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

We investigate a swarm of autonomous mobile robots in the Euclidean plane. A robot has a function called {\em target function} to determine the destination point from the robots' positions. All robots in the swarm conventionally take the same target function, but there is apparent limitation in problem-solving ability. We allow the robots to take different target functions. The number of different target functions necessary and sufficient to solve a problem $\Pi$ is called the {\em minimum algorithm size} (MAS) for $\Pi$. We establish the MASs for solving the gathering and related problems from {\bf any} initial configuration, i.e., in a {\bf self-stabilizing} manner. We show, for example, for $1 \leq c \leq n$, there is a problem $\Pi_c$ such that the MAS for the $\Pi_c$ is $c$, where $n$ is the size of swarm. The MAS for the gathering problem is 2, and the MAS for the fault tolerant gathering problem is 3, when $1 \leq f (< n)$ robots may crash, but the MAS for the problem of gathering all robot (including faulty ones) at a point is not solvable (even if all robots have distinct target functions), as long as a robot may crash.