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arXiv · 2506.22608

On Fine-Grained Distinct Element Estimation

Ilias Diakonikolas·Daniel M. Kane·Jasper C. H. Lee·Thanasis Pittas·David P. Woodruff·Samson Zhou
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Abstract

We study the problem of distributed distinct element estimation, where $α$ 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 $Θ\left(α\log n+\fracα{\varepsilon^2}\right)$ bits, these results rely on assumptions that may not hold in practice. We introduce a new parameterization based on the number $C = \fracβ{\varepsilon^2}$ of pairwise collisions, i.e., instances where the same element appears on multiple servers, and design a protocol that uses only $\mathcal{O}\left(α\log n+\frac{\sqrtβ}{\varepsilon^2} \log n\right)$ 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.

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Ilias Diakonikolas, Daniel M. Kane, Jasper C. H. Lee, Thanasis Pittas, David P. Woodruff, Samson Zhou. 2025-06-27. On Fine-Grained Distinct Element Estimation. https://arxiv.org/abs/2506.22608

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