On ultrametric $1$-median selection

TL;DR

This paper presents a Monte Carlo algorithm for efficiently approximating the 1-median in ultrametric spaces, achieving near-optimal solutions with provable guarantees in sublinear time.

Contribution

It introduces a novel Monte Carlo algorithm that provides a $(1+ ext{epsilon})$-approximation for the ultrametric 1-median problem with a specific time complexity.

Findings

The algorithm runs in $O(( ext{log}^2(1/ ext{epsilon}))/ ext{epsilon}^3)$ time.

It guarantees a $(1+ ext{epsilon})$-approximate solution for all $ ext{epsilon}>0$.

The approach is efficient for large ultrametric datasets.

Abstract

Consider the problem of finding a point in an ultrametric space with the minimum average distance to all points. We give this problem a Monte Carlo $O((\log^2(1/\epsilon))/\epsilon^3)$-time $(1+\epsilon)$-approximation algorithm for all $\epsilon>0$.