# Optimal Gradient Clock Synchronization in Dynamic Networks

We study the problem of clock synchronization in highly dynamic networks, where communication links can appear or disappear at any time. The nodes in the network are equipped with hardware clocks, but the rate of the hardware clocks can vary arbitrarily within specific bounds, and the estimates that nodes can obtain about the clock values of other nodes are inherently inaccurate. Our goal in this setting is to output a logical clock at each node, such that the logical clocks of any two nodes are not too far apart, and nodes that remain close to each other in the network for a long time are better synchronized than distant nodes. This property is called gradient clock synchronization. Gradient clock synchronization has been widely studied in the static setting, where the network topology does not change. We show that the asymptotically optimal bounds obtained for the static case also apply to our highly dynamic setting: if two nodes remain at distance d from each other for sufficiently long, it is possible to upper bound the difference between their clock values by O(d*log(D/d)), where D is the diameter of the network. This is known to be optimal for static networks, and since a static network is a special case of a dynamic network, it is optimal for dynamic networks as well. Furthermore, we show that our algorithm has optimal stabilization time: when a path of length d appears between two nodes, the time required until the clock skew between the two nodes is reduced to O(d*log(D/d)) is O(D), which we prove is optimal.

**Comments:**37 pages; conference version: 29th Annual ACM Symposium on Principles of Distributed Computing (PODC 2010)

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**Affiliations:**

^{1}for the ALICE Collaboration,

^{2}for the ALICE Collaboration,

^{3}for the ALICE Collaboration

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