  In this paper we consider the question of sensor network coverage for a
2-dimensional domain. We seek to compute the probability that a set of sensors
fails to cover given only non-metric, local (who is talking to whom)
information and a probability distribution of failure of each node. This builds
on the work of de Silva and Ghrist who analyzed this problem in the
deterministic situation. We first show that a it is part of a slightly larger
class of problems which is #P-complete, and thus fast algorithms likely do not
exist unless P$=$NP. We then give a deterministic algorithm which is feasible
in the case of a small set of sensors, and give a dynamic algorithm for an
arbitrary set of sensors failing over time which utilizes a new criterion for
coverage based on the one proposed by de Silva and Ghrist. These algorithms
build on the theory of topological persistence.

Computational Geometry