Boolean Unateness Testing with O(n^3/4) Adaptive Queries
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We give an adaptive algorithm which 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 using O(n^3/4/ϵ^2) queries. This improves on the best adaptive O(n/ϵ)-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri when 1/ϵ≪ n^1/4. Combined with the Ω(n)-query lower bound for non-adaptive algorithms with one-sided error of [CWX17, BCPRS17], 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.
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