# A Las Vegas approximation algorithm for metric $1$-median selection

## TL;DR

This paper introduces a randomized approximation algorithm for the metric 1-median problem that guarantees a solution within a factor of (2+ε) in expected linear time, with a high-probability near-optimal solution.

## Contribution

It presents a novel Las Vegas approximation algorithm for the metric 1-median problem with improved expected runtime and approximation guarantees.

## Key findings

- Always outputs a (2+ε)-approximate solution in expected O(n/ε^2) time.
- Inherits properties from Indyk's algorithm to produce a (1+ε)-approximate median with constant probability.
- Provides a practical randomized approach for efficient median approximation in metric spaces.

## Abstract

Given an $n$-point metric space, consider the problem of finding a point with the minimum sum of distances to all points. We show that this problem has a randomized algorithm that {\em always} outputs a $(2+\epsilon)$-approximate solution in an expected $O(n/\epsilon^2)$ time for each constant $\epsilon>0$. Inheriting Indyk's algorithm, our algorithm outputs a $(1+\epsilon)$-approximate $1$-median in $O(n/\epsilon^2)$ time with probability $\Omega(1)$.