Low distortion euclidean embeddings of trees

Nathan Linial*, Avner Magen, Michael E. Saks

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

We consider the problem of embedding a certain finite metric space to the Euclidean space, trying to keep the bi-Lipschitz constant as small as possible. We introduce the notation c2(X, d) for the least distortion with which the metric space (X, d) may be embedded in a Euclidean space It is known that if (X, d) is a metric space with n points, then c2(X, d) ≤ 0(log n) and the bound is tight. Let T be a tree with n vertices, and d be the metric induced by it. We show that c2(T, d) ≤ 0(log log n), that is we provide an embedding f of its vertices to the Euclidean space, such that d(x, y) ≤ ∥f(x) - f(y)∥ ≤ c log log nd(x, y) for some constant c.

Original languageAmerican English
Pages (from-to)339-348
Number of pages10
JournalIsrael Journal of Mathematics
Volume106
DOIs
StatePublished - 1998

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