For positive integers n, d, consider the hypergrid [n]~d with the coordinate-wise product partial ordering denoted by γ. A function f : [n]~d → N is monotone if {arbitrary}_x γ y, f(x) ≤ f(y). A function f is ε-far from monotone if at least an ε-fraction of values must be changed to make f monotone. Given a parameter ε, a monotonicity tester must distinguish with high probability a monotone function from one that is ε-far. We prove that any (adaptive, two-sided) monotonicity tester for functions f: [n]~d → N must make Ω(ε~(-1) d log n-ε~(-1) log ε~(-1)) queries. Recent upper bounds show the existence of O(ε~(-1) d log n) query monotonicity testers for hypergrids. This closes the question of monotonicity testing for hypergrids over arbitrary ranges. The previous best lower bound for general hypergrids was a non-adaptive bound of Ω(d log n).
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