Optimal quantile estimation: beyond the comparison model (2404.03847v1)

Abstract: Estimating quantiles is one of the foundational problems of data sketching. Given $n$ elements $x_1, x_2, \dots, x_n$ from some universe of size $U$ arriving in a data stream, a quantile sketch estimates the rank of any element with additive error at most $\varepsilon n$. A low-space algorithm solving this task has applications in database systems, network measurement, load balancing, and many other practical scenarios. Current quantile estimation algorithms described as optimal include the GK sketch (Greenwald and Khanna 2001) using $O(\varepsilon^{-1} \log n)$ words (deterministic) and the KLL sketch (Karnin, Lang, and Liberty 2016) using $O(\varepsilon^{-1} \log\log(1/\delta))$ words (randomized, with failure probability $\delta$). However, both algorithms are only optimal in the comparison-based model, whereas most typical applications involve streams of integers that the sketch can use aside from making comparisons. If we go beyond the comparison-based model, the deterministic q-digest sketch (Shrivastava, Buragohain, Agrawal, and Suri 2004) achieves a space complexity of $O(\varepsilon^{-1}\log U)$ words, which is incomparable to the previously-mentioned sketches. It has long been asked whether there is a quantile sketch using $O(\varepsilon^{-1})$ words of space (which is optimal as long as $n \leq \mathrm{poly}(U)$). In this work, we present a deterministic algorithm using $O(\varepsilon^{-1})$ words, resolving this line of work.

• The paper introduces a deterministic algorithm that transcends traditional comparison-based models by achieving optimal space complexity in quantile estimation.
• It employs a forest-based, tree-like data structure combined with approximate counting to reduce overhead in data stream processing.
• The approach enhances practical applications in systems like Apache DataSketches, ensuring robust, reliable performance in adversarial environments.

Optimal Quantile Estimation: Beyond the Comparison Model

This paper presents a new quantile estimation algorithm that breaks through the traditional comparison-based model to achieve optimal space complexity in terms of additive error approximation. Specifically, it addresses the problem of quantile sketching with a focus on optimizing space usage. Existing solutions to this problem, namely the GK sketch and KLL sketch, are described as optimal but only within the confines of a comparison-based model, which limits their application to scenarios involving only comparisons between data elements rather than dealing directly with the universe from which the elements are drawn.

The proposed algorithm innovatively utilizes a deterministic approach to achieve space complexity of $O(^{-1})$ words by incorporating structural modifications to the tree-like data representation. The use of a forest instead of a single tree in the recursive layer allows the algorithm to save substantial space, especially at lower layers. Leveraging this new tree structure, the authors demonstrate a concise method for storing non-empty nodes, which when combined with a strategy for approximate counting of elements, reduces unnecessary overhead.

Theoretical implications of this research center on the considered model where typical data streams consist of integers—potentially having elements from universe sizes much smaller than the streams themselves. This context allows for improvements over purely comparison-based sketches by taking advantage of properties characterizing streams of integers. By achieving the space complexity of $O(^{-1})$, the authors essentially resolve a longstanding open question regarding the optimality of quantile sketches beyond the bounds of comparison-based limitations.

The deterministic nature of this algorithm not only contributes to reducing the complexity in realistic data stream environments but also enhances robustness against adversarial data streams—a feature less reliably supported by randomized sketches such as KLL. Furthermore, this deterministic characteristic ensures reliability across a broader spectrum of practical applications where there is a need for consistent accuracy and performance, such as large database management systems and network traffic analysis platforms.

From a practical standpoint, this work may enable the efficient integration of accurate quantile estimation in onboarded systems like Apache DataSketches or GoogleSQL. The implications include not only improved metrics for operation efficiency or network load management but also a shift towards embedding more sophisticated data processing techniques directly within consumer-grade systems, possibly democratizing access to complex statistical computations.

The role of data universality is particularly interesting in this paper, suggesting potential exploration in adaptation or integration of this work with existing frameworks that manage diverse multi-source data streams. Additionally, the partially mergeable nature of the algorithm presents a further avenue for research on its extensive applicability to distributed systems and big data platforms where scalability and resource optimization are paramount.

In conclusion, the contribution this paper makes highlights an advancement in our understanding of quantile estimation beyond comparison-based models, potentially setting a new baseline in the field with improved structural designs and deterministic methodologies. The potential for further development and integration suggests this work could form a foundational base for future research exploring the extension of statistical estimation into dynamic and resource-constrained environments.