# A Linear-Time Algorithm for Radius-Optimally Augmenting Paths in a Metric Space

## TL;DR

This paper introduces an optimal linear-time algorithm for adding a single edge to a path graph in a metric space to minimize the graph's radius, addressing a previously unexplored problem.

## Contribution

It presents the first linear-time algorithm for radius minimization in path graphs within metric spaces, improving upon the prior diameter minimization approach.

## Key findings

- The algorithm runs in O(n) time, which is proven to be optimal.
- It effectively minimizes the radius of the augmented path graph.
- The problem of radius minimization was not previously studied.

## Abstract

Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ to minimize the radius of the resulting graph. Previously, a similar problem for minimizing the diameter of the graph was solved in $O(n\log n)$ time. To the best of our knowledge, the problem of minimizing the radius has not been studied before. In this paper, we present an $O(n)$ time algorithm for the problem, which is optimal.