Gradient Clock Synchronization with Practically Constant Local Skew
Christoph Lenzen
TL;DR
The paper refines Gradient Clock Synchronization by exploiting stability in measurement errors and oscillator frequencies, replacing worst-case bounds with a δ(T)-based model that yields tighter local skew guarantees. It introduces a GCS algorithm using adaptive triggers, level-$s$ potentials, and a δ(T)-driven analysis to bound local skew by $O(Δ+δ(T)\log D)$, while preserving self-stabilization and enabling extension to external synchronization through a virtual reference. Nonlocal estimation is handled via flooding-based routines and a shortest-path structure to ensure stabilization within $O(ΔD/μ)$ time, and the external-sync variant achieves comparable skew bounds with a tunable stabilization time through a slowdown factor $ζ$. Finally, syntonization via PLLs ties local oscillators to a reference, reducing effective frequency error to $1+O(ν(P)+W^s/P)$ and adapting the stabilization dynamics accordingly. Collectively, the results provide practical, adaptive, and self-stabilizing GCS guarantees with broad applicability to hardware clock distribution and wireless networks.
Abstract
Gradient Clock Synchronization (GCS) is the task of minimizing the 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 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 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 change of frequency of an individual oscillator on relevant time scales, rather than a worst-case bound over all oscillators and the lifetime of the system. 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.