An Optimal Lower Bound for Monotonicity Testing over Hypergrids

For positive integers n,d, the hypergrid [n]d is equipped with the coordinatewise 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−1d logn− ε−1 logε−1) queries. Recent upper bounds show the existence of O(ε−1d logn) 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 logn). ACM Classification: F.1.2, F.2.2 AMS Classification: 68Q17, 68W20