## G. Cormode, Z. Karnin, E. Liberty, J. Thaler, and Veselý.
Relative error streaming quantiles.
*SIGMOD Record*, 51(1):66-79, Mar. 2022.

Estimating 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 equipped
with a total order, the task is to compute a sketch (data
structure) of size polylogarithmic in *n*. Given the sketch and
a query item *y*, 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 ε*n* error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This paper investigates multiplicative (1ε)-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(ε^{2}*n*)/ε^{2}) or *O*(log^{3}(ε*n*)/ε) universe items. We present a randomized sketch storing *O*(log^{1.5}(ε*n*)/ε) items,
which is within an *O*(sqrt(log(ε*n*))) factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments.

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