Differentially Private Space-Efficient Algorithms for Counting Distinct Elements in the Turnstile Model

TL;DR

This paper introduces the first space-efficient differentially private algorithms for counting distinct elements in turnstile streams, significantly reducing space requirements while maintaining accuracy, and addressing open research questions.

Contribution

It presents the first sublinear space differentially private algorithms for counting distinct elements in turnstile streams, improving upon previous linear-space methods.

Findings

Achieves $ ilde{O}_{ heta}(T^{1/3})$ space with additive error for arbitrary streams.

Provides $ ilde{O}_{ heta}( oot{2} ot W)$ space and error when a bound on item frequency is known.

Establishes a space lower bound of $ ilde{ ot heta}(T^{1/3})$ for this problem.

Abstract

The turnstile continual release model of differential privacy captures scenarios where a privacy-preserving real-time analysis is sought for a dataset evolving through additions and deletions. In typical applications of real-time data analysis, both the length of the stream $T$ and the size of the universe $|U|$ from which data come can be extremely large. This motivates the study of private algorithms in the turnstile setting using space sublinear in both $T$ and $|U|$. In this paper, we give the first sublinear space differentially private algorithms for the fundamental problem of counting distinct elements in the turnstile streaming model. Our algorithm achieves, on arbitrary streams, $\tilde{O}_{\eta}(T^{1/3})$ space and additive error, and a $(1+\eta)$-relative approximation for all $\eta \in (0,1)$. Our result significantly improves upon the space requirements of the state-of-the-art algorithms for this problem, which is linear, approaching the known $\Omega(T^{1/4})$ additive error lower bound for arbitrary streams. Moreover, when a bound $W$ on the number of times an item appears in the stream is known, our algorithm provides $\tilde{O}_{\eta}(\sqrt{W})$ additive error, using $\tilde{O}_{\eta}(\sqrt{W})$ space. This additive error asymptotically matches that of prior work which required instead linear space. Our results address an open question posed by [Jain, Kalemaj, Raskhodnikova, Sivakumar, Smith, Neurips23] about designing low-memory mechanisms for this problem. We complement these results with a space lower bound for this problem, which shows that any algorithm that uses similar techniques must use space $\tilde{\Omega}(T^{1/3})$ on arbitrary streams.