Near Isometric Terminal Embeddings for Doubling Metrics

Abstract

Given a metric space (X,d), a set of terminals K subseteq X, and a parameter t >= 1, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in K x X up to a factor of t, 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, i.e., t=1+epsilon for some small 0<epsilon<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 size s(|X|) has its terminal counterpart, with distortion 1+O(epsilon) and size s(|K|)+1. In particular, for any doubling metric on n points, a set of k=o(n) terminals, and constant 0<epsilon<1, there exists
- A spanner with stretch 1+epsilon for pairs in K x X, with n+o(n) edges.
- A labeling scheme with stretch 1+epsilon for pairs in K x X, with label size ~~ log k.
- An embedding into l_infty^d with distortion 1+epsilon for pairs in K x X, where d=O(log k). Moreover, surprisingly, the last two results apply if only K is a doubling metric, while X can be arbitrary.