Relative Error Streaming Quantiles

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