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arXiv · 2208.10578

Simpler and Better Cardinality Estimators for HyperLogLog and PCSA

Seth Pettie·Dingyu Wang
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Abstract

\emph{Cardinality Estimation} (aka \emph{Distinct Elements}) is a classic problem in sketching with many industrial applications. Although sketching \emph{algorithms} are fairly simple, analyzing the cardinality \emph{estimators} is notoriously difficult, and even today the state-of-the-art sketches such as HyperLogLog and (compressed) \PCSA{} are not covered in graduate level Big Data courses. In this paper we define a class of \emph{generalized remaining area} (\tGRA) estimators, and observe that HyperLogLog, LogLog, and some estimators for PCSA are merely instantiations of \tGRA{} for various integral values of $τ$. We then analyze the limiting relative variance of \tGRA{} estimators. It turns out that the standard estimators for HyperLogLog and PCSA can be improved by choosing a \emph{fractional} value of $τ$. The resulting estimators come \emph{very} close to the Cramér-Rao lower bounds for HyperLogLog{} and PCSA derived from their Fisher information. Although the Cramér-Rao lower bound \emph{can} be achieved with the Maximum Likelihood Estimator (MLE), the MLE is cumbersome to compute and dynamically update. In contrast, \tGRA{} estimators are trivial to update in constant time. Our presentation assumes only basic calculus and probability, not any complex analysis~\cite{FlajoletM85,DurandF03,FlajoletFGM07}.

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Seth Pettie, Dingyu Wang. 2022-08-22. Simpler and Better Cardinality Estimators for HyperLogLog and PCSA. https://arxiv.org/abs/2208.10578

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