Boolean Unateness Testing with Õ(n3/4) Adaptive Queries.

We give an adaptive algorithm that tests whether an unknown Boolean function f : {0, 1} ⁿ → {0, 1} is unate (i.e. every variable of f is either non-decreasing or non-increasing) or ε-far from unate with one-sided error and Õ(n ^3/4 /ϵ ² ) many queries. This improves on the best adaptive O(n/ϵ)-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri [1] when 1/ϵ <;<; 4 n ^1/4 . Combined with the Ω̃(n)query lower bound for non-adaptive algorithms with one-sided error of [2], [3], we conclude that adaptivity helps for the testing of unateness with one-sided error. A crucial component of our algorithm is a new subroutine for finding bi-chromatic edges in the Boolean hypercube called adaptive edge search.