Matching colored points in the plane: Some new results

Abstract

Let S be a set with n=w+b points in general position in the plane, w of them white, and b of them black. We consider the problem of computing G(S), a largest non-crossing matching of pairs of points of the same color, using straight line segments. We present two new algorithms which compute a large matching, with an improved guarantee in the number of matched points. The first one runs in O(n 2) time and finds a matching of at least 85.71% of the points. The second algorithm runs in O(n logn) time and achieves a performance guarantee as close as we want to that of the first algorithm. On the other hand, we show that there exist configurations of points such that any matching with the above properties matches fewer than 98.95% of the points. We further extend these results to point sets with a prescribed ratio of the sizes of the two color classes. In the end, we discuss the more general problem when the points are colored with any fixed number of colors.