A o(d) ·polylog n Monotonicity Tester for Boolean Functions over the Hypergrid [n]^d
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We study monotonicity testing of Boolean functions over the hypergrid [n]^d and design a non-adaptive tester with 1-sided error whose query complexity is Õ(d^5/6)·poly( n,1/ϵ). Previous to our work, the best known testers had query complexity linear in d but independent of n. We improve upon these testers as long as n = 2^d^o(1). 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.
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