices]: Modes of Computation – Probabilistic computation