We study monotonicity testing of Boolean functions over the hypergrid @math and design a non-adaptive tester with @math -sided error whose query complexity is @math . Previous to our work, the best known testers had query complexity linear in @math but independent of @math . We improve upon these testers as long as @math . To obtain our results, we work with what we call the augmented hypergrid, which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure.
We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm that makes @math nonadaptive queries to a function @math , always accepts a monotone function, and rejects a function that is @math -far from being monotone with constant probability. The primary problem in property testing is to decide whether a given function satisfies a certain property or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions. In this article, we provide simple and optimal testers for bounded derivative properties over arbitrary product distributions. Bounded derivative properties include fundamental properties, such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing over arbitrary ranges. We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show that this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem for all bounded derivative properties that relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best-known result. In property testing, we are given oracle access to a function f, and we wish to test if the function satisfies a given property P, or it is e-far from having that property. In a more general setting, the domain on which the function is defined is equipped with a probability distribution, which assigns different weight to different elements in the domain. This paper relates the complexity of testing the monotonicity of a function over the d-dimensional cube to the Shannon entropy of the underlying distribution. We provide an improved upper bound on the query complexity of the property tester. 