Yoann Dieudonné

Circle formation of weak mobile robots

We consider distributed systems made of <i>weak mobile</i> robots, that is, mobile devices, equipped with sensors, that are <i>anonymous</i>, <i>autonomous</i>, <i>disoriented</i>, and <i>oblivious</i>. The <i>Circle Formation Problem</i> (CFP) consists of the design of a protocol insuring that, starting from an initial arbitrary configuration where no two robots are at the same position, all the robots eventually form a <i>regular n-gon</i>—the robots take place on the circumference of a circle <i>C</i> with equal spacing between any two adjacent robots on <i>C</i>. CFP is known to be unsolvable by arranging the robots evenly along the circumference of a circle <i>C</i> without leaving <i>C</i>—that is, starting from a configuration where the robots are on the boundary of <i>C</i>. We circumvent this impossibility result by designing a scheme based on <i>concentric circles</i>. This is the first scheme that deterministically solves CFP. We present our method with two different implementations working in the semi-synchronous system (SSM) for any number <i>n</i> ≥ 5 of robots.

Gathering Despite Mischief

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 termination? 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 f+2. More precisely, we show a deterministic polynomial algorithm that gathers all good agents in an arbitrary network, provided that there are at least f+2 of them. We also provide a matching lower bound: we prove that if the number of good agents is at most f+1, then they are not able to gather deterministically with termination in some networks. For strongly Byzantine agents, we give a lower bound of f+1, 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 2f+1 good agents when the size of the network is known and for at least 4f+2 good agents when it is unknown.

How to meet asynchronously at polynomial cost

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 problem versus pattern formation problem

Leader election and arbitrary pattern formation are fundamental 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 as 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 elect a leader for n ≥ 3 robots if it is possible to form any pattern 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.

Circle formation of weak robots and Lyndon words

A Lyndon word is a non-empty word strictly smaller in the lexicographic order than any of its suffixes, except itself and the empty word. In this paper, we show how Lyndon words can be used in the distributed control of a set of n weak mobile robots. By weak, we mean that the robots are anonymous, memoryless, without any common sense of direction, and unable to communicate in an other way than observation. An efficient and simple deterministic protocol to form a regular n-gon is presented and proven for n prime.

Self-stabilizing gathering with strong multiplicity detection

In this paper, we investigate the possibility to deterministically solve the gathering problem starting from an arbitrary configuration with weak robots, i.e., anonymous, autonomous, disoriented, oblivious, and devoid of means of communication. By starting from an arbitrary configuration, we mean that robots are not required to be located at distinct positions in the initial configuration. We introduce strong multiplicity detection as the ability for the robots to detect the exact number of robots located at a given position. We show that with strong multiplicity detection, there exists a deterministic algorithm solving the gathering problem starting from an arbitrary configuration for n robots if, and only if, n is odd.

Deterministic Robot-Network Localization is Hard

This paper provides a complexity study of the deterministic localization problem in robot networks using local and relative observations only. This is an important issue in collective and cooperative robotics where global positioning systems (GPS) are not available, and the basic premise is the localization ability of the group. We prove that given a set of relative observations made by the robots, the unique unambiguous pose estimation of the robot network in a deterministic way is an NP-hard problem. This means that no polynomial-time algorithm can deterministically solve the unique pose estimation problem based on relative observations unless P=NP. The consequence is that no guarantee can be provided, in a polynomial time, that the possibly estimated poses of the robots will correspond to the effective (actual) ones. The proof is based on complexity theory where we build appropriate polynomial-time reductions interrelating the multirobot localization problem to a well-known NP-complete problem (the partition problem). This NP -hardness result opens questions and perspectives for research into approximations to overcome its intractability.

Anonymous meeting in networks

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 a common upper bound

Scatter of Robots

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