Minimizing the Aggregate Movements for Interval Coverage

Abstract

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 $$O(n^2)$$O(n2) time. In this paper, by discovering many interesting observations and developing new algorithmic techniques, we present an $$O(n\log n)$$O(nlogn) time algorithm. We also show an $$\varOmega (n\log n)$$Ω(nlogn) time lower bound for this problem, which implies the optimality of our algorithm.