---
title: Minimum Dilation Stars
url: https://www.emergentmind.com/papers/0412025
type: paper
arxiv_id: '0412025'
arxiv_url: https://arxiv.org/abs/0412025
published: '2004-12-07'
categories:
- cs.CG
---

# Minimum Dilation Stars

## Abstract

The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R^d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(n log n)-time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expected-time algorithm for finding an optimal center in R^d; and for the case d=2, a randomized O(n 2^(alpha(n)) log^2 n) expected-time algorithm for finding an optimal center among the input points.