The zero-error randomized query complexity of the pointer function

TL;DR

This paper establishes an almost tight lower bound on the zero-error randomized query complexity of the pointer function, resolving a previously unknown aspect of its complexity and contributing to the understanding of separations among complexity measures.

Contribution

It provides the first near-optimal lower bound on the zero-error randomized query complexity of the original pointer function.

Findings

Lower bound of rac{1}{4} ilde{\u220a} \, n^{3/4} on zero-error randomized query complexity

Matching known upper bound up to polylogarithmic factors

Clarifies the complexity separation for the pointer function

Abstract

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 $\polylog\, n$.