Gradient Clock Synchronization (GCS) is the task of minimizing the \emph{local skew,} i.e., the clock offset between neighboring clocks, in a larger network. While asymptotically optimal bounds are known, from a practical perspective they have crucial shortcomings: - Local skew bounds are determined by upper bounds on offset estimation that need to be guaranteed throughout the entire lifetime of the system. - Worst-case frequency deviations of local oscillators from their nominal rate are assumed, yet frequencies tend to be much more stable in the (relevant) short term. State-of-the-art deployed synchronization methods adapt to the true offset measurement and frequency errors, but achieve no non-trivial guarantees on the local skew. In this work, we provide a refined model and novel analysis of existing techniques for solving GCS in this model. By requiring only \emph{stability} of measurement and frequency errors, we can circumvent existing lower bounds, leading to dramatic improvements under very general conditions. For example, if links exhibit a uniform worst-case estimation error of $Δ$ and a \emph{change} in estimation errors of $δ\ll Δ$ on relevant time scales, we bound the local skew by $O(Δ+δ\log D)$ for networks of diameter $D$, effectively ``breaking'' the established $Ω(Δ\log D)$ lower bound, which holds when $δ=Δ$. Similarly, we show how to limit the influence of local oscillators on $δ$ to scale with the \emph{change} of frequency of an individual oscillator on relevant time scales. Moreover, we show how to ensure self-stabilization in this challenging setting. Last, but not least, we extend all of our results to the scenario of external synchronization, at the cost of a limited increase in stabilization time.

翻译：梯度时钟同步（GCS）旨在大型网络中最小化相邻时钟间的局部偏斜（即时钟偏移量）。尽管已知渐近最优解的理论边界，但实际应用中存在关键缺陷：局部偏斜边界由偏移估计的上界决定，该上界需在整个系统生命周期内得到保证；同时，尽管实际振荡器在短期内的频率稳定性远高于预期，现有方案仍假设本地振荡器存在最坏情况下的频率偏差。当前主流同步方法虽能自适应真实偏移测量值与频率误差，却无法为局部偏斜提供非平凡性能保证。本文对现有GCS求解技术进行了精细化建模与创新性分析：通过仅要求测量误差与频率误差的稳定性，我们规避了传统下界限制，在极宽松条件下实现了显著性能提升。例如，当链路在相关时间尺度上具有一致的极端估计误差Δ且误差变化量δ≪Δ时，对于直径为D的网络，我们可将局部偏斜约束在O(Δ+δ log D)范围内，有效“打破”了δ=Δ场景下Ω(Δ log D)的已知下界。类似地，我们证明了如何将本地振荡器对δ的影响限制在个体振荡器在相关时间尺度上的频率变化量范围内。此外，本方法在该挑战性场景下仍能保证自稳定特性。最后，我们将所有结论拓展至外同步场景，仅需付出有限稳定时间增加的代价。