Testing unateness nearly optimally

Abstract

We present an Õ (n2/3/ε2)-query algorithm that tests whether an unknown Boolean function f : (0, 1)n → (0, 1) is unate (i.e., every variable is either non-decreasing or non-increasing) or ε-far from unate. The upper bound is nearly optimal given the Ω (n2/3) lower bound of Chen, Waingarten and Xie (2017). The algorithm builds on a novel use of the binary search procedure and its analysis over long random paths.