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Estimating the distribution and quantiles of data is a foundational task indata mining and data science. We study algorithms which provide accurateresults for extreme quantile queries using a small amount of space, thushelping to understand the tails of the input distribution. Namely, we focus ontwo recent state-of-the-art solutions: $t$-digest and ReqSketch. While$t$-digest is a popular compact summary which works well in a variety ofsettings, ReqSketch comes with formal accuracy guarantees at the cost of itssize growing as new observations are inserted. In this work, we provide insightinto which conditions make one preferable to the other. Namely, we show how toconstruct inputs for $t$-digest that induce an almost arbitrarily large errorand demonstrate that it fails to provide accurate results even on i.i.d.samples from a highly non-uniform distribution. We propose practicalimprovements to ReqSketch, making it faster than $t$-digest, while its errorstays bounded on any instance. Still, our results confirm that $t$-digestremains more accurate on the ``non-adversarial'' data encountered in practice.

Theory meets Practice at the Median: a worst case comparison of relative error quantile algorithms