text: We investigate initial information, unbounded memory and randomization in gathering mobile agents on a grid. We construct a state machine, such that it is possible to gather, with probability 1, all configurations of its copies. This machine has initial input, unbounded memory, and is randomized. We show that no machine having any two of these capabilities but not the third, can be used to gather, with high probability, all configurations. We construct deterministic Turing Machines that are used to gather all connected configurations, and we construct deterministic finite automata that are used to gather all contractible connected configurations.
  @cite_7: 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.
  @cite_28: 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_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: Consider a set of @math identical mobile computational entities in the plane, called robots, operating in Look-Compute-Move cycles, without any means of direct communication. The Gathering Problem is the primitive task of all entities gathering in finite time at a point not fixed in advance, without any external control. The problem has been extensively studied in the literature under a variety of strong assumptions (e.g., synchronicity of the cycles, instantaneous movements, complete memory of the past, common coordinate system, etc.). In this paper we consider the setting without those assumptions, that is, when the entities are oblivious (i.e., they do not remember results and observations from previous cycles), disoriented (i.e., have no common coordinate system), and fully asynchronous (i.e., no assumptions exist on timing of cycles and activities within a cycle). The existing algorithmic contributions for such robots are limited to solutions for @math or for restricted sets of initial configura...
  @cite_16: 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.
  @cite_34: We revisit the problem of gathering autonomous robots in the plane. In particular, we consider non-transparent unit-disc robots (i.e., fat) in an asynchronous setting with vision as the only means of coordination and robots only make local decisions. We use a state-machine representation to formulate the gathering problem and develop a distributed algorithm that solves the problem for any number of fat robots. The main idea behind the algorithm is to enforce the robots to reach a configuration in which all the following hold: (i) The robots' centers form a convex hull in which all robots are on the convex hull's boundary; (ii) Each robot can see all other robots; (iii) The configuration is connected: every robot touches another robot and all robots form together a connected formation. We show that starting from any initial configuration, the fat robots eventually reach such a configuration and terminate yielding a solution to the gathering problem.
  @cite_20: 
summary: Several authors have investigated asynchronous rendezvous in the plane @cite_34 @cite_32 @cite_7 and in network environments @cite_20 @cite_1 @cite_28 . In the latter scenario it was assumed that 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, and hence the rendezvous requirement is relaxed to permit the agents to meet inside an edge. In @cite_20 the authors studied rendezvous in grids of two agents sharing a common coordinate system. The feasibility of asynchronous rendezvous of two anonymous agents in arbitrary graphs was discussed in @cite_16 , both in the deterministic and in the randomized version.