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A collection of k mobile robots, initially placed at the origin of the plane, are searching for a stationary target. Each robot r i has a unit visibility range and can move no faster than its maximal speed v i , for i = 1 , 2 , . . . , k . The robots can communicate between themselves. We consider two communication models: wireless, in which a message sent by a robot can reach all other robots immediately, regardless of their positions, and face-to-face (F2F), in which robots can only exchange information when they are meeting. We assume that up to f robots, where f k , may be unreliable. We consider two models of unreliability: (1) crash faults, in which we deal with an absence of some of the robots' capabilities (communication, perception, motion, etc.), and (2) byzantine faults, in which the robots may be malicious in that they may execute a wrong algorithm (e.g., transmitting wrong information). The goal is to minimize the group search time, which is equal to the time of arrival to the target of the last reliable robot. This is expressed as a function of d, the distance from the origin to the target. Our proposed algorithms for crash faults are asymptotically optimal in d in both communication models. For byzantine faults, our algorithm is asymptotically optimal for the wireless model. In the F2F model, we propose two algorithms: the first one has a competitive ratio of 2, while the second algorithm works for k ≥ 2 f + 2 and is optimal when the robots speeds are all equal.

A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most $$D>0$$ from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than $$2\pi$$ whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent’s trajectory. It is well known that without any hint the optimal (worst case) cost is $$\varTheta (D^2)$$ . We show that if all angles given as hints are at most $$\pi$$ , then the cost can be lowered to O(D), which is the optimal complexity. If all angles are at most $$\beta$$ , where $$\beta 0$$ . For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than $$2\pi$$ , then we show that cost complexity $$\varTheta (D^2)$$ cannot be beaten.

International audience; A team of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node and declare that they have all met. Agents have different labels which are positive integers, and move in synchronous rounds along links of the network. The above task is known as gathering and was traditionally considered under the assumption that when some agents are at the same node then they can talk, i.e., exchange currently available information. In this paper we ask the question of whether this ability of talking is needed for gathering. The answer turns out to be no. Our main contribution are two deterministic algorithms that always accomplish gathering in a much weaker model. We only assume that at any time an agent knows how many agents are at the node that it currently occupies but agents do not see the labels of other co-located agents and cannot exchange any information with them. They also do not see other nodes than the current one. Our first algorithm works under the assumption that agents know a priori some upper bound N on the size of the network, and it works in time polynomial in N and in the length of the smallest label. Our second algorithm does not assume any a priori knowledge about the network but its complexity is exponential in the size of the network and in the labels of agents. Its purpose is to show feasibility of gathering under this harsher scenario. As a by-product of our techniques we obtain, in the same weak model, the solution of the fundamental problem of leader election among agents: One agent is elected a leader and all agents learn its identity. As an application of our result we also solve, in the same model, the well-known gossiping problem: if each agent has a message at the beginning, we show how to make all messages known to all agents, even without any a priori knowledge about the network. If agents know an upper bound N on the size of the network then our gossiping algorithm works in time polynomial in N , in the length of the smallest label and in the length of the largest message. This result about gossiping is perhaps our most surprising finding: agents devoid of any transmitting devices can solve the most general information exchange problem, as long as they can count the number of agents present at visited nodes.

International audience; Two mobile agents represented by points freely moving in the plane and starting at two different positions, have to meet. The meeting, called rendezvous, occurs when agents are at distance at most r of each other and never move after this time, where r is a positive real unknown to them, called the visibility radius. Agents are anonymous and execute the same deterministic algorithm. Each agent has a set of private attributes, some or all of which can differ between agents. These attributes are: the initial position of the agent, its system of coordinates (orientation and chirality), the rate of its clock, its speed when it moves, and the time of its wake-up. If all attributes (except the initial positions) are identical and agents start at distance larger than r then they can never meet, as the distance between them can never change. However, differences between attributes make it sometimes possible to break the symmetry and accomplish rendezvous. Such instances of the rendezvous problem (formalized as lists of attributes), are called feasible.Our contribution is three-fold. We first give an exact characterization of feasible instances. Thus it is natural to ask whether there exists a single algorithm that guarantees rendezvous for all these instances. We give a strong negative answer to this question: we show two sets S1 and S2 of feasible instances such that none of them admits a single rendezvous algorithm valid for all instances of the set. On the other hand, we construct a single algorithm that guarantees rendezvous for all feasible instances outside of sets S1 and S2. We observe that these exception sets S1 and S2 are geometrically very small, compared to the set of all feasible instances: they are included in low-dimension subspaces of the latter. Thus, our rendezvous algorithm handling all feasible instances other than these small sets of exceptions can be justly called almost universal.

It is well-known inside the Formal Concept Analysis (FCA) community that a concept lattice could have an exponential size with respect to the input data. Hence, the size of concept lattices is a critical issue in large real-life data sets. In this paper, we propose to investigate congruence relations as a tool to get meaningful parts of the whole lattice or its implication basis. This paper presents two main theoretical contributions, namely two context (or lattice) decompositions based on congruence relations and new results about implication computation after decomposition.

International audience; We consider the task of rendezvous in networks modeled as undirected graphs. Two mobile agents with different labels, starting at different nodes of an anonymous graph, have to meet. This task has been considered in the literature under two alternative scenarios: weak and strong. Under the weak scenario, agents may meet either at a node or inside an edge. Under the strong scenario, they have to meet at a node, and they do not even notice meetings inside an edge. Rendezvous algorithms under the strong scenario are known for synchronous agents. For asynchronous agents, rendezvous under the strong scenario is impossible even in the two-node graph, and hence only algorithms under the weak scenario were constructed. In this paper we show that rendezvous under the strong scenario is possible for agents with asynchrony restricted in the following way: agents have the same measure of time but the adversary can impose, for each agent and each edge, the speed of traversing this edge by this agent. The speeds may be different for different edges and different agents but all traversals of a given edge by a given agent have to be at the same imposed speed. We construct a deterministic rendezvous algorithm for such agents, working in time polynomial in the size of the graph, in the length of the smaller label, and in the largest edge traversal time.

International audience; Given a boolean predicate on labeled networks (e.g., the network is acyclic, or the network is properly colored, etc.), deciding in a distributed manner whether a given labeled network satisfies that predicate typically consists, in the standard setting, of every node inspecting its close neighborhood, and outputting a boolean verdict, such that the network satisfies the predicate if and only if all nodes output true. In this paper, we investigate a more general notion of distributed decision in which every node is allowed to output a constant number $b\geq 1$ of bits, which are gathered by a central authority emitting a global boolean verdict based on these outputs, such that the network satisfies the predicate if and only if this global verdict equals true. We analyze the power and limitations of this extended notion of distributed decision.

International audience; A pair of agents (robots) are moving in a graph with the goal of meeting at the same node or while traversing the same edge. An asynchronous adversary knows the prescribed walks of the two agents and is in complete control of the speed of each agent during its walk. We provide a complete characterization of pairs of walks that enforce rendezvous against an asynchronous adversary after traversing a given number of edges. The characterization is efficient in that it can be checked in polynomial time. We argue that the certificate of rendezvous enforcement that is produced by the checking algorithm contains a wealth of information on why rendezvous is enforced.

Two mobile robots are initially placed at the same point on an infinite line. Each robot may move on the line in either direction not exceeding its maximal speed. The robots need to find a stationary target placed at an unknown location on the line. The search is completed when both robots arrive at the target point. The target is discovered at the moment when either robot arrives at its position. The robot knowing the placement of the target may communicate it to the other robot. We look for the algorithm with the shortest possible search time (i.e. the worst-case time at which both robots meet at the target) measured as a function of the target distance from the origin (i.e. the time required to travel directly from the starting point to the target at unit velocity). We consider two standard models of communication between the robots, namely wireless communication and communication by meeting. In the case of communication by meeting, a robot learns about the target while sharing the same location with a robot possessing this knowledge. We propose here an optimal search strategy for two robots including the respective lower bound argument, for the full spectrum of their maximal speeds. This extends the main result of Chrobak et al. (in: Italiano, Margaria-Steffen, Pokorný, Quisquater, Wattenhofer (eds) Current trends in theory and practice of computer science, SOFSEM, 2015) referring to the exact complexity of the problem for the case when the speed of the slower robot is at least one third of the faster one. In the wireless communication model, a message sent by one robot is instantly received by the other robot, regardless of their current positions on the line. For this model, we design a strategy which is optimal whenever the faster robot is at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{17}+4\approx 8.123$$\end{document}17+4≈8.123 times faster than the slower one. We also prove that otherwise the wireless communication offers no advantage over communication by meeting.

A graph environment must be explored by a collection of mobile robots. Some of the robots, a priori unknown, may turn out to be unreliable. The graph is weighted and each node is assigned a deadline. The exploration is successful if each node of the graph is visited before its deadline by a reliable robot. The edge weight corresponds to the time needed by a robot to traverse the edge. Given the number of robots which may crash, is it possible to design an algorithm, which will always guarantee the exploration, independently of the choice of the subset of unreliable robots by the adversary? We find the optimal time, during which the graph may be explored. Our approach permits to find the maximal number of robots, which may turn out to be unreliable, and the graph is still guaranteed to be explored.

We establish a classification of decision problems that are to be solved by mobile agents operating in unlabeled graphs, using a deterministic protocol. The classification is with respect to the ability of a team of agents to solve the problem, possibly with the aid of additional information. In particular, our focus is on studying differences between the decidability of a decision problem by agents and its verifiability when a certificate for a positive answer is provided to the agents. We show that the class MAV of mobile agents verifiable problems is much wider than the class MAD of mobile agents decidable problems. Our main result shows that there exist natural MAV-complete problems: the most difficult problems in this class, to which all problems in MAV are reducible. Our construction of a MAV-complete problem involves two main ingredients in mobile agents computability: the topology of the quotient graph and the number of operating agents. Beyond the class MAV we show that, for a single agent, three natural oracles yield a strictly increasing chain of relative decidability classes.

Mining association rules, including implications, is an important topic in Knowledge Discovery research area and in Formal Concept Analysis (FCA). In this paper, we present a novel algorithm that computes in a parallel way the canonical direct unit basis of a formal context in FCA. To that end, the algorithm first performs a horizontal split of the initial context into subcontexts and then exploits the notion of minimal dual transversal to merge the canonical direct unit bases generated from subcontexts.

A team consisting of an unknown number of mobile agents starting from different nodes of an unknown network, possibly at different times, have to explore the network: Every node must be visited by at least one agent, and all agents must eventually stop. Agents are anonymous (identical), execute the same deterministic algorithm, and move in synchronous rounds along links of the network. They are silent: They cannot send any messages to other agents or mark visited nodes in any way. In the absence of any additional information, exploration with termination of an arbitrary network in this model, devoid of any means of communication between agents, is impossible. Our aim is to solve the exploration problem by giving to agents very restricted local traffic reports . Specifically, an agent that is at a node v in a given round is provided with three bits of information answering the following questions: Am I alone at v ? Did any agent enter v in this round? Did any agent exit v in this round? We show that this small amount of information permits us to solve the exploration problem in arbitrary networks. More precisely, we give a deterministic terminating exploration algorithm working in arbitrary networks for all initial configurations that are not perfectly symmetric ; that is, in which there are agents with different views of the network. The algorithm works in polynomial time in the (unknown) size of the network. A deterministic terminating exploration algorithm working for all initial configurations in arbitrary networks does not exist.

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 2 f +1 good agents when the size of the network is known and for at least 4 f +2 good agents when it is unknown.

Two mobile agents with range of vision 1 start at arbitrary points in the plane and have to accomplish the task of approach, which consists in getting at distance at most one from each other, i.e., in getting within each other's range of vision. An adversary chooses the initial positions of the agents, their possibly different starting times, and assigns a different positive integer label and a possibly different speed to each of them. Each agent is equipped with a compass showing the cardinal directions, with a measure of length and a clock. Each agent knows its label and speed but not those of the other agent and it does not know the initial position of the other agent relative to its own. Agents do not have any global system of coordinates and they cannot communicate. Our main result is a deterministic algorithm to accomplish the task of approach, working in time polynomial in the unknown initial distance between the agents, in the length of the smaller label and in the inverse of the larger speed. The distance travelled by each agent until approach is polynomial in the first two parameters and does not depend on the third. The problem of approach in the plane reduces to a network problem: that of rendezvous in an infinite grid.

International audience; A set of mobile agents is placed at different nodes of a $n$-node network. The agents synchronously move along the network edges in a {\em collision-free} way, i.e., in no round may two agents occupy the same node. In each round, an agent may choose to stay at its currently occupied node or to move to one of its neighbors. An agent has no knowledge of the number and initial positions of other agents. We are looking for the shortest possible time required to complete the collision-free {\em network exploration}, i.e., to reach a configuration in which each agent is guaranteed to have visited all network nodes and has returned to its starting location. We first consider the scenario when each mobile agent knows the map of the network, as well as its own initial position. We establish a connection between the number of rounds required for collision-free exploration and the degree of the minimum-degree spanning tree of the graph. We provide tight (up to a constant factor) lower and upper bounds on the collision-free exploration time in general graphs, and the exact value of this parameter for trees. For our second scenario, in which the network is unknown to the agents, we propose collision-free exploration strategies running in $O(n^2)$ rounds for tree networks and in $O(n^5\log n)$ rounds for general networks.

Two mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds using a deterministic algorithm. Each agent has a different label, which it can use in the execution of the algorithm, but it does not know the label of the other agent. Agents do not know any bound on the size of the network. In each round an agent decides if it remains idle or if it wants to move to one of the adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault in a given round, it remains in the current node, regardless of its decision. If it planned to move and the fault happened, the agent is aware of it. We consider three scenarios of fault distribution: random (independently in each round and for each agent with constant probability 0 < p < 1), unbounded adver- sarial (the adversary can delay an agent for an arbitrary finite number of consecutive rounds) and bounded adversarial (the adversary can delay an agent for at most c consecutive rounds, where c is unknown to the agents). The quality measure of a rendezvous algorithm is its cost, which is the total number of edge traversals. For random faults, we show an algorithm with cost polynomial in the size n of the network and polylogarithmic in the larger label L, which achieves rendezvous with very high probability in arbitrary networks. By contrast, for unbounded adversarial faults we show that rendezvous is not feasible, even in the class of rings. Under this scenario we give a rendezvous algorithm with cost O(nl), where l is the smaller label, working in arbitrary trees, and we show that ��(l) is the lower bound on rendezvous cost, even for the two-node tree. For bounded adversarial faults, we give a rendezvous algorithm working for arbitrary networks, with cost polynomial in n, and logarithmic in the bound c and in the larger label L.

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 $$N$$N on the size of the network is known to all agents. 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 $$N$$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 (non-leader) 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.

A mobile agent has to explore an unknown network with unlabeled nodes: it must visit all nodes by walking along the links of the network, and eventually stop. If no upper bound on the size of the network is known and nodes of the network cannot be marked, then this exploration task cannot be accomplished for arbitrary networks by a deterministic terminating algorithm. On the other hand, it is feasible, if there is one unmovable token at the starting node of the agent. We investigate the exploration problem in arbitrary networks in the presence of identical unmovable tokens, some of which are Byzantine. A Byzantine token can be visible or invisible to the agent whenever the latter visits the node where the token is located, and visibility is decided by the adversary at each visit of the agent. If no upper bound on the number of tokens is known to the agent, deterministic exploration of all networks is impossible, even if all tokens are fault free. It is also impossible if all tokens are Byzantine, even if their number is known. Our main result is a deterministic exploration algorithm with cost polynomial in the (unknown) size of the network, which works in arbitrary networks, provided that the agent knows some upper bound on the total number of tokens, and that at least one token is fault free.

Two mobile agents, modeled as points starting at different locations of an unknown terrain, have to meet. The terrain is a polygon with polygonal holes. We consider two versions of this rendezvous problem: exact RV, when the points representing the agents have to coincide at some time, and ?-RV, when these points have to get at distance less than ? in the terrain. In any terrain, each agent chooses its trajectory, but the movements of the agent on this trajectory are controlled by an adversary that may, e.g., speed up or slow down the agent. Agents have bounded memory: their computational power is that of finite state machines. Our aim is to compare the feasibility of exact and of ?-RV when agents are anonymous vs. when they are labeled. We show classes of polygonal terrains which distinguish all the studied scenarios from the point of view of feasibility of rendezvous. The features which influence the feasibility of rendezvous include symmetries present in the terrains, boundedness of their diameter, and the number of vertices of polygons in the terrains.

International audience; We consider the problem of exploring an anonymous line by a team of k identical, oblivious, asynchronous deterministic mobile robots that can view the environment but cannot communicate. We completely characterize sizes of teams of robots capable of exploring a n-node line. For k= 5, or k=4 and n is odd. For all values of k for which exploration is possible, we give an exploration algorithm. For all others, we prove an impossibility result.

International audience; In the effort to understand the algorithmic limitations of computing by a swarm of robots, the research has focused on the minimal capabilities that allow a problem to be solved. The weakest of the commonly used models is {\sc Asynch} where the autonomous mobile robots, endowed with visibility sensors (but otherwise unable to communicate), operate in Look-Compute-Move cycles performed asynchronously for each robot. The robots are often assumed (or required to be) oblivious: they keep no memory of observations and computations made in previous cycles. We consider the setting when the robots are dispersed in an anonymous and unlabeled graph, and they must perform the very basic task of {\em exploration}: within finite time every node must be visited by at least one robot and the robots must enter a quiescent state. The complexity measure of a solution is the number of robots used to perform the task. We study the case when the graph is an arbitrary tree and establish some unexpected results. We first prove that, in general, exploration cannot be done efficiently. More precisely we prove that there are $n$-node trees where $\Omega(n)$ robots are necessary; this holds even if the maximum degree is $4$. On the other hand, we show that if the maximum degree is $3$, it is possible to explore with only $O(\frac{\log n} {\log\log n})$ robots. The proof of the result is constructive. We also prove that the size of the team used in our solution is asymptotically {\em optimal}: there are trees of degree $3$, whose exploration requires $\Omega(\frac{\log n}{\log\log n})$ robots. Our final result shows that the difficulty in tree exploration comes in fact from the symmetries of the tree. Indeed, we show that, in order to explore trees that do not have any non-trivial automorphisms, 4 robots are always sufficient and often necessary.

International audience; We study deterministic broadcasting in radio networks in the recently introduced framework of network algorithms with {\em advice}. We concentrate on the problem of trade-offs between the number of bits of information (size of advice) available to nodes and the time in which broadcasting can be accomplished. In particular, we ask what is the minimum number of bits of information that must be available to nodes of the network, in order to broadcast very fast. For networks in which constant time broadcast is possible under complete knowledge of the network we give a tight answer to the above question: $O(n)$ bits of advice are sufficient but $o(n)$ bits are not, in order to achieve constant broadcasting time in all these networks. This is in sharp contrast with geometric radio networks of constant broadcasting time: we show that in these networks a constant number of bits suffices to broadcast in constant time. For arbitrary radio networks we present a broadcasting algorithm whose time is inverse-proportional to the size of advice.

International audience; Blin {\it et al.} (2006) proposed a distributed protocol that enables the smallest number of searchers to clear any unknown asynchronous graph in a decentralized manner. {\it Unknown} means that the searchers are provided no {\it a priori} information about the graph. However, the strategy that is actually performed lacks of an important property, namely the monotonicity. That is, the clear part of the graph may decrease at some steps of the execution of the protocol. Actually, the protocol of Blin {\it et al.} is executed in exponential time. Nisse and Soguet (2007) proved that, in order to ensure the smallest number of searchers to clear any $n$-node graph in a monotone way, it is necessary and sufficient to provide $\Theta(n \log n)$ bits of information to the searchers by putting short labels on the nodes of the graph. This paper deals with the smallest number of searchers that are necessary and sufficient to monotoneously clear any graph in a decentralized manner, when the searchers have no a priori information about the graph. The distributed graph searching problem considers a team of searchers that is aiming at clearing any connected contaminated graph. The clearing of the graph is required to be {\it connected}, i.e., the clear part of the graph must remain permanently connected, and {\it monotone}, i.e., the clear part of the graph only grows. The {\it search number} $\mcs(G)$ of a graph $G$ is the smallest number of searchers necessary to clear $G$ in a monotone connected way in centralized settings. We prove that any distributed protocol aiming at clearing any unknown n-node graph in a monotone connected way, in decentralized settings, has {\it competitive ratio} $\Theta(\frac{n}{\log n})$. That is, we prove that, for any distributed protocol $\cal P$, there exists a constant $c$ such that for any sufficiently large $n$, there exists a $n$-node graph $G$ such that $\cal P$ requires at least $c\frac{n}{\log n}\, \mcs(G)$ searchers to clear $G$. Moreover, we propose a distributed protocol that allows $O(\frac{n}{\log n})\, \mcs(G)$ searchers to clear any unknown asynchronous $n$-node graph $G$ in a monotone connected way.

International audience; A mobile agent (robot), modeled as a finite automaton, has to visit all nodes of a regular graph. How does the memory size of the agent (the number of states of the automaton) influence its exploration capability? In particular, does every increase of the memory size enable an agent to explore more graphs? We give a partial answer to this problem by showing that a strict gain of the exploration power can be obtained by a polynomial increase of the number of states. We also show that, for automata with few states, the increase of memory by even one state results in the capability of exploring more graphs.

Leader election is one of the basic problems in distributed computing. This is a symmetry breaking problem: all nodes of a network must agree on a single node, called the leader. If the nodes of the network have distinct labels, then such an agreement means that all nodes have to output the label of the elected leader. For anonymous networks, the task of leader election is formulated as follows: every node v of the network must output a simple path, which is coded as a sequence of port numbers, such that all these paths end at a common node, the leader. In this paper, we study deterministic leader election in arbitrary anonymous networks. It is well known that deterministic leader election is impossible in some networks, regardless of the allocated amount of time, even if nodes know the map of the network. This is due to possible symmetries in it. However, even in networks in which it is possible to elect a leader knowing the map, the task may be still impossible without any knowledge, regardless of the allocated time. On the other hand, for any network in which leader election is possible knowing the map, there is a minimum time, called the election index, in which this can be done. Informally, the election index of a network is the minimum depth at which views of all nodes are distinct. Our aim is to establish tradeoffs between the allocated time $$\tau $$ and the amount of information that has to be given a priori to the nodes to enable leader election in time $$\tau $$ in all networks for which leader election in this time is at all possible. Following the framework of algorithms with advice, this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire network. The length of this string is called the size of advice. For a given time $$\tau $$ allocated to leader election, we give upper and lower bounds on the minimum size of advice sufficient to perform leader election in time $$\tau $$ . We focus on the two sides of the time spectrum. For the smallest possible time, which is the election index of the network, we show that the minimum size of advice is linear in the size n of the network, up to polylogarithmic factors. On the other hand, we consider large values of time: larger than the diameter D by a summand, respectively, linear, polynomial, and exponential in the election index; for these values, we prove tight bounds on the minimum size of advice, up to multiplicative constants. We also show that constant advice is not sufficient for leader election in all graphs, regardless of the allocated time.

A set of identical, mobile agents is deployed in a weighted network. Each agent has a battery -- a power source allowing it to move along network edges. An agent uses its battery proportionally to the distance traveled. We consider two tasks : convergecast, in which at the beginning, each agent has some initial piece of information, and information of all agents has to be collected by some agent; and broadcast in which information of one specified agent has to be made available to all other agents. In both tasks, the agents exchange the currently possessed information when they meet. The objective of this paper is to investigate what is the minimal value of power, initially available to all agents, so that convergecast or broadcast can be achieved. We study this question in the centralized and the distributed settings. In the centralized setting, there is a central monitor that schedules the moves of all agents. In the distributed setting every agent has to perform an algorithm being unaware of the network. In the centralized setting, we give a linear-time algorithm to compute the optimal battery power and the strategy using it, both for convergecast and for broadcast, when agents are on the line. We also show that finding the optimal battery power for convergecast or for broadcast is NP-hard for the class of trees. On the other hand, we give a polynomial algorithm that finds a 2-approximation for convergecast and a 4-approximation for broadcast, for arbitrary graphs. In the distributed setting, we give a 2-competitive algorithm for convergecast in trees and a 4-competitive algorithm for broadcast in trees. The competitive ratio of 2 is proved to be the best for the problem of convergecast, even if we only consider line networks. Indeed, we show that there is no (2 -- $\epsilon$)-competitive algorithm for convergecast or for broadcast in the class of lines, for any $\epsilon$ \textgreater{} 0.

A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots move perpetually along the domain, represented by all points belonging to the graph edges, without exceeding their maximum speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e., they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolman? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by $${\mathfrak {I}}_k^f (G)$$ the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing $${\mathfrak {I}}_k^f (G)$$ . We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness $${\mathfrak {I}}^f_k(G) = (f+1) |E|{/}k$$ , where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs—a known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs.

International audience; A collection of $n$ anonymous mobile robots is deployed on a unit-perimeter ring or a unit-length line segment. Every robot starts moving at constant speed, and bounces each time it meets any other robot or segment endpoint, changing its walk direction. We study the problem of {\em position discovery}, in which the task of each robot is to detect the presence and the initial positions of all other robots. The robots cannot communicate or perceive information about the environment in any way other than by bouncing. Each robot has a clock allowing it to observe the times of its bounces. The robots have no control on their walks, which are determined by their initial positions and the starting directions. Each robot executes the same \emph{position detection algorithm}, which receives input data in real-time about the times of the bounces, and terminates when the robot is assured about the existence and the positions of all the robots. Some initial configuration of robots are shown to be {\em infeasible} --- no position detection algorithm exists for them. We give complete characterizations of all infeasible initial configurations for both the ring and the segment, and we design optimal position detection algorithms for all feasible configurations. For the case of the ring, we show that all robot configurations in which not all the robots have the same initial direction are feasible. We give a position detection algorithm working for all feasible configurations. The cost of our algorithm depends on the number of robots starting their movement in each direction. If the less frequently used initial direction is given to $k \leq n/2$ robots, the time until completion of the algorithm by the last robot is $\frac{1}{2}\lceil \frac{n}{k} \rceil$. We prove that this time is optimal. By contrast to the case of the ring, for the unit segment we show that the family of infeasible configurations is exactly the set of so-called {\em symmetric configurations}. We give a position detection algorithm which works for all feasible configurations on the segment in time $2$, and this algorithm is also proven to be optimal.

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.

International audience; A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in {\em walking mode} or in {\em searching mode}. It is assumed that each robot's maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers' Problem consists in developing efficient {\em schedules} (algorithms) for the robots which collectively search all the points of the given domain as fast as possible. We consider searching schedules in the following one-dimensional geometric domains: the cycle of a known circumference $L$, the finite straight line segment of a known length $L$, and the semi-infinite line $[0,+\infty)$.We first consider the {\em online} Beachcombers' Problem (i.e.~the scenario when the robots do not know in advance the length of the segment to be searched), where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule $A$ with maximum {\em speed} $S$, defined as $S = \inf_{\ell}{\frac{\ell}{t_A(\ell)}}$, where $t_A(\ell)$ denotes the time when the search of the segment $[0,\ell]$ is completed under $A$. We consider a {\em discrete} and a {\em continuous} version of the problem, depending on whether the infimum is takenover $\ell \in \mathbb{N}^*$ or $\ell \geq 1$. We prove that the $\mathtt{LeapFrog}$ algorithm, which was proposed in [Czyzowicz et al., SIROCCO 2014, LNCS 8576, pp. 23--36 (2014)], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting.For the {\em offline} version of the Beachcombers' Problem (i.e.~the scenario when the robots know in advance the length of the segment to be searched), we consider the \emph{$t$-source} Beachcombers' Problem (i.e.~all robots start from a fixed number $t \geq 1$ of starting positions) on the cycle and on the finite segment. For the \emph{$t$-source} Beachcombers' Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the $2t$-source Beachcombers' Problem on a finite segment. In consequence, by using results from [Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp.~3--21 (2014)], we prove that the \emph{1-source} Beachcombers' Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the \emph{$t$-source} variant of the Beachcombers' Problem on the cycle and on the finite segment, we also derive efficient approximation algorithms.One important contribution of our work is that, in all variants of the offline Beachcombers' Problem that we discuss, we allow the robots to \emph{change direction of movement} and search points of the domain on both sides of their respective starting positions. This represents a significant generalization compared to the model considered in~[Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp. 3--21 (2014)], in which each robot had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of direction do not help the robots achieve optimality.

International audience; We study the problem of mobile robots with distinct visibility ranges patrolling a curve. Assume a set of $k$ mobile robots (patrolmen) $a_1, a_2,..., a_k$ walking along a unit-length curve in any of the two directions, not exceeding their maximal speeds. Every robot $a_i$ has a range of visibility $r_i$, representing the distance from its current position at which the robot can see in each direction along the curve. The goal of the patrolling problem is to find the perpetual movement of the robots minimizing the maximal time when a point of the curve remains unseen by any robot. We give the optimal patrolling algorithms for the case of close curve environment (known as the boundary patrolling problem in the robotics literature) and open curve (fence patrolling), when all robots have the same maximal speed. We briefly discuss the case of distinct speeds, showing that the boundary patrolling problem for robots with distinct visibility ranges is essentially different than the case of point visibility robots. We also give the optimal algorithm for fence patrolling by two robots with distinct speeds and visibility ranges. For the case when the environment in which the robots operate is a general graph, we show that the patrolling problem for robots with distinct visibility ranges is NP-hard, while it is known that the same problem for point-visibility robots has been known to have a polynomial-time solution.

International audience; A mobile robot represented by a point moving in the plane has to explore an unknown flat terrain with impassable obstacles. Both the terrain and the obstacles are modeled as arbitrary polygons. We consider two scenarios: the unlimited vision, when the robot situated at a point p of the terrain explores (sees) all points q of the terrain for which the segment pq belongs to the terrain, and the limited vision, when we require additionally that the distance between p and q is at most 1. All points of the terrain (except obstacles) have to be explored and the performance of an exploration algorithm, called its complexity, is measured by the length of the trajectory of the robot. For unlimited vision we show an exploration algorithm with complexity View the MathML source, where P is the total perimeter of the terrain (including perimeters of obstacles), D is the diameter of the convex hull of the terrain, and k is the number of obstacles. We do not assume knowledge of these parameters. We also prove a matching lower bound showing that the above complexity is optimal, even if the terrain is known to the robot. For limited vision we show exploration algorithms with complexity View the MathML source, where A is the area of the terrain (excluding obstacles). Our algorithms work either for arbitrary terrains (if one of the parameters A or k is known) or for c-fat terrains, where c is any constant (unknown to the robot) and no additional knowledge is assumed. (A terrain T with obstacles is c-fat if R/r⩽c, where R is the radius of the smallest disc containing T and r is the radius of the largest disc contained in T.) We also prove a matching lower bound View the MathML source on the complexity of exploration for limited vision, even if the terrain is known to the robot.

International audience; We consider the problem of exploring an anonymous unoriented ring by a team of k identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimensional plane, but (with one exception) has not been investigated before for networks. Our results imply that, although these weak capabilities of robots render the problem considerably more difficult, ring exploration by a small team of robots is still possible. We first show that, when k and n are not co-prime, the problem is not solvable in general, e.g., if k divides n there are initial placements of the robots for which gathering is impossible. We then prove that the problem is always solvable provided that n and k are co-prime, for k >= 17, by giving an exploration algorithm that always terminates, starting from arbitrary initial configurations. Finally, we consider the minimum number rho(n) of robots that can explore a ring of size n. As a consequence of our positive result we show that rho(n) is O(log n). We additionally prove that Omega(log n) robots are necessary for infinitely many n.

International audience; A set of mobile robots is deployed on a simple curve of finite length, composed of a finite set of vital segments separated by neutral segments. The robots have to patrol the vital segments by perpetually moving on the curve, without exceeding their maximum speed. The quality of patrolling is measured by the idleness, i.e., the longest time period during which any vital point on the curve is not visited by any robot. Given a configuration of vital segments, our goal is to provide algorithms describing the movement of the robots along the curve so as to minimize the idleness. Our main contribution is a proof that the optimal solution to the patrolling problem is attained either by the cyclic strategy, in which all the robots move in one direction around the curve, or by the partition strategy, in which the curve is partitioned into sections which are patrolled separately by individual robots. These two fundamental types of strategies were studied in the past in the robotics community in different theoretical and experimental settings. However, to our knowledge, this is the first theoretical analysis proving optimality in such a general scenario. Throughout the paper we assume that all robots have the same maximum speed. In fact, the claim is known to be invalid when this assumption does not hold, cf. [Czyzowicz et al., Proc. ESA 2011].

International audience; We are dealing with the design of algorithms using few resources, enabling to decide whether or not any given $n$-node graph $G$ belongs to some given graph class $\caC$. Our model borrows from property testing the way the decision is taken, by an \emph{unconstrained interpretation function} applied to the set of outputs produced by individual queries (instead of an interpretation function limited to the conjunction operator as in local distributed decision). It borrows from local distributed decision the fact that \emph{all nodes are involved} in the decision (instead of $o(n)$ nodes as in property testing). The unique, but severe restriction we impose to the nodes is a limitation on the amount of information they are enabled to output: every node is bounded to output a \emph{constant} number of bits. In this paper, we provide separation results between distributed decision and verification classes, and we analyze the size of the certificates enabling to verify distributed languages.

International audience; We consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all n nodes of an input simple, connected, undirected graph in a periodic manner. Graphs are assumed to be anonymous, that is, nodes are unlabeled. While visiting a node, the agent may distinguish between the edges incident to it; for each node v, the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in algorithms for assigning the port numbers together with traversal algorithms for agents using these port numbers to obtain short traversal periods. Periodic graph exploration is unsolvable if the port numbers are set arbitrarily; see Budach (1978) [1]. However, surprisingly small periods can be achieved by carefully assigning the port numbers. Dobrev et al. (2005) [4] described an algorithm for assigning port numbers and an oblivious agent (i.e., an agent with no memory) using it, such that the agent explores any graph with n nodes within the period 10n. When the agent has access to a constant number of memory bits, the optimal length of the period was proved in Gąsieniec et al. (2008) [7] to be no more than 3.75n−2 (using a different assignment of the port numbers and a different traversal algorithm). In this paper, we improve both these bounds. More precisely, we show how to achieve a period length of at most for oblivious agents and a period length of at most 3.5n−2 for agents with constant memory. To obtain our results, we introduce a new, fast graph decomposition technique called a three-layer partition that may also be useful for solving other graph problems in the future. Finally, we present the first non-trivial lower bound, 2.8n−2, on the period length for the oblivious case.

Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We consider deterministic algorithms for this rendezvous task. The main result of this paper is a tight trade-off between the optimal time of completing rendezvous and the size of memory of the agents. For agents with k memory bits, we show that optimal rendezvous time is Θ(n+n2/k) in n-node trees. More precisely, if k ≥ c log n, for some constant c, we design agents accomplishing rendezvous in arbitrary trees of unknown size n in time O(n+n2/k), starting with arbitrary delay. We also show that no pair of agents can accomplish rendezvous in time o(n+n2/k), even in the class of lines of known length and even with simultaneous start. Finally, we prove that at least logarithmic memory is necessary for rendezvous, even for agents starting simultaneously in a n-node line.

A set of identical, mobile agents is deployed in a weighted network. Each agent possesses a battery - a power source allowing the agent to move along network edges. Agents use their batteries proportionally to the distance traveled. At the beginning, each agent has its initial information. Agents exchange the actually possessed information when they meet. The agents collaborate in order to perform an efficient convergecast , where the initial information of all agents must be eventually transmitted to some agent. The objective of this paper is to investigate what is the minimal value of power, initially available to all agents, so that convergecast may be achieved. We study the question in the centralized and the distributed settings. In the distributed setting every agent has to perform an algorithm being unaware of the network. We give a linear-time centralized algorithm solving the problem for line networks. We give a 2-competitive distributed algorithm achieving convergecast for tree networks. The competitive ratio of 2 is proved to be the best possible for this problem, even if we only consider line networks. We show that already for the case of tree networks the centralized problem is strongly NP-complete. We give a 2-approximation centralized algorithm for general graphs.

Two mobile agents (robots) have to meet in an a priori unknown bounded terrain modeled as a polygon, possibly with polygonal obstacles. Agents are modeled as points, and each of them is equipped with a compass. Compasses of agents may be incoherent. Agents construct their routes, but the actual walk of each agent is decided by the adversary: the movement of the agent can be at arbitrary speed, the agent may sometimes stop or go back and forth, as long as the walk of the agent in each segment of its route is continuous, does not leave it and covers all of it. We consider several scenarios, depending on three factors: (1) obstacles in the terrain are present, or not, (2) compasses of both agents agree, or not, (3) agents have or do not have a map of the terrain with their positions marked. The cost of a rendezvous algorithm is the worst-case sum of lengths of the agents' trajectories until their meeting. For each scenario we design a deterministic rendezvous algorithm and analyze its cost. We also prove lower bounds on the cost of any deterministic rendezvous algorithm in each case. For all scenarios these bounds are tight.

International audience; A set of k mobile agents are placed on the boundary of a simply connected planar object represented by a cycle of unit length. Each agent has its own predefined maximal speed, and is capable of moving around this boundary without exceeding its maximal speed. The agents are required to protect the boundary from an intruder which attempts to penetrate to the interior of the object through a point of the boundary, unknown to the agents. The intruder needs some time interval of length τ to accomplish the intrusion. Will the intruder be able to penetrate into the object, or is there an algorithm allowing the agents to move perpetually along the boundary, so that no point of the boundary remains unprotected for a time period τ? Such a problem may be solved by designing an algorithm which defines the motion of agents so as to minimize the idle time I, i.e., the longest time interval during which any fixed boundary point remains unvisited by some agent, with the obvious goal of achieving I

The aim of rendezvous in a graph is meeting of two mobile agents at some node of an unknown anonymous connected graph. In this article, we focus on rendezvous in trees, and, analogously to the efforts that have been made for solving the exploration problem with compact automata, we study the size of memory of mobile agents that permits to solve the rendezvous problem deterministically. We assume that the agents are identical, and move in synchronous rounds. We first show that if the delay between the starting times of the agents is arbitrary , then the lower bound on memory required for rendezvous is Ω (log n ) bits, even for the line of length n . This lower bound meets a previously known upper bound of O (log n ) bits for rendezvous in arbitrary graphs of size at most n . Our main result is a proof that the amount of memory needed for rendezvous with simultaneous start depends essentially on the number ℓ of leaves of the tree, and is exponentially less impacted by the number n of nodes. Indeed, we present two identical agents with O (log ℓ + log log n ) bits of memory that solve the rendezvous problem in all trees with at most n nodes and at most ℓ leaves. Hence, for the class of trees with polylogarithmically many leaves, there is an exponential gap in minimum memory size needed for rendezvous between the scenario with arbitrary delay and the scenario with delay zero. Moreover, we show that our upper bound is optimal by proving that Ω (log ℓ + log log n ) bits of memory are required for rendezvous, even in the class of trees with degrees bounded by 3.

International audience; A mobile robot, represented by a point moving along a polygonal line in the plane, has to explore an unknown polygon and return to the starting point. The robot has a sensing area which can be a circle or a square centered at the robot. This area shifts while the robot moves inside the polygon, and at each point of its trajectory the robot "sees" (explores) all points for which the segment between the robot and the point is contained in the polygon and in the sensing area. We focus on two tasks: exploring the entire polygon and exploring only its boundary. We consider several scenarios: both shapes of the sensing area and the Manhattan and the Euclidean metrics. We focus on two quality benchmarks for exploration performance: optimality (the length of the trajectory of the robot is equal to that of the optimal robot knowing the polygon) and competitiveness (the length of the trajectory of the robot is at most a constant multiple of that of the optimal robot knowing the polygon). Most of our results concern rectilinear polygons. We show that optimal exploration is possible in only one scenario, that of exploring the boundary by a robot with square sensing area, starting at the boundary and using the Manhattan metric. For this case we give an optimal exploration algorithm, and in all other scenarios we prove impossibility of optimal exploration. For competitiveness the situation is more optimistic: we show a competitive exploration algorithm for rectilinear polygons whenever the sensing area is a square, for both tasks, regardless of the metric and of the starting point. Finally, we show a competitive exploration algorithm for arbitrary convex polygons, for both shapes of the sensing area, regardless of the metric and of the starting point.

International audience; We study rendezvous of two anonymous agents, where each agent knows its own initial position in the environment. Their task is to meet each other as quickly as possible. The time of the rendezvous is measured by the number of synchronous rounds that agents need to use in the worst case in order to meet. In each round, an agent may make a simple move or it may stay motionless. We consider two types of environments, finite or infinite graphs and Euclidean spaces. A simple move traverses a single edge (in a graph) or at most a unit distance (in Euclidean space). The rendezvous consists in visiting by both agents the same point of the environment simultaneously (in the same round). In this paper, we propose several asymptotically optimal rendezvous algorithms. In particular, we show that in the line and trees as well as in multi-dimensional Euclidean spaces and grids the agents can rendezvous in time O(d), where d is the distance between the initial positions of the agents. The problem of location-aware rendezvous was studied before in the asynchronous model for Euclidean spaces and multi-dimensional grids, where the emphasis was on the length of the adopted rendezvous trajectory. We point out that, contrary to the asynchronous case, where the cost of rendezvous is dominated by the size of potentially large neighborhoods, the agents are able to meet in all graphs of at most n nodes in time almost linear in d, namely, O(d log^2 n). We also determine an infinite family of graphs in which synchronized rendezvous takes time Ω(d).

Two identical (anonymous) mobile agents start from arbitrary nodes in an a priori unknown graph and move synchronously from node to node with the goal of meeting. This rendezvous problem has been thoroughly studied, both for anonymous and for labeled agents, along with another basic task, that of exploring graphs by mobile agents. The rendezvous problem is known to be not easier than graph exploration. A well-known recent result on exploration, due to Reingold, states that deterministic exploration of arbitrary graphs can be performed in log-space, i.e., using an agent equipped with O(log n) bits of memory, where n is the size of the graph. In this paper we study the size of memory of mobile agents that permits us to solve the rendezvous problem deterministically. Our main result establishes the minimum size of the memory of anonymous agents that guarantees deterministic rendezvous when it is feasible. We show that this minimum size is Θ(log n), where n is the size of the graph, regardless of the delay between the starting times of the agents. More precisely, we construct identical agents equipped with Θ(log n) memory bits that solve the rendezvous problem in all graphs with at most n nodes, if they start with any delay τ, and we prove a matching lower bound Ω(log n) on the number of memory bits needed to accomplish rendezvous, even for simultaneous start. In fact, this lower bound is achieved already on the class of rings. This shows a significant contrast between rendezvous and exploration: e.g., while exploration of rings (without stopping) can be done using constant memory, rendezvous, even with simultaneous start, requires logarithmic memory. As a by-product of our techniques introduced to obtain log-space rendezvous we get the first algorithm to find a quotient graph of a given unlabeled graph in polynomial time, by means of a mobile agent moving around the graph.

Two mobile agents (robots) with distinct labels have to meet in an arbitrary, possibly infinite, unknown connected graph or in an unknown connected terrain in the plane. Agents are modeled as points, and the route of each of them only depends on its label and on the unknown environment. The actual walk of each agent also depends on an asynchronous adversary that may arbitrarily vary the speed of the agent, stop it, or even move it back and forth, as long as the walk of the agent is continuous, does not leave its route and covers all of it. Meeting in a graph means that both agents must be at the same time in some node or in some point inside an edge of the graph, while meeting in a terrain means that both agents must be at the same time in some point of the terrain. Does there exist a deterministic algorithm that allows any two agents to meet in any unknown environment in spite of this very powerful adversary? We give deterministic rendezvous algorithms for agents starting at arbitrary nodes of any anonymous connected graph (finite or infinite) and for agents starting at any interior points with rational coordinates in any closed region of the plane with path-connected interior. In the geometric scenario agents may have different compasses and different units of length. While our algorithms work in a very general setting -- agents can, indeed, meet almost everywhere -- we show that none of these few limitations imposed on the environment can be removed. On the other hand, our algorithm also guarantees the following approximate rendezvous for agents starting at arbitrary interior points of a terrain as previously stated agents will eventually get to within an arbitrarily small positive distance from each other.

International audience; We study the amount of knowledge about a communication network that must be given to its nodes in order to efficiently disseminate information. Our approach is {\em quantitative}: we investigate the minimum total number of bits of information (minimum size of advice) that has to be available to nodes, regardless of the type of information provided. We compare the size of advice needed to perform broadcast and wakeup (the latter is a broadcast in which nodes can transmit only after getting the source information), both using a linear number of messages (which is optimal). We show that the minimum size of advice permitting the {\em wakeup} with a linear number of messages in a $n$-node network, is $\Theta (n \log n)$, while the {\em broadcast} with a linear number of messages can be achieved with advice of size $O(n)$. We also show that the latter size of advice is almost optimal: no advice of size $o(n)$ can permit to broadcast with a linear number of messages. Thus an efficient wakeup requires strictly more information about the network than an efficient broadcast.

International audience; A mobile robot represented by a point moving in the plane has to explore an unknown flat terrain with impassable obstacles. Both the terrain and the obstacles are modeled as arbitrary polygons. We consider two scenarios: the unlimited vision, when the robot situated at a point p of the terrain explores (sees) all points q of the terrain for which the segment pq belongs to the terrain, and the limited vision, when we require additionally that the distance between p and q be at most 1. All points of the terrain (except obstacles) have to be explored and the performance of an exploration algorithm, called its complexity, is measured by the length of the trajectory of the robot. For unlimited vision we show an exploration algorithm with complexity $O(P+D\sqrt{k})$, where P is the total perimeter of the terrain (including perimeters of obstacles), D is the diameter of the convex hull of the terrain, and k is the number of obstacles. We do not assume knowledge of these parameters. We also prove a matching lower bound showing that the above complexity is optimal, even if the terrain is known to the robot. For limited vision we show exploration algorithms with complexity $O(P+A+\sqrt{Ak})$, where A is the area of the terrain (excluding obstacles). Our algorithms work either for arbitrary terrains, if one of the parameters A or k is known, or for c-fat terrains, where c is any constant (unknown to the robot) and no additional knowledge is assumed. (A terrain ${\mathcal T}$ with obstacles is c-fat if R/r ≤ c, where R is the radius of the smallest disc containing ${\mathcal T}$ and r is the radius of the largest disc contained in ${\mathcal T}$.) We also prove a matching lower bound $\Omega(P+A+\sqrt{Ak})$ on the complexity of exploration for limited vision, even if the terrain is known to the robot.

International audience; Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension $\dime > 0$, starting from two arbitrary positions at distance at most $d$. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a $\dime$-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length $O(d^\dime{\rm polylog\ }d)$. This bound for the case of {\sc 2d}~-grids subsumes the main result of \cite{CCGL}. The algorithm is almost optimal, since the $\Omega(d^\dime)$ lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the $\dime$-dimensional grid have to be set such that two anonymous agents starting at distance at most $d$ from each other will always meet, moving in an asynchronous manner, after traversing a $O(d^\dime{\rm polylog\ }d)$ length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the $\dime$-dimensional Euclidean space. The agents have the radii of visibility of $r_1$ and $r_2$, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of $O((\frac{d}{r})^\dime {\rm polylog}(\frac{d}{r}))$, where $r = \min(r_1, r_2)$ and for $r\geq 1$.

International audience; We study the problem of the amount of information (advice) about a graph that must be given to its nodes in order to achieve fast distributed computations. The required size of the advice enables to measure the information sensitivity of a network problem. A problem is information sensitive if little advice is enough to solve the problem rapidly (i.e., much faster than in the absence of any advice), whereas it is information insensitive if it requires giving a lot of information to the nodes in order to ensure fast computation of the solution. In this paper, we study the information sensitivity of distributed graph coloring.