# Optimized mixing by cutting-and-shuffling

## TL;DR

This paper improves mixing efficiency in cutting-and-shuffling systems by optimizing variable protocols, demonstrating analytical solutions for simple cases and proposing a computationally inexpensive heuristic for more complex systems.

## Contribution

It introduces a variable approach to cutting-and-shuffling, enhancing mixing beyond fixed protocols, with analytical solutions and a practical heuristic for complex systems.

## Key findings

- Optimized variable protocols significantly improve mixing.
- Analytical solutions are available for simple cases.
- Heuristic methods can approximate optimal mixing efficiently.

## Abstract

Mixing by cutting-and-shuffling can be understood and predicted using dynamical systems based tools and techniques. In existing studies, mixing is generated by maps that repeat the same cut-and-shuffle process at every iteration, in a "fixed" manner. However, mixing can be greatly improved by varying the cut-and-shuffle parameters at each step, using a "variable" approach. To demonstrate this approach, we show how to optimize mixing by cutting-and-shuffling on the one-dimensional line interval, known as an interval exchange transformation (IET). Mixing can be significantly improved by optimizing variable protocols, especially for initial conditions more complex than just a simple two-color line interval. While we show that optimal variable IETs can be found analytically for arbitrary numbers of iterations, for more complex cutting-and-shuffling systems, computationally expensive numerical optimization methods would be required. Furthermore, the number of control parameters grows linearly with the number of iterations in variable systems. Therefore, optimizing over large numbers of iterations is generally computationally prohibitive. We demonstrate an ad hoc approach to cutting-and-shuffling that is computationally inexpensive and guarantees the mixing metric is within a constant factor of the optimum. This ad hoc approach yields significantly better mixing than fixed IETs which are known to produce weak-mixing, because cut pieces never reconnect. The heuristic principles of this method can be applied to more general cutting-and-shuffling systems.