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.
For the deterministic setting a lot of effort has been dedicated to the study of the feasibility of rendezvous, and to the time required to achieve this task, when feasible. For instance, deterministic rendezvous with agents equipped with tokens used to mark nodes was considered, e.g., in @cite_28 . Deterministic rendezvous of two agents that cannot mark nodes but have unique labels was discussed in @cite_17 @cite_9 . These papers are concerned with the time of rendezvous in arbitrary graphs. In @cite_17 the authors show a rendezvous algorithm polynomial in the size of the graph, in the length of the shorter label and in the delay between the starting time of the agents. In @cite_9 rendezvous time is polynomial in the first two of these parameters and independent of the delay.
@cite_28 @cite_9 @cite_17
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In the rendezvous search problem, two mobile agents must move along the n nodes of a network so as to minimize the time required to meet or rendezvous. When the mobile agents are identical and the network is anonymous, however, the resulting symmetry can make the problem impossible to solve. Symmetry is typically broken by having the mobile agents run either a randomized algorithm or different deterministic algorithms. We investigate the use of identical tokens to break symmetry so that the two mobile agents can run the same deterministic algorithm. After deriving the explicit conditions under which identical tokens can be used to break symmetry on the n node ring, we derive the lower and upper bounds for the time and memory complexity of the rendezvous search problem with various parameter sets. While these results suggest a possible tradeoff between the mobile agents' memory and the time complexity of the rendezvous search problem, we prove that this tradeoff is limited. We obtain several improved solutions for the deterministic rendezvous problem in general undirected graphs. Our solutions answer several problems left open in a recent paper by 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.