The Zero-Error Randomized Query Complexity of the Pointer Function

Abstract

The pointer function of Goos, Pitassi and Watson and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and quantum query complexity, exact and approximate polynomial degree, etc. In particular, Ambainis et al. (STOC 2016) obtained 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 by considering variants of this pointer function.

However, as Ambainis et al. pointed out in their work, the precise zero-error complexity of the original pointer function was
not known.  We show a lower bound of ~Omega(n^{3/4}) on the zero-error randomized query complexity of the pointer function on Theta(n * log(n)) bits; since an ~O(n^{3/4}) upper bound was already shown by Mukhopadhyay and Sanyal (FSTTCS 2015), our lower bound is optimal up to polylog factors. We, in fact, consider a generalization of the original function and obtain lower bounds for it that are optimal up to polylog factors.