Self-deployment of mobile sensors on a ring

Mobile sensors can self-deploy in a purely decentralized and distributed fashion, so as to reach in a finite time a state of static equilibrium in which they uniformly cover the environment. We consider the self-deployment problem in a ring (e.g., a circular rim); in particular we investigate under what conditions the problem is solvable by a collection of identical sensors without a global coordinate system, however capable of determining the location (in their local coordinate system) of the other sensors within a fixed distance (called visibility radius). A self-deployment is exact   if within finite time the distance between any two consecutive sensors along the ring is the same, d$d$; it is ?$?$-approximate   if within finite time the distance between two consecutive sensors is between d−?$d-?$ and d+?$d+?$.