Two mobile agents, starting from different nodes of a network at possibly different times, have to meet at the same node. This problem is known as @math . Agents move in synchronous rounds. Each agent has a distinct integer label from the set @math . Two main efficiency measures of rendezvous are its @math (the number of rounds until the meeting) and its @math (the total number of edge traversals). We investigate tradeoffs between these two measures. A natural benchmark for both time and cost of rendezvous in a network is the number of edge traversals needed for visiting all nodes of the network, called the exploration time. Hence we express the time and cost of rendezvous as functions of an upper bound @math on the time of exploration (where @math and a corresponding exploration procedure are known to both agents) and of the size @math of the label space. We present two natural rendezvous algorithms. Algorithm @math has cost @math (and, in fact, a version of this algorithm for the model where the agents start simultaneously has cost exactly @math ) and time @math . Algorithm @math has both time and cost @math . Our main contributions are lower bounds showing that, perhaps surprisingly, these two algorithms capture the tradeoffs between time and cost of rendezvous almost tightly. We show that any deterministic rendezvous algorithm of cost asymptotically @math (i.e., of cost @math ) must have time @math . On the other hand, we show that any deterministic rendezvous algorithm with time complexity @math must have cost @math .
Apart from the synchronous model used in this paper, several authors investigated asynchronous rendezvous in the plane @cite_44 @cite_13 and in network environments @cite_31 @cite_1 @cite_40 @cite_21 . 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_21 @cite_1 @cite_44 @cite_40 @cite_31 @cite_13
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We study rendezvous of two anonymous agents, where each agent knows its own initial position in the environment. Their task is to meet each other as quickly as possible. The time of the rendezvous is measured by the number of synchronous rounds that agents need to use in the worst case in order to meet. In each round, an agent may make a simple move or it may stay motionless. We consider two types of environments, finite or infinite graphs and Euclidean spaces. A simple move traverses a single edge (in a graph) or at most a unit distance (in Euclidean space). The rendezvous consists in visiting by both agents the same point of the environment simultaneously (in the same round). In this paper, we propose several asymptotically optimal rendezvous algorithms. In particular, we show that in the line and trees as well as in multidimensional Euclidean spaces and grids the agents can rendezvous in time O(d), where d is the distance between the initial positions of the agents. The problem of location-aware rendezvous was studied before in the asynchronous model for Euclidean spaces and multi-dimensional grids, where the emphasis was on the length of the adopted rendezvous trajectory. We point out that, contrary to the asynchronous case, where the cost of rendezvous is dominated by the size of potentially large neighborhoods, the agents are able to meet in all graphs of at most n nodes in time almost linear in d, namely, O(d log2 n). We also determine an infinite family of graphs in which synchronized rendezvous takes time Ω(d). Two identical (anonymous) mobile agents have to meet in an arbitrary, possibly infinite, unknown connected graph. Agents are modeled as points, they start at nodes of the graph chosen by the adversary and the route of each of them only depends on the already traversed portion of the graph and, in the case of randomized rendezvous, on the result of coin tossing. 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 means that both agents must be at the same time in some node or in some point inside an edge of the graph. In the deterministic scenario we characterize the initial positions of the agents for which rendezvous is feasible and we provide an algorithm guaranteeing asynchronous rendezvous from all such positions in an arbitrary connected graph. In the randomized scenario we show an algorithm that achieves asynchronous rendezvous with probability 1, for arbitrary initial positions in an arbitrary connected graph. In both cases the graph may be finite or (countably) infinite. In this paper we study efficient rendezvous of two mobile agents moving asynchronously in the Euclidean 2d-space. Each agent has limited visibility, permitting it to see its neighborhood at unit range from its current location. Moreover, it is assumed that each agent knows its own initial position in the plane given by its coordinates. The agents, however, are not aware of each others position. The agents possess coherent compasses and the same unit of length, which permit them to consider their current positions within the same system of coordinates. The cost of the rendezvous algorithm is the sum of lengths of the trajectories of both agents. This cost is taken as the maximum over all possible asynchronous movements of the agents, controlled by the adversary. We propose an algorithm that allows the agents to meet in a local neighborhood of diameter O(d), where d is the original distance between the agents. This seems rather surprising since each agent is unaware of the possible location of the other agent. In fact, the cost of our algorithm is O(d2+e), for any constant e > 0. This is almost optimal, since a lower bound of Ω(d2) is straightforward. The only up to date paper [12] on asynchronous rendezvous of bounded-visibility agents in the plane provides the feasibility proof for rendezvous, proposing a solution exponential in the distance d and in the labels of the agents. In contrast, we show here that, when the identity of the agent is based solely on its original location, an almost optimal solution is possible. An integral component of our solution is the construction of a novel type of non-simple space-filling curves that preserve locality. An infinite curve of this type visits specific grid points in the plane and provides a route that can be adopted by the mobile agents in search for one another. This new concept may also appear counter-intuitive in view of the result from [22] stating that for any simple space-filling curve, there always exists a pair of close points in the plane, such that their distance along the space-filling curve is arbitrarily large. 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...