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 way to circumvent the impossibility result in @cite_37 consists in assuming unique agent identifiers: by tagging each initial value @math with @math 's identifier, at each step of the algorithm, the agents can compute the average of the input values that they have heard of so far, and thus compute the global average @math after @math communication steps when the topology is continuously strongly connected. Unfortunately, the price to pay in this simple average consensus algorithm is messages of size in @math bits. By repeated leader election on the basis of agent identifiers and a shared bound on the network diameter, the quadratic time algorithm in @cite_42 also achieves average consensus, using message and memory size in only @math bits with a fixed, strongly connected network.
@cite_37 @cite_42
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We consider deterministic anonymous distributed systems with broadcast communications where each node has some initial value, and the goal is to compute a function of all these values. We show that only a very restricted set of functions can be computed if the nodes do not know (and cannot use) the number of their out-neighbors. Our results remain valid even if nodes know the precise structure of the network but do not know where they lie within the structure. They also remain valid if nodes know their out-degree up to an uncertainty of 1. These results are a variation of those obtained by Boldi and Vigna (1997) for a weaker computation model. As a consequence, computing more complex functions in the context of broadcast communications requires the explicit or implicit knowledge or use of either (a) the out-degree of each node, (b) global node identifiers, (c) randomization, or (d) asynchronous updates with specific properties. Consensus is a fundamental feature of distributed systems, and it is the prerequisite for several complex tasks, such as flocking of mobile robots, localization in wireless-sensor networks, or decentralized control of smart grids. Average consensus, in particular, is quite challenging, because it is typically obtained asymptotically, while few finite-time algorithms are available. In this paper, we provide a methodology to achieve distributed average consensus in finite time, while maintaining low computational and memory requirements, and small completion times. The provided solution, namely, finite-time average-consensus by iterated max-consensus (FAIM) is based on several runs of the max-consensus algorithm, and has low memory requirements for each node. Compared to existing Flooding approaches, the proposed algorithm requires less memory, at the cost of a slight increase in the number of steps required for termination. The FAIM algorithm assumes that the nodes are aware of an upper bound on the network diameter. To relax this assumption, we complement this paper with a novel distributed algorithm that, in the case of undirected graphs, provides an upper bound on the network diameter which, in the worst case, is twice the actual diameter. A comparison of the proposed finite-time algorithm against the state of the art concludes this paper.