Molecular pattern formation on grids in the Moblot model

In the theoretical studies on distributed algorithms for swarm robotics, the complexity and capabilities of the robots are usually reduced to their minimum. Recently, the Moblot model has been introduced in order to deal with robots considered silent, anonymous, and oblivious but capable to aggregate into more complex structures, called molecules. We study the case where robots move along a graph based on a square lattice and we formally define the Molecular Pattern Formation (MPF) problem, where a specific configuration of robots assembled into molecules must be reached. As a preliminary general result, we provide a necessary condition for its solvability. Then, we actually show that dealing with molecules can resolve in some cases the symmetry breaking issue on grids where otherwise robots cannot. Finally, we introduce an interesting case study, representative of the MPF problem, in which the molecules can be formed by the set of the seven tetrominoes (aka Tetris blocks). We provide a complete characterization of this specific problem, providing a distributed algorithm able to form a molecular pattern whenever the necessary condition for the solvability of MPF is verified.