We present an efficient algorithm for solving an interval coverage problem. Given n intervals of the same length on a line L and a line segment B on L, we want to move the intervals along L such that every point of B is covered by at least one interval and the sum of the moving distances of all intervals is minimized. As a fundamental computational geometry problem, it has applications in mobile sensor barrier coverage in wireless sensor networks. The previous work gave an O(n2) time algorithm for it. In this thesis, by discovering many interesting observations and developing new algorithmic techniques, we present an O(nlogn) time algorithm for this problem. We also show that Ω(n log n) is the lower bound for the time complexity. Therefore, our algorithm is optimal. Further, our observations and algorithmic techniques may be useful for solving other related problems.