Near Isometric Terminal Embeddings for Doubling Metrics

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