Near Isometric Terminal Embeddings for Doubling Metrics
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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 in K× 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+ϵ for some small 0<ϵ<1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1+ϵ and size s(|X|) has its terminal counterpart, with distortion 1+O(ϵ) and size s(|K|)+1. In particular, for any doubling metric on n points, a set of k=o(n) terminals, and constant 0<ϵ<1, there exists: (1) A spanner with stretch 1+ϵ for pairs in K× X, with n+o(n) edges. (2) A labeling scheme with stretch 1+ϵ for pairs in K× X, with label size ≈ k. (3) An embedding into ℓ_∞^d with distortion 1+ϵ for pairs in K× X, where d=O( k). Moreover, surprisingly, the last two results apply if only K is a doubling metric, while X can be arbitrary.
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