# Testing Unateness Nearly Optimally

## TL;DR

This paper introduces a nearly optimal algorithm for testing whether a Boolean function is unate, using a query complexity close to the theoretical lower bound, by leveraging a novel binary search approach.

## Contribution

The paper presents a new unateness testing algorithm with nearly optimal query complexity, improving upon previous methods with a novel binary search technique.

## Key findings

- Query complexity of $ ilde{O}(n^{2/3}/psilon^2)$ for unateness testing
- Nearly matches the known lower bound of $ ilde{
Omega}(n^{2/3})$
- Uses a novel binary search method over random paths

## Abstract

We present an $\tilde{O}(n^{2/3}/\epsilon^2)$-query algorithm that tests whether an unknown Boolean function $f\colon\{0,1\}^n\rightarrow \{0,1\}$ is unate (i.e., every variable is either non-decreasing or non-increasing) or $\epsilon$-far from unate. The upper bound is nearly optimal given the $\tilde{\Omega}(n^{2/3})$ lower~bound of [CWX17a]. The algorithm builds on a novel use of the binary search procedure and its analysis over long random paths.