Algorithms for Covering Multiple Barriers

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 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 log n log log n + nm logm)-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.