Simpler and Better Cardinality Estimators for HyperLogLog and PCSA

Seth Pettie; Dingyu Wang

Home/Archive/arXiv

arXiv · 2208.10578

Simpler and Better Cardinality Estimators for HyperLogLog and PCSA

Seth Pettie·Dingyu Wang

View original source ↗Read source document ↗Export metadata ↓

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 $\tau$. 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 $\tau$. The resulting estimators come \emph{very} close to the Cram\'{e}r-Rao lower bounds for HyperLogLog{} and PCSA derived from their Fisher information. Although the Cram\'{e}r-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}.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗