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WADS 2017: Algorithms and Data Structures pp 193-204

# Maximum Plane Trees in Multipartite Geometric Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

## Abstract

A geometric graph is a graph whose vertices are points in the plane and whose edges are straight-line segments between the points. A plane spanning tree in a geometric graph is a spanning tree that is non-crossing. 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 plane spanning tree is a plane spanning tree in the complete bipartite geometric graph with bipartition (RB). In this paper we consider the maximum bichromatic plane spanning tree problem, which is the problem of computing a bichromatic plane spanning tree of maximum total edge length.

1. 1.

For the maximum bichromatic plane spanning tree problem, we present an approximation algorithm with ratio 1/4 that runs in $$O(n\log n)$$ time.

2. 2.

We also consider the multicolored version of this problem where the input points are colored with $$k>2$$ colors. We present an approximation algorithm that computes a plane spanning tree in a complete k-partite geometric graph, and whose ratio is 1/6 if $$k=3$$, and 1/8 if $$k\geqslant 4$$.

3. 3.

We also revisit the special case of the problem where $$k=n$$, i.e., the problem of computing a maximum plane spanning tree in a complete geometric graph. For this problem, we present an approximation algorithm with ratio 0.503; this is an extension of the algorithm presented by Dumitrescu and Tóth (2010) whose ratio is 0.502.

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## References

1. 1.
Alon, N., Rajagopalan, S., Suri, S.: Long non-crossing configurations in the plane. Fundamenta Informaticae 22(4), 385–394 (1995). Also in Proceedings of the 9th ACM Symposium on Computational Geometry (SoCG), pp. 257–263 (1993)
2. 2.
Biniaz, A., Bose, P., Eppstein, D., Maheshwari, A., Morin, P., Smid, M.: Spanning trees in multipartite geometric graphs. CoRR, abs/1611.01661 (2016). Also submitted to AlgorithmicaGoogle Scholar
3. 3.
Borgelt, M.G., van Kreveld, M.J., Löffler, M., Luo, J., Merrick, D., Silveira, R.I., Vahedi, M.: Planar bichromatic minimum spanning trees. Journal of Discrete Algorithms 7(4), 469–478 (2009)
4. 4.
Dumitrescu, A., Tóth, C.D.: Long non-crossing configurations in the plane. Discrete & Computational Geometry 44(4), 727–752 (2010). Also in Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 311–322 (2010)
5. 5.
Monma, C.L., Paterson, M., Suri, S., Yao, F.F.: Computing Euclidean maximum spanning trees. Algorithmica 5(3), 407–419 (1990)

## Copyright information

© Springer International Publishing AG 2017

## Authors and Affiliations

• Ahmad Biniaz
• 1
Email author
• Prosenjit Bose
• 1
• Kimberly Crosbie
• 1
• Jean-Lou De Carufel
• 2
• David Eppstein
• 3
• Anil Maheshwari
• 1
• Michiel Smid
• 1
1. 1.Carleton UniversityOttawaCanada
2. 2.University of OttawaOttawaCanada
3. 3.University of CaliforniaIrvineUSA