Space and move-optimal Arbitrary Pattern Formation on infinite rectangular grid by Oblivious Robot Swarm

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.