Maximum Plane Trees in Multipartite Geometric Graphs
- 54 Downloads
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.
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 \(k\geqslant 4\).
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 (Discrete Comput Geom 44(4):727–752, 2010) whose ratio is 0.502.
For points that are in convex position, the maximum bichromatic plane spanning tree problem can be solved in \(O(n^3)\) time. We present an \(O(n^5)\)-time algorithm that solves this problem for the case where the red points lie on a line and the blue points lie on one side of the line.
KeywordsMultipartite geometric graphs Plane spanning trees Maximum spanning trees Approximation algorithms
We would like to thank an anonymous referee whose comments improved the readability of the paper. Funding was provide by NSERC and NSF (Grant CCF-1228639)
- 2.Biniaz, A., Bose, P., Eppstein, D., Maheshwari, A., Morin, P., Smid, M.: Spanning trees in multipartite geometric graphs. Algorithmica (2017). https://doi.org/10.1007/s00453-017-0375-4