On Fine-Grained Distinct Element Estimation

We study the problem of distributed distinct element estimation, where $\alpha$ servers each receive a subset of a universe $[n]$ and aim to compute a $(1+\varepsilon)$-approximation to the number of distinct elements using minimal communication. While prior work establishes a worst-case bound of $\Theta\left(\alpha\log n+\frac{\alpha}{\varepsilon^2}\right)$ bits, these results rely on assumptions that may not hold in practice. We introduce a new parameterization based on the number $C = \frac{\beta}{\varepsilon^2}$ of pairwise collisions, i.e., instances where the same element appears on multiple servers, and design a protocol that uses only $\mathcal{O}\left(\alpha\log n+\frac{\sqrt{\beta}}{\varepsilon^2} \log n\right)$ bits, breaking previous lower bounds when $C$ is small. We further improve our algorithm under assumptions on the number of distinct elements or collisions and provide matching lower bounds in all regimes, establishing $C$ as a tight complexity measure for the problem. Finally, we consider streaming algorithms for distinct element estimation parameterized by the number of items with frequency larger than $1$. Overall, our results offer insight into why statistical problems with known hardness results can be efficiently solved in practice.