Showing papers by "Andrzej Pelc published in 2012"

Deterministic rendezvous in networks: A comprehensive survey

Abstract: Two or more mobile entities, called agents or robots, starting at distinct initial positions, have to meet. This task is known in the literature as rendezvous. Among many alternative assumptions that have been used to study the rendezvous problem, two most significantly influence the methodology appropriate for its solution. The first of these assumptions concerns the environment in which the mobile entities navigate: it can be either a terrain in the plane, or a network modeled as an undirected graph. The second assumption concerns the way in which the entities move: it can be either deterministic or randomized. In this article, we survey results on deterministic rendezvous in networks. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012 © 2012 Wiley Periodicals, Inc.

How to meet when you forget: log-space rendezvous in arbitrary graphs

Abstract: 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.

How to meet asynchronously (almost) everywhere

Abstract: 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 adversaryq 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.

Electing a Leader in Multi-hop Radio Networks

Abstract: We consider the task of electing a leader in a distributed manner in ad hoc multi-hop radio networks Radio networks represent the class of wireless networks in which one frequency is used for transmissions, network’s topology can be represented by a simple undirected graph with some n nodes, and there is no collision detection We give a randomized algorithm electing a leader in \(\mathcal{O}(n)\) expected time and prove that this time bound is optimal We give a deterministic algorithm electing a leader in \(\mathcal{O}(n\log^{3/2}n \sqrt{\log\log n})\) time By way of application, we show how to perform gossiping with combined messages in \(\mathcal{O}(n\log^{3/2} n \sqrt{\log\log n})\) time by a deterministic algorithm, and in \(\mathcal{O}(n)\) expected time by a randomized algorithm

Decidability classes for mobile agents computing

Abstract: 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. Our main result shows that there exists a natural complete problem for mobile agent verification. We also show that, for a single agent, three natural oracles yield a strictly increasing chain of relative decidability classes.

Collecting information by power-aware mobile agents

Abstract: 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.

Gathering despite mischief

Abstract: 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 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 permit to solve the problem for networks of known size. If the size is unknown, then this minimum number is f + 2. More precisely, we show a deterministic polynomial algorithm that gathers all good agents in an arbitrary network, provided that there are at least f + 2 of them. We also provide a matching lower bound: we prove that if the number of good agents is at most f + 1, then they are not able to gather deterministically with termination in some networks.For strongly Byzantine agents we give a lower bound of f + 1, even when the graph is known: we show that f good agents cannot gather deterministically in the presence of f Byzantine agents even in a ring of known size. On the positive side we give deterministic gathering algorithms for at least 2f + 1 good agents when the size of the network is known, and for at least 4f + 2 good agents when it is unknown.

Knowledge, level of symmetry, and time of leader election

Abstract: We study the time needed for deterministic leader election in the $\mathcal{LOCAL}$ model, where in every round a node can exchange any messages with its neighbors and perform any local computations. The topology of the network is unknown and nodes are unlabeled, but ports at each node have arbitrary fixed labelings which, together with the topology of the network, can create asymmetries to be exploited in leader election. We consider two versions of the leader election problem: strong LE in which exactly one leader has to be elected, if this is possible, while all nodes must terminate declaring that leader election is impossible otherwise, and weak LE, which differs from strong LE in that no requirement on the behavior of nodes is imposed, if leader election is impossible. We show that the time of leader election depends on three parameters of the network: its diameter D, its size n, and its level of symmetryλ, which, when leader election is feasible, is the smallest depth at which some node has a unique view of the network. It also depends on the knowledge by the nodes, or lack of it, of parameters D and n. Optimal time of weak LE is shown to be Θ(D+λ) if either D or n is known to the nodes. (If none of these parameters is known, even weak LE is impossible.) For strong LE, knowing only D is insufficient to perform it. If only n is known then optimal time is Θ(n), and if both n and D are known, then optimal time is Θ(D+λ).

Time vs. space trade-offs for rendezvous in trees

Abstract: 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.

Time of anonymous rendezvous in trees: determinism vs. randomization

Abstract: Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and move along its edges with the goal of meeting 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 study optimal time of completing this rendezvous task. For deterministic rendezvous we seek algorithms that achieve rendezvous whenever possible, while for randomized rendezvous we seek almost safe algorithms, which achieve rendezvous with probability at least 1−1/n in n-node trees, for sufficiently large n. We construct a deterministic algorithm that achieves rendezvous in time O(n) in n-node trees, whenever rendezvous is feasible, and we show that this time cannot be improved in general, even when agents start at distance 1 in bounded degree trees. We also show an almost safe algorithm that achieves rendezvous in time O(n) for arbitrary starting positions in any n-node tree. We then analyze when randomization can help to speed up rendezvous. For n-node trees of known constant maximum degree and for a known constant upper bound on the initial distance between the agents, we show an almost safe algorithm achieving rendezvous in time O(logn). By contrast, we show that for some trees, every almost safe algorithm must use time Ω(n), even for initial distance 1. This shows an exponential gap between randomized rendezvous time in trees of bounded degree and in arbitrary trees. Such a gap does not occur for deterministic rendezvous. All our upper bounds hold when agents start with an arbitrary delay, controlled by the adversary, and all our lower bounds hold even when agents start simultaneously.

Deterministic network exploration by anonymous silent agents with local traffic reports

Abstract: 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 weak model is impossible. Our aim is to solve the exploration problem 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 information permits 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, i.e., in which there are agents with different views of the network. The algorithm works in time polynomial in the (unknown) size of the network. A deterministic terminating exploration algorithm working for all initial configurations in arbitrary networks does not exist.

Tree Exploration by a Swarm of Mobile Agents

Abstract: A swarm of mobile agents starting at the root of a tree has to explore it: every node of the tree has to be visited by at least one agent. In every round, each agent can remain idle or move to an adjacent node. In any round all agents have to be at distance at most d, where d is a parameter called the range of the swarm. The goal is to explore the tree as fast as possible.

Distributed tree comparison with nodes of limited memory

Abstract: We consider the task of comparing two rooted trees with port labels. Roots of the trees are joined by an edge and the comparison has to be performed distributedly, by exchanging messages among nodes. If the two trees are isomorphic, all nodes must finish in a state YES; otherwise they have to finish in a state NO and break symmetry, nodes of one tree getting label 0 and nodes of the other getting label 1. Nodes are modeled as identical automata, and our goal is to establish trade-offs between the memory size of such an automaton and the efficiency of distributed tree comparison, measured either by the time or by the number of messages used for communication between nodes. We consider both the synchronous and the asynchronous communication and establish exact trade-offs in both scenarios. For the synchronous scenario, we are concerned with memory versus time trade-offs. We show that if the automaton has x bits of memory, where x ≥ c log n, for a small constant c, then the optimal time to accomplish the comparison task in the class of trees of size at most n and of height at most h > 1 is Θ(h + n/x). For the asynchronous scenario, we study memory versus number of messages trade-offs. We show that if the automaton has x bits of memory, where n ≥ x ≥ c log n, then the optimal number of messages to accomplish the comparison task in the class of trees of size at most n is Θ(n2/x). © 2012 Wiley Periodicals, Inc. NETWORKS, Vol. 2012 (A preliminary version of this article appeared in the Proceedings of the 17th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2010), LNCS 6058. This work was done during the visit of Emanuele G. Fusco at the Research Chair in Distributed Computing of the Universite du Quebec en Outaouais.)

Leader Election for Anonymous Asynchronous Agents in Arbitrary Networks

Abstract: We study the problem of leader election among mobile agents operating in an arbitrary network modeled as an undirected graph. Nodes of the network are unlabeled and all agents are identical. Hence the only way to elect a leader among agents is by exploiting asymmetries in their initial positions in the graph. Agents do not know the graph or their positions in it, hence they must gain this knowledge by navigating in the graph and share it with other agents to accomplish leader election. This can be done using meetings of agents, which is difficult because of their asynchronous nature: an adversary has total control over the speed of agents. When can a leader be elected in this adversarial scenario and how to do it? We give a complete answer to this question by characterizing all initial configurations for which leader election is possible and by constructing an algorithm that accomplishes leader election for all configurations for which this can be done.