Abstract

We study continuous partitioning problems on tree network spaces whose edges and nodes are points in Euclidean spaces. A continuous partition of this space into p connected components is a collection of p subtrees, such that no pair of them intersect at more than one point, and their union is the tree space. An edge-partition is a continuous partition defined by selecting p−1 cut points along the edges of the underlying tree, which is assumed to have n nodes. These cut points induce a partition into p subtrees (connected components). The objective is to minimize (maximize) the maximum (minimum) “size” of the components (the min–max (max–min) problem). When the size is the length of a subtree, the min–max and the max–min partitioning problems are NP-hard. We present O(n² log(min(p,n))) algorithms for the edge-partitioning versions of the problem. When the size is the diameter, the min–max problems coincide with the continuous p-center problem. We describe O(n log³ n) and O(n log² n) algorithms for the max–min partitioning and edge-partitioning problems, respectively, where the size is the diameter of a component.

Author Keywords: Tree partitioning; Continuous p-center problems; Bottleneck problems; Parametric search

Article Outline

1. Introduction

2. Formulation of the continuous bottleneck tree edge-partitioning problems

Notation

3. The continuous max–min tree length edge-partitioning problem

3.1. Algorithm 1: Computation of M(l)

3.2. Characterizing the optimal value

3.3. The parametric algorithm

3.4. An example

4. The continuous min–max tree length edge-partitioning problem

4.1. Algorithm 2: Computation of m(l)

5. Continuous bottleneck models involving the component diameters

5.1. Maximizing the minimum diameter using edge-partitioning

5.2. Maximizing the minimum diameter using partitioning

6. Remarks on non-crossing problems

References