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Asynchronous Approximate Agreement with Quadratic Communication

Main:18 Pages
2 Figures
Bibliography:3 Pages
Appendix:6 Pages
Abstract

We consider an asynchronous network of nn message-sending parties, up to tt 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 R\mathbb{R} with the optimal resilience t<n3t < \frac{n}{3} with a protocol where each party reliably broadcasts a value in every iteration. This takes Θ(n2)\Theta(n^2) messages per reliable broadcast, or Θ(n3)\Theta(n^3) messages per iteration.In this work, we forgo reliable broadcast to achieve asynchronous approximate agreement against t<n3t < \frac{n}{3} faults with quadratic communication. In trees of diameter DD and maximum degree Δ\Delta, we achieve edge agreement in 6log2D\lceil{6\log_2 D}\rceil rounds with O(n2)\mathcal{O}(n^2) messages of size O(logΔ+loglogD)\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 DD to edge agreement in a tree of diameter D2\frac{D}{2}. Then, we achieve edge agreement in the infinite path Z\mathbb{Z}, again with the help of 2-graded consensus. Finally, by reducing ε\varepsilon-agreement in R\mathbb{R} to edge agreement in Z\mathbb{Z}, we show that our edge agreement protocol enables ε\varepsilon-agreement in R\mathbb{R} in 6log2(Mε+1)+O(loglogMε)6\log_2(\frac{M}{\varepsilon} + 1) + \mathcal{O}(\log \log \frac{M}{\varepsilon}) rounds with O(n2logMε)\mathcal{O}(n^2 \log \frac{M}{\varepsilon}) messages and O(n2logMεloglogMε)\mathcal{O}(n^2\log \frac{M}{\varepsilon}\log \log \frac{M}{\varepsilon}) bits of communication, where MM is the maximum input magnitude.

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@article{erbes2025_2408.05495,
  title={ Asynchronous Approximate Agreement with Quadratic Communication },
  author={ Mose Mizrahi Erbes and Roger Wattenhofer },
  journal={arXiv preprint arXiv:2408.05495},
  year={ 2025 }
}
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