text: In rendezvous, two agents traverse network edges in synchronous rounds and have to meet at some node. In treasure hunt, a single agent has to find a stationary target situated at an unknown node of the network. We study tradeoffs between the amount of information ( @math ) available @math to the agents and the cost (number of edge traversals) of rendezvous and treasure hunt. Our goal is to find the smallest size of advice which enables the agents to solve these tasks at some cost @math in a network with @math edges. This size turns out to depend on the initial distance @math and on the ratio @math , which is the @math due to advice. For arbitrary graphs, we give upper and lower bounds of @math and @math , respectively, on the optimal size of advice. For the class of trees, we give nearly tight upper and lower bounds of @math and @math , respectively.
  @cite_38: We consider rendezvous problems in which two players move on the plane and wish to cooperate to minimise their first meeting time. We begin by considering the case where both players are placed such that the vector difference is chosen equiprobably from a finite set. We also consider a situation in which they know they are a distanced apart, but they do not know the direction of the other player. Finally, we give some results for the case in which player 1 knows the initial position of player 2, while player 2 is given information only on the initial distance of player 1.
  @cite_26: The author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time. This time is called the rendezvous value of the region. It is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise. A general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players. These concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network. The discrete rendezvous problem on a cycle graph for players restricted to symmetric Markovian strategies is then solved. Finally, the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other.
  @cite_33: Two players A and B are randomly placed on a line. The distribution of the distance between them is unknown except that the expected initial distance of the (two) players does not exceed some constant @math The players can move with maximal velocity 1 and would like to meet one another as soon as possible. Most of the paper deals with the asymmetric rendezvous in which each player can use a different trajectory. We find rendezvous trajectories which are efficient against all probability distributions in the above class. (It turns out that our trajectories do not depend on the value of @math ) We also obtain the minimax trajectory of player A if player B just waits for him. This trajectory oscillates with a geometrically increasing amplitude. It guarantees an expected meeting time not exceeding @math We show that, if player B also moves, then the expected meeting time can be reduced to @math The expected meeting time can be further reduced if the players use mixed strategies. We show that if player B rests, then the optimal strategy of player A is a mixture of geometric trajectories. It guarantees an expected meeting time not exceeding @math This value can be reduced even more (below @math ) if player B also moves according to a (correlated) mixed strategy. We also obtain a bound for the expected meeting time of the corresponding symmetric rendezvous problem.
  @cite_22: Search Theory is one of the original disciplines within the field of Operations Research. It deals with the problem faced by a Searcher who wishes to minimize the time required to find a hidden object, or “target. ” The Searcher chooses a path in the “search space” and finds the target when he is sufficiently close to it. Traditionally, the target is assumed to have no motives of its own regarding when it is found; it is simply stationary and hidden according to a known distribution (e. g. , oil), or its motion is determined stochastically by known rules (e. g. , a fox in a forest). The problems dealt with in this book assume, on the contrary, that the “target” is an independent player of equal status to the Searcher, who cares about when he is found. We consider two possible motives of the target, and divide the book accordingly. Book I considers the zero-sum game that results when the target (here called the Hider) does not want to be found. Such problems have been called Search Games (with the “ze- sum” qualifier understood). Book II considers the opposite motive of the target, namely, that he wants to be found. In this case the Searcher and the Hider can be thought of as a team of agents (simply called Player I and Player II) with identical aims, and the coordination problem they jointly face is called the Rendezvous Search Problem.
  @cite_15: 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.
  @cite_41: 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 © 2012 Wiley Periodicals, Inc.
  @cite_6: Two players are independently placed on a commonly labelled network X. They cannot see each other but wish to meet in least expected time. We consider continuous and discrete versions, in which they may move at unit speed or between adjacent distinct nodes, respectively. There are two versions of the problem (asymmetric or symmetric), depending on whether or not we allow the players to use different strategies. After obtaining some optimality conditions for general networks, we specialize to the interval and circle networks. In the first setting, we extend the work of J. V. Howard; in the second we prove a conjecture concerning the optimal symmetric strategy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 256–274, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002 nav.10011
  @cite_23: Leaving marks at the starting points in a rendezvous search problem may provide the players with important information. Many of the standard rendezvous search problems are investigated under this new framework which we call markstart rendezvous search. Somewhat surprisingly, the relative difficulties of analysing problems in the two scenarios differ from problem to problem. Symmetric rendezvous on the line seems to be more tractable in the new setting whereas asymmetric rendezvous on the line when the initial distance is chosen by means of a convex distribution appears easier to analyse in the original setting. Results are also obtained for markstart rendezvous on complete graphs and on the line when the players' initial distance is given by an unknown probability distribution. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 722–731, 2001
  @cite_47: Two friends have become separated in a building or shopping mall and and wish to meet as quickly as possible. There are n possible locations where they might meet. However, the locations are identical and there has been no prior agreement where to meet or how to search. Hence they must use identical strategies and must treat all locations in a symmetrical fashion. Suppose their search proceeds in discrete time. Since they wish to avoid the possibility of never meeting, they will wish to use some randomizing strategy. If each person searches one of the n locations at random at each step, then rendezvous will require n steps on average. It is possible to do better than this: although the optimal strategy is difficult to characterize for general n, there is a strategy with an expected time until rendezvous of less than 0.829 n for large enough n. For n = 2 and 3 the optimal strategy can be established and on average 2 and 8 3 steps are required respectively. There are many tantalizing variations on this problem, which we discuss with some conjectures. DYNAMIC PROGRAMMING; SEARCH PROBLEMS
  @cite_16: 
summary: The problem of rendezvous has been studied both under randomized and deterministic scenarios. In the framework of networks, it is usually assumed that the nodes do not have distinct identities. An extensive survey of randomized rendezvous in various models can be found in @cite_22 , cf. also @cite_26 @cite_6 @cite_47 @cite_33 . Deterministic rendezvous in networks has been surveyed in @cite_41 . Several authors considered geometric scenarios (rendezvous in an interval of the real line, e.g., @cite_33 @cite_23 , or in the plane, e.g., @cite_16 @cite_38 ). Gathering more than two agents was studied, e.g., in @cite_15 .