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

##### Graham Cormode, Abhinav Mishra, Joseph Ross, Pavel Veselý

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

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