基于梯度TRIX的时钟分配
Clock Distribution with Gradient TRIX
AI总结：
本文提出一种面向类网格有向图的自稳定梯度时钟同步算法，以最优入出度3容忍1个故障入邻居，在节点独立故障概率$p\in o(n^{-1/2})$下以高概率实现$Θ(\log D)$的渐近最优局部时钟偏差，从而大幅提升大规模同步SoC的可靠性与可扩展性。
摘要：
梯度时钟同步（GCS）算法最小化直径为$D$、规模为$n$的分布式网络中节点之间的最坏情况时钟偏移。它们实现了局部最优偏移$Θ(\log D)$（即相邻节点之间，如Lenzen等人所示）和全局最优偏移$Θ(D)$（如Biaz和Welch所示）。正如Bund等人的工作所证明的，对于大规模同步片上系统（SoC）的改进时钟方案而言，这是一种极具前景的方法。遗憾的是，在大型系统中，故障阻碍了其实际应用。Bund等人提出的最先进的容错方案存在一个在此环境中致命的缺陷：它依赖于节点和边的复制。对于$f=1$，这意味着至少16倍的边复制和高节点度，远未达到容忍最多$f$个故障邻居所需的$2f+1=3$的最优值。\n\n在这项工作中，我们提出了一种适用于类网格有向图的自稳定GCS算法，其节点入度和出度均为最优值$3$，能够容忍$1$个故障入邻居。如果节点以独立概率$p\in o(n^{-1/2})$发生故障，则该算法以概率$1-o(1)$实现渐近最优的局部时钟偏差$Θ(\log D)$；这一结论在链路延迟和时钟速度变化的一般最坏情况假设下成立，前提是它们相对于系统速度变化缓慢。该故障概率是确保以概率$1-o(1)$每个节点至多有一个入邻居发生故障的最大可能值。由于现代硬件的时钟频率达到千兆赫兹级别，且该算法能够在每个时钟周期内同时承受由故障引起的恒定数量的任意变化，这带来了足够的鲁棒性，从而能够大幅扩大可靠同步时钟SoC的规模。
英文摘要：
Gradient clock synchronization (GCS) algorithms minimize the worst-case clock offset between the nodes in a distributed network of diameter $D$ and size $n$. They achieve optimal offsets of $Θ(\log D)$ locally, i.e. between adjacent nodes as shown by Lenzen et al., and $Θ(D)$ globally as shown by Biaz and Welch. As demonstrated in the work of Bund et al., this is a highly promising approach for improved clocking schemes for large-scale synchronous Systems-on-Chip (SoC). Unfortunately, in large systems, faults hinder their practical use. State of the art fault-tolerant, as presented by Bund et al., has a drawback that is fatal in this setting: It relies on node and edge replication. For $f=1$, this translates to at least $16$-fold edge replication and high degree nodes, far from the optimum of $2f+1=3$ for tolerating up to $f$ faulty neighbors. In this work, we present a self-stabilizing GCS algorithm for a grid-like directed graph with optimal node in- and out-degrees of $3$ that tolerates $1$ faulty in-neighbor. If nodes fail with independent probability $p\in o(n^{-1/2})$, it achieves asymptotically optimal local skew of $Θ(\log D)$ with probability $1-o(1)$; this holds under general worst-case assumptions on link delay and clock speed variations, provided they change slowly relative to the speed of the system. The failure probability is the largest possible ensuring that with probabity $1-o(1)$ for each node at most one in-neighbor fails. As modern hardware is clocked at gigahertz speeds and the algorithm can simultaneously sustain a constant number of arbitrary changes due to faults in each clock cycle, this results in sufficient robustness to dramatically increase the size of reliable synchronously clocked SoCs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
