A d^1/2+o(1) Monotonicity Tester for Boolean Functions on d-Dimensional Hypergrids
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Monotonicity testing of Boolean functions on the hypergrid, f:[n]^d →{0,1}, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary n, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity O(ε^-4/3d^5/6). This complexity is independent of n, but has a suboptimal dependence on d. Recently, [Braverman-Khot-Kindler-Minzer, ITCS 2023] and [Black-Chakrabarty-Seshadhri, STOC 2023] describe O(ε^-2 n^3√(d)) and O(ε^-2 n√(d))-query testers, respectively. These testers have an almost optimal dependence on d, but a suboptimal polynomial dependence on n. In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity O(ε^-2 d^1/2 + o(1)), independent of n. Up to the d^o(1)-factors, our result resolves the non-adaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of n yields a non-adaptive, one-sided O(ε^-2 d^1/2 + o(1))-query monotonicity tester for Boolean functions f:ℝ^d →{0,1} associated with an arbitrary product measure.
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