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Distance Reconstruction of Sparse Random Graphs

Paul Bastide

TL;DR

It is shown 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 $G$.

Abstract

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.

Distance Reconstruction of Sparse Random Graphs

TL;DR

It is shown that there exists an algorithm that reconstructs G \sim G(n,p) using queries in expectation, where is the expected average degree of .

Abstract

In the distance query model, we are given access to the vertex set of a -vertex graph , and an oracle that takes as input two vertices and returns the distance between these two vertices in . We study how many queries are needed to reconstruct the edge set of when is sampled according to the Erdős-Renyi-Gilbert distribution. Our approach applies to a large spectrum of values for starting slightly above the connectivity threshold: . We show that there exists an algorithm that reconstructs using queries in expectation, where is the expected average degree of . In particular, for the algorithm uses queries.
Paper Structure (9 sections, 7 theorems, 19 equations, 1 figure)

This paper contains 9 sections, 7 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Distance reconstruction setup
  3. Reconstruction of random graphs
  4. Our contribution
  5. Preliminaries
  6. A Simple Algorithm
  7. Tools
  8. Proof of \ref{['thm:main']}
  9. Acknowledgment

Key Result

Theorem 1.1

For any $\varepsilon \geqslant 0$, and every $n \in \mathbb{N}$, for every $\frac{2000\log n}{n} \leqslant p \leqslant n^{-\frac{1}{2} - \varepsilon}$ , there exists an algorithm that reconstructs $G \sim G(n,p)$ using $O(\Delta^2 n\log n)$ queries in expectation, where $\Delta = (n-1)p$ is the exp

Figures (1)

  • Figure 1: Representation of the three parts $\mathcal{B}^k_1,\mathcal{B}^k_2,\mathcal{B}^k_3$ that compose $B_k$

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1: Chernoff's bound chernoff1952measure
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Lemma 3.2: Main Lemma
  • proof
  • ...and 13 more