Revised: December 3, 2014
Published: December 24, 2014
Abstract: [Plain Text Version]
For positive integers $n, d$, the hypergrid $[n]^d$ is equipped with the coordinatewise product partial ordering denoted by $\prec$. A function $f: [n]^d \to \NN$ is monotone if $\forall x \prec y$, $f(x) \leq f(y)$. A function $f$ is $\eps$-far from monotone if at least an $\eps$ fraction of values must be changed to make $f$ monotone. Given a parameter $\eps$, a monotonicity tester must distinguish with high probability a monotone function from one that is $\eps$-far.
We prove that any (adaptive, two-sided) monotonicity tester for functions $f:[n]^d \to \NN$ must make $\Omega(\eps^{-1}d\log n - \eps^{-1}\log \eps^{-1})$ queries. Recent upper bounds show the existence of $O(\eps^{-1}d \log n)$ 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 $\Omega(d \log n)$.
A conference version of this paper appeared in the Proceedings of the 17th Internat. Workshop o Randomization and Computation (RANDOM 2013).
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