# Dispersion on Trees

## TL;DR

This paper presents optimal algorithms for the k-dispersion problem on trees, improving previous solutions in both unweighted and weighted cases, with significant reductions in computational complexity.

## Contribution

It introduces an optimal $O(n)$ algorithm for unweighted trees and an improved $O(n ext{log}^2 n)$ algorithm for weighted trees, advancing the computational efficiency of dispersion problems.

## Key findings

- Optimal $O(n)$ algorithm for unweighted trees
- Improved $O(n ext{log}^2 n)$ algorithm for weighted trees
- Tight bounds for the search version of the problem

## Abstract

In the $k$-dispersion problem, we need to select $k$ nodes of a given graph so as to maximize the minimum distance between any two chosen nodes. This can be seen as a generalization of the independent set problem, where the goal is to select nodes so that the minimum distance is larger than 1. We design an optimal $O(n)$ time algorithm for the dispersion problem on trees consisting of $n$ nodes, thus improving the previous $O(n\log n)$ time solution from 1997. We also consider the weighted case, where the goal is to choose a set of nodes of total weight at least $W$. We present an $O(n\log^2n)$ algorithm improving the previous $O(n\log^4 n)$ solution. Our solution builds on the search version (where we know the minimum distance $\lambda$ between the chosen nodes) for which we present tight $\Theta(n\log n)$ upper and lower bounds.