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Vectors clocks seem to be a common way to synchronize the partial ordering of events in a distributed, peer-to-peer, system across all clients.

Is there any benefit to using them in a centralized system, where one node in the system has the power to order all events anyway, to order events? If one computer can decide the order anyway, there would be no need for vector clocks, right?

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No, there's no need for a vector clock in a centralized system.

A vector clock uses a $N$-vector of timestamps, where $N$ is the number of computers in the distributed system and the $i$th component of the vector block is a timestamp chosen by the $i$th computer. In a centralized system you'd use a $N$-vector with $N=1$, so it just reduces to a single number instead of a vector. In other words, instead of a vector, all you need is a single timestamp, where the timestamp is chosen by the central computer.

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