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

Abstract

AbstractWe 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 $$\mathcal {O}$$ O be a set of $$k\ge 2$$ k ≥ 2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of $$\mathcal {O}$$ O for which the $$\mathcal {O}$$ O -convex hull of R contains no points of B. For $$k=2$$ k = 2 orthogonal lines we have the rectilinear convex hull. In optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space, $$n = \vert R \vert + \vert B \vert $$ n = | R | + | B | , we compute the set of rotation angles such that, after simultaneously rotating the lines of $$\mathcal {O}$$ 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 $$\mathcal {O}$$ O is formed by $$k \ge 2$$ k ≥ 2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $$\mathcal {O}$$ O , let $$\alpha _i$$ α 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 $$O({1}/{\Theta }\cdot N \log N)$$ O ( 1 / Θ · N log N ) time and $$O({1}/{\Theta }\cdot N)$$ O ( 1 / Θ · N ) space, where $$\Theta = \min \{ \alpha _1,\ldots ,\alpha _k \}$$ Θ = min { α 1 , … , α k } and $$N=\max \{k,\vert R \vert + \vert B \vert \}$$ N = max { k , | R | + | B | } . We finally consider the case in which $$\mathcal {O}$$ O is formed by $$k=2$$ k = 2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to $$\pi $$ π . We show that this last case can also be solved in optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space, where $$n = \vert R \vert + \vert B \vert $$ n = | R | + | B | .