🤖 AI Summary
This study addresses the problem of efficiently approximating the number of distinct elements in large-scale data streams. Revisiting the HyperLogLog algorithm from a probabilistic perspective, it establishes—for the first time—the non-asymptotic, explicit exponential concentration inequality for the estimator using elementary probability methods. This work breaks away from the traditional asymptotic analysis framework that relies on Poissonization and Mellin transforms, instead deriving rigorous finite-sample error bounds. The resulting bounds significantly enhance the theoretical reliability and error controllability of HyperLogLog in practical applications.
📝 Abstract
HyperLogLog is a now classic probabilistic algorithm that provides an approximation of the number of distinct elements in a massive dataset, using only one pass over the data. In the original article, Flajolet, Fusy, Gandouet, Meunier (2007) provided a sharp analysis of the expectation and variance of the output, using explicit formulas analyzed using poissonization and Mellin transform. In this short article, we revisit the analysis of HyperLogLog with a more probabilistic viewpoint. This allows us to establish exponential deviation inequalities for the HyperLogLog estimator. The methods are elementary, but the estimates are non-asymptotic and totally explicit.