Planar bichromatic minimum spanning trees

Authors:
Magdalene G. Borgelt;Marc Van Kreveld;Maarten LöFfler;Jun Luo;Damian Merrick;Rodrigo I. Silveira;Mostafa Vahedi
Affiliations:
European Centre for Soft Computing, Mieres, Asturias, Spain;Department of Information and Computing Science, Utrecht University, the Netherlands;Department of Information and Computing Science, Utrecht University, the Netherlands;Department of Information and Computing Science, Utrecht University, the Netherlands;National ICT Australia, School of Information Technologies, University of Sydney, Australia;Department of Information and Computing Science, Utrecht University, the Netherlands;Department of Information and Computing Science, Utrecht University, the Netherlands
Venue:
Journal of Discrete Algorithms
Year:
2009

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Abstract

Given a set S of n red and blue points in the plane, a planar bichromatic minimum spanning tree is the shortest possible spanning tree of S, such that every edge connects a red and a blue point, and no two edges intersect. We show that computing this tree is NP-hard in general. For points in convex position, a cubic-time algorithm can be easily designed using dynamic programming. We adapt such an algorithm for the special case where the number of red points (m) is much smaller than the number of blue points (n), resulting in an O(nm^2) time algorithm. For the general case, we present a factor O(n) approximation algorithm that runs in O(nlognloglogn) time. Finally, we show that if the number of points in one color is bounded by a constant, the optimal tree can be computed in polynomial time.