Space optimal and asymptotically move optimal Arbitrary Pattern Formation on rectangular grid by asynchronous robot swarm

Arbitrary pattern formation (\textsc{Apf}) is well studied problem in swarm robotics. The problem has been considered in two different settings so far; one is in plane and another is in infinite grid. This work deals the problem in infinite rectangular grid setting. The previous works in literature dealing with \textsc{Apf} problem in infinite grid had a fundamental issue. These deterministic algorithms use a lot space of the grid to solve the problem mainly because of maintaining asymmetry of the configuration or to avoid collision. These solution techniques can not be useful if there is a space constrain in the application field. In this work, we consider luminous robots (with one light that can take two colors) in order to avoid symmetry, but we carefully designed a deterministic algorithm which solves the \textsc{Apf} problem using minimal required space in the grid. The robots are autonomous, identical, anonymous and they operate in Look-Compute-Move cycles under a fully asynchronous scheduler. The \textsc{Apf} algorithm proposed in [WALCOM'2019] by Bose et al. can be modified using luminous robots so that it uses minimal space but that algorithm is not move optimal. The algorithm proposed in this paper not only uses minimal space but also asymptotically move optimal. The algorithm proposed in this work is designed for infinite rectangular grid but it can be easily modified to work in a finite grid as well.