Consider a set of autonomous computational entities, called , operating inside a polygonal enclosure (possibly with holes), that have to perform some collaborative tasks. The boundary of the polygon obstructs both visibility and mobility of a robot. Since the polygon is initially unknown to the robots, the natural approach is to first explore and construct a map of the polygon. For this, the robots need an unlimited amount of persistent memory to store the snapshots taken from different points inside the polygon. However, it has been shown by Di [DISC 2017] that map construction can be done even by oblivious robots by employing a positional encoding strategy where a robot carefully positions itself inside the polygon to encode information in the binary representation of its distance from the closest polygon vertex. Of course, to execute this strategy, it is crucial for the robots to make accurate movements. In this paper, we address the question whether this technique can be implemented even when the movements of the robots are unpredictable in the sense that the robot can be stopped by the adversary during its movement before reaching its destination. However, there exists a constant @math , unknown to the robot, such that the robot can always reach its destination if it has to move by no more than @math amount. This model is known in literature as movement. We give a partial answer to the question in the affirmative by presenting a map construction algorithm for robots with non-rigid movement, but having @math bits of persistent memory and ability to make circular moves.
The work that is closest to ours is @cite_7 , where the problem was studied for a set of anonymous, oblivious and asynchronous robots in a polygon. The problem asks to design a distributed movement strategy for @math robots, so that eventually at least two of them come to see each other and become mutually aware'. by two robots in polygons have been studied in @cite_9 @cite_4 @cite_2 @cite_13 , where the two robots have to meet at a point or get arbitrarily close to each other. However, their model is significantly different from @cite_7 . Another problem related to our map construction problem is that of constructing the visibility graph of a polygon by mobile robots @cite_12 @cite_18 @cite_16 @cite_19 . The paper is organized as follows. In Section , the basic model and relevant definitions are presented. In Section , a brief overview of the positional encoding technique is presented. Then in Section , we describe our main algorithm. In Section , we briefly discuss how our algorithm and the techniques used in it can be used to solve some distributed coordination problems in polygonal environment.
@cite_18 @cite_4 @cite_7 @cite_9 @cite_19 @cite_2 @cite_16 @cite_13 @cite_12
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Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ > 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(dδ polylog d). This bound for the case of 2D-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(dδ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(dδ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O((d r)δ polylog(d r)), where r = min(r1, r2) and for r ≥ 1. The Meeting problem for (k 2 ) searchers in a polygon P (possibly with holes) consists in making the searchers move within P, 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 P, we minimize the number of searchers k for which the Meeting problem is solvable. Specifically, if P has a rotational symmetry of order ( ) (where ( =1 ) corresponds to no rotational symmetry), we prove that (k= +1 ) searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with (k=2 ) 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 (provided that the coordinates of the polygon’s vertices are algebraic real numbers in some global coordinate system), and each searcher can geometrically construct its own destination point at each cycle using only a compass and a straightedge. 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. Consider a set of mobile robots placed on distinct nodes of a discrete, anonymous, and bidirectional ring. Asynchronously, each robot takes a snapshot of the ring, determining the size of the ring and which nodes are either occupied by robots or empty. Based on the observed configuration, it decides whether to move to one of its adjacent nodes or not. In the first case, it performs the computed move, eventually. This model of computation is known as Look-Compute-Move. The computation depends on the required task. In this paper, we solve both the well-known Gathering and Exclusive Searching tasks. In the former problem, all robots must simultaneously occupy the same node, eventually. In the latter problem, the aim is to clear all edges of the graph. An edge is cleared if it is traversed by a robot or if both its endpoints are occupied. We consider the exclusive searching where it must be ensured that two robots never occupy the same node. Moreover, since the robots are oblivious, the clearing is perpetual, i.e., the ring is cleared infinitely often. In the literature, most contributions are restricted to a subset of initial configurations. Here, we design two different algorithms and provide a characterization of the initial configurations that permit the resolution of the problems under very weak assumptions. More precisely, we provide a full characterization (except for few pathological cases) of the initial configurations for which gathering can be solved. The algorithm relies on the necessary assumption of the local-weak multiplicity detection. This means that during the Look phase a robot detects also whether the node it occupies is occupied by other robots, without acquiring the exact number. For the exclusive searching, we characterize all (except for few pathological cases) aperiodic configurations from which the problem is feasible. We also provide some impossibility results for the case of periodic configurations. Anonymous mobile robots are often classified into synchronous, semi-synchronous, and asynchronous robots when discussing the pattern formation problem. For semi-synchronous robots, all patterns formable with memory are also formable without memory, with the single exception of forming a point (i.e., the gathering) by two robots. (All patterns formable with memory are formable without memory for synchronous robots, and little is known for asynchronous robots.) However, the gathering problem for two semi-synchronous robots without memory (called oblivious robots in this paper) is trivially solvable when their local coordinate systems are consistent, and the impossibility proof essentially uses the inconsistencies in their coordinate systems. Motivated by this, this paper investigates the magnitude of consistency between the local coordinate systems necessary and sufficient to solve the gathering problem for two oblivious robots under semi-synchronous and asynchronous models. To discuss the magnitude of consistency, we assume that each robot is equipped with an unreliable compass, the bearings of which may deviate from an absolute reference direction, and that the local coordinate system of each robot is determined by its compass. We consider two families of unreliable compasses, namely, static compasses with (possibly incorrect) constant bearings and dynamic compasses the bearings of which can change arbitrarily (immediately before a new look-compute-move cycle starts and after the last cycle ends). For each of the combinations of robot and compass models, we establish the condition on deviation @math that allows an algorithm to solve the gathering problem, where the deviation is measured by the largest angle formed between the @math -axis of a compass and the reference direction of the global coordinate system: @math for semi-synchronous and asynchronous robots with static compasses, @math for semi-synchronous robots with dynamic compasses, and @math for asynchronous robots with dynamic compasses. Except for asynchronous robots with dynamic compasses, these sufficient conditions are also necessary. Swarms of mobile robots recently attracted the focus of the Distributed Computing community. One of the fundamental problems in this context is that of exploration: the robots must coordinate to visit all locations that are reachable from their initial positions. Despite its apparent simplicity, this problem proved quite hard to characterise fully, due to many model variants, leading to informal error-prone reasoning. Over the past few years, a significant effort permitted to set up a formal framework, relying on the Coq proof assistant, which was used to provide certified results when robots evolve in a continuous bi-dimensional Euclidean space. However, the most challenging issues with exploration arise in the discrete setting (a.k.a. graph), where locations are modeled as vertices and where edges between vertices denote the ability for a robot to move from one location to the next. We present a formal model to tackle problems and reason about robot algorithms arising in the discrete setting. Our approach extends and generalises previous research efforts focusing on the continuous model. As case studies, we consider fundamental impossibility results for exploration with stop in the discrete model. To our knowledge, those are the first certified results in this context. This framework paves the way for a general certification workflow dedicated to mobile robots on graphs. In this paper we study the problem of gathering a collection of identical oblivious mobile robots in the same location of the plane. Previous investigations have focused mostly on the unlimited visibility setting, where each robot can always see all the others regardless of their distance.In the more difficult and realistic setting where the robots have limited visibility, the existing algorithmic results are only for convergence (towards a common point, without ever reaching it) and only for semi-synchronous environments, where robots' movements are assumed to be performed instantaneously.In contrast, we study this problem in a totally asynchronous setting, where robots' actions, computations, and movements require a finite but otherwise unpredictable amount of time. We present a protocol that allows anonymous oblivious robots with limited visibility to gather in the same location in finite time, provided they have orientation (i.e., agreement on a coordinate system).Our result indicates that, with respect to gathering, orientation is at least as powerful as instantaneous movements. The LOOK-COMPUTE-MOVE model for a set of autonomous robots has been thoroughly studied for over two decades. Each robot repeatedly LOOKS at its surroundings and obtains a snapshot containing the positions of all robots; based on this information, the robot COMPUTES a destination and then MOVES to it. Previous work assumed all robots are present at the beginning of the computation. What would be the effect of robots appearing asynchronously? This paper studies thisquestion, for problems of bringing the robots close together, andexposes an intimate connection with combinatorial topology. A central problem in the mobile robots area is the gathering problem. In its discrete version, the robots start at vertices in some graph G known to them, move towards the same vertex and stop. The paper shows that if robots are asynchronous and may crash, then gathering is impossible for any graph G with at least two vertices, even if robots can have unique IDs, remember the past, know the same names for the vertices of G and use an arbitrary number of lights to communicate witheach other. Next, the paper studies two weaker variants of gathering: edge gathering and 1-gathering. For both problems we present possibility and impossibility results. The solvability of edge gathering is fully characterized: it is solvable for three or more robots on a given graph if and only if the graph is acyclic. Finally, general robot tasks in a graph are considered. A combinatorial topology characterization for the solvable tasks is presented, by a reduction of the asynchronous fault-tolerant LOOK-COMPUTE-MOVE model to a wait-free read write shared-memory computing model, bringing together two areas that have been independently studied for a long time into a common theoretical foundation. Consider a set of n finite set of simple autonomous mobile robots (asynchronous, no common coordinate system, no identities, no central coordination, no direct communication, no memory of the past, non-rigid, deterministic) initially in distinct locations, moving freely in the plane and able to sense the positions of the other robots. We study the primitive task of the robots arranging themselves on the vertices of a regular n-gon not fixed in advance (Uniform Circle Formation). In the literature, the existing algorithmic contributions are limited to conveniently restricted sets of initial configurations of the robots and to more powerful robots. The question of whether such simple robots could deterministically form a uniform circle has remained open. In this paper, we constructively prove that indeed the Uniform Circle Formation problem is solvable for any initial configuration in which the robots are in distinct locations, without any additional assumption (if two robots are in the same location, the problem is easily seen to be unsolvable). In addition to closing a long-standing problem, the result of this paper also implies that, for pattern formation, asynchrony is not a computational handicap, and that additional powers such as chirality and rigidity are computationally irrelevant. We consider the problem of localizing a robot in a known environment modeled by a simple polygon P. We assume that the robot has a map of P but is placed at an unknown location inside P. From its initial location, the robot sees a set of points called the visibility polygon V of its location. In general, sensing at a single point will not suffice to uniquely localize the robot, since the set H of points in P with visibility polygon V may have more than one element. Hence, the robot must move around and use range sensing and a compass to determine its position (i.e., localize itself). We seek a strategy that minimizes the distance the robot travels to determine its exact location. We show that the problem of localizing a robot with minimum travel is NP-hard. We then give a polynomial time approximation scheme that causes the robot to travel a distance of at most (k - 1)d, where k = |H|, which is no greater than the number of reflex vertices of P, and d is the length of a minimum length tour that would allow the robot to verify its true initial location by sensing. We also show that this bound is the best possible.