Minimum algorithm sizes for self-stabilizing gathering and related problems of autonomous mobile robots

We investigate swarms of autonomous mobile robots in the Euclidean plane. Each robot has a target function to determine a destination point from the robots' positions. All robots in a swarm conventionally take the same target function. We allow the robots in a swarm to take different target functions, and investigate the effects of the number of distinct target functions on the problem-solving ability. Specifically, we are interested in how many distinct target functions are necessary and sufficient to solve some known problems which are not solvable when all robots take the same target function, regarding target function as a resource to solve a problem, like time and message. The number of distinct target functions necessary and sufficient to solve a problem $\Pi$ is called the minimum algorithm size (MAS) for $\Pi$. (The MAS is $\infty$, if $\Pi$ is not solvable even for the robots with unique target functions.) We establish the MASs for solving the gathering and related problems from any initial configuration, i.e., in a self-stabilizing manner. Our results include: There is a family of the scattering problems $c$SCT $(1 \leq c \leq n)$ such that the MAS for the $c$SCAT is $c$, where $n$ is the size of the swarm. The MAS for the gathering problem is 2. It is 3, for the problem of gathering all non-faulty robots at a single point, regardless of the number $(< n)$ of crash failures. It is however $\infty$, for the problem of gathering all robots at a single point, in the presence of at most one crash failure.