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 $O(\frac{1}{\epsilon} \log^2 N)$ 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 $\Omega(\frac{1}{\epsilon}\log^2 N)$ 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…