# An Optimal Algorithm for Online Freeze-tag

## TL;DR

This paper presents a new online algorithm for the freeze-tag problem that achieves a competitive ratio of approximately 2.414, improving upon previous bounds and matching the lower limit, in the context of robots activating frozen peers.

## Contribution

The paper introduces a novel $(1+\sqrt{2})$-competitive algorithm for online freeze-tag, establishing the best possible competitive ratio across all metric spaces.

## Key findings

- The algorithm achieves a competitive ratio of about 2.414.
- No online algorithm can do better than this ratio in all metric spaces.
- The result answers an open question about the existence of $O(1)$-competitive algorithms.

## Abstract

In the freeze-tag problem, one active robot must wake up many frozen robots. The robots are considered as points in a metric space, where active robots move at a constant rate and activate other robots by visiting them. In the (time-dependent) online variant of the problem, frozen robots are not revealed until a specified time. Hammar, Nilsson, and Persson have shown that no online algorithm can achieve a competitive ratio better than $7/3$ for online freeze-tag, and asked whether there is any $O(1)$-competitive algorithm. In this paper, we provide a $(1+\sqrt{2})$-competitive algorithm for online time-dependent freeze-tag, and show that no algorithm can achieve a lower competitive ratio on every metric space.