Reconstructing random graphs from distance queries

TL;DR

This paper determines the minimal number of distance queries needed to reconstruct binomial random graphs with constant diameter, revealing non-monotone behavior of query complexity relative to graph diameter.

Contribution

It provides tight bounds on query complexity for reconstructing random graphs and introduces an optimal non-adaptive reconstruction algorithm.

Findings

Query complexity equals Θ(n^{4-d}p^{2-d}) for certain p values.

Query complexity exhibits non-monotone behavior with respect to graph diameter.

Existence of an optimal non-adaptive reconstruction algorithm with O(n^{4-d}p^{2-d} log n) queries.

Abstract

We estimate the minimum number of distance queries that is sufficient to reconstruct the binomial random graph $G(n,p)$ with constant diameter with high probability. We get a tight (up to a constant factor) answer for all $p>n^{-1+o(1)}$ outside "threshold windows" around $n^{-k/(k+1)+o(1)}$, $k\in\mathbb{Z}_{>0}$: with high probability the query complexity equals $\Theta(n^{4-d}p^{2-d})$, where $d$ is the diameter of the random graph. This demonstrates the following non-monotone behaviour: the query complexity jumps down at moments when the diameter gets larger; yet, between these moments the query complexity grows. We also show that there exists a non-adaptive algorithm that reconstructs the random graph with $O(n^{4-d}p^{2-d}\ln n)$ distance queries with high probability, and this is best possible.