Computing the approximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream of elements x1,x2,. ..,xn and a query x, a relative-error quantile estimation algorithm can estimate the rank of x with respect to the stream, up to a multiplicative ±∈ · rank(x ) error. Notably, this requires the sketch to obtain more precise estimates for the ranks of elements on the tails of the distribution, as compared to the additive ±en error regime. This is particularly favorable for some practical applications, such as anomaly detection.

The majority of streaming problems are defined and analyzed in a static setting, where the data stream is any worst-case sequence of insertions and deletions which is fixed in advance. However, many real-world applications require a more flexible model, where an adaptive adversary may select future stream elements after observing the previous outputs of the algorithm. Over the last few years, there has been increased interest in proving lower bounds for natural problems in the adaptive streaming model. In this work, we give the first known adaptive attack against linear sketches for the well-studied$\ell_{0}$-estimation problem over turnstile, integer streams. For any linear streaming algorithm$\mathcal{A}$which uses sketching matrix$\mathbf{A}\varepsilon \mathbb{Z}^{r\times n}$, this attack makes$\tilde{\mathcal{O}}(r^{8})$queries and succeeds with high constant probability in breaking the sketch. Additionally, we give an adaptive attack against linear sketches for the$\ell_{0}$-estimation problem over finite fields$\mathbb{F}_{p}$, which requires a smaller number of$\tilde{\mathcal{O}}(r^{3})$queries. Finally, we provide an adaptive attack over$\mathbb{R}^{n}$against linear sketches A$\in \mathbb{R}^{r\times \mathfrak{n}}$for$\ell_{0}$-estimation, in the setting where A has all nonzero subdeterminants at least$\frac{1}{\text{poly}(r)}$. Our results provide an exponential improvement over the previous number of queries known to break an$\ell_{0}$-estimation sketch.

Estimating the ε-approximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream x_1,..., x_n from a universe \mathcalU with total order, an additive-error quantile sketch \mathcalM allows us to approximate the rank of any query y\in \mathcalU up to additive ε n error. In 2001, Greenwald and Khanna gave a deterministic algorithm (GK sketch) that solves the ε-approximate quantiles estimation problem using O(ε^-1 łog(ε n)) space \citegreenwald2001space ; recently, this algorithm was shown to be optimal by Cormode and Vesleý in 2020 \citecormode2020tight. However, due to the intricacy of the GK sketch and its analysis, over-simplified versions of the algorithm are implemented in practical applications, often without any known theoretical guarantees. In fact, it has remained an open question whether the GK sketch can be simplified while maintaining the optimal space bound. In this paper, we resolve this open question by giving a simplified deterministic algorithm that stores at most (2 + o(1))ε^-1 łog (ε n) elements and solves the additive-error quantile estimation problem; as a side benefit, our algorithm achieves a smaller constant factor than the \frac11 2 ε^-1 łog(ε n) space bound in the original GK sketch~\citegreenwald2001space. Our algorithm features an easier analysis and still achieves the same optimal asymptotic space complexity as the original GK sketch. Lastly, our simplification enables an efficient data structure implementation, with a worst-case runtime of O(łog(1/ε) + łog łog (ε n)) per-element for the ordinary ε-approximate quantile estimation problem. Also, for the related "weighted'' quantile estimation problem, we give efficient data structures for our simplified algorithm which guarantee a worst-case per-element runtime of O(łog(1/ε) + łog łog (ε W_n/w_\textrmmin )), achieving an improvement over the previous upper bound of \citeassadi2023generalizing.