TY - JOUR

T1 - Near Isometric Terminal Embeddings for Doubling Metrics

AU - Elkin, Michael

AU - Neiman, Ofer

N1 - Funding Information:
Supported in part by the ISF Grant 1817/17 and BSF grant 2015813.
Funding Information:
This research was supported by the ISF Grant No. (2344/19)
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/11/1

Y1 - 2021/11/1

N2 - Given a metric space (X, d), a set of terminals K⊆ X, and a parameter 0 < ϵ< 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 1 + ϵ, 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 + ϵ and space s(|X|) has its terminal counterpart, with distortion 1 + O(ϵ) and space s(| K|) + n. In particular, for any doubling metric on n points, a set of k terminals, and constant 0 < ϵ< 1 , there existsA spanner with stretch 1 + ϵ for pairs in K× X, with n+ O(k) edges.A labeling scheme with stretch 1 + ϵ for pairs in K× X, with label size ≈ log k.An embedding into ℓ∞d with distortion 1 + ϵ for pairs in K× 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.

AB - Given a metric space (X, d), a set of terminals K⊆ X, and a parameter 0 < ϵ< 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 1 + ϵ, 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 + ϵ and space s(|X|) has its terminal counterpart, with distortion 1 + O(ϵ) and space s(| K|) + n. In particular, for any doubling metric on n points, a set of k terminals, and constant 0 < ϵ< 1 , there existsA spanner with stretch 1 + ϵ for pairs in K× X, with n+ O(k) edges.A labeling scheme with stretch 1 + ϵ for pairs in K× X, with label size ≈ log k.An embedding into ℓ∞d with distortion 1 + ϵ for pairs in K× 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.

UR - http://www.scopus.com/inward/record.url?scp=85111307728&partnerID=8YFLogxK

U2 - 10.1007/s00453-021-00843-6

DO - 10.1007/s00453-021-00843-6

M3 - Article

AN - SCOPUS:85111307728

VL - 83

SP - 3319

EP - 3337

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 11

ER -