论文信息 - Embedding Tree Metrics into Low-Dimensional Euclidean Spaces

Embedding Tree Metrics into Low-Dimensional Euclidean Spaces

Abstract. We consider embedding metrics induced by trees into Euclidean spaces with a restricted number of dimensions. We show that any weighted tree T with n vertices and L leaves can be embedded into d -dimensional Euclidean space with Õ (L1/(d-1)) distortion. Furthermore, we exhibit an embedding with almost the same distortion which can be computed efficiently. This distortion substantially improves the previous best upper bound of \tilde O (n2/d) and almost matches the best known lower bound of Ω(L1/d) .

Anupam Gupta

[1] J. Matousek,et al. On embedding trees into uniformly convex Banach spaces , 1999 .

[2] Michael E. Saks,et al. Trees and Euclidean metrics , 1998, STOC '98.

[3] J. Matousek,et al. Bi-Lipschitz embeddings into low-dimensional Euclidean spaces , 1990 .

[4] Piotr Indyk,et al. Approximate nearest neighbors: towards removing the curse of dimensionality , 1998, STOC '98.

[5] Noga Alon,et al. The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[6] J. Bourgain. The metrical interpretation of superreflexivity in banach spaces , 1986 .

[7] N. J. A. Sloane,et al. Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.