A team of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node and declare that they have all met. Agents have different labels and move in synchronous rounds along links of the network. The above task is known as gathering and was traditionally considered under the assumption that when some agents are at the same node then they can talk. In this paper we ask the question of whether this ability of talking is needed for gathering. The answer turns out to be no. Our main contribution are two deterministic algorithms that always accomplish gathering in a much weaker model. We only assume that at any time an agent knows how many agents are at the node that it currently occupies but agents do not see the labels of other co-located agents and cannot exchange any information with them. They also do not see other nodes than the current one. Our first algorithm works under the assumption that agents know a priori some upper bound N on the network size, and it works in time polynomial in N and in the length l of the smallest label. Our second algorithm does not assume any a priori knowledge about the network but its complexity is exponential in the network size and in the labels of agents. Its purpose is to show feasibility of gathering under this harsher scenario. As a by-product of our techniques we obtain, in the same weak model, the solution of the fundamental problem of leader election among agents. As an application of our result we also solve, in the same model, the well-known gossiping problem: if each agent has a message at the beginning, we show how to make all messages known to all agents, even without any a priori knowledge about the network. If agents know an upper bound N on the network size then our gossiping algorithm works in time polynomial in N, in l and in the length of the largest message.
In @cite_32 , the authors considered the problem of network exploration by many agents that could not communicate between them. However, the information available to an agent in each round was much different than in the present paper. Indeed, in @cite_32 , agents were getting local traffic reports consisting of answers to three questions: Am I alone in the node?'', Did any agent enter this node in this round?'', Did any agent leave this node in this round?''. To see that this feedback cannot be derived from our present assumption of knowing the number of agents co-located with an agent in a given round, consider the situation when an agent @math stays at a node, and in a given round one other agent leaves the node and another agent enters it. In our present model, agent @math does not notice any change, while in the model from @cite_32 it gets reports about somebody leaving the node and somebody entering it.
@cite_32
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A team consisting of an unknown number of mobile agents starting from different nodes of an unknown network, possibly at different times, have to explore the network: Every node must be visited by at least one agent, and all agents must eventually stop. Agents are anonymous (identical), execute the same deterministic algorithm, and move in synchronous rounds along links of the network. They are silent: They cannot send any messages to other agents or mark visited nodes in any way. In the absence of any additional information, exploration with termination of an arbitrary network in this model, devoid of any means of communication between agents, is impossible. Our aim is to solve the exploration problem by giving to agents very restricted local traffic reports. Specifically, an agent that is at a node v in a given round is provided with three bits of information answering the following questions: Am I alone at vq Did any agent enter v in this roundq Did any agent exit v in this roundq We show that this small amount of information permits us to solve the exploration problem in arbitrary networks. More precisely, we give a deterministic terminating exploration algorithm working in arbitrary networks for all initial configurations that are not perfectly symmetric; that is, in which there are agents with different views of the network. The algorithm works in polynomial time in the (unknown) size of the network. A deterministic terminating exploration algorithm working for all initial configurations in arbitrary networks does not exist. 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. In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model, this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is well-connected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of O(φ-1 log n). In this paper, we show that a fairly simple modification of the protocol gives an algorithm that solves the information dissemination problem in at most O(D + polylog (n)) rounds in a network of diameter D, with no dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of D, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires T rounds in the LOCAL model can be simulated in O(T + polylog(n)) rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent. A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds. Each agent has a different label. Up to f of the agents are Byzantine. We consider two levels of Byzantine behavior. A strongly Byzantine agent can choose an arbitrary port when it moves and it can convey arbitrary information to other agents, while a weakly Byzantine agent can do the same, except changing its label. What is the minimum number of good agents that guarantees deterministic gathering of all of them, with terminationq We solve exactly this Byzantine gathering problem in arbitrary networks for weakly Byzantine agents and give approximate solutions for strongly Byzantine agents, both when the size of the network is known and when it is unknown. It turns out that both the strength versus the weakness of Byzantine behavior and the knowledge of network size significantly impact the results. For weakly Byzantine agents, we show that any number of good agents permits solving the problem for networks of known size. If the size is unknown, then this minimum number is fp2. More precisely, we show a deterministic polynomial algorithm that gathers all good agents in an arbitrary network, provided that there are at least fp2 of them. We also provide a matching lower bound: we prove that if the number of good agents is at most fp1, then they are not able to gather deterministically with termination in some networks. For strongly Byzantine agents, we give a lower bound of fp1, even when the graph is known: we show that f good agents cannot gather deterministically in the presence of f Byzantine agents even in a ring of known size. On the positive side, we give deterministic gathering algorithms for at least 2fp1 good agents when the size of the network is known and for at least 4fp2 good agents when it is unknown.