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Spanning Trees in Multipartite Geometric Graphs

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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).

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Authors and Affiliations

  1. School of Computer Science, Carleton University, Ottawa, Canada

    Ahmad Biniaz, Prosenjit Bose, Anil Maheshwari, Pat Morin & Michiel Smid

  2. Computer Science Department, University of California, Irvine, CA, USA

    David Eppstein

Authors
  1. Ahmad Biniaz
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  2. Prosenjit Bose
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  3. David Eppstein
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  4. Anil Maheshwari
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  5. Pat Morin
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  6. Michiel Smid
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Corresponding author

Correspondence to Ahmad Biniaz.

Additional information

A. Biniaz, P. Bose, A. Maheshwari, P. Morin, M.Smid supported by NSERC. D. Eppstein supported by NSF Grant CCF-1228639.

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Biniaz, A., Bose, P., Eppstein, D. et al. Spanning Trees in Multipartite Geometric Graphs. Algorithmica 80, 3177–3191 (2018). https://doi.org/10.1007/s00453-017-0375-4

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  • DOI: https://doi.org/10.1007/s00453-017-0375-4

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