Min-Max Gathering on Infinite Grid

TL;DR

This paper studies the optimal gathering problem on an infinite grid, introducing a deterministic algorithm for most initial configurations to minimize maximum travel distance, and characterizes the subgraph of min-max nodes.

Contribution

It identifies unsolvable initial configurations and presents a new deterministic algorithm for efficient gathering on an infinite grid.

Findings

Characterization of initial configurations where gathering is impossible.

A deterministic algorithm that gathers robots efficiently for most configurations.

Complete description of the subgraph of min-max nodes in the grid.

Abstract

Gathering is a fundamental coordination problem in swarm robotics, where the objective is to bring robots together at a point not known to them at the beginning. While most research focuses on continuous domains, some studies also examine the discrete domain. This paper addresses the optimal gathering problem on an infinite grid, aiming to improve the energy efficiency by minimizing the maximum distance any robot must travel. The robots are autonomous, anonymous, homogeneous, identical, and oblivious. We identify all initial configurations where the optimal gathering problem is unsolvable. For the remaining configurations, we introduce a deterministic distributed algorithm that effectively gathers $n$ robots ($n\ge 9$). The algorithm ensures that the robots gathers at one of the designated min-max nodes in the grid. Additionally, we provide a comprehensive characterization of the subgraph formed by the min-max nodes in this infinite grid model.