# On the monotone complexity of the shift operator

## TL;DR

This paper analyzes the complexity of minimal monotone circuits implementing a permutation operator on boolean vectors, providing new insights and alternative proofs for known bounds in monotone computational complexity.

## Contribution

It establishes the exact asymptotic complexity of the monotone permutation operator and offers an alternative proof for the complexity of the monotone shift operator.

## Key findings

- Complexity of monotone permutation operator is Θ(qn log n).
- Provides an alternative proof for the Θ(n log n) bound of the monotone shift operator.
- Enhances understanding of monotone circuit complexity for permutation functions.

## Abstract

We show that the complexity of minimal monotone circuits implementing a monotone version of the permutation operator on $n$ boolean vectors of length $q$ is $\Theta(qn\log n)$. In particular, we obtain an alternative way to prove the known complexity bound $\Theta(n\log n)$ for the monotone shift operator on $n$ boolean inputs.