Springer Science and Business Media LLC

Stratified random sampling (SRS) is a widely used sampling technique for approximate query processing. We consider SRS on continuously arriving data streams and statically stored data sets. We present a tight lower bound showing that any streaming algorithm for SRS over the entire stream must have, in the worst case, a variance that is $$\varOmega (r)$$ factor away from the optimal, where r is the number of strata. We present S-VOILA, a practical streaming algorithm for SRS over the entire stream that is locally variance-optimal. We prove that any sliding window-based streaming SRS needs a workspace of $$\varOmega (rM\log W)$$ in the worst case, to maintain a variance-optimal SRS of size M, where W is the number of elements in the sliding window. Due to the inherent high workspace needs for sliding window-based SRS, we present SW-VOILA, a multi-layer practical sampling algorithm that uses only O(M) workspace but can maintain an SRS of size close to M in practice over a sliding window. Experiments show that both S-VOILA and SW-VOILA result in a variance that is typically close to their optimal offline counterparts, which was given the entire input beforehand. We also present VOILA, a variance-optimal offline algorithm for stratified random sampling. VOILA is a strict generalization of the well-known Neyman allocation, which is optimal only under the assumption that each stratum is abundant. Experiments show that VOILA can have significantly smaller variance (1.4x to 50x) than Neyman allocation on real-world data.