## Abstract

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to J. Bourgain and S. Rao. We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O(√α _{X} · log n), where α _{X} is a geometric estimate on the decomposability of X. An an immediate corollary, we obtain an O(√log λ _{X} · log n ) distortion embedding, where λ _{X} is the doubling constant of X. Since λ _{X} ≤ n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, "low-dimensional," improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(logn)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in l _{∞} ^{O(log n)},with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O(log ^{2} n).

Original language | English (US) |
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Pages (from-to) | 434-443 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

State | Published - 2004 |

Externally published | Yes |

Event | Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 - Rome, Italy Duration: Oct 17 2004 → Oct 19 2004 |

## All Science Journal Classification (ASJC) codes

- General Engineering