Maintaining stream statistics over sliding windows

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 1 s in the last N elements seen from the stream. We show that, using O(1/epsilon log(2) N) bits of memory, we can estimate the number of 1 s to within a factor of 1 + epsilon. We also give a matching lower bound of Omega(1/epsilon 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 and provide matching upper and lower bounds for this more general problem as well. We also show how to efficiently compute the L-p norms (p is an element of[1, 2]) of vectors in the sliding window model using our techniques. Using our algorithm, one can adapt many other techniques to work for the sliding window model with a multiplicative overhead of O(1/epsilon log N) in memory and a 1 + epsilon factor loss in accuracy. These include maintaining approximate histograms, hash tables, and statistics or aggregates such as sum and averages.