Breaking Symmetries on Tessellation Graphs via Asynchronous Robots: The Line Formation Problem as a Case Study

Concerning the coordination of autonomous mobile robots, the main focus has been on the important class of
Pattern Formation
problems, where the robots are required to arrange themselves to form a given geometric shape. This class of problems has been extensively studied in the continuous environment (e.g., the Euclidean plane), whereas few results exist when robots move in a discretization of the plane, like infinite grids. In this environment, to form any pattern, the problem of breaking symmetries emerges. Breaking the symmetry by moving some leader robot is not a straightforward task due to the movement restrictions as all the adjacent nodes of the leader may be occupied. It may even happen that before obtaining the requested asymmetric configuration, most of the robots must be moved. Due to the asynchrony of robots, this fact greatly increases the difficulty of the problem. We assume very weak robots moving on any regular tessellation graph as a discretization of the Euclidean plane, and we devise an algorithm
$\mathcal {A}_{{break}}$
able to solve the
Symmetry Breaking
problem on both the square and triangular grids. It is important to note that
$\mathcal {A}_{{break}}$
is proposed so that it can be used as a module for solving more general problems. As a case study, we use
$\mathcal {A}_{{break}}$
to deal with the
Line Formation
problem, where
$n\ge 3$
robots must arrange themselves to occupy
$n$
contiguous vertices along a grid line. In this respect, we first provide an algorithm
$\mathcal {A}_{{LF}^{-}}$
able to partially solve this problem (it works with configurations in which it is not necessary to break symmetries), and then we show how
$\mathcal {A}_{{break}}$
and
$\mathcal {A}_{{LF}^{-}}$
can be combined to form
$\mathcal {A}_{{LF}}$
. We provide a complete characterization of the solvability of the
Line Formation
problem on the considered topologies by showing that
$\mathcal {A}_{{LF}}$
solves the problem in each configuration where this is possible.