A randomized online quantile summary in O(frac1varepsilon log frac1varepsilon) words

A quantile summary is a data structure that approximates to varepsilon-relative 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(frac1varepsilon log frac1varepsilon) words of memory. This improves upon the previous best upper bound of O(frac1varepsilon log3/2 frac1varepsilon) 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(frac1varepsilon log frac1varepsilon) words suffice to ensure that the success probability is 1 - e-poly(1/varepsilon).