An $\varepsilon$-approximate quantile sketch over a stream of $n$ inputs approximates the rank of any query point $q$ - that is, the number of input points less than $q$ - up to an additive error of $\varepsilon n$, generally with some probability of at least $1 - 1/\mathrm{poly}(n)$, while consuming $o(n)$ space. While the celebrated KLL sketch of Karnin, Lang, and Liberty achieves a provably optimal quantile approximation algorithm over worst-case streams, the approximations it achieves in practice are often far from optimal. Indeed, the most commonly used technique in practice is Dunning's t-digest, which often achieves much better approximations than KLL on real-world data but is known to have arbitrarily large errors in the worst case. We apply interpolation techniques to the streaming quantiles problem to attempt to achieve better approximations on real-world data sets than KLL while maintaining similar guarantees in the worst case.

翻译：一个针对 $n$ 个输入流的 $\varepsilon$-近似分位数草图，能以加性误差 $\varepsilon n$ 近似任意查询点 $q$ 的秩（即小于 $q$ 的输入点数量），通常具有至少 $1 - 1/\mathrm{poly}(n)$ 的概率，同时消耗 $o(n)$ 空间。尽管 Karnin、Lang 和 Liberty 提出的著名 KLL 草图在最坏情况流上实现了可证明最优的分位数逼近算法，但其实际逼近效果往往远非最优。事实上，实践中使用最广泛的技术是 Dunning 的 t-digest，它在真实数据上通常比 KLL 获得好得多的近似值，但已知在最坏情况下会产生任意大的误差。我们将插值技术应用于流式分位数问题，旨在在保持类似于 KLL 的最坏情况保证的同时，在真实数据集上实现比 KLL 更好的近似效果。