Testing Unateness Nearly Optimally

Abstract

We present an $\tilde{O}(n^{2/3}/\epsilon^2)$-query algorithm that tests whether an unknown Boolean function $f\colon\{0,1\}^n\rightarrow \{0,1\}$ is unate (i.e., every variable is either non-decreasing or non-increasing) or $\epsilon$-far from unate. The upper bound is nearly optimal given the $\tilde{\Omega}(n^{2/3})$ lower~bound of [CWX17a]. The algorithm builds on a novel use of the binary search procedure and its analysis over long random paths.