On the Solvability of Anonymous Partial Grids Exploration by Mobile Robots

Given an arbitrary partial anonymous grid (a finite grid with possibly missing vertices or edges), this paper focuses on the exploration of such a grid by a set of mobile anonymous agents (called robots). Assuming that the robots can move synchronously, but cannot communicate with each other, the aim is to design an algorithm executed by each robot that allows, as many robots as possible (let

k

be this maximal number), to visit infinitely often all the vertices of the grid, in such a way that no vertex hosts more than one robot at a time, and each edge is traversed by at most one robot at a time.

The paper addresses this problem by considering a central parameter, denoted

ρ

, that captures the view of each robot. More precisely, it is assumed that each robot sees the part of the grid (and its current occupation by other robots, if any) centered at the vertex it currently occupies and delimited by the radius

ρ

. Based on such a radius notion, a previous work has investigated the cases

ρ

= 0 and

ρ

 = + ∞, and shown that, while there is no solution for

ρ

= 0,

k

 ≤ 

p

 − 

q

is a necessary and sufficient requirement when

ρ

 = + ∞, where

p

is the number of vertices of the grid, and

q

a parameter whose value depends on the actual topology of the partial grid. This paper completes our previous results by addressing the more difficult case, namely

ρ

= 1. It shows that

k

 ≤ 

p

 − 1 when

q

 = 0, and

k

 ≤ 

p

 − 

q

otherwise, is a necessary and sufficient requirement for solving the problem. More generally, the paper shows that this case is the borderline from which the considered problem can be solved.