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 even for static networks. 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 to be optimal. Finally, the techniques employed for the more intricate analysis of the algorithm for dynamic graphs provide additional insights that are also of interest for the static setting. In particular, we establish self-stabilization of the gradient property within $O(D)$ time.
A core algorithmic problem in distributed computing is to establish coordination among the participants of a distributed system, which is often achieved through a common notion of time. Typically, every node in a network has its own local hardware clock, which can be used for this purpose; however, hardware clocks of different nodes run at slightly different rates, and the rates can change over time. This clock drift causes clocks to drift out of synch, requiring periodic communication to restore synchronization. However, communication is typically subject to delay, and although an upper bound on the delay may be known, specific message delays are unpredictable. Consequently, estimates for the current local time at other nodes are inherently inaccurate.
A distributed clock synchronization algorithm computes at each node a logical clock, and the goal is to synchronize these clocks as tightly as possible. Traditionally, distributed clock synchronization algorithms focus on minimizing the clock skew between the logical clocks of any two nodes in the network. The clock skew between two clocks is simply the difference between the two clock values. The maximum clock skew that may occur in the worst case between any two nodes at any time is called the global skew of a clock synchronization algorithm. A well-known result states that no algorithm can guarantee a global skew better than Ω(D), where D denotes the diameter of the network [1]. However, in many cases it is more important to tightly synchronize the logical clocks of nearby nodes in the network than it is to minimize the global skew. For example, if a time division multiple access (TDMA) protocol is used to coordinate access to a shared communication medium in a wireless sensor network, it suffices to synchronize the clocks of nodes that interfere with each other when transmitting. The problem of providing better guarantees on the synchronization quality between nodes that are closer is called gradient clock synchronization. The problem was introduced in a seminal paper by Fan and Lynch [7], where the authors show that a clock skew of Ω(log D/ log log D) cannot be prevented between immediate neighbors in the network. The largest possible clock skew that may occur between the logical clocks of any two adjacent nodes at any time is called the local skew of a clock synchronization algorithm. For static networks, it has been proved that the best possible local skew that an algorithm can achieve is bounded by Θ(log D) [15,16].
While tight bounds have been shown for the static model, the dynamic case has not been as well understood. A dynamic network arises in many natural contexts: for example, when nodes are mobile, or when communication links are unreliable and may fail and recover. The dynamic network model we consider in this article is general: it allows communication links to appear and disappear arbitrarily, subject only to a global connectivity constraint (which is required to maintain a bounded global skew). Hence the model is suitable for modeling various types of dynamic networks which remain connected over time.
In a dynamic network the distances between nodes change over time as communication links appear and disappear. Consequently, we divide the synchronization guarantee into two parts: a global skew guarantee bounds the skew between any two nodes in the network at any time, and a dynamic gradient skew guarantee that bounds the skew between two nodes as a function of the distance between them and how long they remain at that distance.
In [11], three of the authors showed that a clock synchronization algorithm cannot react immediately to the formation of new links, and that a certain stabilization time is required before the clocks of newly-adjacent nodes can be brought into synch. The stabilization time is inversely related to the synchronization guarantee: the tighter the synchronization required in stable state, the longer the time to reach that state. Intuitively, this is because when strict synchronization guarantees are imposed, the algorithm cannot change clock values quickly without violating the guarantee, and hence it takes longer to react. The algorithm given in [11] achieves the optimal trade-off between skew bound and stabilization time; however, its local skew bound is O( √ D), which is far from optimal.
In this article, we propose an algorithm, referred to as A OPT , that achieves the same asymptotically optimal skew bounds as in the static model: if two nodes remain at distance d for sufficiently long, the skew between them is reduced to O(d log(D/d)), where D is the dynamic diameter of the network (corresponding roughly to the time it takes for information to propagate from one end of the network to the other). The stabilization time of the algorithm, that is, the time to reach this guarantee, is O(D).
The fundamental problem of synchronizing clocks in distributed systems has been studied extensively and many results have be
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