Mobile agent rendezvous: A survey

Abstract. Recent results on the problem of mobile agent rendezvous on distributed networks are surveyed with an emphasis on outlining the various approaches taken by researchers in the theoretical computer science community.

Move-optimal partial gathering of mobile agents in asynchronous trees

In this paper, we consider the partial gathering problem of mobile agents in asynchronous tree networks. The partial gathering problem is a generalization of the classical gathering problem, which requires that all the agents meet at the same node. The partial gathering problem requires, for a given positive integer g, that each agent should move to a node and terminate so that at least g agents should meet at each of the nodes they terminate at. The requirement for the partial gathering problem is weaker than that for the (well-investigated) classical gathering problem, and thus, we clarify the difference on the move complexity between them. We consider two multiplicity detection models: weak multiplicity detection and strong multiplicity detection models. In the weak multiplicity detection model, each agent can detect whether another agent exists at the current node or not but cannot count the exact number of the agents. In the strong multiplicity detection model, each agent can count the number of agents at the current node. In addition, we consider two token models: non-token model and removable token model. In the non-token model, agents cannot mark the nodes or the edges in any way. In the removable-token model, each agent initially leaves a token on its initial node, and agents can remove the tokens. Our contribution is as follows. First, we show that for the non-token model agents require Ω(kn) total moves to solve the partial gathering problem, where n is the number of nodes and k is the number of agents. Second, we consider the weak multiplicity detection and non-token model. In this model, for asymmetric trees, by a previous result agents can achieve the partial gathering in O(kn) total moves, which is asymptotically optimal in terms of total moves. In addition, for symmetric trees we show that there exist no algorithms to solve the partial gathering problem. Third, we consider the strong multiplicity detection and non-token model. In this model, for any trees we propose an algorithm to achieve the partial gathering in O(kn) total moves, which is asymptotically optimal in terms of total moves. At last, we consider the weak multiplicity detection and removable-token model. In this model, we propose an algorithm to achieve the partial gathering in O(gn) total moves. Note that in this model, agents require Ω(gn) total moves to solve the partial gathering problem. Hence, the second proposed algorithm is also asymptotically optimal in terms of total moves

Uniform deployment of mobile agents in asynchronous rings

In this paper, we consider the uniform deployment problem of mobile agents in asynchronous unidirectional rings, which requires the agents to uniformly spread in the ring. The uniform deployment problem is in striking contrast to the rendezvous problem which requires the agents to meet at the same node. While rendezvous aims to break the symmetry, uniform deployment aims to attain the symmetry. It is well known that the symmetry breaking is difficult in distributed systems and the rendezvous problem cannot be solved from some initial configurations. Hence, we are interested in clarifying what difference the uniform deployment problem has on the solvability and the number of agent moves compared to the rendezvous problem. We consider two problem settings, with knowledge of k (or n) and without knowledge of k or n where k is the number of agents and n is the number of nodes. First, we consider agents with knowledge of k (or n since k and n can be easily obtained if one of them is given). In this case, we propose two algorithms. The first algorithm solves the uniform deployment problem with termination detection. This algorithm requires O(k log n) memory space per agent, O(n) time, and O(kn) total moves. The second algorithm also solves the uniform deployment problem with termination detection. This algorithm reduces the memory space per agent to O(log n), but uses O(n log k) time, and requires O(kn) total moves. Both algorithms are asymptotically optimal in terms of total moves since there are some initial configurations such that agents re- quire Ω(kn) total moves to solve the problem. Next, we consider agents with no knowledge of k or n. In this case, we show that, when termination detection is required, there exists no algorithm to solve the uniform deployment problem. For this reason, we consider the relaxed uniform deployment problem that does not require termination detection, and we propose an algorithm to solve the relaxed uniform deployment problem. This algorithm requires O((k/l) log(n/l)) memory space per agent, O(n/l) time, and O(kn/l) total moves when the initial configuration has symmetry degree l. This means that the algorithm can solve the problem more efficiently when the initial configuration has higher symmetric degree (i.e., is closer to uniform deployment). Note that all the proposed algorithms achieve uniform deployment from any initial configuration, which is a striking difference from the rendezvous problem because the rendezvous problem is not solvable from some initial configurations

Move-optimal partial gathering of mobile agents without identifiers or global knowledge in asynchronous unidirectional rings

In this paper, we consider the partial gathering problem of mobile agents in asynchronous unidirectional ring networks. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all the k agents distributed in the network terminate at a single node. The partial gathering problem requires, for a given positive integer g(<k), that all the agents terminate in a configuration such that either at least g agents or no agent exists at each node. The requirement for the partial gathering problem is strictly weaker than that for the total gathering problem, and thus it is interesting to clarify the difference on the move complexity between them. In this paper, we aim to solve the partial gathering problem for agents without identifiers or any global knowledge such as the number k of agents or the number n of nodes. We consider deterministic and randomized cases. First, in the deterministic case, we show that the set of unsolvable initial configurations is the same as that for the case of agents with knowledge of k. In addition, we propose an algorithm that solves the problem from any solvable initial configuration in a total number of O(gn) moves. Next, in the randomized case, we propose an algorithm that solves the problem in a total number of O(gn) moves in expectation from any initial configuration. Note that g<k holds and agents require a total number of Ω(gn) (resp., Ω(kn)) moves to solve the partial (resp., total) gathering problem. Thus, our algorithms can solve the partial gathering problem in asymptotically optimal total number of moves without identifiers or global knowledge, and the total number of O(gn) moves is strictly smaller than that for the total gathering problem

Randomized Byzantine Gathering in Rings

We study the problem of gathering k anonymous mobile agents on a ring with n nodes. Importantly, f out of the k anonymous agents are Byzantine. The agents operate synchronously and in an autonomous fashion. In each round, each agent can communicate with other agents co-located with it by broadcasting a message. After receiving all the messages, each agent decides to either move to a neighbouring node or stay put. We begin with the k agents placed arbitrarily on the ring, and the task is to gather all the good agents in a single node. The task is made harder by the presence of Byzantine agents, which are controlled by a single Byzantine adversary. Byzantine agents can deviate arbitrarily from the protocol. The Byzantine adversary is computationally unbounded. Additionally, the Byzantine adversary is adaptive in the sense that it can capitalize on information gained over time (including the current round) to choreograph the actions of Byzantine agents. Specifically, the entire state of the system, which includes messages sent by all the agents and any random bits generated by the agents, is known to the Byzantine adversary before all the agents move. Thus the Byzantine adversary can compute the positioning of good agents across the ring and choreograph the movement of Byzantine agents accordingly. Moreover, we consider two settings: standard and visual tracking setting. With visual tracking, agents have the ability to track other agents that are moving along with them. In the standard setting, agents do not have such an ability. In the standard setting we can achieve gathering in ?(nlog nlog k) rounds with high probability and can handle ?(k/(log k)) number of Byzantine agents. With visual tracking, we can achieve gathering faster in ?(n log n) rounds whp and can handle any constant fraction of the total number of agents being Byzantine

Leader Election for Anonymous Asynchronous Agents in Arbitrary Networks

We study the problem of leader election among mobile agents operating in an arbitrary network modeled as an undirected graph. Nodes of the network are unlabeled and all agents are identical. Hence the only way to elect a leader among agents is by exploiting asymmetries in their initial positions in the graph. Agents do not know the graph or their positions in it, hence they must gain this knowledge by navigating in the graph and share it with other agents to accomplish leader election. This can be done using meetings of agents, which is difficult because of their asynchronous nature: an adversary has total control over the speed of agents. When can a leader be elected in this adversarial scenario and how to do it? We give a complete answer to this question by characterizing all initial configurations for which leader election is possible and by constructing an algorithm that accomplishes leader election for all configurations for which this can be done

Partial gathering of mobile agents in dynamic rings

In this paper, we consider the partial gathering problem of mobile agents in synchronous dynamic bidirectional rings. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all k agents distributed in the network terminate at a non-predetermined single node. The partial gathering problem requires, for a given positive integer g(<k), that agents terminate in a configuration such that either at least g agents or no agent exists at each node. The requirement for the partial gathering problem is strictly weaker than that for the total gathering problem, and thus it is interesting to clarify the difference in the move complexity between them. So far, partial gathering has been considered in static graphs. In this paper, we consider this problem in 1-interval connected rings, that is, one of the links in the ring may be missing at each time step. In such networks, we aim to clarify the solvability of the partial gathering problem and the move complexity, focusing on the relationship between values of k and g. First, we consider the case of 3g≤k≤8g−2. In this case, we show that our algorithm can solve the problem with the total number of O(kn) moves, where n is the number of nodes. Since k=O(g) holds when 3g≤k≤8g−2, the move complexity O(kn) in this case can be represented also as O(gn). Next, we consider the case of k≥8g−3. In this case, we show that our algorithm can also solve the problem and its move complexity is O(gn). These results mean that, when k≥3g, the partial gathering problem can be solved also in dynamic rings. In addition, agents require a total number of Ω(gn) (resp., Ω(kn)) moves to solve the partial (resp., total) gathering problem. Thus, the both proposed algorithms can solve the partial gathering problem with the asymptotically optimal total number of O(gn) moves, which is strictly smaller than that for the total gathering problem.23rd International Symposium on Stabilization, Safety, and Security of Distributed Systems, November 17-20, 2021, Virtual Conferenc

Partial Gathering of Mobile Agents in Arbitrary Networks

In this paper, we consider the partial gathering problem of mobile agents in arbitrary networks. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all the agents meet at the same node. The partial gathering problem requires, for a given positive integer g, that each agent should move to a node and terminate so that at least g agents should meet at each of the nodes they terminate at. The requirement for the partial gathering problem is no stronger than that for the total gathering problem, and thus, we clarify the difference on the move complexity between them. First, we show that agents require Ω(gn+m) total moves to solve the partial gathering problem, where n is the number of nodes and m is the number of communication links. Next, we propose a deterministic algorithm to solve the partial gathering problem in O(gn+m) total moves, which is asymptotically optimal in terms of total moves. Note that, it is known that agents require Ω(kn+m) total moves to solve the total gathering problem in arbitrary networks, where k is the number of agents. Thus, our result shows that the partial gathering problem is solvable with strictly fewer total moves compared to the total gathering problem in arbitrary networks