Closest distance between two points(disjoint set)

Question

This problem is a kind of closest pair between two disjoint set. Upperside picture

Answer 1

This problem is usually called the closest bichromatic pair problem. Here are a couple approaches.

1. Delaunay triangulation. (This certainly works with L₂ (= Euclidean) distances; I think the steps generalize to L₁.) For every Delaunay triangulation (there can be more than one in degenerate cases), there exists a minimum spanning tree whose edges all belong to the triangulation. In turn, this minimum spanning tree contains a shortest edge crossing the cut between the color classes.

2. Nearest neighbor data structures.

3. If it is given that the red points are separated in x from the blue points, then you may be able to adapt the O(n) merge step of the Shamos–Hoey divide-and-conquer algorithm for the closest (monochromatic) pair problem, described here.