Near Isometric Terminal Embeddings for Doubling Metrics

Abstract

Given a metric space (X, d), a set of terminals $K\subseteq X$, and a parameter $0<\epsilon <1$, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in $K\times X$ up to a factor of $1+\epsilon$, and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion $1+\epsilon$ and space s(|X|) has its terminal counterpart, with distortion $1+O(\epsilon )$ and space $s(|K|)+n$. In particular, for any doubling metric on n points, a set of k terminals, and constant $0<\epsilon <1$, there exists

• A spanner with stretch $1+\epsilon$ for pairs in $K\times X$, with $n+O(k)$ edges.

• A labeling scheme with stretch $1+\epsilon$ for pairs in $K\times X$, with label size $\approx \log k$.

• An embedding into $\ell _\infty ^d$ with distortion $1+\epsilon$ for pairs in $K\times X$, where $d=O(\log k)$.

Moreover, surprisingly, the last two results apply if only the metric space on K is doubling, while the metric on X can be arbitrary.