Fast Multidimensional Asymptotic and Approximate Consensus

We study the problems of asymptotic and approximate consensus in which agents have to get their values arbitrarily close to each others' inside the convex hull of initial values, either without or with an explicit decision by the agents. In particular, we are concerned with the case of multidimensional data, i.e., the agents' values are $d$-dimensional vectors. We introduce two new algorithms for dynamic networks, subsuming classical failure models like asynchronous message passing systems with Byzantine agents. The algorithms are the first to have a contraction rate and time complexity independent of the dimension $d$. In particular, we improve the time complexity from the previously fastest approximate consensus algorithm in asynchronous message passing systems with Byzantine faults by Mendes et al. [Distrib. Comput. 28] from $\Omega\!\left( d \log\frac{d\Delta}{\varepsilon} \right)$ to $O\!\left( \log\frac{\Delta}{\varepsilon} \right)$, where $\Delta$ is the initial and $\varepsilon$ is the terminal diameter of the set of vectors of correct agents.