Feasible Sampling of Non-strict Turnstile Data Streams
  We present the first feasible method for sampling a dynamic data stream with deletions, where the sample consists of pairs $(k,C_k)$ of a value $k$ and its exact total count $C_k$. Our algorithms are for both Strict Turnstile data streams and the most general Non-strict Turnstile data streams, where each element may have a negative total count. Our method improves by an order of magnitude the known processing time of each element in the stream, which is extremely crucial for data stream applications. For example, for a sample of size $O(\epsilon^{-2} \log{(1/\delta)})$ in Non-strict streams, our solution requires $O((\log\log(1/\epsilon))^2 + (\log\log(1/\delta)) ^ 2)$ operations per stream element, whereas the best previous solution requires $O(\epsilon^{-2} \log^2(1/\delta))$ evaluations of a fully independent hash function per element. Here $1-\delta$ is the success probability and $\epsilon$ is the additive approximation error.   We achieve this improvement by constructing a single data structure from which multiple elements can be extracted with very high success probability. The sample we generate is useful for calculating both forward and inverse distribution statistics, within an additive error, with provable guarantees on the success probability. Furthermore, our algorithms can run on distributed systems and extract statistics on the union or difference between data streams. They can be used to calculate the Jaccard similarity coefficient as well. 
Neta Barkay, Ely Porat, Bar Shalem
cs.DS