Optimal Embedding Into Star Metrics

TL;DR

This paper introduces an efficient algorithm for embedding finite metric spaces into star-topology networks with minimal dilation, and also solves a related parametric negative cycle detection problem.

Contribution

It provides an O(n^3 log^2 n)-time algorithm for optimal star embedding of metric spaces and addresses the parametric negative cycle detection problem within the same complexity.

Findings

Algorithm achieves minimal dilation star embedding efficiently.

Solves parametric negative cycle detection in the same time bound.

Advances understanding of metric space embeddings and cycle detection.

Abstract

We present an O(n^3 log^2 n)-time algorithm for the following problem: given a finite metric space X, create a star-topology network with the points of X as its leaves, such that the distances in the star are at least as large as in X, with minimum dilation. As part of our algorithm, we solve in the same time bound the parametric negative cycle detection problem: given a directed graph with edge weights that are increasing linear functions of a parameter lambda, find the smallest value of lambda such that the graph contains no negative-weight cycles.