Approximate Weighted Farthest Neighbors and Minimum Dilation Stars

TL;DR

This paper introduces an efficient method to approximate weighted farthest neighbors in metric spaces, enabling fast queries and applications like optimizing star network centers to minimize maximum dilation.

Contribution

It presents a reduction from weighted to unweighted farthest neighbor queries and combines it with core-sets for efficient approximate solutions.

Findings

Achieves (1+epsilon)-approximate farthest neighbor queries in O(log n) time in Euclidean space.

Provides an O(n log n) expected time algorithm for selecting star network centers to minimize dilation.

Demonstrates practical application in network topology optimization.

Abstract

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find (1+epsilon)-approximate farthest neighbors in time O(log n) per query in D-dimensional Euclidean space for any constants D and epsilon. As an application, we find an O(n log n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.