Distance Reconstruction of Sparse Random Graphs
<|reference_start|>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 $O( \Delta^2 n \log n )$ queries in expectation, where $\Delta$ is the expected average degree of $G$. In particular, for $p \in [\frac{2000 \log n}{n}, \frac{\log^2 n}{n}]$ the algorithm uses $O(n \log^5 n)$ queries.<|reference_end|>
@article{bastide2024distance,
    title={Distance Reconstruction of Sparse Random Graphs},
    author={Paul Bastide},
    journal={arXiv preprint arXiv:2407.17376},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.17376},
    primaryClass={math.CO cs.DS}
}