Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking. One of the fundamental problems in sensor networks is the calculation of the coverage. In Meguerdichian et al. (2001), it is assumed that the sensor has uniform sensing ability. In this paper, we give efficient distributed algorithms to optimally solve the best-coverage problem raised in Meguerdichian. Here, we consider the sensing model: the sensing ability diminishes as the distance increases. As energy conservation is a major concern in wireless (or sensor) networks, we also consider how to find an optimum best-coverage-path with the least energy consumption. We also consider how to find an optimum best-coverage-path that travels a small distance. In addition, we justify the correctness of the method proposed in Meguerdichian, that uses the Delaunay triangulation to solve the best coverage problem. Moreover, we show that the search space of the best coverage problem can be confined to the relative neighborhood graph, which can be constructed locally.