Minimum Plane Bichromatic Spanning Trees

TL;DR

It is proved that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and the maximum number of crossings is determined.

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.

Figures (5)

• Figure 1: A bicolored point set and its minimum bichromatic spanning tree (MinBST).
• Figure 2: (a) Shaded square contains $S$. (b) Translated subdivision of $Q$. (c) Randomly shifted quadtree on points of $S$ with respect to the subdivision of $Q$.
• Figure 3: Merging two squares in the same row: (a) case 1, (b) case 2, and (c) case 3.
• Figure 4: Illustration of the proof of Lemma \ref{['endpoint-lemma']}. Uncrossing a pair of crossing edges.
• Figure 5: (a) Replacing $e_1,e_2,e_3$ by $(r_1,b_3), (r_3,b_2),(r_2,b_1)$; the highlighted path is $\delta(e_1,e_2)$. (b) Getting a cycle in the union of $\delta(e_1,e_2)$, $\delta(e_2,e_3)$, $\delta(e_1,e_3)$, together with $e_1$ or $e_3$. Gray paths represent the two possible choices for $\delta(e_1,e_3)$.

Theorems & Definitions (17)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 2