Close the space gap for fully mergeable relative-error quantile sketches

Determine whether the $\log\log(\varepsilon N)$ and $\log \varepsilon^{-1}$ factors in the space bound of the fully mergeable randomized comparison-based relative-error quantile sketch can be eliminated.

Background

The proposed sketch is within multiplicative factors of the known lower bound for randomized comparison-based relative-error quantile summaries, but its space complexity still contains additional factors involving log log(εN), log ε^{-1}, and log log δ^{-1}.

The authors regard removal of the log log(εN) and log ε^{-1} factors as plausible but unresolved. They distinguish this question from the dependence on δ, whose improvement may require progress on deterministic comparison-based algorithms.