Improved Testing Algorithms for Monotonicity.

TL;DR: Improved algorithms for testing monotonicity of functions are presented, given the ability to query an unknown function f: Σ n ↦ Ξ, and the test always accepts a monotone f, and rejects f with high probability if it is e-far from being monotones.

Abstract: We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f: Σ n ↦ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f, and rejects f with high probability if it is e-far from being monotone (i.e., every monotone function differs from f on more than an e fraction of the domain). For any e > 0, the query complexity of the test is O((n/e) · log ∣Σ ∣ · log ∣Ξ∣). The previous best known bound was \(\tilde{O}((n^2/\epsilon) \cdot \vert\Sigma\vert^2 \cdot \vert\Xi\vert)\).