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.
Average consensus algorithms built around the update rule ) typically require bidirectional communication links with some stability and assume that agents have access to global information. This is to be expected, as they operate by broadcast and over anonymous networks, and thus have to bypass the impossibility result in @cite_37 : they do so through the use, at least implicitly, of the out-degree of the agents.
@cite_37
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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.