Spanning Trees in Multipartite Geometric Graphs

TL;DR

This paper introduces efficient algorithms for computing minimum and maximum bichromatic spanning trees in geometric graphs, extending to multicolored point sets with improved time complexities.

Contribution

It presents new algorithms with optimal and near-optimal time bounds for bichromatic and multicolored spanning tree problems in geometric graphs.

Findings

Algorithms run in $O(n ext{log}^3 n)$ time for minimum and maximum bichromatic spanning trees.

Optimal $\Theta(n\text{log} n)$-time algorithms are developed for these problems.

Multicolored spanning tree algorithms operate in $O(n\text{log} n\text{log} k)$ time, where $k$ is the number of colors.

Abstract

Let $R$ and $B$ be two disjoint sets of points in the plane where the points of $R$ are colored red and the points of $B$ are colored blue, and let $n=|R\cup B|$. A bichromatic spanning tree is a spanning tree in the complete bipartite geometric graph with bipartition $(R,B)$. The minimum (respectively maximum) bichromatic spanning tree problem is the problem of computing a bichromatic spanning tree of minimum (respectively maximum) total edge length. 1. We present a simple algorithm that solves the minimum bichromatic spanning tree problem in $O(n\log^3 n)$ time. This algorithm can easily be extended to solve the maximum bichromatic spanning tree problem within the same time bound. It also can easily be generalized to multicolored point sets. 2. We present $\Theta(n\log n)$-time algorithms that solve the minimum and the maximum bichromatic spanning tree problems. 3. We extend the bichromatic spanning tree algorithms and solve the multicolored version of these problems in $O(n\log n\log k)$ time, where $k$ is the number of different colors (or the size of the multipartition in a complete multipartite geometric graph).