Deterministic Rendezvous at a Node of Agents with Arbitrary Velocities

We consider the task of rendezvous in networks modeled as undirected graphs. Two mobile agents with different labels, starting at different nodes of an anonymous graph, have to meet. This task has been considered in the literature under two alternative scenarios: weak and strong. Under the weak scenario, agents may meet either at a node or inside an edge. Under the strong scenario, they have to meet at a node, and they do not even notice meetings inside an edge. Rendezvous algorithms under the strong scenario are known for synchronous agents. For asynchronous agents, rendezvous under the strong scenario is impossible even in the two-node graph, and hence only algorithms under the weak scenario were constructed. In this paper we show that rendezvous under the strong scenario is possible for agents with restricted asynchrony: agents have the same measure of time but the adversary can arbitrarily impose the speed of traversing each edge by each of the agents. We construct a deterministic rendezvous algorithm for such agents, working in time polynomial in the size of the graph, in the length of the smaller label, and in the largest edge traversal time.Comment: arXiv admin note: text overlap with arXiv:1704.0888

Deterministic meeting of sniffing agents in the plane

Two mobile agents, starting at arbitrary, possibly different times from arbitrary locations in the plane, have to meet. Agents are modeled as discs of diameter 1, and meeting occurs when these discs touch. Agents have different labels which are integers from the set of 0 to L-1. Each agent knows L and knows its own label, but not the label of the other agent. Agents are equipped with compasses and have synchronized clocks. They make a series of moves. Each move specifies the direction and the duration of moving. This includes a null move which consists in staying inert for some time, or forever. In a non-null move agents travel at the same constant speed, normalized to 1. We assume that agents have sensors enabling them to estimate the distance from the other agent (defined as the distance between centers of discs), but not the direction towards it. We consider two models of estimation. In both models an agent reads its sensor at the moment of its appearance in the plane and then at the end of each move. This reading (together with the previous ones) determines the decision concerning the next move. In both models the reading of the sensor tells the agent if the other agent is already present. Moreover, in the monotone model, each agent can find out, for any two readings in moments t1 and t2, whether the distance from the other agent at time t1 was smaller, equal or larger than at time t2. In the weaker binary model, each agent can find out, at any reading, whether it is at distance less than \r{ho} or at distance at least \r{ho} from the other agent, for some real \r{ho} > 1 unknown to them. Such distance estimation mechanism can be implemented, e.g., using chemical sensors. Each agent emits some chemical substance (scent), and the sensor of the other agent detects it, i.e., sniffs. The intensity of the scent decreases with the distance.Comment: A preliminary version of this paper appeared in the Proc. 23rd International Colloquium on Structural Information and Communication Complexity (SIROCCO 2016), LNCS 998

Positional Encoding by Robots with Non-Rigid Movements

Consider a set of autonomous computational entities, called \emph{robots}, 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 Luna et al. [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 $\delta > 0$, unknown to the robot, such that the robot can always reach its destination if it has to move by no more than $\delta$ amount. This model is known in literature as \emph{non-rigid} 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 $O(1)$ bits of persistent memory and ability to make circular moves

Asynchronous approach in the plane: A deterministic polynomial algorithm

In this paper we study the task of approach of two mobile agents having the same limited range of vision and moving asynchronously in the plane. This task consists in getting them in finite time within each other's range of vision. The agents execute the same deterministic algorithm and are assumed to have a compass showing the cardinal directions as well as a unit measure. On the other hand, they do not share any global coordinates system (like GPS), cannot communicate and have distinct labels. Each agent knows its label but does not know the label of the other agent or the initial position of the other agent relative to its own. The route of an agent is a sequence of segments that are subsequently traversed in order to achieve approach. For each agent, the computation of its route depends only on its algorithm and its label. An adversary chooses the initial positions of both agents in the plane and controls the way each of them moves along every segment of the routes, in particular by arbitrarily varying the speeds of the agents. A deterministic approach algorithm is a deterministic algorithm that always allows two agents with any distinct labels to solve the task of approach regardless of the choices and the behavior of the adversary. The cost of a complete execution of an approach algorithm is the length of both parts of route travelled by the agents until approach is completed. Let $\Delta$ and $l$ be the initial distance separating the agents and the length of the shortest label, respectively. Assuming that $\Delta$ and $l$ are unknown to both agents, does there exist a deterministic approach algorithm always working at a cost that is polynomial in $\Delta$ and $l$? In this paper, we provide a positive answer to the above question by designing such an algorithm

Meeting in a Polygon by Anonymous Oblivious Robots

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 sigma (where sigma=1 corresponds to no rotational symmetry), we prove that k=sigma+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\u27s 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. 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

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ 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 $\sigma$ (where $\sigma=1$ corresponds to no rotational symmetry), we prove that $k=\sigma+1$ 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, 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.Comment: 37 pages, 9 figure

Asynchronous Approach in the Plane: A Deterministic Polynomial Algorithm

In this paper we study the task of approach of two mobile agents having the same limited range of vision and moving asynchronously in the plane. This task consists in getting them in finite time within each other\u27s range of vision. The agents execute the same deterministic algorithm and are assumed to have a compass showing the cardinal directions as well as a unit measure. On the other hand, they do not share any global coordinates system (like GPS), cannot communicate and have distinct labels. Each agent knows its label but does not know the label of the other agent or the initial position of the other agent relative to its own. The route of an agent is a sequence of segments that are subsequently traversed in order to achieve approach. For each agent, the computation of its route depends only on its algorithm and its label. An adversary chooses the initial positions of both agents in the plane and controls the way each of them moves along every segment of the routes, in particular by arbitrarily varying the speeds of the agents. Roughly speaking, the goal of the adversary is to prevent the agents from solving the task, or at least to ensure that the agents have covered as much distance as possible before seeing each other. A deterministic approach algorithm is a deterministic algorithm that always allows two agents with any distinct labels to solve the task of approach regardless of the choices and the behavior of the adversary. The cost of a complete execution of an approach algorithm is the length of both parts of route travelled by the agents until approach is completed. Let Delta and l be the initial distance separating the agents and the length of (the binary representation of) the shortest label, respectively. Assuming that Delta and l are unknown to both agents, does there exist a deterministic approach algorithm whose cost is polynomial in Delta and l? Actually the problem of approach in the plane reduces to the network problem of rendezvous in an infinite oriented grid, which consists in ensuring that both agents end up meeting at the same time at a node or on an edge of the grid. By designing such a rendezvous algorithm with appropriate properties, as we do in this paper, we provide a positive answer to the above question. Our result turns out to be an important step forward from a computational point of view, as the other algorithms allowing to solve the same problem either have an exponential cost in the initial separating distance and in the labels of the agents, or require each agent to know its starting position in a global system of coordinates, or only work under a much less powerful adversary

Approche asynchrone dans le plan : un algorithme déterministe polynomial

International audienceNous étudions la tâche de l'approche entre deux agents mobiles ayant la même portée de vision limitée et se déplaçant de façon asynchrone dans le plan. En partant de positions initiales quelconques, cette tâche consiste à les amener au moyen d'un algorithme déterministe à portée de vision l'un de l'autre en un temps fini. Ils sont autonomes et ne peuvent pas communiquer entre eux. Ils disposent cependant d'identifiants distincts et d'une boussole leur indiquant les points cardinaux. Chaque agent ne connaît que son identifiant. En particulier, il ignore l'identifiant et la position initiale de l'autre agent. Un adversaire choisit les identifiants et les positions initiales des deux agents dans le plan et modifie à volonté la vitesse à laquelle chacun se déplace. Le coût d'une exécution complète d'un algorithme pour cette tâche est la somme des distances parcourues par les deux agents jusqu'à accomplir l'approche. Dans ces conditions, certains algorithmes déterministes résolvent cette tâche avec un coût exponentiel en fonction de la distance initiale ∆ séparant les deux robots et de la longueur des identifiants. D'autres nécessitent que chaque agent connaisse sa position de départ dans un système de coordonnées commun ; d'autres encore fonctionnent avec un adversaire plus faible. Ce papier présente un algorithme déterministe pour la tâche de l'approche dont le coût est polynomial en fonction de ∆ et de la longueur (de la représentation binaire) du plus petit identifiant