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 -far from unate with one-sided error and ̃ O(n/ ) many queries. This improves on the best adaptive O(n/ )-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri [1] when 1/ n. Combined with the ̃ Ω(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. Keywords-property testing; unateness;