Optimal Adaptive Detection of Monotone Patterns

TL;DR

This paper develops an optimal adaptive algorithm for detecting monotone patterns in arrays, significantly improving over non-adaptive methods and matching known lower bounds, with applications to testing the length of the longest increasing subsequence.

Contribution

It introduces an adaptive algorithm with $O( ext{log } n)$ queries for detecting length-$k$ increasing subsequences, breaking previous non-adaptive bounds and matching lower bounds for monotonicity testing.

Findings

Adaptive algorithm achieves $O( ext{log } n)$ query complexity.

Breaks non-adaptive lower bounds for pattern detection.

Query complexity for testing LIS length is $ heta( ext{log } n)$.

Abstract

We investigate adaptive sublinear algorithms for detecting monotone patterns in an array. Given fixed $2 \leq k \in \mathbb{N}$ and $\varepsilon > 0$, consider the problem of finding a length-$k$ increasing subsequence in an array $f \colon [n] \to \mathbb{R}$, provided that $f$ is $\varepsilon$-far from free of such subsequences. Recently, it was shown that the non-adaptive query complexity of the above task is $\Theta((\log n)^{\lfloor \log_2 k \rfloor})$. In this work, we break the non-adaptive lower bound, presenting an adaptive algorithm for this problem which makes $O(\log n)$ queries. This is optimal, matching the classical $\Omega(\log n)$ adaptive lower bound by Fischer [2004] for monotonicity testing (which corresponds to the case $k=2$), and implying in particular that the query complexity of testing whether the longest increasing subsequence (LIS) has constant length is $\Theta(\log n)$.