Time-optimal geodesic mutual visibility of robots on grids within minimum area

TL;DR

This paper addresses the problem of moving robots on grid graphs to achieve mutual visibility along shortest paths without collisions, providing algorithms with proven correctness and optimized time complexity for both finite and infinite grids.

Contribution

It introduces algorithms for geodesic mutual visibility on grid graphs, with formal proofs and time optimization for robots operating under the Look-Compute-Move model.

Findings

Algorithms for finite and infinite grids with correctness proofs.

Optimized time complexity for mutual visibility achievement.

Handling peculiarities of grid contexts in robot movement.

Abstract

The \textsc{Mutual Visibility} is a well-known problem in the context of mobile robots. For a set of $n$ robots disposed in the Euclidean plane, it asks for moving the robots without collisions so as to achieve a placement ensuring that no three robots are collinear. For robots moving on graphs, we consider the \textsc{Geodesic Mutual Visibility} ($\GMV$) problem. Robots move along the edges of the graph, without collisions, so as to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means that there is a shortest path (i.e., a "geodesic") between each pair of robots along which no other robots reside. We study this problem in the context of finite and infinite square grids, for robots operating under the standard Look-Compute-Move model. In both scenarios, we provide resolution algorithms along with formal correctness proofs, highlighting the most relevant peculiarities arising within the different contexts, while optimizing the time complexity.