On selecting a fraction of leaves with disjoint neighborhoods in a plane tree
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...
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| Published in: | Discrete Applied Mathematics Vol. 319; pp. 141 - 148 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
15.10.2022
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| Subjects: | |
| ISSN: | 0166-218X, 1872-6771 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| ISSN: | 0166-218X 1872-6771 |
| DOI: | 10.1016/j.dam.2021.02.002 |