An optimal lower bound for monotonicity testing over hypergrids

Abstract

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. 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). © 2013 Springer-Verlag.