Boolean Unateness Testing with $\widetilde{O}(n^{3/4})$ Adaptive Queries

We give an adaptive algorithm which tests whether an unknown Boolean function f : {0, 1} → {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 using Õ(n/ε) queries. This improves on the best adaptive O(n/ε)-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri [BCP17b] when 1/ε n. Combined with the Ω̃(n)-query lower bound for non-adaptive algorithms with one-sided error of [CWX17, BCP17a], 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. ∗Columbia University, email: [email protected]. †Columbia University, email: [email protected]. ‡Columbia University, email: [email protected] ar X iv :1 70 8. 05 78 6v 1 [ cs .C C ] 1 9 A ug 2 01 7