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Minimum Plane Bichromatic Spanning Trees

Authors Hugo A. Akitaya , Ahmad Biniaz, Erik D. Demaine , Linda Kleist , Frederick Stock, Csaba D. Tóth

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Bichromatic Spanning Tree
  • Minimum Spanning Tree
  • Plane Tree

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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(√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.

Cite As Get BibTex

Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth. Minimum Plane Bichromatic Spanning Trees. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.4

Author Details

Hugo A. Akitaya
  • Miner School of Computer & Information Sciences, University of Massachusetts, Lowell, MA, USA
Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Erik D. Demaine
  • Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, Cambridge, MA, USA
Linda Kleist
  • Institute of Computer Science, Universität Potsdam, Germany
Frederick Stock
  • Miner School of Computer & Information Sciences, University of Massachusetts, Lowell, MA, USA
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA

Funding

  • A. Akitaya, Hugo: Research supported by the NSF award CCF-2348067.
  • Biniaz, Ahmad: Research supported by NSERC.
  • Tóth, Csaba D.: Research supported in part by the NSF award DMS-2154347.

Acknowledgements

This work was initiated at the Eleventh Annual Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Holetown, Barbados in March 2024. The authors thank the organizers and the participants.

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