Asynchronous Approximate Agreement with Quadratic Communication (2408.05495v3)

Abstract: 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 quadratic communication. In trees of diameter $D$ and maximum degree $\Delta$, we achieve edge agreement in $\lceil{6\log_2 D}\rceil$ rounds with $\mathcal{O}(n^2)$ messages of size $\mathcal{O}(\log \Delta + \log\log D)$ per round. We do this by designing a 6-round multivalued 2-graded consensus protocol, and by repeatedly using it to reduce edge agreement in a tree of diameter $D$ to edge agreement in a tree of diameter $\frac{D}{2}$. Then, we achieve edge agreement in the infinite path $\mathbb{Z}$, again with the help of 2-graded consensus. Finally, by reducing $\varepsilon$-agreement in $\mathbb{R}$ to edge agreement in $\mathbb{Z}$, we show that our edge agreement protocol enables $\varepsilon$-agreement in $\mathbb{R}$ in $6\log_2(\frac{M}{\varepsilon} + 1) + \mathcal{O}(\log \log \frac{M}{\varepsilon})$ 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 input magnitude.