The zero-error randomized query complexity of the pointer function

The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson \cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and quantum query complexities, exact and approximate polynomial degrees, etc. In particular, the widest possible (quadratic) separations between deterministic and zero-error randomized query complexity, as well as between bounded-error and zero-error randomized query complexity, have been obtained by considering {\em variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function. However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the precise zero-error complexity of the original pointer function was not known. We show a lower bound of $\widetilde{\Omega}(n^{3/4})$ on the zero-error randomized query complexity of the pointer function on $\Theta(n \log n)$ bits; since an $\widetilde{O}(n^{3/4})$ upper bound is already known \cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a factor of \polylog\, n

Towards Better Separation between Deterministic and Randomized Query Complexity

We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$ and randomized one-sided error query complexity $(R_1(F))$: $R_1(F) = \widetilde{O}(\sqrt{D(F)})$ and $R_0(F)=\widetilde{O}(D(F))^{3/4}$. This refutes the conjecture made by Saks and Wigderson that for any Boolean function $f$, $R_0(f)=\Omega({D(f)})^{0.753..}$. This also shows widest separation between $R_1(f)$ and $D(f)$ for any Boolean function. The function $F$ was defined by G{\"{o}}{\"{o}}s, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic 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 $R_0(f)$, and polynomial separation between $R_0(f)$ and $R_1(f)$. Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considerd in the work of Ambainis et al are different variants of $F$, we work with the original function $F$ itself.Comment: Reference adde