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The MapReduce/Hadoop architecture has become very important and effective in cloud systems because many data-intensive applications are usually required to process big data. In such environments, big data is partitioned and stored over several data nodes; thus, the total completion time of a task would be delayed if the maximum access latency among all pairs of a data node and its assigned computation node is not bounded. Moreover, the computation nodes usually need to communicate with each other for aggregating the computation results; therefore, the maximum access latency among all pairs of assigned computation nodes also needs to be bounded. In the literature, it has been proved that the placement problem of computation nodes (virtual machines) to minimize the maximum access latency among all pairs of a data node and its assigned computation node and among all pairs of assigned computation nodes does not admit any approximation algorithm with a factor smaller than two, whereas no approximation algorithms have been proposed so far. In this paper, we first propose a 3-approximation algorithm for resolving the problem. Subsequently, we close the gap by proposing a 2-approximation algorithm, that is, an optimal approximation algorithm, for resolving the problem in the price of higher time complexity. Finally, we conduct simulations for evaluating the performance of our algorithms.

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Previous research on SDN traffic engineering mostly focuses on static traffic, whereas dynamic traffic, though more practical, has drawn much less attention. Especially, online SDN multicast that supports IETF dynamic group membership (i.e., any user can join or leave at any time) has not been explored. Different from traditional shortest-path trees (SPT) and graph theoretical Steiner trees (ST), which concentrate on routing one tree at any instant, online SDN multicast traffic engineering is more challenging because it needs to support dynamic group membership and optimize a sequence of correlated trees without the knowledge of future join and leave, whereas the scalability of SDN due to limited TCAM is also crucial. In this paper, therefore, we formulate a new optimization problem, named Online Branch-aware Steiner Tree (OBST), to jointly consider the bandwidth consumption, SDN multicast scalability, and rerouting overhead. We prove that OBST is NP-hard and does not have a |Dmax| 1− -competitive algorithm for any > 0, where |Dmax| is the largest group size at any time. We design a |Dmax|competitive algorithm equipped with the notion of the budget, the deposit, and Reference Tree to achieve the tightest bound. The simulations and implementation on real SDNs with YouTube traffic manifest that the total cost can be reduced by at least 25% compared with SPT and ST, and the computation time is small for massive SDN.

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