Let R be a connected closed region in the plane and let S be a set of n points (representing mobile sensors) in the interior of R. We think of R's boundary as a barrier which needs to be monitored. This gives rise to the barrier coverage problem, where one needs to move the sensors to the boundary of R, so that every point on the boundary is covered by one of the sensors. We focus on the variant of the problem where the goal is to place the sensors on R's boundary, such that the distance (along R's boundary) between any two adjacent sensors is equal to R's perimeter divided by n and the sum of the distances traveled by the sensors is minimum. In this paper, we consider the cases where R is either a circle or a convex polygon. We present a PTAS for the circle case and explain how to overcome the main difficulties that arise when trying to adapt it to the convex polygon case. Our PTASs are significantly faster than the previous ones due to Bhattacharya et al. [4]. Moreover, our PTASs require efficient solutions to problems, which, as we observe, are equivalent to the circle-restricted and line-restricted Weber problems. Thus, we also devise efficient PTASs for these Weber problems.
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