We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information that needs to be transmitted is f(j). One natural and important (though not necessarily comprehensive) class of functions is those which are concave, non-decreasing, and satisfy f(0) = 0. Our goal is to find a tree which is a good approximation simultaneously to the optimum trees for all such functions. This problem is motivated by aggregation in sensor networks, as well as by buy-at-bulk network design. We present a randomized tree construction algorithm that guarantees E[maxf Cf/C*(f)] ≤ 1 + log k, where Cf is a random variable denoting the cost of the tree for function f and C*(f) is the cost of the optimum tree for function f. To the best of our knowledge, this is the first result regarding simultaneous optimization for concave costs. We also show how to derandomize this result to obtain a deterministic algorithm that guarantees max_f Cf/C*(f) = O(log k). Both these results are much stronger than merely obtaining a guarantee on max_f E[Cf/C*(f)]. A guarantee on maxf E[Cf/C*(f)] can be obtained using existing techniques, but this does not capture simultaneous optimization since no one tree is guaranteed to be a good approximation for all f simultaneously. While our analysis is quite involved, the algorithm itself is very simple and may well find practical use. We also hope that our techniques will prove useful for other problems where one needs simultaneous optimization for concave costs.
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Goel, A., Estrin, D. Simultaneous Optimization for Concave Costs: Single Sink Aggregation or Single Source Buy-at-Bulk. Algorithmica 43, 5–15 (2005). https://doi.org/10.1007/s00453-005-1155-0