Abstract
We present a generalization of a combinatorial result from Aggarwal, Guibas, Saxe and Shor [1] on selecting a fraction of leaves, with pairwise disjoint neighborhoods, in a tree embedded in the plane. This result has been used by linear-time algorithms to compute certain tree-like Voronoi diagrams, such as the Voronoi diagram of points in convex position. Our generalization allows that only a fraction of the tree leaves is considered: Given is a plane tree T of n leaves, m of which have been marked. Each marked leaf is associated with a neighborhood (a subtree of T) and any topologically consecutive marked leaves have disjoint neighborhoods. We show how to select in linear time a constant fraction of the marked leaves that have pairwise disjoint neighborhoods.
Research supported in part by the Swiss National Science Foundation, project SNF 200021E-154387.
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Junginger, K., Mantas, I., Papadopoulou, E. (2019). On Selecting Leaves with Disjoint Neighborhoods in Embedded Trees. In: Pal, S., Vijayakumar, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2019. Lecture Notes in Computer Science(), vol 11394. Springer, Cham. https://doi.org/10.1007/978-3-030-11509-8_16
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