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In this paper, we study the problem of detecting and eliminating redundancy in a sensor network with a view to improving energy efficiency, while preserving the network's coverage. We also examine the impact of redundancy elimination on the related problem of coverage-boundary detection. We reduce both problems to the computation of Voronoi diagrams, prove and achieve lower bounds on the solution of these problems, and present efficient distributed algorithms for computing and maintaining solutions in cases of sensor failures or insertion of new sensors. We prove the correctness and termination properties of our distributed algorithms, and analytically characterize the time complexity and the traffic generated by our algorithms. Our simulations show that the traffic generated per sensor insertion or removal (failure) experiences a dramatic decrease with an increase in sensor density, (up to 300% when the number of sensors deployed in the same 1000 × 1000 m2 area increases from 150 to 800), and with an increase in radio transmission range (up to 200% when the sensor's transmission range increases from 70 m to 200 m).