An embarrassingly parallel optimal-space cardinality estimation algorithm

TL;DR

This paper introduces a parallelizable algorithm for cardinality estimation that matches the optimal space complexity of previous sequential solutions, enabling efficient distributed processing while simplifying implementation.

Contribution

It presents a new mergeable algorithm that retains optimal space bounds and improves on prior solutions by reducing complexity and resource requirements.

Findings

Achieves optimal space complexity in parallel and sequential settings.

Enables distributed processing through mergeability.

Simplifies implementation with fewer pseudo-random objects.

Abstract

In 2020 Blasiok (ACM Trans. Algorithms 16(2) 3:1-3:28) constructed an optimal space streaming algorithm for the cardinality estimation problem with the space complexity of $\mathcal O(\varepsilon^{-2} \ln(\delta^{-1}) + \ln n)$ where $\varepsilon$, $\delta$ and $n$ denote the relative accuracy, failure probability and universe size, respectively. However, his solution requires the stream to be processed sequentially. On the other hand, there are algorithms that admit a merge operation; they can be used in a distributed setting, allowing parallel processing of sections of the stream, and are highly relevant for large-scale distributed applications. The best-known such algorithm, unfortunately, has a space complexity exceeding $\Omega(\ln(\delta^{-1}) (\varepsilon^{-2} \ln \ln n + \ln n))$. This work presents a new algorithm that improves on the solution by Blasiok, preserving its space complexity, but with the benefit that it admits such a merge operation, thus providing an optimal solution for the problem for both sequential and parallel applications. Orthogonally, the new algorithm also improves algorithmically on Blasiok's solution (even in the sequential setting) by reducing its implementation complexity and requiring fewer distinct pseudo-random objects.