Mutual visibility by luminous robots without collisions

We consider the Mutual Visibility problem for anonymous dimensionless robots with obstructed visibility moving in a plane: starting from distinct locations, the robots must reach, without colliding, a configuration where no three of them are collinear. We study this problem in the luminous robots model, in which each robot has a visible light that can assume colors from a fixed set. Among other results, we prove that Mutual Visibility can be solved in SSynch with 2 colors and in ASynch with 3 colors. If an adversary can interrupt and stop a robot moving to its computed destination, Mutual Visibility is still solvable in SSynch with 3 colors and, if the robots agree on the direction of one axis, also in ASynch. As a byproduct, we provide the first obstructed-visibility solutions to two classical problems for oblivious robots: collision-less convergence to a point (also known as near-gathering) and circle formation

Pattern Formation for Fat Robots with Memory

Given a set of $n\geq 1$ autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small $O(1)$-sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in $O(n) + O(q \log n)$ rounds, where $q>0$ is related to leader election, which takes $O(q \log n)$ rounds with probability at least $1-n^{-q}$. The algorithms are collision-free and do not require the knowledge of the number of robots.Comment: arXiv admin note: text overlap with arXiv:2306.1444

Circle formation by asynchronous opaque robots on infinite grid

This paper presents a distributed algorithm for circle formation problem under the infinite grid environment by asynchronous mobile opaque robots. Initially all the robots are acquiring distinct positions and they have to form a circle over the grid. Movements of the robots are restricted only along the grid lines. They do not share any global co-ordinate system. Robots are controlled by an asynchronous adversarial scheduler that operates in Look-Compute-Move cycles. The robots are indistinguishable by their nature, do not have any memory of their past configurations and previous actions. We consider the problem under luminous model, where robots communicate via lights, other than that they do not have any external communication system. Our protocol solves the  circle formation problem using seven colors. A subroutine of our algorithm also solves the line formation problem using three colors

Total mutual-visibility in Hamming graphs

If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $\mu_{\rm t}(G)$ of $G$. In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values $\mu_{\rm t}(K_{n_1}\,\square\, K_{n_2}\,\square\, K_{n_3})$ are determined. It is proved that $\mu_{\rm t}(K_{n_1} \,\square\, \cdots \,\square\, K_{n_r}) = O(N^{r-2})$, where $N = n_1+\cdots + n_r$, and that $\mu_{\rm t}(K_s^{\,\square\,, r}) = \Theta(s^{r-2})$ for every $r\ge 3$, where $K_s^{\,\square\,, r}$ denotes the Cartesian product of $r$ copies of $K_s$. The main theorems are also reformulated as Tur\'an-type results on hypergraphs

Forming Sequences of Patterns with Luminous Robots

The extensive studies on computing by a team of identical mobile robots operating in the plane in Look-Compute-Move cycles have been carried out mainly in the traditional {mathcal{ OBLOT}} model, where the robots are silent (have no communication capabilities) and oblivious (in a cycle, they have no memory previous cycles). To partially overcome the limits of obliviousness and silence while maintaining some of their advantages, the stronger model of luminous robots, {mathcal{ LUMI}} , has been introduced where the robots, otherwise oblivious and silent, carry a visible light that can take a number of different colors; a color can be seen by observing robots, and persists from a cycle to the next. In the study of the computational impact of lights, an immediate concern has been to understand and determine the additional computational strength of {mathcal{ LUMI}} over {mathcal{ OBLOT}}. Within this line of investigation, we examine the problem of forming a sequence of geometric patterns, PatternSequenceFormation. A complete characterization of the sequences of patterns formable from a given starting configuration has been determined in the {mathcal{ OBLOT}} model. In this paper, we study the formation of sequences of patterns in the {mathcal{ LUMI}} model and provide a complete characterization. The characterization is constructive: our universal protocol forms all formable sequences, and it does so asynchronously and without rigidity. This characterization explicitly and clearly identifies the computational strength of {mathcal{ LUMI}} over {mathcal{ OBLOT}} with respect to the Pattern Sequence Formation problem

Finding Water on Poleless Using Melomaniac Myopic Chameleon Robots

In 2042, the exoplanet exploration program, launched in 2014 by NASA, finally discovers a new exoplanet so-called Poleless, due to the fact that it is not subject to any magnetism. A new generation of autonomous mobile robots, called M2C (for Melomaniac Myopic Chameleon), have been designed to find water on Poleless. To address this problem, we investigate optimal (w.r.t., visibility range and number of used colors) solutions to the infinite grid exploration problem (IGE) by a small team of M2C robots. Our first result shows that minimizing the visibility range and the number of used colors are two orthogonal issues: it is impossible to design a solution to the IGE problem that is optimal w.r.t. both parameters simultaneously. Consequently, we address optimality of these two criteria separately by proposing two algorithms; the former being optimal in terms of visibility range, the latter being optimal in terms of number of used colors. It is worth noticing that these two algorithms use a very small number of robots, respectively six and eight