Separating bichromatic point sets in the plane by restricted orientation convex hulls

Abstract

We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and mathcal {O} be a set of kge 2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of mathcal {O} for which the mathcal {O}-convex hull of R contains no points of B. For k=2 orthogonal lines we have the rectilinear convex hull. In optimal O(nlog n) time and O(n) space, n = vert R vert + vert B vert , we compute the set of rotation angles such that, after simultaneously rotating the lines of mathcal {O} around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where mathcal {O} is formed by k ge 2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of mathcal {O}, let alpha _i be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in O({1}/{Theta }cdot N log N) time and O({1}/{Theta }cdot N) space, where Theta = min { alpha _1,ldots ,alpha _k } and N=max {k,vert R vert + vert B vert }. We finally consider the case in which mathcal {O} is formed by k=2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to pi . We show that this last case can also be solved in optimal O(nlog n) time and O(n) space, where n = vert R vert + vert B vert .