Minimum Dilation Stars

TL;DR

This paper investigates the problem of positioning the center of a star graph in Euclidean space to minimize dilation, providing algorithms with different complexities for evaluating and finding optimal centers.

Contribution

It introduces new algorithms for evaluating and optimizing star center placement to minimize dilation, including deterministic and randomized approaches with improved complexities.

Findings

Deterministic O(n log n) algorithm for dilation evaluation.

Randomized O(n log n) algorithm for optimal center in R^d.

Specialized algorithm for d=2 with expected O(n 2^(alpha(n)) log^2 n) complexity.

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.