Jérémie Chalopin

Constructing a map of an anonymous graph: applications of universal sequences

We study the problem of mapping an unknown environment represented as an unlabelled undirected graph. A robot (or automaton) starting at a single vertex of the graph G has to traverse the graph and return to its starting point building a map of the graph in the process. We are interested in the cost of achieving this task (whenever possible) in terms of the number of edge traversal made by the robot. Another optimization criteria is to minimize the amount of information that the robot has to carry when moving from node to node in the graph. We present efficient algorithms for solving map construction using a robot that is not allowed to mark any vertex of the graph, assuming the knowledge of only an upper bound on the size of the graph. We also give universal algorithms (independent of the size of the graph) for map construction when only the starting location of the robot is marked. Our solutions apply the technique of universal exploration sequences to solve the map construction problem under various constraints. We also show how the solution can be adapted to solve other problems such as the gathering of two identical robots dispersed in an unknown graph.

Network exploration by silent and oblivious robots

In this paper we investigate the basic problem of Exploration of a graph by a group of identical mobile computational entities, called robots, operating autonomously and asynchronously. In particular we are concerned with what graphs can be explored, and how, if the robots do not remember the past and have no explicit means of communication. This model of robots is used when the spatial universe in which the robots operate is continuous (e.g., a curve, a polygonal region, a plane, etc.). The case when the spatial universe is discrete (i.e., a graph) has been also studied but only for the classes of acyclic graphs and of simple cycles. In this paper we consider networks of arbitrary topology modeled as connected graphs with local orientation (locally distinct edge labels). We concentrate on class Hk of asymmetric configurations with k robots. Our results indicate that the explorability of graphs in this class depends on the number k of robots participating in the exploration. In particular, exploration is impossible for k 4 all networks in Hk can be explored by presenting a general algorithm. The determination of which networks can be explored when k > 4 is even, is still open but can be reduced to the existence of a gathering algorithm for Hk.

Every planar graph is the intersection graph of segments in the plane: extended abstract

Given a set S of segments in the plane, the intersection graph of S is the graph with vertex set S in which two vertices are adjacent if and only if the corresponding two segments intersect. We prove a conjecture of Scheinerman (PhD Thesis, Princeton University, 1984) that every planar graph is the intersection graph of some segments in the plane.

Mobile agent algorithms versus message passing algorithms

In this paper, we are interested in the computational power of a mobile agent system and, more particularly, in the comparison with a message passing system. First we give formal definitions. Then we explain how a mobile agent algorithm can be simulated by a message passing algorithm.We also prove that any message passing algorithm can be implemented by a mobile agent algorithm. As a consequence of this result, known characterisations of solvable tasks by message passing algorithms can be translated into characterisations of solvable tasks by mobile agent algorithms. We illustrate this result with the election problem.

A bridge between the asynchronous message passing model and local computations in graphs

A distributed system is a collection of processes that can interact. Three major process interaction models in distributed systems have principally been considered: – the message passing model, – the shared memory model, – the local computation model. In each model the processes are represented by vertices of a graph and the interactions are represented by edges. In the message passing model and the shared memory model, processes interact by communication primitives: messages can be sent along edges or atomic read/write operations can be performed on registers associated with edges. In the local computation model interactions are defined by labelled graph rewriting rules; supports of rules are edges or stars. These models (and their sub-models) reflect different system architectures, different levels of synchronization and different levels of abstraction. Understanding the power of various models, the role of structural network properties and the role of the initial knowledge enhances our understanding of basic distributed algorithms. This is done with some typical problems in distributed computing: election, naming, spanning tree construction, termination detection, network topology recognition, consensus, mutual exclusion. Furthermore, solutions to these problems constitute primitive building blocks for many other distributed algorithms. A survey may be found in [FR03], this survey presents some links with several parameters of the models including synchrony, communication media and randomization. An important goal in the study of these models is to understand some relationships between them. This paper is a contribution to this goal; more precisely we establish a bridge between tools and results presented in [YK96] for the message passing model and tools and results presented in [Ang80, BCG+96, Maz97, CM04, CMZ04, Cha05] for the local computation model.

Election, Naming and Cellular Edge Local Computations

We examine the power and limitations of the weakest vertex relabelling system which allows to change a label of a vertex in function of its own label and of the label of one of its neighbours. We characterise the graphs for which two important distributed algorithmic problems are solvable in this model: naming and election.

Election and Local Computations on Edges

The point of departure and the motivation for this paper are the results of Angluin [1] which has introduced a tool to analyze the election algorithm: the coverings, Yamashita and Kameda [21] and Mazurkiewicz [15] which have obtained characterizations of graphs in which election is possible under two different models of distributed computations. Our aim is twofold. First it is to obtain characterizations of graphs in which election is possible under intermediate models between the models of Yamashita-Kameda and of Mazurkiewicz. Our second aim is to understand the implications of the models for the borderline between positive and negative results for distributed computations. In this work, characterizations are obtained under three different models.

Local computations on closed unlabelled edges: the election problem and the naming problem

The different local computations mechanisms are very useful for delimiting the borderline between positive and negative results in distributed computations. Indeed, they enable to study the importance of the synchronization level and to understand how important is the initial knowledge. A high level of synchronization involved in one atomic computation step makes a model powerful but reduces the degree of parallelism. Charron-Bost et al. [1] study the difference between synchronous and asynchronous message passing models. The model studied in this paper involves more synchronization than the message passing model: an elementary computation step modifies the states of two neighbours in the network, depending only on their current states. The information the processors initially have can be global information about the network, such as the size, the diameter or the topology of the network. The initial knowledge can also be local: each node can initially know its own degree for example. Another example of local knowledge is the existence of a port numbering: each processor locally gives numbers to its incident edges and in this way, it can consistently distinguish its neighbours. In Angluins model [2], it is assumed that a port numbering exists, whereas it is not the case in our model. In fact, we obtain a model with a strictly lower power of computation by relaxing the hypothesis on the existence of a port numbering.

Tight bounds for scattered black hole search in a ring

We study the problem of locating a particularly dangerous node, the so-called black hole in a synchronous anonymous ring network with mobile agents. A black hole destroys all mobile agents visiting that node without leaving any trace. Unlike most previous research on the black hole search problem which employed a colocated team of agents, we consider the more challenging scenario when the agents are identical and initially scattered within the network. Moreover, we solve the problem with agents that have constant-sized memory and carry a constant number of identical tokens, which can be placed at nodes of the network. In contrast, the only known solutions for the case of scattered agents searching for a black hole, use stronger models where the agents have non-constant memory, can write messages in whiteboards located at nodes or are allowed to mark both the edges and nodes of the network with tokens. We are interested in the minimum resources (number of agents and tokens) necessary for locating all links incident to the black hole. In fact, we provide matching lower and upper bounds for the number of agents and the number of tokens required for deterministic solutions to the black hole search problem, in oriented or unoriented rings, using movable or unmovable tokens.

Data Delivery by Energy-Constrained Mobile Agents

We consider mobile agents of limited energy, which have to collaboratively deliver data from specified sources of a network to a central repository. Every move consumes energy that is proportional to the travelled distance. Thus, every agent is limited in the total distance it can travel. We ask whether there is a schedule of agents’ movements that accomplishes the delivery. We provide hardness results, as well as exact, approximation, and resource-augmented algorithms for several variants of the problem. Among others, we show that the decision problem is NP-hard already for a single source, and we present a 2-approximation algorithm for the problem of finding the minimum energy that can be assigned to each agent such that the agents can deliver the data.