Minimum Plane Bichromatic Spanning Trees

TL;DR

This paper studies minimum bichromatic spanning trees in the plane, proving structural properties of such trees, and introduces an approximation algorithm for the NP-hard problem of finding a minimum plane bichromatic spanning tree.

Contribution

It proves that MinBSTs are quasi-plane and provides bounds on crossings, and presents an $O(rac{ ext{log} n})$-factor approximation algorithm for MinPBST.

Findings

MinBSTs are quasi-plane with no three pairwise crossing edges.

Maximum number of crossings in a MinBST is determined.

An $O( ext{log} n)$-approximation algorithm for MinPBST is developed.

Abstract

For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in $O(n\log n)$ time where $n$ is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of $O(\sqrt{n})$. It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an $O(\log n)$-factor approximation algorithm for the general case.