Minimum Polygonal Separation

Abstract In this paper we study the problem of polygonal separation in the plane, i.e., finding a convex polygon with minimum number k of sides separating two given finite point sets ( k -separator), if it exists. We show that for k = Θ ( n ), Ω(n log n) is a lower bound to the running time of any algorithm for this problem, and exhibit two algorithms of distinctly different flavors. The first relies on an O ( n log n )-time preprocessing task, which constructs the convex hull of the internal set and a nested star-shaped polygon determined by the external set; the k -separator is contained in the annulus between the boundaries of these two polygons and is constructed in additional linear time. The second algorithm adapts the prune-and-search approach, and constructs, in each iteration, one side of the separator; its running time is O ( kn ), but the separator may have one more side than the minimum.