Relative Error Streaming Quantiles

Approximating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of $n$ items from a data universe $\mathcal{U}$ equipped with a total order, the task is to compute a sketch (data structure) of size poly$(\log(n), 1/\varepsilon)$. Given the sketch and a query item $y \in \mathcal{U}$, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to $y$. Most works to date focused on additive $\varepsilon n$ error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This paper investigates multiplicative $(1\pm\varepsilon)$-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms due to prior work store either $O(\log(\varepsilon^2 n)/\varepsilon^2)$ or $O(\log^3(\varepsilon n)/\varepsilon)$ universe items. This paper presents a randomized algorithm storing $O(\log^{1.5}(\varepsilon n)/\varepsilon)$ items, which is within an $O(\sqrt{\log(\varepsilon n)})$ factor of optimal. The algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments.