Adaptive Lower Bound for Testing Monotonicity on the Line.

In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. It is an active area of research. Recently, Pallavoor, Raskhodnikova and Varma (ITCSu002717) proposed an $varepsilon$-tester for monotonicity of a function $fcolon [n]to[r]$, whose complexity depends on the size of the range as $O(frac{log r}{varepsilon})$. In this paper, we prove a nearly matching lower bound of $Omega(frac{log r}{log log r})$ for adaptive two-sided testers. Additionally, we give an alternative proof of the $Omega(varepsilon^{-1}dlog n - varepsilon^{-1}logvarepsilon^{-1})$ lower bound for testing monotonicity on the hypergrid $[n]^d$ due to Chackrabarty and Seshadhri (RANDOMu002713).