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 Ω(n log n) time lower bound for this problem, which implies the optimality of our algorithm.
Similarly, in a homogeneous wireless sensor network, Wang et al. REF present an O(n log n) algorithm to minimize sensors' total movement on a line interval.
Minimizing the Aggregate Movements for Interval Coverage
Algorithmica
Algorithmica
Mathematics Computer Science