Fault Tolerant Gradient Clock Synchronization
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Synchronizing clocks in distributed systems is well-understood, both in terms of fault-tolerance in fully connected systems and the dependence of local and global worst-case skews (i.e., maximum clock difference between neighbors and arbitrary pairs of nodes, respectively) on the diameter of fault-free systems. However, so far nothing non-trivial is known about the local skew that can be achieved in topologies that are not fully connected even under a single Byzantine fault. Put simply, in this work we show that the most powerful known techniques for fault-tolerant and gradient clock synchronization are compatible, in the sense that the best of both worlds can be achieved simultaneously. Concretely, we combine the Lynch-Welch algorithm [Welch1988] for synchronizing a clique of n nodes despite up to f<n/3 Byzantine faults with the gradient clock synchronization (GCS) algorithm by Lenzen et al. [Lenzen2010] in order to render the latter resilient to faults. As this is not possible on general graphs, we augment an input graph G by replacing each node by 3f+1 fully connected copies, which execute an instance of the Lynch-Welch algorithm. We then interpret these clusters as supernodes executing the GCS algorithm, where for each cluster its correct nodes' Lynch-Welch clocks provide estimates of the logical clock of the supernode in the GCS algorithm. By connecting clusters corresponding to neighbors in G in a fully bipartite manner, supernodes can inform each other about (estimates of) their logical clock values. This way, we achieve asymptotically optimal local skew, granted that no cluster contains more than f faulty nodes, at factor O(f) and O(f^2) overheads in terms of nodes and edges, respectively. Note that tolerating f faulty neighbors trivially requires degree larger than f, so this is asymptotically optimal as well.
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