Sidorenko-Inspired Pessimistic Estimation

Recently, Abo Khamis et al. showed how to upper bound the size of a join of multiple tables, a problem essential to query optimization in database theory. They unified earlier works by the following information-theoretical framework. 1. Let $(X_1,..., X_n)$ be a row selected from the join uniformly at random. 2. The size of the join is now $\exp(H(X_1,..., X_n))$. 3. To upper bound $H(X_1,..., X_n)$, break it into several $\textit{local entropies}$, such as $H(X_1)$, $H(X_2, X_3)$, and $H(X_4|X_5)$, using Shannon-type inequalities. 4. Upper bound local entropies using statistics of the tables being joined. The statistics Abo Khamis et al. considered are the counts of graph homomorphisms from stars to the tables. In a follow-up work, we generalized stars to bi-stars. In this paper, we generalize bi-stars to caterpillars, an even larger class of graphs inspired by Sidorenko's conjecture. Simulations show that, while Abo Khamis et al.'s star bound overestimates the join size by $m$, our bi-star bound overestimates by about $m^{3/4}$, and this paper's new caterpillar bound overestimates by about $m^{3/5}$. These exponents are obtained by log-log regressions with R-square $> 0.98$. All homomorphisms are counted in time linear in the size of the tables being joined.