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. In their seminal work, Abraham, Amit and Dolev [OPODIS '04] solve this problem in $\mathbb{R}$ with the optimal resilience $t < \frac{n}{3}$ with a protocol where each party reliably broadcasts a value in every iteration. This takes $\Theta(n^2)$ messages per reliable broadcast, or $\Theta(n^3)$ messages per iteration. In this work, we forgo reliable broadcast to achieve asynchronous approximate agreement against $t < \frac{n}{3}$ faults with a quadratic communication. In a tree with the maximum degree $\Delta$ and the centroid decomposition height $h$, we achieve edge agreement in at most $6h + 1$ rounds with $\mathcal{O}(n^2)$ messages of size $\mathcal{O}(\log \Delta + \log h)$ per round. We do this by designing a 6-round multivalued 2-graded consensus protocol and using it to recursively reduce the task to edge agreement in a subtree with a smaller centroid decomposition height. Then, we achieve edge agreement in the infinite path $\mathbb{Z}$, again with the help of 2-graded consensus. Finally, we show that our edge agreement protocol enables $\varepsilon$-agreement in $\mathbb{R}$ in $6\log_2\frac{M}{\varepsilon} + \mathcal{O}(\log \log \frac{M}{\varepsilon})$ rounds with $\mathcal{O}(n^2 \log \frac{M}{\varepsilon})$ messages and $\mathcal{O}(n^2\log \frac{M}{\varepsilon}\log \log \frac{M}{\varepsilon})$ bits of communication, where $M$ is the maximum non-byzantine input magnitude.