On graphically local versions of metric embeddings

We consider the problem of graphically local metric embedding, i.e. embedding points from an arbitrary finite metric space into a target metric space while preserving, up to a small distortion, only a subset of the pairwise distances specified by a bounded degree graph $G$. We provide a general reduction showing that, in many cases, this is no easier than embedding the points while approximately preserving all pairwise distances. As an illustration of our general reduction, we show that there exists a Euclidean metric space $X$ on $n$ points along with a graph $G = (X,E)$ of maximum degree $3$ such that any embedding of $X$ into $\ell_2^m$ which only preserves distances specified by $E$ up to a relative error of $(1+\varepsilon)$ must satisfy $m = Ω(\varepsilon^{-2}\log n)$. Our lower bound matches the upper bound on the dimension coming from the Johnson-Lindenstrauss lemma for approximately preserving all pairwise distances; previously, such a lower bound was known only for the class of noncontracting embeddings [Schechtman-Shraibman, Discrete & Computational Geometry, 2009]. Moreover, the condition that the maximum degree of the graph is $3$ is best possible: for graphs $G$ of maximum degree $2$ (or more generally, treewidth at most $2$), any metric space embeds $G$-isometrically into any two-dimensional normed space.