Low Distortion Euclidean Embeddings of Trees
نویسنده
چکیده
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(logn) 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(loglogn), that is we provide an embedding f of its vertices to the Euclidean space, such that d(x,y) <_ Ill(x) f(Y)l[ -< c log log nd(x, y) for some constant c. 1. I n t r o d u c t i o n E m b e d d i n g s of f in i te m e t r i c spaces in to n o r m e d spaces a r e of in t e res t in t h e local t h e o r y of B a n a c h spaces ([2, 3, 4]), in c o m b i n a t o r i c s (e.g. [1] a n d t h e re fe rences t he r e in ) a n d in t h e t h e o r y of a l g o r i t h m s [5]. In o rde r to s t u d y a gene ra l me t r i c , * Suppor ted in par t by grants from the Israeli Academy of Sciences and the US-Israe l Binat ional Science Foundation. ** Suppor ted in part by NSF under grants CCR-9215293 and by DIMACS, which is suppor ted by NSF grant STC-91-19999 and by the New Jersey Commission on Science and Technology. Received June 3, 1997
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