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Spanning Trees in Multipartite Geometric Graphs

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).

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References

  1. 1.

    Aggarwal, A., Edelsbrunner, H., Raghavan, P., Tiwari, P.: Optimal time bounds for some proximity problems in the plane. Inf. Process. Lett. 42(1), 55–60 (1992)

  2. 2.

    Aggarwal, A., Guibas, L.J., Saxe, J.B., Shor, P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete Comput. Geom. 4, 591–604 (1989)

  3. 3.

    Avis, D.: Lower bounds for geometric problems. In: 18th Allerton Conference, Urbana, IL, pp. 35–40 (1980)

  4. 4.

    Chazelle, B., Devillers, O., Hurtado, F., Mora, M., Sacristán, V., Teillaud, M.: Splitting a Delaunay triangulation in linear time. Algorithmica 34(1), 39–46 (2002)

  5. 5.

    Cherkassky, B.V., Goldberg, A.V., Silverstein, C.: Buckets, heaps, lists, and monotone priority queues. SIAM J. Comput. 28(4), 1326–1346 (1999)

  6. 6.

    Edelsbrunner, H.: Computing the extreme distances between two convex polygons. J. Algorithms 6(2), 213–224 (1985)

  7. 7.

    Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P., Sharir, M.: Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (\(SODA\)), pp. 2495–2504 (2017)

  8. 8.

    Kirkpatrick, D.G.: Efficient computation of continuous skeletons. In: 20th Annual Symposium on Foundations of Computer Science, pp. 18–27 (1979)

  9. 9.

    Löffler, M., Mulzer, W.: Triangulating the square and squaring the triangle: quadtrees and Delaunay triangulations are equivalent. SIAM J. Comput. 41(4), 941–974 (2012)

  10. 10.

    Monma, C.L., Paterson, M., Suri, S., Yao, F.F.: Computing Euclidean maximum spanning trees. Algorithmica 5(3), 407–419 (1990)

  11. 11.

    Preparata, F.P., Shamos, M.I.: Computational Geometry—An Introduction. Texts and Monographs in Computer Science. Springer, Berlin (1985)

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Author information

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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Keywords

  • Multipartite geometric graphs
  • Minimum spanning tree
  • Maximum spanning tree