Thursday, 19.12.2019, 12:30
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 U (equipped with a total order), the task is to compute a sketch (data structure) of size poly(log(n), 1/ε). Given the sketch and a query item y ∈ U, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than y.
Most works to date focused on additive εn error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This paper investigates multiplicative (1±ε)-error approximations to the rank. The motivation stems from practical demand to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. Despite its great practical importance, the most space-efficient algorithm that can be derived from prior work stores O(polylog(n)/ε^2) universe items. Typically, ε is small enough to render an algorithm with squared dependence impractical. This paper presents the first sketch of size O(polylog(n)/ε) that achieves a 1±ε multiplicative error guarantee, without prior knowledge of the stream length or dependence on the size of the data universe.