Abstract
Given a set of points in the plane, we are interested in matching them with straight line segments. We focus on perfect (all points are matched) non-crossing (no two edges intersect) matchings. Apart from the well known MinMax variant, where the length of the longest edge is minimized, we extend work by looking into different optimization variants such as MaxMin, MinMin and MaxMax. We consider both the monochromatic and bichromatic versions of these problems and by employing diverse techniques we provide efficient algorithms for various input point configurations.
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Acknowledgements
M. S. was partially supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, project 174019, and H. S. by the German Research Foundation, DFG grant FE-340/11-1.
Initial discussions took place at the Intensive Research Program in Discrete, Combinatorial and Computational Geometry which took place in Barcelona in 2018. We are grateful to CRM, UAB for hosting the event and to the organizers for providing the platform to meet and collaborate. We would like to thank Carlos Alegría, Carlos Hidalgo Toscano, Oscar Iglesias Valiño, and Leonardo Martínez Sandoval for preliminary discussions, and Carlos Seara for raising a question that motivated this work. Finally, we would like to thank an anonymous reviewer for bringing to our attention the halfplane range queries.
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Mantas, I., Savić, M., Schrezenmaier, H. (2021). New Variants of Perfect Non-crossing Matchings. In: Mudgal, A., Subramanian, C.R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2021. Lecture Notes in Computer Science(), vol 12601. Springer, Cham. https://doi.org/10.1007/978-3-030-67899-9_12
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