In rendezvous, two agents traverse network edges in synchronous rounds and have to meet at some node. In treasure hunt, a single agent has to find a stationary target situated at an unknown node of the network. We study tradeoffs between the amount of information ( @math ) available @math to the agents and the cost (number of edge traversals) of rendezvous and treasure hunt. Our goal is to find the smallest size of advice which enables the agents to solve these tasks at some cost @math in a network with @math edges. This size turns out to depend on the initial distance @math and on the ratio @math , which is the @math due to advice. For arbitrary graphs, we give upper and lower bounds of @math and @math , respectively, on the optimal size of advice. For the class of trees, we give nearly tight upper and lower bounds of @math and @math , respectively.
Providing nodes or agents with information of arbitrary type that can be used to perform network tasks more efficiently has been proposed in @cite_46 @cite_40 @cite_44 @cite_3 @cite_21 @cite_4 @cite_2 @cite_17 @cite_19 @cite_28 @cite_10 @cite_25 @cite_43 @cite_14 @cite_35 @cite_9 @cite_7 @cite_11 . This approach was referred to as algorithms with advice . The advice is given either to nodes of the network or to mobile agents performing some network task. Several of the authors cited above studied the minimum size of advice required to solve the respective network problem in an efficient way.
@cite_35 @cite_14 @cite_4 @cite_11 @cite_7 @cite_28 @cite_9 @cite_21 @cite_3 @cite_44 @cite_19 @cite_40 @cite_43 @cite_2 @cite_46 @cite_10 @cite_25 @cite_17
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We study the problem of the amount of information required to draw a complete or a partial map of a graph with unlabeled nodes and arbitrarily labeled ports. A mobile agent, starting at any node of an unknown connected graph and walking in it, has to accomplish one of the following tasks: draw a complete map of the graph, i.e., find an isomorphic copy of it including port numbering, or draw a partial map, i.e., a spanning tree, again with port numbering. The agent executes a deterministic algorithm and cannot mark visited nodes in any way. None of these map drawing tasks is feasible without any additional information, unless the graph is a tree. Hence we investigate the minimum number of bits of information (minimum size of advice) that has to be given to the agent to complete these tasks. It turns out that this minimum size of advice depends on the number n of nodes or the number m of edges of the graph, and on a crucial parameter @m, called the multiplicity of the graph, which measures the number of nodes that have an identical view of the graph. We give bounds on the minimum size of advice for both above tasks. For @m=1 our bounds are asymptotically tight for both tasks and show that the minimum size of advice is very small. For @m>1 the minimum size of advice increases abruptly. In this case our bounds are asymptotically tight for topology recognition and asymptotically almost tight for spanning tree construction. Consider a team of mobile software agents deployed to capture a (possibly hostile) intruder in a network. All agents, including the intruder move along the network links; the intruder could be arbitrarily fast, and aware of the positions of all the agents. The problem is to design the agents' strategy for capturing the intruder. The main efficiency parameter is the size of the team. This is an instance of the well known graph-searching problem whose many variants have been extensively studied in the literature. In all existing solutions, and in all the variants of the problem, it is assumed that agents can be removed from their current location and placed in another network site arbitrarily and at any time. As a consequence, the existing optimal strategies cannot be employed in situations for which agents cannot access the network at any point, or cannot "jump" across the network, or cannot reach an arbitrary point of the network via an internal travel through insecure zones. This motivates the contiguous search problem in which agents cannot be removed from the network, and clear links must form a connected sub-network at any time, providing safety of movements. This new problem is NP-complete in general. We study it for tree networks, and we consider its more general version, the weighted case, which arises naturally when considering networks whose nodes and links are of different nature and thus require a different number of agents to be explored. We give a linear-time algorithm that computes, for any tree @math , the minimum number of agents to capture the intruder, and the corresponding search strategy. Beside its optimality in time, our algorithm is naturally distributed: if @math is a processor-network, then the minimal search strategy for @math can be computed by @math in a decentralized manner, using a linear number of messages. 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 vq Did any agent enter v in this roundq Did any agent exit v in this roundq 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. We study the amount of knowledge about the network that is required in order to efficiently solve a task concerning this network. The impact of available information on the efficiency of solving network problems, such as communication or exploration, has been investigated before but assumptions concerned availability of particular items of information about the network, such as the size, the diameter, or a map of the network. In contrast, our approach is quantitative: we investigate the minimum number of bits of information (bits of advice) that has to be given to an algorithm in order to perform a task with given efficiency. We illustrate this quantitative approach to available knowledge by the task of tree exploration. A mobile entity (robot) has to traverse all edges of an unknown tree, using as few edge traversals as possible. The quality of an exploration algorithm A is measured by its competitive ratio, i.e., by comparing its cost (number of edge traversals) to the length of the shortest path containing all edges of the tree. Depth-First-Search has competitive ratio 2 and, in the absence of any information about the tree, no algorithm can beat this value. We determine the minimum number of bits of advice that has to be given to an exploration algorithm in order to achieve competitive ratio strictly smaller than 2. Our main result establishes an exact threshold number of bits of advice that turns out to be roughly loglogD, where D is the diameter of the tree. More precisely, for any constant c, we construct an exploration algorithm with competitive ratio smaller than 2, using at most loglogD-c bits of advice, and we show that every algorithm using loglogD-g(D) bits of advice, for any function g unbounded from above, has competitive ratio at least 2. 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 in each segment of its route is continuous, does not leave it 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. While our algorithms work in a very general setting - agents can, indeed, meet almost everywhere - we show that none of the above 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 above: agents will eventually get at an arbitrarily small positive distance from each other. 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 terminationq 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 fp2. More precisely, we show a deterministic polynomial algorithm that gathers all good agents in an arbitrary network, provided that there are at least fp2 of them. We also provide a matching lower bound: we prove that if the number of good agents is at most fp1, then they are not able to gather deterministically with termination in some networks. For strongly Byzantine agents, we give a lower bound of fp1, 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 2fp1 good agents when the size of the network is known and for at least 4fp2 good agents when it is unknown. 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 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 wakeup with a linear number of messages in an n-node network, is @Q(nlogn), while the 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. We consider the problem of gathering identical, memoryless, mobile agents in one node of an anonymous graph. Agents start from different nodes of the graph. They operate in Look-Compute-Move cycles and have to end up in the same node. In one cycle, an agent takes a snapshot of its immediate neighborhood (Look), makes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case makes an instantaneous move to this neighbor (Move). Cycles are performed asynchronously for each agent. The novelty of our model with respect to the existing literature on gathering asynchronous oblivious agents in graphs is that the agents have very restricted perception capabilities: they can only see their immediate neighborhood. An initial configuration of agents is called gatherable if there exists an algorithm that gathers all the agents of the configuration in one node and keeps them idle from then on, regardless of the actions of the asynchronous adversary. (The algorithm can be even tailored to gather this specific configuration.) The gathering problem is to determine which configurations are gatherable and find a (universal) algorithm which gathers all gatherable configurations. We give a complete solution of the gathering problem for regular bipartite graphs. Our main contribution is the proof that the class of gatherable initial configurations is very small: it consists only of ''stars'' (an agent A with all other agents adjacent to it) of size at least 3. On the positive side we give an algorithm accomplishing gathering for every gatherable configuration. 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. Efficient graph search is a central issue in many aspects of AI. In most of existing work there is a distinction between the active “searcher”, which both executes the algorithm and holds the memory, and the passive “searched graph”, over which the searcher has no control at all. Large dynamic networks like the Internet, where the nodes are powerful computers and the links have narrow bandwidth and are heavilydloaded, call for a different paradigm, in which most of the burden of computing and memorizing is moved from the searching agent to the nodes of the network. In this paper we suggest a method for searching an undirected, connected graph using the VertexdAntdWalk method, where an a(ge)nt walks along the edges of a graph G, occasionally leaving “pheromone” traces at nodes, and using those traces to guide its exploration. We show that the ant can cover the graph within time O (nd), where n is the number of vertices and d the diameter of G. The use of traces achieves a tradedoff between random and selfdavoiding walks, as it dictates a lower priority for alreadydvisited neighbors. Further properties of the suggested method are: (a) modularity: a group of searching agents, each applying the same protocol, can cooperate on a mission of covering a graph with minimal explicit communication between thems (b) possible convergence to a limit cycle: a Hamiltonian path in G (if one exists) is a possible limit cycle of the process. Designing robust algorithms for mobile agents with reliable communication is difficult due to the distributed nature of computation, in mobile ad hoc networks (MANETs) the matter is exacerbated by the need to ensure connectivity. Existing distributed algorithms provide coordination but typically assume connectivity is ensured by other means. We present a connectivity service that encapsulates an arbitrary motion planner and can refine any plan to preserve connectivity (the graph of agents remains connected) and ensure progress (the agents advance towards their goal). The service is realized by a distributed algorithm that is modular in that it makes no assumptions of the motion-planning mechanism except the ability for an agent to query its position and intended goal position, local in that it uses 1-hop broadcast to communicate with nearby agents but doesn't need any network routing infrastructure, and oblivious in that it does not depend on previous computations. We prove the progress of the algorithm in one round is at least Ω(min(d, r)), where d is the minimum distance between an agent and its target and r is the communication radius. We characterize the worst case configuration and show that when d ≥ r this bound is tight and the algorithm is optimal, since no algorithm can guarantee greater progress. Finally we show all agents get Ɛ-close to their targets within O(D0 r+n2 Ɛ) rounds where n is the number of agents and D0 is the sum of the initial distances to the targets. 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) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O(D|Lmin|2) when D is known and O((D + |Lmax|)3) if D is unknown, where |Lmin| and |Lmax| are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost O(n|Lmin|), if the size n of the ring is known, and of cost O(n|Lmax|), if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|Lmin|) if the topology of the graph and the initial positions are known to agents. In rendezvous, two agents traverse network edges in synchronous rounds and have to meet at some node. In treasure hunt, a single agent has to find a stationary target situated at an unknown node of the network. We study tradeoffs between the amount of information (advice) available a priori to the agents and the cost (number of edge traversals) of rendezvous and treasure hunt. Our goal is to find the smallest size of advice which enables the agents to solve these tasks at some cost C in a network with e edges. This size turns out to depend on the initial distance D and on the ratio e C , which is the relative cost gain due to advice. For arbitrary graphs, we give upper and lower bounds of O ( D log ( D ? e C ) + log log e ) and ? ( D log e C ) , respectively, on the optimal size of advice. For the class of trees, we give nearly tight upper and lower bounds of O ( D log e C + log log e ) and ? ( D log e C ) , respectively. In rendezvous, two agents traverse edges in rounds and have to meet at some node.In treasure hunt, an agent must find a fixed target at some node of the network.Objective: tradeoffs between the advice available to the agents and the cost.Results: bounds on the size of advice to achieve a given cost. [L. Blin, P. Fraigniaud, N. Nisse, S. Vial, Distributing chasing of network intruders, in: 13th Colloquium on Structural Information and Communication Complexity, SIROCCO, in: LNCS, vol. 4056, Springer-Verlag, 2006, pp. 70-84] introduced a new measure of difficulty for a distributed task in a network. The smallest number of bits of advice of a distributed problem is the smallest number of bits of information that has to be available to nodes in order to accomplish the task efficiently. Our paper deals with the number of bits of advice required to perform efficiently the graph searching problem in a distributed setting. In this variant of the problem, all searchers are initially placed at a particular node of the network. The aim of the team of searchers is to clear a contaminated graph in a monotone connected way, i.e., the cleared part of the graph is permanently connected, and never decreases while the search strategy is executed. Moreover, the clearing of the graph must be performed using the optimal number of searchers, i.e. the minimum number of searchers sufficient to clear the graph in a monotone connected way in a centralized setting. We show that the minimum number of bits of advice permitting the monotone connected and optimal clearing of a network in a distributed setting is @Q(nlogn), where n is the number of nodes of the network. More precisely, we first provide a labelling of the vertices of any graph G, using a total of O(nlogn) bits, and a protocol using this labelling that enables the optimal number of searchers to clear G in a monotone connected distributed way. Then, we show that this number of bits of advice is optimal: any distributed protocol requires @W(nlogn) bits of advice to clear a network in a monotone connected way, using an optimal number of searchers. 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... This thesis addresses problems of optimization to be solved collectively (distributely, we say) by groups of networked agents having limited communication and computation capabilities. The focus is on two paradigmatic problems, average consensus and coverage control. The former consists in computing the average of values which are priory known to the agents, while the latter consists in optimally deploying robotic agents in a given environment. Their solution is possible thanks to iterative algorithms which exploit the available communications among agents, described by a graph. Research has been very intense in latest years on these problems, due to strong applicative motivations: these algorithms are crucial for the design of networks of sensors, and for the coordinated motion of groups of unmanned vehicles. Namely, the ultimate goal of these algorithms is to make such artificial networks able to self-organize and perform specific tasks without a centralized supervision. In the thesis networks have been considered, which allow only communication in pairs, or limited to messages belonging to a discrete alphabet. Several novel algorithms for the solution of the above optimization problems are proposed: they are proved to lead the agents to the desired solution, and their performance is studied. The mathematical tools required for the analysis are novel, or need to be used in a non-standard way. They are drawn from very different branches of pure and applied mathematics: spectral graph theory, Markov chains, combinatorics, linear algebra, set-valued analysis, convex and discrete geometry, topology of hyper-spaces of sets. We study the problem of the amount of information required to perform fast broadcasting in tree networks. The source located at the root of a tree has to disseminate a message to all nodes. In each round each informed node can transmit to one child. Nodes do not know the topology of the tree but an oracle knowing it can give a string of bits of advice to the source which can then pass it down the tree with the source message. The quality of a broadcasting algorithm with advice is measured by its competitive ratio: the worst case ratio, taken over n-node trees, between the time of this algorithm and the optimal broadcasting time in the given tree. Our goal is to find a trade-off between the size of advice and the best competitive ratio of a broadcasting algorithm for n-node trees. We establish such a trade-off with an approximation factor of O(n e ), for an arbitrarily small positive constant e. This is the first communication problem for which a trade-off between the size of advice and the efficiency of the solution is shown for arbitrary size of advice.