Asynchronous Approximate Agreement with Quadratic Communication

We consider an asynchronous network of $n$ message-sending parties, up to $t$ of which are byzantine. We study approximate agreement, where the parties obtain approximately equal outputs in the convex hull of their inputs. The seminal protocol of Abraham, Amit and Dolev [OPODIS '04] achieves approximate agreement in $\mathbb{R}$ with the optimal resilience $t < \frac{n}{3}$ by making each party reliably broadcast its input. This takes $\Omega(n^2)$ messages per reliable broadcast, or $\Omega(n^3)$ messages in total. In this work, we present optimally resilient asynchronous approximate agreement protocols which forgo reliable broadcast and thus require communication proportional to $n^2$ instead of $n^3$. First, we achieve $\omega$-dimensional barycentric agreement with $\mathcal{O}(\omega n^2)$ small messages. Then, we achieve edge agreement in a tree of diameter $D$ with $\lceil \log_2 D \rceil$ iterations of a multivalued graded consensus variant for which we design an efficient protocol. This results in a $\mathcal{O}(\log\frac{1}{\varepsilon})$-round protocol for $\varepsilon$-agreement in $[0, 1]$ with $\mathcal{O}(n^2\log\frac{1}{\varepsilon})$ messages and $\mathcal{O}(n^2\log\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon})$ bits of communication, improving over the state of the art which matches this complexity only when the inputs are all either $0$ or $1$. Finally, we extend our edge agreement protocol to achieve edge agreement in $\mathbb{Z}$ and thus $\varepsilon$-agreement in $\mathbb{R}$ with quadratic communication, in $\mathcal{O}(\log\frac{M}{\varepsilon})$ rounds where $M$ is the maximum honest input magnitude.