Reconstructing random graphs from distance 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 $\hat{I}˜(n^{4-d}{p}^{2-d})$, 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.