Bottleneck segment matching

Let R be a set of n red segments and B a set of n blue segments, we wish to find the minimum value d-, such that there exists a perfect matching between R and B with bottleneck d-, i.e., the maximum distance between a matched red-blue pair is d-. We first solve the corresponding decision problem: Given R, B and a distance d > 0, find a maximum matching between R and B with bottleneck at most d. We begin with the simpler case where d = 0 and then extend our solution to the case where d > 0. We focus on the settings for which we are able to solve the decision problem efficiently, i.e., in roughly O(n1.5) time. The most general of these, is when one of the sets consists of disjoint arbitrary segments and the other of vertical segments. We apply similar ideas to find a matching in the setting in which the vertical segments are replaced by points in the plane. After solving the decision problem, we explain how to find the minimum value d-. Finally, we show how to compute a shortest path tree for a given set of n orthogonal segments and a designated root segment in O(n log2 n) time.