Non-Crossing Matching of Online Points

We consider the non-crossing matching problem in the online setting. In the monochromatic setting, a sequence of points in general position in the plane is revealed in an online manner, and the goal is to create a maximum matching of these points such that the line segments connecting pairs of matched points do not cross. The problem is online in the sense that the decisions to match each arriving point are irrevocable and should be taken without prior knowledge about forthcoming points. The bichromatic setting is defined similarly, except that half of the points are red and the rest are blue, and each matched pair consists of one red point and one blue point. Inspired by the online bipartite matching problem [15], where vertices on one side of a bipartite graph appear in an online manner, we assume red points are given a priory and blue points arrive in an online manner. In the offline setting, both the monochromatic and bichromatic problems can be solved optimally with all pairs matched [11]. In the online setting of the monochromatic version, we show that a greedy family of algorithms matches 2⌈(n - 1)/3⌉ points, where n is the number of input points. Meanwhile, we prove that no deterministic online algorithm can match more than 2⌈(n-1)/3⌉ points, i.e., the greedy strategy is optimal. In the bichromatic setting, we introduce an algorithm that matches log n - o(log n) points for instances consisting of n red and n blue points, and show that no deterministic algorithm can do better. We also consider the problem under the advice setting, where an online algorithm receives some bits of advice about the input sequence, and provide lower and upper bounds for the amount of advice that is required and sufficient to match all points.