TY - JOUR

T1 - Low distortion euclidean embeddings of trees

AU - Linial, Nathan

AU - Magen, Avner

AU - Saks, Michael E.

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0032415613&partnerID=8YFLogxK

U2 - 10.1007/BF02773475

DO - 10.1007/BF02773475

M3 - Article

AN - SCOPUS:0032415613

SN - 0021-2172

VL - 106

SP - 339

EP - 348

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -