Minimizing the Aggregate Movements for Interval Coverage

We consider 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 basic geometry problem, it has applications in mobile sensor barrier coverage in wireless sensor networks. The previous work solved the problem in $O(n^2)$ time. In this paper, by discovering many interesting observations and developing new algorithmic techniques, we present an $O(n\log n)$ time algorithm. We also show an $\mathrm{\Omega }(n\log n)$ time lower bound for this problem, which implies the optimality of our algorithm.