  Single time-scale distributed estimation of dynamic systems via a network of
sensors/estimators is addressed in this letter. In single time-scale
distributed estimation, the two fusion steps, consensus and measurement
exchange, are implemented only once, in contrast to, e.g., a large number of
consensus iterations at every step of the system dynamics. We particularly
discuss the problem of failure in the sensor/estimator network and how to
recover for distributed estimation by adding new sensor measurements from
equivalent states. We separately discuss the recovery for two types of sensors,
namely \alpha and \beta sensors. We propose polynomial order algorithms to find
equivalent state nodes in graph representation of system to recover for
distributed observability. The polynomial order solution is particularly
significant for large-scale systems.
