Distance Reconstruction of Sparse Random Graphs

In the distance query model, we are given access to the vertex set of a $n$-vertex graph $G$, and an oracle that takes as input two vertices and returns the distance between these two vertices in $G$. We study how many queries are needed to reconstruct the edge set of $G$ when $G$ is sampled according to the $G(n,p)$ Erd\H{o}s-Renyi-Gilbert distribution. Our approach applies to a large spectrum of values for $p$ starting slightly above the connectivity threshold: $p \geq \frac{2000 \log n}{n}$. We show that there exists an algorithm that reconstructs $G \sim G(n,p)$ using $O( \Delta² n \log n )$ queries in expectation, where $\Delta$ is the expected average degree of $G$. In particular, for $p \in [\frac{2000 \log n}{n}, \frac{\log² n}{n}]$ the algorithm uses $O(n \log⁵ n)$ queries.