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Mayur Datar Aristides Gionis y Piotr Indyk z Rajeev Motwani x Abstract We consider the problem of maintaining aggregates and statistics over data streams, with respect to the last N data elements seen so far. We refer to this model as the sliding window model. We consider the following basic problem: Given a stream of bits, maintain a count of the number of 1s in the last N elements seen from the stream. We show that using O (1 ffl log 2 N) bits of memory, we can estimate the number of 1s to within a factor of 1 + ffl. We also give a matching lower bound of \\ 1 ffl log 2 N) memory bits for any deterministic or randomized algorithms. We extend our scheme to maintain the sum of the last N positive integers. We provide matching upper and lower bounds for this more general problem as well. We apply our techniques to obtain efficient algorithms for the Lp norms (for p 2 1; 2) of vectors under the sliding window model. Using the algorithm for the basic counting problem, one can adapt many other techniques to work for the sliding window model, with a multiplicative overhead of O (1 ffl log N) in memory and a 1 + ffl factor loss in accuracy. These include maintaining approximate histograms, hash tables, and statistics or aggregates such as sum and averages.
Datar et al. (Sun,) studied this question.