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.

翻译：聚集是群体机器人学中的一个基本协调问题，其目标是将机器人聚集到一个在开始时未知的点。尽管大多数研究集中于连续域，但也有一些研究探讨离散域。本文研究无限网格上的最优聚集问题，旨在通过最小化任意机器人必须行进的最大距离来提高能量效率。机器人是自主、匿名、同质、相同且无记忆的。我们识别了所有最优聚集问题不可解的初始配置。对于其余配置，我们提出一种确定性分布式算法，能有效聚集 $n$ 个机器人（$n\ge 9$）。该算法确保机器人聚集在网格中指定的某个最小-最大节点处。此外，我们完整刻画了该无限网格模型中由最小-最大节点构成的子图结构。