Two mobile agents, starting from different nodes of an @math -node network at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds using a deterministic algorithm. In each round, an agent decides to either remain idle or to move to one of the adjacent nodes. Each agent has a distinct integer label from the set @math , which it can use in the execution of the algorithm, but it does not know the label of the other agent. If @math is the distance between the initial positions of the agents, then @math is an obvious lower bound on the time of rendezvous. However, if each agent has no initial knowledge other than its label, time @math is usually impossible to achieve. We study the minimum amount of information that has to be available a priori to the agents to achieve rendezvous in optimal time @math . This information is provided to the agents at the start by an oracle knowing the entire instance of the problem, i.e., the network, the starting positions of the agents, their wake-up rounds, and both of their labels. The oracle helps the agents by providing them with the same binary string called advice, which can be used by the agents during their navigation. The length of this string is called the size of advice. Our goal is to find the smallest size of advice which enables the agents to meet in time @math . We show that this optimal size of advice is @math . The upper bound is proved by constructing an advice string of this size, and providing a natural rendezvous algorithm using this advice that works in time @math for all networks. The matching lower bound, which is the main contribution of this paper, is proved by exhibiting classes of networks for which it is impossible to achieve rendezvous in time @math with smaller advice.
Apart from the synchronous model used in this paper, several authors investigated asynchronous rendezvous in the plane @cite_42 @cite_12 and in network environments @cite_26 @cite_1 @cite_34 @cite_18 . In the latter scenario, the agent chooses the edge to traverse, but the adversary controls the speed of the agent. Under this assumption, rendezvous at a node cannot be guaranteed even in very simple graphs. Hence the rendezvous requirement is relaxed to permit the agents to meet inside an edge.
@cite_18 @cite_26 @cite_42 @cite_1 @cite_34 @cite_12
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Two mobile agents starting at different nodes of an unknown network have to meet. This task is known in the literature as rendezvous. Each agent has a different label which is a positive integer known to it, but unknown to the other agent. Agents move in an asynchronous way: the speed of agents may vary and is controlled by an adversary. The cost of a rendezvous algorithm is the total number of edge traversals by both agents until their meeting. The only previous deterministic algorithm solving this problem has cost exponential in the size of the graph and in the larger label. In this paper we present a deterministic rendezvous algorithm with cost polynomial in the size of the graph and in the length of the smaller label. Hence we decrease the cost exponentially in the size of the graph and doubly exponentially in the labels of agents. As an application of our rendezvous algorithm we solve several fundamental problems involving teams of unknown size larger than 1 of labeled agents moving asynchronously in unknown networks. Among them are the following problems: team size, in which every agent has to find the total number of agents, leader election, in which all agents have to output the label of a single agent, perfect renaming in which all agents have to adopt new different labels from the set 1,...,k , where k is the number of agents, and gossiping, in which each agent has initially a piece of information (value) and all agents have to output all the values. Using our rendezvous algorithm we solve all these problems at cost polynomial in the size of the graph and in the smallest length of all labels of participating agents. Consider a set of @math identical mobile computational entities in the plane, called robots, operating in Look-Compute-Move cycles, without any means of direct communication. The Gathering Problem is the primitive task of all entities gathering in finite time at a point not fixed in advance, without any external control. The problem has been extensively studied in the literature under a variety of strong assumptions (e.g., synchronicity of the cycles, instantaneous movements, complete memory of the past, common coordinate system, etc.). In this paper we consider the setting without those assumptions, that is, when the entities are oblivious (i.e., they do not remember results and observations from previous cycles), disoriented (i.e., have no common coordinate system), and fully asynchronous (i.e., no assumptions exist on timing of cycles and activities within a cycle). The existing algorithmic contributions for such robots are limited to solutions for @math or for restricted sets of initial configura... Two mobile agents (robots) with distinct labels have to meet in an arbitrary, possibly infinite, unknown connected graph or in an unknown connected terrain in the plane. Agents are modeled as points, and the route of each of them only depends on its label and on the unknown environment. The actual walk of each agent also depends on an asynchronous adversary that may arbitrarily vary the speed of the agent, stop it, or even move it back and forth, as long as the walk of the agent in each segment of its route is continuous, does not leave it and covers all of it. Meeting in a graph means that both agents must be at the same time in some node or in some point inside an edge of the graph, while meeting in a terrain means that both agents must be at the same time in some point of the terrain. Does there exist a deterministic algorithm that allows any two agents to meet in any unknown environment in spite of this very powerful adversary? We give deterministic rendezvous algorithms for agents starting at arbitrary nodes of any anonymous connected graph (finite or infinite) and for agents starting at any interior points with rational coordinates in any closed region of the plane with path-connected interior. While our algorithms work in a very general setting - agents can, indeed, meet almost everywhere - we show that none of the above few limitations imposed on the environment can be removed. On the other hand, our algorithm also guarantees the following approximate rendezvous for agents starting at arbitrary interior points of a terrain as above: agents will eventually get at an arbitrarily small positive distance from each other. Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O(D|Lmin|2) when D is known and O((D + |Lmax|)3) if D is unknown, where |Lmin| and |Lmax| are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost O(n|Lmin|), if the size n of the ring is known, and of cost O(n|Lmax|), if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|Lmin|) if the topology of the graph and the initial positions are known to agents. 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.