Algorithms for Covering Multiple Barriers

Abstract

In this paper, we consider the problems for covering multiple intervals on a line. Given a set B of m line segments (called “barriers”) on a horizontal line L and another set S of n horizontal line segments of the same length in the plane, we want to move all segments of S to L so that their union covers all barriers and the maximum movement of all segments of S is minimized. Previously, an \(O(n^3\log n)\)-time algorithm was given for the problem but only for the special case \(m=1\). In this paper, we propose an \(O(n^2\log n\log \log n+nm\log m)\)-time algorithm for any m, which improves the previous work even for \(m=1\). We then consider a line-constrained version of the problem in which the segments of S are all initially on the line L. Previously, an \(O(n\log n)\)-time algorithm was known for the case \(m=1\). We present an algorithm of \(O((n+m)\log (n+ m))\) time for any m. These problems may have applications in mobile sensor barrier coverage in wireless sensor networks.