Terminal embeddings

Michael Elkin, Arnold Filtser, Ofer Neiman

Research output: Contribution to journalArticlepeer-review

31 Scopus citations


In this paper we study terminal embeddings, in which one is given a finite metric (X,dX) (or a graph G=(V,E)) and a subset K⊆X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve ≈|K|⋅|X| pairs, the distortion depends only on |K|, rather than on |X|. We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X×X and with respect to K×X. Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [10] devised an O˜(log⁡r)-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an O˜(log⁡|K|)- approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K|≤r, our bound generalizes that of [10].

Original languageEnglish
Pages (from-to)1-36
Number of pages36
JournalTheoretical Computer Science
StatePublished - 12 Oct 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 Elsevier B.V.


  • Distortion
  • Embedding
  • Terminals


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