Many problems in distributed control reduce to the distributed computation of the average of initial values in a networked system of autonomous agents, known as the average consensus problem. We present a randomized algorithm that solves this problem in networks with directed, time-varying communication topologies, in linear time in the size of the network. This algorithm leverages properties of exponential random variables, which allows for approximating sums by computing minima. It is completely decentralized, in the sense that it does not rely on agent identifiers or global information of any kind. Besides, the agents do not need to know their out-degree; hence, our algorithm demonstrates how randomization can be used to circumvent the impossibility result established in [1]. Using a logarithmic rounding rule, we show that this algorithm can be used under the additional constraints of finite memory and channel capacity. We furthermore extend the algorithm with a termination test, by which the agents can decide irrevocably in finite time — rather than simply converge — on an estimate of the average.
Another example is to be found in the algorithm @cite_25 in which agents make explicit use of their out-degrees in the messages they send. This method converges on fixed strongly connected graphs @cite_1 , and on continuously strongly connected dynamic graphs @cite_11 .
@cite_1 @cite_25 @cite_11
2158401151 1516427227 1972775782 2100580556
This paper is concerned with the convergence of a class of continuous-time nonlinear consensus algorithms for single integrator agents. In the consensus algorithms studied here, the control input of each agent is assumed to be a state-dependent combination of the relative positions of its neighbors in the information flow graph. Using a novel approach based on the smallest order of the nonzero derivative, it is shown that under some mild conditions the convex hull of the agents has a contracting property. A set-valued LaSalle-like approach is subsequently employed to show the convergence of the agents to a common point. The results are shown to be more general than the ones reported in the literature in some cases. An illustrative example demonstrates how the proposed convergence conditions can be verified. 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. 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 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.