Covering a set of line segments with a few squares

We study three covering problems in the plane. Our original motivation for these problems comes from trajectory analysis. The first is to decide whether a given set of line segments can be covered by up to k = 4unit-sized, axis-parallel squares. We give linear time algorithms for k <= 3and an O(n log n) time algorithm for k = 4. The second is to build a data structure on a trajectory to efficiently answer whether any query subtrajectory is coverable by up to three unit-sized axis-parallel squares. For k = 2 and k = 3we construct data structures of size O(n alpha(n) logn) in O(n alpha(n) logn) time, so that we can test if an arbitrary subtrajectory can be k-covered in O(logn) time. The third problem is to compute a longest subtrajectory of a given trajectory that can be covered by up to two unit-sized axis-parallel squares. We give O(n2(alpha(n)) log(2)n) time algorithms for k <= 2. (c) 2022 Elsevier B.V. All rights reserved.