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Two mobile agents starting at different nodes of an unknown network have to meet. This task is known in the literature as rendezvous. Each agent has a different label which is a positive integer known to it, but unknown to the other agent. Agents move in an asynchronous way: the speed of agents may vary and is controlled by an adversary. The cost of a rendezvous algorithm is the total number of edge traversals by both agents until their meeting. The only previous deterministic algorithm solving this problem has cost exponential in the size of the graph and in the larger label. In this paper we present a deterministic rendezvous algorithm with cost polynomial in the size of the graph and in the length of the smaller label. Hence we decrease the cost exponentially in the size of the graph and doubly exponentially in the labels of agents.As an application of our rendezvous algorithm we solve several fundamental problems involving teams of unknown size larger than 1 of labeled agents moving asynchronously in unknown networks. Among them are the following problems: team size, in which every agent has to find the total number of agents, leader election, in which all agents have to output the label of a single agent, perfect renaming in which all agents have to adopt new different labels from the set {1, . . . , k}, where k is the number of agents, and gossiping, in which each agent has initially a piece of information (value) and all agents have to output all the values. Using our rendezvous algorithm we solve all these problems at cost polynomial in the size of the graph and in the smallest length of all labels of participating agents.

Leader election and arbitrary pattern formation are fundammental tasks for a set of autonomous mobile robots. The former consists in distinguishing a unique robot, called the leader. The latter aims in arranging the robots in the plane to form any given pattern. The solvability of both these tasks turns out to be necessary in order to achieve more complex tasks.In this paper, we study the relationship between these two tasks in a model, called CORDA, wherein the robots are weak in several aspects. In particular, they are fully asynchronous and they have no direct means of communication. They cannot remember any previous observation nor computation performed in any previous step. Such robots are said to be oblivious. The robots are also uniform and anonymous, i.e, they all have the same program using no global parameter (such that an identity) allowing to differentiate any of them. Moreover, none of them share any kind of common coordinate mechanism or common sense of direction, except that they agree on a common handedness (chirality).In such a system, Flochini et al. proved in [9] that it is possible to solve the leader election problem for n ≥ 3 robots if the arbitrary pattern formation is solvable for n ≥ 3. In this paper, we show that the converse is true for n ≥ 4 and thus, we deduce that both problems are equivalent for n ≥ 4 in CORDA provided the robots share the same chirality. The possible equivalence for n = 3 remains an open problem in CORDA.

Two mobile agents starting at different nodes of an unknown network have to meet. This task is known in the literature as rendezvous. Each agent has a different label which is a positive integer known to it, but unknown to the other agent. Agents move in an asynchronous way: the speed of agents may vary and is controlled by an adversary. The cost of a rendezvous algorithm is the total number of edge traversals by both agents until their meeting. The only previous deterministic algorithm solving this problem has cost exponential in the size of the graph and in the larger label. In this paper we present a deterministic rendezvous algorithm with cost polynomial in the size of the graph and in the length of the smaller label. Hence we decrease the cost exponentially in the size of the graph and doubly exponentially in the labels of agents.As an application of our rendezvous algorithm we solve several fundamental problems involving teams of unknown size larger than 1 of labeled agents moving asynchronously in unknown networks. Among them are the following problems: team size, in which every agent has to find the total number of agents, leader election, in which all agents have to output the label of a single agent, perfect renaming in which all agents have to adopt new different labels from the set {1, . . . , k}, where k is the number of agents, and gossiping, in which each agent has initially a piece of information (value) and all agents have to output all the values. Using our rendezvous algorithm we solve all these problems at cost polynomial in the size of the graph and in the smallest length of all labels of participating agents.

The contribution is twofold. We first show the validity of the conjecture of Défago and Konagaya in [DK02], i.e., there exists no deterministic oblivious algorithm solving the Uniform Transformation Problem for any number of robots * . Next, a protocol which solves deterministically the Circle Formation Problem in finite time for any number n of weak robots-n / ∈ {4, 6, 8}-is proposed. The robots are assumed to be uniform, anonymous, oblivious, and they share no kind of coordinate system nor common sense of direction.

This paper investigates an open problem introduced in [14]. Two or more mobile agents start from different nodes of a network and have to accomplish the task of gathering which consists in getting all together at the same node at the same time. An adversary chooses the initial nodes of the agents and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label but does not know the labels of the other agents or their positions relative to its own. Agents move in synchronous rounds and can communicate with each other only when located at the same node. Up to f of the agents are Byzantine. A Byzantine agent can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. What is the minimum number M of good agents that guarantees deterministic gathering of all of them, with termination?We provide exact answers to this open problem by considering the case when the agents initially know the size of the network and the case when they do not. In the former case, we prove M = f + 1 while in the latter, we prove M = f + 2. More precisely, for networks of known size, we design a deterministic algorithm gathering all good agents in any network provided that the number of good agents is at least f + 1. For networks of unknown size, we also design a deterministic algorithm ensuring the gathering of all good agents in any network but provided that the number of good agents is at least f + 2. Both of our algorithms are optimal in terms of required number of good agents, as each of them perfectly matches the respective lower bound on M shown in [14], which is of f + 1 when the size of the network is known and of f + 2 when it is unknown.Perhaps surprisingly, our results highlight an interesting feature when put in perspective with known results concerning a relaxed variant of this problem in which the Byzantine agents cannot change their initial labels. Indeed under this variant M = 1 for networks of known size and M = f + 2 for networks of unknown size. Following this perspective, it turns out that when the size of the network is known, the ability for the Byzantine agents to change their labels significantly impacts the value of M. However, the relevance for M of such an ability completely disappears in the most general case where the size of the network is unknown, as M = f + 2 regardless of whether Byzantine agents can change their labels or not.

In this paper, we investigate the scatter problem, which is defined as follows: Given a set of n robots, regardless of the initial position of the robots on the plane, eventually, no two robots are located at the same position forever. We show that this problem cannot be deterministically solved. Next, we propose a randomized algorithm. The proposed solution is trivially self-stabilizing. We then show how to design a self-stabilizing version of any deterministic solution for the Pattern Formation and the Gathering problems for any number n > 2 of robots.

A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node. Agents are anonymous (identical), execute the same deterministic algorithm and move in synchronous rounds along links of the network. An initial configuration of agents is called gatherable if there exists a deterministic algorithm (even dedicated to this particular configuration) that achieves meeting of all agents in one node. Which configurations are gatherable and how to gather all of them deterministically by the same algorithm?We give a complete solution of this gathering problem in arbitrary networks. We characterize all gatherable configurations and give two universal deterministic gathering algorithms, i.e., algorithms that gather all gatherable configurations. The first algorithm works under the assumption that an upper bound n on the size of the network is known. In this case our algorithm guarantees gathering with detection, i.e., the existence of a round for any gatherable configuration, such that all agents are at the same node and all declare that gathering is accomplished. If no upper bound on the size of the network is known, we show that a universal algorithm for gathering with detection does not exist. Hence, for this harder scenario, we construct a second universal gathering algorithm, which guarantees that, for any gatherable configuration, all agents eventually get to one node and stop, although they cannot tell if gathering is over. The time of the first algorithm is polynomial in the upper bound n on the size of the network, and the time of the second algorithm is polynomial in the (unknown) size itself.Our results have an important consequence for the leader election problem for anonymous agents in arbitrary graphs. Leader election is a fundamental symmetry breaking problem in distributed computing. Its goal is to assign, in some common round, value 1 (leader) to one of the entities and value 0 (nonleader) to all others. For anonymous agents in graphs, leader election turns out to be equivalent to gathering with detection. Hence, as a by-product, we obtain a complete solution of the leader election problem for anonymous agents in arbitrary graphs.