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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 (L 1/(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 (n 2/d ) and almost matches the best known lower bound of Ω(L 1/d ) .

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Received August 17, 1999, and in revised form January 25, 2000.

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Gupta, A. Embedding Tree Metrics into Low-Dimensional Euclidean Spaces . Discrete Comput Geom 24, 105–116 (2000). https://doi.org/10.1007/s004540010020

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Keywords

  • Euclidean Space
  • Restricted Number
  • Tree Metrics
  • Dimensional Euclidean Space
  • Weighted Tree