  We study the problem of distributed coverage control in a network of mobile
agents arranged on a line. The goal is to design distributed dynamics for the
agents to achieve optimal coverage positions with respect to a scalar density
field that measures the relative importance of each point on the line. Unlike
previous work, which has implicitly assumed the agents know this density field,
we only assume that each agent can access noisy samples of the field at points
close to its current location. We provide a simple randomized protocol wherein
every agent samples the scalar field at three nearby points at each step and
which guarantees convergence to the optimal positions. We further analyze the
convergence time of this protocol and show that, under suitable assumptions,
the squared distance to the optimal coverage configuration decays as $O(1/t)$
with the number of iterations $t$, where the constant scales polynomially with
the number of agents $n$. We illustrate these results with simulations.
