Optimal Physical Sorting of Mobile Agents

TL;DR

This paper presents an optimal centralized algorithm for physically sorting mobile agents by color on a grid, matching the theoretical lower bound on sorting time, and compares it to a decentralized approach.

Contribution

It derives an exact lower bound on sorting time and introduces an instance-optimal algorithm that achieves this bound, improving understanding of mobile agent sorting efficiency.

Findings

The optimal algorithm matches the lower bound on sorting time.

Decentralized algorithms are nearly as efficient in specific configurations.

The study provides a theoretical foundation for mobile agent sorting efficiency.

Abstract

Given a collection of red and blue mobile agents located on two grid rows, we seek to move all the blue agents to the far left side and all the red agents to the far right side, thus \textit{physically sorting} them according to color. The agents all start on the bottom row. They move simultaneously at discrete time steps and must not collide. Our goal is to design a centralized algorithm that controls the agents so as to sort them in the least number of time steps. We derive an \textbf{exact} lower bound on the amount of time any algorithm requires to sort a given initial configuration of agents. We find an instance optimal algorithm that provably matches this lower bound, attaining the best possible sorting time for any initial configuration. Surprisingly, we find that whenever the leftmost agent is red and the rightmost agent is blue, a straightforward decentralized and local sensing-based algorithm is at most $1$ time step slower than the centralized instance-optimal algorithm.