Distance Reconstruction of Sparse Random Graphs
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916335243493376 |
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| author | Bastide, Paul |
| author_facet | Bastide, Paul |
| contents | 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 $G(n,p)$ Erdős-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 $O( Δ^2 n \log n )$ queries in expectation, where $Δ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17376 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distance Reconstruction of Sparse Random Graphs Bastide, Paul Combinatorics Data Structures and Algorithms 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 $G(n,p)$ Erdős-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 $O( Δ^2 n \log n )$ queries in expectation, where $Δ$ 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. |
| title | Distance Reconstruction of Sparse Random Graphs |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.17376 |