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

• URL: http://arxiv.org/abs/2505.23682v1
• Date: Thu, 29 May 2025 17:21:20 GMT
• Title: Differentially Private Space-Efficient Algorithms for Counting Distinct Elements in the Turnstile Model
• Authors: Rachel Cummings, Alessandro Epasto, Jieming Mao, Tamalika Mukherjee, Tingting Ou, Peilin Zhong,
• Abstract summary: We give the first sublinear space differentially private algorithms for the fundamental problem of counting distinct elements in the turnstile streaming model.<n>Our result significantly improves upon the space requirements of the state-of-the-art algorithms for this problem, which is linear.<n>When a bound $W$ on the number of times an item appears in the stream is known, our algorithm provides $tildeO_eta(sqrtW)$ additive error using $tildeO_eta(sqrtW)$ space.
• Score: 61.40326886123332
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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