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This paper deals with optimal strategies to place replicas in tree networks, with the double objective to minimize the total cost of the servers, and/or to optimize power consumption. The client requests are known beforehand, and some servers are assumed to pre-exist in the tree. Without power consumption constraints, the total cost is an arbitrary function of the number of existing servers that are reused, and of the number of new servers. Whenever creating and operating a new server has higher cost than reusing an existing one (which is a very natural assumption), cost optimal strategies have to trade-off between reusing resources and load-balancing requests on new servers. We provide an optimal dynamic programming algorithm that returns the optimal cost, thereby extending known results without pre-existing servers. With power consumption constraints, we assume that servers operate under a set of M different modes depending upon the number of requests that they have to process. In practice M is a small number, typically 2 or 3, depending upon the number of allowed voltages. Power consumption includes a static part, proportional to the total number of servers, and a dynamic part, proportional to a constant exponent of the server mode, which depends upon the model for power. The cost function becomes a more complicated function that takes into account reuse and creation as before, but also upgrading or downgrading an existing server from one mode to another. We show that with an arbitrary number of modes, the power minimization problem is NP-complete, even without cost constraint, and without static power. Still, we provide an optimal dynamic programming algorithm that returns the minimal power, given a threshold value on the total cost; it has exponential complexity in the number of modes M, and its practical usefulness is limited to small values of M. Still, experiments conducted with this algorithm show that it can process large trees in reasonable time, despite its worst-case complexity.
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