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Abstract

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. We show that the bounds 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 synchronize their clocks to within 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 skew between the two nodes is reduced to O(d log(D/d)) is O(D), which we prove is optimal.

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There is a growing demand for live, on-the-fly processing of increasingly large amounts of data. In order to ensure the timely and reliable processing of streaming data, a variety of distributed stream processing architectures and platforms have been developed, which handle the fundamental tasks of (dynamically) assigning processing tasks to the currently available physical resources and routing streaming data between these resources. However, while there are plenty of platforms offering such functionality, the theory behind it is not well understood. In particular, it is unclear how to best allocate the processing tasks to the given resources.In this paper, we establish a theoretical foundation by formally defining a task allocation problem for distributed stream processing, which we prove to be NP-hard. Furthermore, we propose an approximation algorithm for the class of seriesparallel decomposable graphs, which captures a broad range of common stream processing applications. The algorithm achieves a constant-factor approximation under the assumptions that the number of resources scales at least logarithmically with the number of computational tasks and the computational cost of the tasks dominates the cost of communication.