Asynchronous Approximate Agreement with Quadratic Communication
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908508914450432 |
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| author | Erbes, Mose Mizrahi Wattenhofer, Roger |
| author_facet | Erbes, Mose Mizrahi Wattenhofer, Roger |
| contents | 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 $Θ(n^2)$ messages per reliable broadcast, or $Θ(n^3)$ 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 $Δ$ and the centroid decomposition height $h$, we achieve edge agreement in at most $6h + 1$ rounds with $\mathcal{O}(n^2)$ messages of size $\mathcal{O}(\log Δ+ \log h)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asynchronous Approximate Agreement with Quadratic Communication Erbes, Mose Mizrahi Wattenhofer, Roger Distributed, Parallel, and Cluster Computing Cryptography and Security 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 $Θ(n^2)$ messages per reliable broadcast, or $Θ(n^3)$ 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 $Δ$ and the centroid decomposition height $h$, we achieve edge agreement in at most $6h + 1$ rounds with $\mathcal{O}(n^2)$ messages of size $\mathcal{O}(\log Δ+ \log h)$ 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. |
| title | Asynchronous Approximate Agreement with Quadratic Communication |
| topic | Distributed, Parallel, and Cluster Computing Cryptography and Security |
| url | https://arxiv.org/abs/2408.05495 |