The Meeting problem for @math searchers in a polygon @math (possibly with holes) consists in making the searchers move within @math , according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of @math , we minimize the number of searchers @math for which the Meeting problem is solvable. Specifically, if @math has a rotational symmetry of order @math (where @math corresponds to no rotational symmetry), we prove that @math searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with @math searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.
@cite_9: We consider the following fundamental Mutual Visibility problem: Given a set of n identical autonomous point robots in arbitrary distinct positions in the Euclidean plane, find a schedule to move them such that within finite time they reach, without collisions, a configuration in which they all see each other. The robots operate following Look-Compute-Move cycles and a robot @math can not see other robot @math if there lies a third robot @math in the line segment connecting the positions of @math and @math . Moreover, n is not assumed to be known to the robots. We study this problem in the robots with lights model, where each robot has an externally visible persistent light which can assume colors from a fixed set of colors and the color set is identical to all the robots. This model corresponds to the classical model of oblivious robots when the number of colors @math in the color set. Therefore, we focus here on the objective of minimizing the number of colors required to successfully solve Mutual Visibility. Di [16] presented two algorithms and showed that Mutual Visibility can always be solved without collisions with @math colors for both semi-synchronous and asynchronous robots under both rigid and non-rigid moves. In this paper, we present and analyze an improved algorithm which requires only @math colors and works for both semi-synchronous and asynchronous robots under both rigid and non-rigid moves; this is optimal since any algorithm for Mutual Visibility needs at least 2 colors in the robots with lights model, when n is not known. We employ a non-trivial technique which moves all the interior robots first to the boundary of the initial convex hull and then the robots in the boundary of the hull except the corners to outside of the hull until all the robots eventually become corners, without the need of any third color. Our result is interesting in the sense that asynchronicity and non-rigidity of robot movements have no effect on Mutual Visibility with respect to the number of colors needed to successfully solve it. We also provide an improved solution to the Circle Formation problem.
@cite_18: 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.
@cite_12: We consider the problem of repositioning N autonomous robots on a plane so that each robot is visible to all others (the Complete Visibility problem); a robot cannot see another robot if its visibility is obstructed by a third robot positioned between them on a straight line. This problem is important since it provides a basis to solve many other problems under obstructed visibility. Robots operate following Look-Compute-Move (LCM) cycles and communicate with other robots using colored lights as in the recently proposed robots with lights model. The challenge posed by this model is that each robot has only a constant number of colors for its lights (symbols for communication) and no memory (except for the persistence of lights) between LCM cycles. Our goal is to minimize the number of rounds needed to solve Complete Visibility, where a round is measured as the time duration for all robots to execute at least one complete LCM cycle since the end of the previous round. The best previously known algorithm for Complete Visibility on this robot model has runtime of (O( N) ) rounds. That algorithm has the assumptions of full synchronicity, chirality, and robot paths may collide. In this paper we present the first algorithm for Complete Visibility with O(1) runtime that runs on the semi-synchronous (and also the fully synchronous) model. The proposed algorithm is deterministic, does not have the chirality assumption, and is collision free.
The model in which robots can obstruct each other's view has also been studied. Here, the goal is typically to make all robots see each other by making sure that no three of them are collinear @cite_18 @cite_9 @cite_12 . As with the literature on Gathering, none of these works considers robots in a polygon.