text: The aim of rendezvous in a graph is meeting of two mobile agents at some node of an unknown anonymous connected graph. In this paper, we focus on rendezvous in trees, and, analogously to the efforts that have been made for solving the exploration problem with compact automata, we study the size of memory of mobile agents that permits to solve the rendezvous problem deterministically. We assume that the agents are identical, and move in synchronous rounds. We first show that if the delay between the starting times of the agents is arbitrary, then the lower bound on memory required for rendezvous is Omega(log n) bits, even for the line of length n. This lower bound meets a previously known upper bound of O(log n) bits for rendezvous in arbitrary graphs of size at most n. Our main result is a proof that the amount of memory needed for rendezvous with simultaneous start depends essentially on the number L of leaves of the tree, and is exponentially less impacted by the number n of nodes. Indeed, we present two identical agents with O(log L + loglog n) bits of memory that solve the rendezvous problem in all trees with at most n nodes and at most L leaves. Hence, for the class of trees with polylogarithmically many leaves, there is an exponential gap in minimum memory size needed for rendezvous between the scenario with arbitrary delay and the scenario with delay zero. Moreover, we show that our upper bound is optimal by proving that Omega(log L + loglog n)$ bits of memory are required for rendezvous, even in the class of trees with degrees bounded by 3.
  @cite_1: 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.
  @cite_32: 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.
  @cite_23: We investigate the relation between the time complexity and the space complexity for the rendezvous problem with k agents in asynchronous tree networks. The rendezvous problem requires that all the agents in the system have to meet at a single node within finite time. First, we consider asymptotically time-optimal algorithms and investigate the minimum memory requirement per agent for asymptotically time-optimal algorithms. We show that there exists a tree with n nodes in which Ω(n) bits of memory per agent is required to solve the rendezvous problem in O(n) time (asymptotically time-optimal). Then, we present an asymptotically time-optimal rendezvous algorithm. This algorithm can be executed if each agent has O(n) bits of memory. From this lower upper bound, this algorithm is asymptotically space-optimal on the condition that the time complexity is asymptotically optimal. Finally, we consider asymptotically space-optimal algorithms while allowing slowdown in time required to achieve rendezvous. We present an asymptotically space-optimal algorithm that each agent uses only O(logn) bits of memory. This algorithm terminates in O(Δn8) time where Δ is the maximum degree of the tree.
  @cite_34: Consider a set of n > 2 simple autonomous mobile robots (decentralized, asynchronous, no common coordinate system, no identities, no central coordination, no direct communication, no memory of the past, deterministic) moving freely in the plane and able to sense the positions of the other robots. We study the primitive task of gathering them at a point not fixed in advance (GATHERING PROBLEM). In the literature, most contributions are simulation-validated heuristics. The existing algorithmic contributions for such robots are limited to solutions for n ≤ 4 or for restricted sets of initial configurations of the robots. In this paper, we present the first algorithm that solves the GATHERING PROBLEM for any initial configuration of the robots.
  @cite_10: We consider a collection of robots which are identical (anonymous), have limited visibility of the environment, and no memory of the past (oblivious); furthermore, they are totally asynchronous in their actions, computations, and movements. We show that, even in such a totally asynchronous setting, it is possible for the robots to gather in the same location in finite time, provided they have a compass.
summary: Apart from the synchronous model used in this paper, several authors have investigated asynchronous rendezvous in the plane @cite_34 @cite_10 and in network environments @cite_23 @cite_1 @cite_32 . In the latter scenario the agent chooses the edge which it decides to traverse but the adversary controls the speed of the agent. Under this assumption rendezvous in 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.