MIS Filling and Mutual Visibility with Luminous Robot Swarm

Here, we first discuss the model considered for the MIS filling problem on a graph and the associated problem we solved. Then, we highlight two models for two variants of the mutual visibility problem with the problem definitions. The first one is called MV_Inaccurate, and the second one is MV_Disoriented.

▶ MIS Filling: A connected, port-labelled graph is given with specific vertices, called doors. Each vertex v of the graph contains port numbers corresponding to the incident edges from [1, 2, …, δ(v)], where δ(v) is the degree of the vertex v. The robots enter the graph through the doors, and each of them has a limited visibility range. A robot having z hops of visibility can see all the vertices and the associated ports up to the distance z from its current position. As soon as a robot leaves a door, another one appears at the door. We define the following problem.

Problem 1: Given an anonymous connected port-labelled graph G = (V, E) with k Doors, the objective is to relocate robots that appear at Doors such that at termination, the robots occupy a set of vertices V₁ (V₁ ⊂ V) that forms a maximal independent set of G.

▶ MV_Inaccurate: A set of N luminous, opaque point robots are deployed on the Euclidean plane, each operating under LCM cycles. We introduce a new flavour of challenge to the existing model, where a robot may exhibit inaccurate movement. Inaccuracy is defined as an angular deviation that makes a robot r reach at a point T′ instead of the actual target point T, where \(\overline{rT} = \overline{rT^{\prime }}\) (refer to Fig. 1). We call θ_r = ∠TrT′ the inaccuracy angle of r. We assume that θ_r < 90° and this angle are not necessarily the same for two robots. None of the robots has knowledge about N. The current position of a robot r is considered to be the origin of its local coordinate system. However, the robots are assumed to agree on one coordinate axis (without loss of generality, y-axis). The robots can be susceptible to mobility fault, where a faulty robot becomes immobile thenceforth, but the light remains functional. We define the following two problems.

Problem 2: We are given N anonymous and luminous opaque point robots on the 2D plane with a common coordinate axis. They might exhibit angular inaccuracy in their movement. The robots must reach a configuration starting from any configuration so that no three robots are collinear.

Problem 3: N luminous point robots with a common coordinate axis might exhibit angular inaccuracy in their movement. With that, they are also vulnerable to mobility faults. The goal is to obtain a configuration where all non-faulty robots achieve mutual visibility.

▶ MV_Disoriented: In this model, we want to study the mutual visibility of N luminous point robots from the fault tolerance perspective by revoking the assumption of one-axis agreement as in [4]. The movements of the robots are accurate, but they are completely disoriented, meaning they do not agree on any coordinate system and orientation. A robot is vulnerable to mobility faults, which may occur anytime. So, we define the following problem.

Problem 4: N disoriented luminous mobile robots are deployed on the 2D plane, out of which f(< N) can be faulty anytime. The goal is to achieve mutual visibility among the non-faulty robots.

We propose two algorithms, IND and MULTIND, in [6] for two versions of Problem 1 corresponding to single and multiple doors, respectively. IND algorithm runs under an asynchronous scheduler using robots with 3 hops of visibility, Δ + 8 colors, and O(log Δ) bits memory (Δ is the maximum degree). MULTIND runs under a semi-synchronous scheduler using robots with 5 hops of visibility, Δ + k + 7 colors, and O(log (Δ + k)) bits memory. Both algorithms take O(m²) epochs, where m is the number of robots that form an MIS. The robots move throughout the graph similar to the depth-first search.

We solve Problem 2 in [5] under a semi-synchronous setting using 2 colors (color optimal). We propose another algorithm under the asynchronous setting that needs 3 colors to solve the problem. Both algorithms require O(N) epochs. We also proved that the problem becomes unsolvable of θ_r ≥ 90°. We use a technique to sequentially relocate the robots to break the collinearities, where virtual horizontal lines are imagined through the robots to serialize them. In this process, we ensure that no robot inadvertently creates new collinearities after its movements with two other robots with which it was not collinear. Additionally, we assume y-axis agreement where robots know the north (positive y-axis) and south (negative y-axis) directions. Upon activation, a robot r needs to check whether it is eligible to execute a movement. This can be done by checking the colors of all visible robots located in the northward direction. If all of them have the color FINISH, r checks whether it is a part of some collinearity. We take advantage of the fact that a middle robot of collinearity always detects the collinearity. If r finds itself as a part of multiple collinearity, it chooses a target to move vertically northward. Inaccuracy might make r collinear again. After this movement, we ensure that r is not part of more than one collinearity. When r detects itself in exactly one collinearity \(\overline{r_irr_j}\) for two other robots r_i and r_j, it executes another movement with respect to the line \(\overleftrightarrow{r_ir_j}\) to break the collinearity. In case of no collinearity, r sets its color to FINISH and terminates.

Problem 3 is solved in [5] using 10 colors in asynchronous setting, and the proposed algorithm runs in O(N) epochs. In case of mobility failure, the middle robot of a collinearity might be faulty. So, we move the last robot (based on the common y-axis) of collinearity using the guidance of the faulty visible robots using extra colors. So, with the help of a faulty robot r_f, a non-faulty robot r moves against the line \(\overleftrightarrow{rr_f}\) to break the collinearity. We ensure that any non-faulty r moves at most four times, ensuring O(N) run time.

We solved Problem 4 under the FSYNC setting, and the proposed algorithm runs in O(N²) rounds using 21 colors. When the robots agree on one coordinate axis, it becomes easier to serialize them. In the case of robots, without any agreement on the coordinate system, it becomes very difficult to serialize them. Moreover, it is difficult to achieve the goal for the opaque robots if all the robots are executing movement simultaneously. In that case, the serialization of the robots becomes useful in designing algorithms to achieve mutual visibility. We introduce the concepts of layers, which have a similar concept to the virtual horizontal lines in the case of y-axis agreement. The boundary of the convex hull \(\mathcal {CH}\) of all the robots present on the plane, denoted by \(\mathcal {CH}^0\), is the outer-most layer of the robots. For j ≥ 1, \(\mathcal {CH}^j\), referred to as the j-th layer, is the boundary of the convex hull of all the robots excluding the robots on \(\mathcal {CH}^0 \cup \mathcal {CH}^1 \cup \cdots \cup \mathcal {CH}^{j-1}\). We start moving the non-faulty robots starting from the outermost layer \(\mathcal {CH}^0\) (refer to Fig. 2) towards the innermost layer \(\mathcal {CH}^k\). The challenge comes when all of the robots on a layer are faulty. In that case, the robots on the next layer need to be made to understand that it is their turn to move towards the innermost layer.

First, our aim is to identify the innermost layer of the robots using colors. Starting from \(\mathcal {CH}^0\), a robot r moves to a side of the subsequent inner layer. Our purpose is to eventually reach \(\mathcal {CH}^{k-1}\) (when the inner-most layer is linear) or \(\mathcal {CH}^k\) (when the inner-most layer is non-linear). After this, the robots move in such a way that they either become a corner of a common convex hull inside \(\mathcal {CH}^k\) or reach on some points on an ellipse (for linear \(\mathcal {CH}^k\)) around the layer \(\mathcal {CH}^k\).