Despite decades of research on approximate query processing (AQP), our understanding of sample-based joins has remained limited and, to some extent, even superficial. The common belief in the community is that joining random samples is futile. This belief is largely based on an early result showing that the join of two uniform samples is not an independent sample of the original join, and that it leads to quadratically fewer output tuples. However, unfortunately, this result has little applicability to the key questions practitioners face. For example, the success metric is often the final approximations accuracy, rather than output cardinality. Moreover, there are many non-uniform sampling strategies that one can employ. Is sampling for joins still futile in all of these settings? If not, what is the best sampling strategy in each case? To the best of our knowledge, there is no formal study answering these questions. This paper aims to improve our understanding of sample-based joins and offer a guideline for practitioners building and using real-world AQP systems. We study limitations of offline samples in approximating join queries: given an offline sampling budget, how well can one approximate the join of two tables? We answer this question for two success metrics: output size and estimator variance. We show that maximizing output size is easy, while there is an information-theoretical lower bound on the lowest variance achievable by any sampling strategy. We then define a hybrid sampling scheme that captures all combinations of stratified, universe, and Bernoulli sampling, and show that this scheme with our optimal parameters achieves the theoretical lower bound within a constant factor. Since computing these optimal parameters requires shuffling statistics across the network, we also propose a decentralized variant where each node acts autonomously using minimal statistics.
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