# Evacuating Equilateral Triangles and Squares in the Face-to-Face Model

## TL;DR

This paper studies the problem of evacuating robots from equilateral triangles and squares in a face-to-face communication model, providing bounds and algorithms for various numbers of robots to optimize evacuation time.

## Contribution

It introduces new bounds and algorithms for evacuating multiple robots from geometric regions with face-to-face communication constraints, including optimality results and generalizations.

## Key findings

- Lower bound of 1+2/√3 ≈ 2.154 for 2 robots in a triangle.
- Algorithms achieving evacuation times of 2.3367 (k=2), 2.0887 (k=3), 1.9816 (k=4) in triangles.
- Algorithms for squares with evacuation times of 3.4645 (k=2), 3.1786 (k=3), 2.6646 (k=4).

## Abstract

Consider $k$ robots initially located at a point inside a region $T$. Each robot can move anywhere in $T$ independently of other robots with maximum speed one. The goal of the robots is to \emph{evacuate} $T$ through an exit at an unknown location on the boundary of $T$. The objective is to minimize the \emph{evacuation time}, which is defined as the time the \emph{last} robot reaches the exit. We consider the \emph{face-to-face} communication model for the robots: a robot can communicate with another robot only when they meet in $T$. In this paper, we give upper and lower bounds for the face-to-face evacuation time by $k$ robots that are initially located at the centroid of a unit-sided equilateral triangle or square. For the case of a triangle with $k=2$ robots, we give a lower bound of $1+2/\sqrt{3} \approx 2.154$, and an algorithm with upper bound of 2.3367 on the worst-case evacuation time. We show that for any $k$, any algorithm for evacuating $k\geq 2$ robots requires at least $\sqrt{3}$ time. This bound is asymptotically optimal, as we show that even a straightforward strategy of evacuation by $k$ robots gives an upper bound of $\sqrt{3} + 3/k$. For $k=3$ and $4$, we give better algorithms with evacuation times of 2.0887 and 1.9816, respectively. For the case of the square and $k=2$, we give an algorithm with evacuation time of $3.4645$ and show that any algorithm requires time at least $3.118$ to evacuate in the worst-case. Moreover, for $k=3$, and $4$, we give algorithms with evacuation times 3.1786 and 2.6646, respectively. The algorithms given for $k=3$ and $4$ for evacuation in the triangle or the square can be easily generalized for larger values of $k$.