Arbitrary Pattern Formation (APF) is a fundamental coordination problem in swarm robotics. It requires a set of autonomous robots (mobile computing units) to form any arbitrary pattern (given as input) starting from any initial pattern. The APF problem is well-studied in both continuous and discrete settings. This work concerns the discrete version of the problem. A set of robots is placed on the nodes of an infinite rectangular grid graph embedded in a euclidean plane. The movements of the robots are 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. Here we have considered the classical $\mathcal{OBLOT}$ robot model, i.e., the robots have no persistent memory and no explicit means of communication. The robots have full unobstructed visibility. This work proposes an algorithm that solves the APF problem in a fully asynchronous scheduler under this setting 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. A definition of the space-complexity is presented here. 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：任意图案形成（APF）是群体机器人学中的一个基础协调问题。该问题要求一组自主机器人（移动计算单元）从任意初始图案出发，形成任意给定的输入图案。APF问题在连续与离散设置中均被广泛研究，本文关注其离散版本。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的节点上，其移动限制为从当前位置向四个相邻网格节点之一移动。机器人是自主、匿名、相同且同质的，并执行"观察-计算-移动"循环。本文考虑经典的$\mathcal{OBLOT}$机器人模型，即机器人无持久性内存且无显式通信手段，但具有完全无遮挡的视野。本文提出一种算法，在假设初始配置为非对称的条件下，于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的空间与移动次数。该算法在渐近意义下移动最优。本文给出了空间复杂度的定义，并观察到空间复杂度的显式下界$\mathcal{D}$，同时证明所提算法的空间复杂度为$\mathcal{D}+4$。与以往相关工作对比表明，这是首个考虑$\mathcal{OBLOT}$机器人模型且同时满足渐近移动最优与近乎最优空间复杂度（最小）的算法。