Learning random points from geometric graphs or orderings

Abstract

Suppose that there is a family of $n$ random points $X_v$ for $v \in V$, independently and uniformly distributed in the square $\left[-\sqrt{n}/2,\sqrt{n}/2\right]^2$ of area $n$. We do not see these points, but learn about them in one of the following two ways. Suppose first that we are given the corresponding random geometric graph $G$, where distinct vertices $u$ and $v$ are adjacent when the Euclidean distance $d_E(X_u,X_v)$ is at most $r$. If the threshold distance $r$ satisfies $n^{3/14} \ll r \ll n^{1/2}$, then the following holds with high probability. Given the graph $G$ (without any geometric information), in polynomial time we can approximately reconstruct the hidden embedding, in the sense that, `up to symmetries', for each vertex $v$ we find a point within distance about $r$ of $X_v$; that is, we find an embedding with `displacement' at most about $r$. Now suppose that, instead of being given the graph $G$, we are given, for each vertex $v$, the ordering of the other vertices by increasing Euclidean distance from $v$. Then, with high probability, in polynomial time we can find an embedding with the much smaller displacement error $O(\sqrt{\log n})$.