On selecting a fraction of leaves with disjoint neighborhoods in a plane tree

Abstract

We present a generalization of a combinatorial result by Aggarwal et al. (1989) on a linear-time algorithm that selects a constant fraction of leaves, with pairwise disjoint neighborhoods, from a binary tree embedded in the plane. This result of Aggarwal et al. (1989) is essential to the linear-time framework, which they also introduced, that computes certain Voronoi diagrams of points with a tree structure in linear time. An example is the diagram computed while updating the Voronoi diagram of points after deletion of one site. Our generalization allows that only a fraction of the tree leaves is considered, and it is motivated by research on linear time construction algorithms for Voronoi diagrams of non-point sites. We are given a plane tree T of n leaves, m of which have been marked, and each marked leaf is associated with a neighborhood (a subtree of T ) such that any two topologically consecutive marked leaves have disjoint neighborhoods. We show how to select in linear time a constant fraction of the marked leaves having pairwise disjoint neighborhoods.