Covering points with convex sets of minimum size

Abstract

For a set P of n points in the plane, we present algorithms for finding two bounded convex sets that cover P such that the total area or perimeter of the convex sets is minimized in O(n4log⁡n) and O(n2log⁡n) time, respectively. The former is the first result for minimum total area, and the latter is an improvement on the fastest previous algorithm for minimum total perimeter, which runs in O(n3) time [24]. We also extend our algorithms to find k⩾2 convex sets minimizing area in O(n2k(k−1)log⁡n) time. The algorithms can be applied to detect road intersections from the GPS trajectories of moving vehicles for automated map generation or partial clustering.