Finding monotone patterns in sublinear time

Abstract: We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and $\varepsilon &gt; 0$, we show that the non-adaptive query complexity of finding a length-$k$ monotone subsequence of $f \colon [n] \to \mathbb{R}$, assuming that $f$ is $\varepsilon$-far from free of such subsequences, is $$\Theta((\log n)^{\lfloor \log_2 k \rfloor})$$. Prior to our work, the best algorithm for this problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made $(\log n)<sup>{O(k<sup>2)}$ non-adaptive queries; and the only lower bound known, of $\Omega(\log n)$ queries for the case $k = 2$, followed from that on testing monotonicity due to Erg\"un, Kannan, Kumar, Rubinfeld, and Viswanathan (2000) and Fischer (2004).