Fast deterministic rendezvous in labeled lines

The rendezvous problem in graphs has been extensively studied in the literature, mainly using a randomized approach. Two mobile agents have to meet at some node of a connected graph. We study deterministic algorithms for this problem, assuming that agents have distinct identifiers and are located in nodes of an unknown anonymous connected graph. Startup times of the agents are arbitrarily decided by the adversary. 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 steps since the startup of the later agent until rendezvous is achieved. Deterministic rendezvous has been previously shown feasible in arbitrary graphs [16] but the proposed algorithm had cost exponential in the number n of nodes and in the smaller identifier l, and polynomial in the difference τ between startup times. The following problem was stated in [16]: Does there exist a deterministic rendezvous algorithm with cost polynomial in n, τ and in labels L 1, L 2 of the agents (or even polynomial in n, τ and log L 1, log L 2)? We give a positive answer to both problems: our main result is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l. We also show a lower bound Ω (n 2) on the cost of rendezvous in some family of graphs.KeywordsMobile AgentArbitrary GraphStartup TimePassive PhaseRendezvous ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Two or more mobile entities, called agents or robots, starting at distinct initial positions, have to meet. This task is known in the literature as rendezvous. Among many alternative assumptions that have been used to study the rendezvous problem, two most significantly influence the methodology appropriate for its solution. The first of these assumptions concerns the environment in which the mobile entities navigate: it can be either a terrain in the plane, or a network modeled as an undirected graph. The second assumption concerns the way in which the entities move: it can be either deterministic or randomized. In this article, we survey results on deterministic rendezvous in networks. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

How to meet in anonymous network

Two or more mobile entities, called agents or robots, starting at distinct initial positions in some environment, have to meet. This task is known in the literature as rendezvous. Among many alternative assumptions that have been used to study the rendezvous problem, two most significantly influence the methodology appropriate for its solution. The first of these assumptions concerns the environment in which the mobile entities navigate: it can be either a terrain in the plane, or a network modeled as an undirected graph. In the case of networks, methods and results further depend on whether the agents have the ability to mark nodes in some way. The second assumption concerns the way in which the entities move: it can be either deterministic or randomized. In this paper we survey models and results concerning deterministic rendezvous in networks, where agents cannot mark nodes.Keywordsmobile agentrendezvousdeterministicnetworkgraph

Two mobile agents (robots) have to meet in an a priori unknown bounded terrain modeled as a polygon, possibly with polygonal obstacles. Agents are modeled as points, and each of them is equipped with a compass. Compasses of agents may be incoherent. Agents construct their routes, but the actual walk of each agent is decided by the adversary: the movement of the agent can be at arbitrary speed, the agent may sometimes stop or go 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. We consider several scenarios, depending on three factors: (1) obstacles in the terrain are present, or not, (2) compasses of both agents agree, or not, (3) agents have or do not have a map of the terrain with their positions marked. The cost of a rendezvous algorithm is the worst-case sum of lengths of the agents' trajectories until their meeting. For each scenario we design a deterministic rendezvous algorithm and analyze its cost. We also prove lower bounds on the cost of any deterministic rendezvous algorithm in each case. For all scenarios these bounds are tight.

We obtain several improved solutions for the deterministic rendezvous problem in general undirected graphs. Our solutions answer several problems left open by Dessmark et al. We also introduce an interesting variant of the rendezvous problem, which we call the deterministic treasure hunt problem. Both the rendezvous and the treasure hunt problems motivate the study of universal traversal sequences and universal exploration sequences with some strengthened properties. We call such sequences strongly universal traversal (exploration) sequences . We give an explicit construction of strongly universal exploration sequences. The existence of strongly universal traversal sequences, as well as the solution of the most difficult variant of the deterministic treasure hunt problem, are left as intriguing open problems.

Deterministic rendezvous with different maps

Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. 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 steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is Θ(D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound Ω (n2) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

Asynchronous deterministic rendezvous in bounded terrains

Mobile agent computing is being used in fields as diverse as artificial intelligence, computational economics and robotics. Agents' ability to adapt dynamically and execute asynchronously and autonomously brings potential advantages in terms of fault-tolerance, flexibility and simplicity. This monograph focuses on studying mobile agents as modelled in distributed systems research and in particular within the framework of research performed in the distributed algorithms community. It studies the fundamental question of how to achieve {\em rendezvous}, the gathering of two or more agents at the same node of a network. Like leader election, such an operation is a useful subroutine in more general computations that may require the agents to synchronize, share information, divide up chores, etc. The work provides an introduction to the algorithmic issues raised by the rendezvous problem in the distributed computing setting. For the most part our investigation concentrates on the simplest case of two agents attempting to rendezvous on a ring network. Other situations including multiple agents, faulty nodes and other topologies are also examined. An extensive bibliography provides many pointers to related work not covered in the text. The presentation has a distinctly algorithmic, rigorous, distributed computing flavor and most results should be easily accessible to advanced undergraduate and graduate students in computer science and mathematics departments. Table of Contents: Models for Mobile Agent Computing / Deterministic Rendezvous in a Ring / Multiple Agent Rendezvous in a Ring / Randomized Rendezvous in a Ring / Other Models / Other Topologies

On deterministic rendezvous at a node of agents with arbitrary velocities

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 n is Θ(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 n − 1 links incident to a node. Thus, in the second part of the paper, we focus on the potential existence of pairs of finite agents (i.e., finite automata) capable of accomplishing rendezvous in all bounded degree trees. 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.

In this paper we consider the problem of synchronous rendezvous in which two anonymous mobile entities (robots) A and B are expected to meet at the same time and point in a graph G = (V,E). Most of the work devoted to rendezvous in graphs assumes that robots have access to the same sets of nodes and edges, where the topology of connections may be initially known or unknown. In our work we assume the movement of robots is restricted by the topological properties of the graph space coupled with the intrinsic characteristics of robots preventing them from visiting certain edges in E.

Deterministic Rendezvous in Graphs

We study the rendezvous problem in the asynchronous setting in the graph of infinite line following the model introduced in [13]. We formulate general lemmas about deterministic rendezvous algorithms in this setting which characterize the algorithms in which the agents have the shortest routes. We also improve rendezvous algorithms in the infinite line which formulated in [13]. Two agents have distinct labels L m in,L m ax and |L m in |leq |L m ax |. When the initial distance D between the agents is known, our algorithm has cost \(D |L_min|^2\) which is an improvement in the constant. If the initial distance is unknown we give an algorithm of cost \(O(D\log^2 D+D log D|L_max |+D|L_min |^2+|L_max ||L_min |log|L_min |)\) which is an asymptotic improvement.KeywordsMobile AgentInitial DistanceStarting VertexSkeleton AlgorithmRendezvous ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.