On Selecting Leaves with Disjoint Neighborhoods in Embedded Trees
We present a generalization of a combinatorial result from Aggarwal, Guibas, Saxe and Shor  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.
KeywordsTree Linear-time algorithm Neighborhood Voronoi diagram
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