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Volume 10 (2014) Article 17 pp. 453-464
An Optimal Lower Bound for Monotonicity Testing over Hypergrids
Received: April 2, 2014
Revised: December 3, 2014
Published: December 24, 2014
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Keywords: lower bounds, property testing, monotonicity testing
ACM Classification: K.4.1, I.2.6, F.2.0
AMS Classification: 68Q32, 68Q25, 68W20

Abstract: [Plain Text Version]

$ \newcommand{\eps}{\varepsilon} \newcommand{\NN}{\mathbb{N}} $

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).