Theory of Computing ------------------- Title : A Randomized Online Quantile Summary in $O((1/\varepsilon) \log(1/\varepsilon))$ Words Authors : David Felber and Rafail Ostrovsky Volume : 13 Number : 14 Pages : 1-17 URL : https://theoryofcomputing.org/articles/v013a014 Abstract -------- A quantile summary is a data structure that approximates to $\epsilon$ error the order statistics of a much larger underlying dataset. In this paper we develop a randomized online quantile summary for the cash register data input model and comparison data domain model that uses $O((1/\epsilon)\log(1/\epsilon))$ words of memory. This improves upon the previous best upper bound of $O((1/\epsilon)\log^{3/2}(1/\epsilon))$ by Agarwal et al. (PODS 2012). Further, by a lower bound of Hung and Ting (FAW 2010) no deterministic summary for the comparison model can outperform our randomized summary in terms of space complexity. Lastly, our summary has the nice property that $O((1/\epsilon)\log(1/\epsilon))$ words suffice to ensure that the success probability is at least $1-\exp(-\poly{1/\epsilon})$. A conference version of this paper appeared in the Proceedings of the 19th Internat. Workshop on Randomization and Computation (RANDOM 2015).