For positive integers n, d, consider the hypergrid [n] d with the coordinate-wise product
partial ordering denoted by≺. A function f:[n] d→ ℕ is monotone if∀ x≺ 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 …

Optimal algorithms for testing monotonicity and Lipschitz over hypergrids … Lower bounds
for monotonicity testing … • Our functions are monotone or far from monotone in both worlds …