Evacuation from a Finite 2D Square Grid Field by a Metamorphic Robotic System

We consider evacuation from a finite two-dimensional (2D) square grid field by a metamorphic robotic system (MRS). An MRS is composed of anonymous memoryless modules. Each module of an MRS executes an identical distributed algorithm and moves autonomously while keeping the connectivity of modules. Since the modules are memoryless, an MRS utilizes its shape to remember the progress of execution. The number of available shapes that an MRS can form depends on the number of modules, which is thus an important complexity measure for a behavior of an MRS. In this paper, we investigate the minimum number of modules required to solve the evacuation problem with several conditions. First, we consider a rectangular field surrounded by walls with at least one exit and show that two modules are necessary and sufficient for evacuation from any rectangular field if the modules are equipped with a global compass, which allows the modules to have a common sense of direction. Then, we focus on the case where modules do not have a global compass and show that four (resp.~seven) modules are necessary and sufficient for restricted (resp.~any) initial states of an MRS. We also show that two modules are sufficient in the special case where an MRS is on a wall in an initial configuration. Finally, we extend these results to another type of fields, that is, mazes.