Space and Move-optimal Arbitrary Pattern Formation on a Rectangular Grid by Robot Swarms

Arbitrary pattern formation (\textsc{Apf}) is a well-studied problem in swarm robotics. To the best of our knowledge, the problem has been considered in two different settings: one in a euclidean plane and another in an infinite grid. This work deals with the problem in an infinite rectangular grid setting. The previous works in literature dealing with the \textsc{Apf} problem in an infinite grid had a fundamental issue. These deterministic algorithms use a lot of space in the grid to solve the problem, mainly to maintain the asymmetry of the configuration or to avoid a collision. These solution techniques cannot be useful if there is a space constraint in the application field. In this work, we consider luminous robots (with one light that can take three colors) to avoid symmetry, but we carefully designed a deterministic algorithm that solves the \textsc{Apf} problem using the minimal required space in the grid. The robots are autonomous, identical, and anonymous, and they operate in Look-Compute-Move cycles under a fully asynchronous scheduler. The \textsc{Apf} algorithm proposed in \cite{BOSE2020} 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 is also asymptotically move-optimal. The algorithm proposed in this work is designed for an infinite rectangular grid, but it can be easily modified to work on a finite grid as well.