Abstract
Let P be a set of n points in general position in the plane which is partitioned into color classes. P is said to be color-balanced if the number of points of each color is at most \(\lfloor n/2\rfloor \). Given a color-balanced point set P, a balanced cut is a line which partitions P into two color-balanced point sets, each of size at most \(2n/3 + 1\). A colored matching of P is a perfect matching in which every edge connects two points of distinct colors by a straight line segment. A plane colored matching is a colored matching which is non-crossing. In this paper, we present an algorithm which computes a balanced cut for P in linear time. Consequently, we present an algorithm which computes a plane colored matching of P optimally in \(\Theta (n\log n)\) time.
Research supported by NSERC.
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References
The 1979 Putnam exam. In: Alexanderson, G.L., Klosinski, L.F., Larson, L.C. (eds.) The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965–1984. Mathematical Association of America, USA (1985)
Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput. 29(3), 912–953 (1999)
Aichholzer, O., Cabello, S., Monroy, R.F., Flores-Peñaloza, D., Hackl, T., Huemer, C., Hurtado, F., Wood, D.R.: Edge-removal and non-crossing configurations in geometric graphs. Disc. Math. & Theo. Comp. Sci. 12(1), 75–86 (2010)
Bereg, S., Hurtado, F., Kano, M., Korman, M., Lara, D., Seara, C., Silveira, R.I., Urrutia, J., Verbeek, K.: Balanced partitions of 3-colored geometric sets in the plane. Discrete Applied Mathematics 181, 21–32 (2015)
Bereg, S., Kano, M.: Balanced line for a 3-colored point set in the plane. Electr. J. Comb. 19(1), P33 (2012)
Cherkassky, B.V., Goldberg, A.V., Silverstein, C.: Buckets, heaps, lists, and monotone priority queues. SIAM J. Comput. 28(4), 1326–1346 (1999)
Gabow, H.N.: Data structures for weighted matching and nearest common ancestors with linking. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434–443 (1990)
Hershberger, J., Suri, S.: Applications of a semi-dynamic convex hull algorithm. BIT 32(2), 249–267 (1992)
Kano, M., Suzuki, K., Uno, M.: Properly colored geometric matchings and 3-trees without crossings on multicolored points in the plane. In: Akiyama, J., Ito, H., Sakai, T. (eds.) JCDCGG 2013. LNCS, vol. 8845, pp. 96–111. Springer, Heidelberg (2014)
Lo, C., Matousek, J., Steiger, W.L.: Algorithms for ham-sandwich cuts. Discrete & Computational Geometry 11, 433–452 (1994)
Overmars, M.H., van Leeuwen, J.: Maintenance of configurations in the plane. J. Comput. Syst. Sci. 23(2), 166–204 (1981)
Sitton, D.: Maximum matchings in complete multipartite graphs. Furman University Electronic Journal of Undergraduate Mathematics 2, 6–16 (1996)
Vaidya, P.M.: Geometry helps in matching. SIAM J. Comput. 18(6) (1989)
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Biniaz, A., Maheshwari, A., Nandy, S.C., Smid, M. (2015). An Optimal Algorithm for Plane Matchings in Multipartite Geometric Graphs. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_6
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DOI: https://doi.org/10.1007/978-3-319-21840-3_6
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