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
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A spanner with stretch \(1+\epsilon\) for pairs in \(K\times X\), with \(n+O(k)\) edges.
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A labeling scheme with stretch \(1+\epsilon\) for pairs in \(K\times X\), with label size \(\approx \log k\).
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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.
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Notes
By “high-dimensional” we mean here typically dimension \(\log n\) or greater.
See [32] for definition of tree-width.
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Acknowledgements
We are grateful to Paz Carmi for fruitful discussions.
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A preliminary version of this paper appears in SOCG’18
This research was supported by the ISF Grant No. (2344/19)
Supported in part by the ISF Grant 1817/17 and BSF grant 2015813.
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Elkin, M., Neiman, O. Near Isometric Terminal Embeddings for Doubling Metrics. Algorithmica 83, 3319–3337 (2021). https://doi.org/10.1007/s00453-021-00843-6
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DOI: https://doi.org/10.1007/s00453-021-00843-6