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Algorithmica

, Volume 80, Issue 11, pp 3177–3191 | Cite as

Spanning Trees in Multipartite Geometric Graphs

  • Ahmad BiniazEmail author
  • Prosenjit Bose
  • David Eppstein
  • Anil Maheshwari
  • Pat Morin
  • Michiel Smid
Article

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 (RB). 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).

Keywords

Multipartite geometric graphs Minimum spanning tree Maximum spanning tree 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Ahmad Biniaz
    • 1
    Email author
  • Prosenjit Bose
    • 1
  • David Eppstein
    • 2
  • Anil Maheshwari
    • 1
  • Pat Morin
    • 1
  • Michiel Smid
    • 1
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Computer Science DepartmentUniversity of CaliforniaIrvineUSA

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