Near Isometric Terminal Embeddings for Doubling Metrics

Given a metric space (X, d), a set of terminals K ⊆ 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 inK ×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 + ǫ for some small 0 < ǫ < 1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1+ǫ and size s(|X |) has its terminal counterpart, with distortion 1+O(ǫ) and size s(|K|) + 1. In particular, for any doubling metric on n points, a set of k = o(n) terminals, and constant 0 < ǫ < 1, there exists • A spanner with stretch 1 + ǫ for pairs inK ×X , with n+ o(n) edges. • A labeling scheme with stretch 1 + ǫ for pairs inK ×X , with label size ≈ log k. • An embedding into l ∞ with distortion 1 + ǫ for pairs in K ×X , where d = O(log k). Moreover, surprisingly, the last two results apply if onlyK is a doublingmetric, whileX can be arbitrary.