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.

翻译：在数据分析和监测中, 排序、 量级和流数据分布是一项核心任务 。 如果来自数据宇宙的 $\ mathcal{U} 以总顺序配置的 $\ mathcal{U} $, 任务在于计算一个大小( log( n), 1/\ varepsilon) $ 的草图( 数据结构) 。 鉴于草图和查询项 $y \ mathcal{U}, 一个人应该能够大约其在流中的位置, 也就是说, 流要素小于或等于$$。 大部分工作到日期的重心都集中在 $\ varepal= n$ 错误的添加 $\ vol, 最终的 KLL 草图( 数据结构) 达到最佳的亚值行为。 本文调查的是多复制$(1\ pm\ varepsil) $- orororborization 。 多复制错误的实际动机来自了解发行的( liveralalralalal) ral ral ral2) nqal= dal= dalslus= dalmaxx