In this paper we address the complexity issues of two agreement problems in oblivious robot networks namely gathering and scattering. These abstractions are fundamental coordination problems in cooperative mobile robotics. Moreover, their oblivious characteristics makes them appealing for self-stabilization since they are self-stabilizing with no extra-cost. Given a set of robots with arbitrary initial location and no initial agreement on a global coordinate system, gathering requires that all robots reach the exact same but not predetermined location while scattering aims at scatter robots such that no two robots share the same location. Both deterministic gathering and scattering have been proved impossible under arbitrary schedulers therefore probabilistic solutions have been recently proposed. The contribution of this paper is twofold. First, we propose a detailed complexity analysis of the existent probabilistic gathering algorithms in both fault-free and fault-prone environments. We consider both crash and byzantine-prone environments. Moreover, using Markov chains tools and additional assumptions on the environment we prove that the gathering convergence time can be reduced from O(n^2) (the best known tight bound) to O(nln(n)). Additionally, we prove that in crash-prone environments gathering is achieved in O(nln(n)+2f). Second, using the same technique we prove that the best known scattering strategy converges in fault-free systems is O(n) (which is one to optimal) while in crash-prone environments it needs O(n-f). Finally, we conclude the paper with a discussion related to different strategies to gather oblivious robots.
Consider a system of autonomous mobile robots initially randomly deployed on the nodes of an anonymous finite grid. A gathering algorithm is a sequence of moves to be executed independently by each robot so that all robots meet at a single node after finite time. The robots operate in Look-Compute-Move cycles. In each cycle, a robot takes a snapshot of the current configuration of the grid in terms of occupied nodes (\emph{Look}), then based on the perceived configuration, decides whether to stay put or to move to an adjacent node (\emph{Compute}), and in the later case makes an instantaneous …
Conservatoire national des arts et métiers（法国国立工艺学院）；Collège de France（法兰西公学院）；Sorbonne Universités, UPMC Univ Paris 06（索邦大学，巴黎第六大学（UPMC））；Institut Universitaire de France（法国大学研究院）；École Nat. Sup. d’Informatique pour l’Industrie et l’Entreprise (ENSIIE)（法国国立工业与企业信息学高等学院（ENSIIE））；Université Paris-Sud（巴黎第十一大学）
We present a new algorithm for the problem of universal gathering mobile oblivious robots (that is, starting from any initial configuration that is not bivalent, using any number of robots, the robots reach in a finite number of steps the same position, not known beforehand) without relying on a common chirality. We give very strong guaranties on the correctness of our algorithm by proving formally that it is correct, using the COQ proof assistant. To our knowledge, this is the first certified positive (and constructive) result in the context of oblivious mobile robots. It demonstrates both th…
Heinz Nixdorf Institute（海因茨·尼克斯多夫研究所）；University of Paderborn（帕德博恩大学）
We consider a swarm of $n$ autonomous mobile robots, distributed on a 2-dimensional grid. A basic task for such a swarm is the gathering process: All robots have to gather at one (not predefined) place. A common local model for extremely simple robots is the following: The robots do not have a common compass, only have a constant viewing radius, are autonomous and indistinguishable, can move at most a constant distance in each step, cannot communicate, are oblivious and do not have flags or states. The only gathering algorithm under this robot model, with known runtime bounds, needs $\mathcal{…
Cédric – Conservatoire national des arts et métiers（法国国立工艺学院）；Collège de France（法兰西公学院）；UPMC Sorbonne Universités（巴黎第六大学索邦大学）；LIP6-CNRS 7606（法国国家科学研究中心LIP6实验室（7606号混合研究单位））；Institut Universitaire de France（法国大学研究院）；École Nat. Sup. d’Informatique pour l’Industrie et l’Entreprise (ENSIIE)（法国国立工业企业高等计算机学院（ENSIIE））；LRI, CNRS UMR 8623（法国国家科学研究中心LRI实验室（8623号混合研究单位））；Université Paris-Sud（巴黎南大学）；Université Paris-Saclay（巴黎萨克雷大学）
We present a unified formal framework for expressing mobile robots models, protocols, and proofs, and devise a protocol design/proof methodology dedicated to mobile robots that takes advantage of this formal framework. As a case study, we present the first formally certified protocol for oblivious mobile robots evolving in a two-dimensional Euclidean space. In more details, we provide a new algorithm for the problem of universal gathering mobile oblivious robots (that is, starting from any initial configuration that is not bivalent, using any number of robots, the robots reach in a finite numb…
Nagoya Institute of Technology（名古屋工业大学）；Japan Advanced Institute of Science and Technology（日本北陆先端科学技术大学院大学）；Kyushu University（九州大学）
Anonymous mobile robots are often classified into synchronous, semi-synchronous and asynchronous robots when discussing the pattern formation problem. For semi-synchronous robots, all patterns formable with memory are also formable without memory, with the single exception of forming a point (i.e., the gathering) by two robots. However, the gathering problem for two semi-synchronous robots without memory is trivially solvable when their local coordinate systems are consistent, and the impossibility proof essentially uses the inconsistencies in their coordinate systems. Motivated by this, this …
This paper revisits the widely researched \textit{gathering} problem for two robots in a scenario which allows randomization in the asynchronous scheduling model. The scheduler is considered to be the adversary which determines the activation schedule of the robots. The adversary comes in two flavors, namely, oblivious and adaptive, based on the knowledge of the outcome of random bits. The robots follow \textit{wait-look-compute-move} cycle. In this paper, we classify the problems based on the capability of the adversary to control the parameters such as wait time, computation delay and the sp…
We consider the gathering problem for asynchronous and oblivious robots that cannot communicate explicitly with each other, but are endowed with visibility sensors that allow them to see the positions of the other robots. Most of the investigations on the gathering problem on the discrete universe are done on ring shaped networks due to the number of symmetric configuration. We extend in this paper the study of the gathering problem on torus shaped networks assuming robots endowed with local weak multiplicity detection. That is, robots cannot make the difference between nodes occupied by only …
Gathering is a fundamental coordination problem in cooperative mobile robotics. In short, given a set of robots with arbitrary initial locations and no initial agreement on a global coordinate system, gathering requires that all robots, following their algorithm, reach the exact same but not predetermined location. Gathering is particularly challenging in networks where robots are oblivious (i.e., stateless) and direct communication is replaced by observations on their respective locations. Interestingly any algorithm that solves gathering with oblivious robots is inherently self-stabilizing i…
We propose a gathering protocol for an even number of robots in a ring-shaped network that allows symmetric but not periodic configurations as initial configurations, yet uses only local weak multiplicity detection. Robots are assumed to be anonymous and oblivious, and the execution model is the non- atomic CORDA model with asynchronous fair scheduling. In our scheme, the number of robots k must be greater than 8, the number of nodes n on a network must be odd and greater than k+3. The running time of our protocol is O(n2) asynchronous rounds.
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE（法国国家信息与自动化研究所）；VERIMAG UMR 5104（VERIMAG实验室（法国国家科研中心联合研究单位5104））；Université Joseph Fourier（约瑟夫·傅里叶大学）；INRIA（法国国家信息与自动化研究所）；LIP UMR 5668（LIP实验室（法国国家科研中心联合研究单位5668））；Université de Lyon（里昂大学）；ENS Lyon（里昂高等师范学院）；Université Pierre et Marie Curie - Paris 6（巴黎第六大学（皮埃尔和玛丽·居里大学））；LIP6（LIP6实验室）
We consider a team of $k$ identical, oblivious, asynchronous mobile robots that are able to sense (\emph{i.e.}, view) their environment, yet are unable to communicate, and evolve on a constrained path. Previous results in this weak scenario show that initial symmetry yields high lower bounds when problems are to be solved by \emph{deterministic} robots. In this paper, we initiate research on probabilistic bounds and solutions in this context, and focus on the \emph{exploration} problem of anonymous unoriented rings of any size. It is known that $Θ(\log n)$ robots are necessary and sufficient t…
Nagoya Institute of Technology（名古屋工业大学）；Université Pierre et Marie Curie - Paris 6（巴黎第六大学（皮埃尔和玛丽·居里大学））；CNRS（法国国家科学研究中心）
This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show s…
We consider a swarm of mobile robots evolving in a bidimensional Euclidean space. We study a variant of the crash-tolerant gathering problem: if no robot crashes, robots have to meet at the same arbitrary location, not known beforehand, in finite time; if one or several robots crash at the same location, the remaining correct robots gather at the crash location to rescue them. Motivated by impossibility results in the semi-synchronous setting, we present the first solution to the problem for the fully synchronous setting that operates in the vanilla Look-Compute-Move model with no additional h…
Consider a set of $n$ mobile entities, called robots, located and operating on a continuous circle, i.e., all robots are initially in distinct locations on a circle. The \textit{gathering} problem asks to design a distributed algorithm that allows the robots to assemble at a point on the circle. Robots are anonymous, identical, and homogeneous. Robots operate in a deterministic Look-Compute-Move cycle within the circular path. Robots agree on the clockwise direction. The robot's movement is rigid and they have limited visibility $π$, i.e., each robot can only see the points of the circle which…
We consider a swarm of $n$ robots in \mathbb{R}^d. The robots are oblivious, disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task gathering requires that all robots reach the same, not predefined position. In the related near-gathering task, they must reach distinct positions such that every robot sees the entire swarm. In the considered setting, gathering can be solved in $\mathcal{O}(n + Δ^2)$ synchronous rounds both in two and three dimensions, where $Δ$ denotes the initial maximal distance …
This work deals with the problem of gathering $n$ oblivious mobile entities, called robots, at a point (not known beforehand) placed on an infinite triangular grid. The robots are considered to be myopic, i.e., robots have limited visibility. Earlier works of gathering mostly considered the robots either on a plane or on a circle or on a rectangular grid under both full and limited visibility. In the triangular grid, there are two works to the best of our knowledge. The first one is by Cicerone et al. on arbitrary pattern formation where full visibility is considered. The other one by Shibata …
We consider a set of k autonomous robots that are endowed with visibility sensors (but that are otherwise unable to communicate) and motion actuators. Those robots must collaborate to reach a sin- gle vertex that is unknown beforehand, and to remain there hereafter. Previous works on gathering in ring-shaped networks suggest that there exists a tradeoff between the size of the set of potential initial configurations, and the power of the sensing capabilities of the robots (i.e. the larger the initial configuration set, the most powerful the sensor needs to be). We prove that there is no such t…
This work focuses on the following question related to the Gathering problem of $n$ autonomous, mobile robots in the Euclidean plane: Is it possible to solve Gathering of robots that do not agree on any axis of their coordinate systems (disoriented robots) and see other robots only up to a constant distance (limited visibility) in $o(n^2)$ fully synchronous rounds? The best known algorithm that solves Gathering of disoriented robots with limited visibility assuming oblivious robots needs $Θ(n^2)$ rounds [SPAA'11]. The lower bound for this algorithm even holds in a simplified closed chain model…
Aix-Marseille University（艾克斯-马赛大学）；University of Ottawa（渥太华大学）；Carleton University（卡尔顿大学）；JAIST（北陆先端科学技术大学院大学）
In this paper we investigate the computational power of a set of mobile robots with limited visibility. At each iteration, a robot takes a snapshot of its surroundings, uses the snapshot to compute a destination point, and it moves toward its destination. Each robot is punctiform and memoryless, it operates in $\mathbb{R}^m$, it has a local reference system independent of the other robots' ones, and is activated asynchronously by an adversarial scheduler. Moreover, robots are non-rigid, in that they may be stopped by the scheduler at each move before reaching their destination (but are guarant…
Université Pierre et Marie Curie - Paris 6（巴黎第六大学（皮埃尔和玛丽·居里大学））；Ben Gurion University of the Negev（内盖夫本·古里安大学）
This paper introduces the \emph{RoboCast} communication abstraction. The RoboCast allows a swarm of non oblivious, anonymous robots that are only endowed with visibility sensors and do not share a common coordinate system, to asynchronously exchange information. We propose a generic framework that covers a large class of asynchronous communication algorithms and show how our framework can be used to implement fundamental building blocks in robot networks such as gathering or stigmergy. In more details, we propose a RoboCast algorithm that allows robots to broadcast their local coordinate syste…
Università di Pisa（比萨大学）；University of Ottawa（渥太华大学）
In this paper we study the Near-Gathering problem for a finite set of dimensionless, deterministic, asynchronous, anonymous, oblivious and autonomous mobile robots with limited visibility moving in the Euclidean plane in Look-Compute-Move (LCM) cycles. In this problem, the robots have to get close enough to each other, so that every robot can see all the others, without touching (i.e., colliding with) any other robot. The importance of solving the Near-Gathering problem is that it makes it possible to overcome the restriction of having robots with limited visibility. Hence it allows to exploit…
An oblivious mobile robot is a stateless computational entity located in a spatial universe, capable of moving in that universe. When activated, the robot observes the universe and the location of the other robots, chooses a destination, and moves there. The computation of the destination is made by executing an algorithm, the same for all robots, whose sole input is the current observation. No memory of all these actions is retained after the move. When the universe is a graph, distributed computations by oblivious mobile robots have been intensively studied focusing on the conditions for fea…
The design of distributed gathering and convergence algorithms for tiny robots has recently received much attention. In particular, it has been shown that convergence problems can even be solved for very weak, \emph{oblivious} robots: robots which cannot maintain state from one round to the next. The oblivious robot model is hence attractive from a self-stabilization perspective, where state is subject to adversarial manipulation. However, to the best of our knowledge, all existing robot convergence protocols rely on the assumption that robots, despite being "weak", can measure distances. We i…
We propose a new probabilistic pattern formation algorithm for oblivious mobile robots that operates inthe ASYNC model. Unlike previous work, our algorithm makes no assumptions about the local coordinatesystems of robots (the robots do not share a common "North" nor a common "Right"), yet it preserves theability from any initial configuration that contains at least 5 robots to form any general pattern (and not justpatterns that satisfy symmetricity predicates). Our proposal also gets rid of the previous assumption (in thesame model) that robots do not pause while moving (so, our robots really …
VERIMAG（VERIMAG实验室）；Université Joseph Fourier（约瑟夫·傅里叶大学）；MIS（MIS实验室）；Université de Picardie Jules Verne（皮卡第儒勒·凡尔纳大学）；LIP6（LIP6实验室）；UPMC Sorbonne Universités（巴黎第六大学索邦大学）
We consider a team of {\em autonomous weak robots} that are endowed with visibility sensors and motion actuators. Autonomous means that the team cannot rely on any kind of central coordination mechanism or scheduler. By weak we mean that the robots are devoid of (1) any (observable) IDs allowing to differentiate them (anonymous), (2) means of communication allowing them to communicate directly, and (3) any way to remember any previous observation nor computation performed in any previous step (oblivious). Robots asynchronously operate in cycles of three phases: Look, Compute, and Move. Further…
In this paper, we consider the problem of scattering a swarm of mobile oblivious robots in a continuous space. We consider the fully asynchronous setting where robots may base their computation on past observations, or may be observed by other robots while moving. It turns out that asynchronous scattering is solvable in the most general case when both vision (the ability to see others robots positions) and weak local multiplicity detection are available. In the case of a bidimensional Euclidean space, ASYNC scattering is also solvable with blind robots if moves are rigid. Our approach is const…
