Improved Bounds with a Simple Algorithm for Edge Estimation for Graphs of Unknown Size

Abstract

We propose a randomized algorithm with query access that given a graph $G$ with arboricity $\alpha$, and average degree $d$, makes $\widetilde{O}\left(\frac{\alpha}{\varepsilon^2d}\right)$ \texttt{Degree} and $\widetilde{O}\left(\frac{1}{\varepsilon^2}\right)$ \texttt{Random Edge} queries to obtain an estimate $\widehat{d}$ satisfying $\widehat{d} \in (1\pm\varepsilon)d$. This improves the $\widetilde{O}_{\varepsilon,\log n}\left(\sqrt{\frac{n}{d}}\right)$ query algorithm of [Beretta et al., SODA 2026] that has access to \texttt{Degree}, \texttt{Neighbour}, and \texttt{Random Edge} queries. Our algorithm does not require any graph parameter as input, not even the size of the vertex set, and attains both simplicity and practicality through a new estimation technique. We complement our upper bounds with a lower bound that shows for all valid $n,d$, and $\alpha$, any algorithm that has access to \texttt{Degree}, \texttt{Neighbour}, and \texttt{Random Edge} queries, must make at least $\Omega\left(\min\left(d,\frac{\alpha}{d}\right)\right)$ queries to obtain a $(1\pm\varepsilon)$-multiplicative estimate of $d$, even with the knowledge of $n$ and $\alpha$. We also show that even with \texttt{Pair} and \texttt{FullNbr} queries, an algorithm must make $\Omega\left(\min\left(d,\frac{\alpha}{d}\right)\right)$ queries to obtain a $(1\pm\varepsilon)$-multiplicative estimate of $d$. Our work addresses both the questions raised by the work of [Beretta et al., SODA 2026].