Arbitrary Pattern Formation (APF) is a fundamental coordination problem in swarm robotics. It requires a set of autonomous robots (mobile computing units) to form an arbitrary pattern (given as input) starting from any initial pattern. This problem has been extensively investigated in continuous and discrete scenarios, with this study focusing on the discrete variant. A set of robots is placed on the nodes of an infinite rectangular grid graph embedded in the euclidean plane. The movements of each robot is restricted to one of the four neighboring grid nodes from its current position. The robots are autonomous, anonymous, identical, and homogeneous, and operate Look-Compute-Move cycles. In this work, we adopt the classical $\mathcal{OBLOT}$ robot model, meaning the robots have no persistent memory or explicit communication methods, yet they possess full and unobstructed visibility. This work proposes an algorithm that solves the APF problem in a fully asynchronous scheduler assuming the initial configuration is asymmetric. The considered performance measures of the algorithm are space and number of moves required for the robots. The algorithm is asymptotically move-optimal. Here, we provide a definition of space complexity that takes the visibility issue into consideration. We observe an obvious lower bound $\mathcal{D}$ of the space complexity and show that the proposed algorithm has the space complexity $\mathcal{D}+4$. On comparing with previous related works, we show that this is the first proposed algorithm considering $\mathcal{OBLOT}$ robot model that is asymptotically move-optimal and has the least space complexity which is almost optimal.

翻译：任意模式形成（APF）是群体机器人学中一个基础协调问题。它要求一组自主机器人（移动计算单元）从任意初始模式出发，形成给定输入的目标模式。该问题在连续与离散场景中均得到广泛研究，本文聚焦于离散变体。一组机器人被放置在嵌入欧几里得平面的无限矩形网格图节点上，每个机器人只能从当前位置向四个相邻网格节点之一移动。机器人具备自主性、匿名性、同一性与同质性，并遵循“观察-计算-移动”周期。本文采用经典的$\mathcal{OBLOT}$机器人模型，即机器人没有持久记忆或显式通信手段，但拥有完整无阻碍的视野。我们提出一种算法，在初始配置非对称的前提下，于完全异步调度器中解决APF问题。算法的性能指标为机器人所需空间和移动次数，且该算法在移动次数上渐近最优。本文给出考虑视野问题的空间复杂度定义，观察到空间复杂度的明显下界$\mathcal{D}$，并证明所提算法的空间复杂度为$\mathcal{D}+4$。与先前相关工作对比表明，这是首个在$\mathcal{OBLOT}$机器人模型下兼具渐近移动最优性与最低近优空间复杂度的算法。