Sidorenko-Inspired Pessimistic Estimation
For database query optimization, this provides a more accurate join size upper bound, though the improvement is incremental over prior bi-star bounds.
This paper extends prior work on join size estimation by using caterpillar-shaped graph homomorphisms, inspired by Sidorenko's conjecture. The new bound reduces overestimation from m (star bound) to about m^{3/5} (caterpillar bound), as shown by simulations with R^2 > 0.98.
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.