We study the problem of rendezvous of two mobile agents starting at distinct locations in an unknown graph. The agents have distinct labels and walk in synchronous steps. However the graph is unlabelled and the agents have no means of marking the nodes of the graph and cannot communicate with or see each other until they meet at a node. When the graph is very large we want the time to rendezvous to be independent of the graph size and to depend only on the initial distance between the agents and some local parameters such as the degree of the vertices, and the size of the agent's label. It is well known that even for simple graphs of degree @math , the rendezvous time can be exponential in @math in the worst case. In this paper, we introduce a new version of the rendezvous problem where the agents are equipped with a device that measures its distance to the other agent after every step. We show that these agents are able to rendezvous in any unknown graph, in time polynomial in all the local parameters such the degree of the nodes, the initial distance @math and the size of the smaller of the two agent labels @math . Our algorithm has a time complexity of @math and we show an almost matching lower bound of @math on the time complexity of any rendezvous algorithm in our scenario. Further, this lower bound extends existing lower bounds for the general rendezvous problem without distance awareness.
There have been several studies on the minimum capabilities needed by the agents to solve rendezvous. For example, the minimum memory required by an agent to solve rendezvous is known to be @math for arbitrary graphs. @cite_3 have provided a memory optimal algorithm for rendezvous, and there are studies on the tradeoff between time and space requirements for rendezvous @cite_5 . In some papers, additional capacities are assumed for the agents to overcome other limitations, e.g. global vision is assumed to overcome memory limitations @cite_9 or the capability to mark nodes using tokens @cite_15 or whiteboards @cite_16 is often used to break symmetry. The model used in this paper can be seen as a special case of the oracle model for computation @cite_17 where the agent is allowed to query an oracle that has global knowledge of the environment. However in our case, since the only queries are distance queries, the oracle can be implemented without complete knowledge of the graph topology.
@cite_9 @cite_3 @cite_5 @cite_15 @cite_16 @cite_17
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Two identical (anonymous) mobile agents start from arbitrary nodes in an a priori unknown graph and move synchronously from node to node with the goal of meeting. This rendezvous problem has been thoroughly studied, both for anonymous and for labeled agents, along with another basic task, that of exploring graphs by mobile agents. The rendezvous problem is known to be not easier than graph exploration. A well-known recent result on exploration, due to Reingold, states that deterministic exploration of arbitrary graphs can be performed in log-space, i.e., using an agent equipped with O(log n) bits of memory, where n is the size of the graph. In this paper we study the size of memory of mobile agents that permits us to solve the rendezvous problem deterministically. Our main result establishes the minimum size of the memory of anonymous agents that guarantees deterministic rendezvous when it is feasible. We show that this minimum size is Θ(log n), where n is the size of the graph, regardless of the delay between the starting times of the agents. More precisely, we construct identical agents equipped with Θ(log n) memory bits that solve the rendezvous problem in all graphs with at most n nodes, if they start with any delay τ, and we prove a matching lower bound Ω(log n) on the number of memory bits needed to accomplish rendezvous, even for simultaneous start. In fact, this lower bound is achieved already on the class of rings. This shows a significant contrast between rendezvous and exploration: e.g., while exploration of rings (without stopping) can be done using constant memory, rendezvous, even with simultaneous start, requires logarithmic memory. As a by-product of our techniques introduced to obtain log-space rendezvous we get the first algorithm to find a quotient graph of a given unlabeled graph in polynomial time, by means of a mobile agent moving around the graph. The aim of rendezvous in a graph is meeting of two mobile agents at some node of an unknown anonymous connected graph. In this article, 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 Ω(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 e 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 e + log log n) bits of memory that solve the rendezvous problem in all trees with at most n nodes and at most e 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 Ω(log e + log log n) bits of memory are required for rendezvous, even in the class of trees with degrees bounded by 3. We study the size of memory of mobile agents that permits to solve deterministically the rendezvous problem, i.e., the task of meeting at some node, for two identical agents moving from node to node along the edges of an unknown anonymous connected graph. The rendezvous problem is unsolvable in the class of arbitrary connected graphs, as witnessed by the example of the cycle. Hence we restrict attention to rendezvous in trees, where rendezvous is feasible if and only if the initial positions of the agents are not symmetric. We prove that the minimum memory size guaranteeing rendezvous in all trees of size at most nis i¾?(logn) bits. The upper bound is provided by an algorithm for abstract state machines accomplishing rendezvous in all trees, and using O(logn) bits of memory in trees of size at most n. The lower bound is a consequence of the need to distinguish between up to ni¾? 1 links incident to a node. Thus, in the second part of the paper, we focus on the potential existence of pairs of finiteagents (i.e., finite automata) capable of accomplishing rendezvous in all bounded degreetrees. We show that, as opposed to what has been proved for the graph exploration problem, there are no finite agents capable of accomplishing rendezvous in all bounded degree trees. 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).