## Abstract

We study the metric properties of finite subsets of L_{1}. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L_{1}. We present some new observations concerning the relation of L_{1} to dimension, topology, and Euclidean distortion. We show that every n-point subset of L_{1} embeds into L_{2} with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in L_{p} for p ∈. (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the ℓ_{1} norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251-260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in ℓ_{1}, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514-523], cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.

Original language | English |
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Pages (from-to) | 1180-1190 |

Number of pages | 11 |

Journal | European Journal of Combinatorics |

Volume | 26 |

Issue number | 8 |

DOIs | |

State | Published - Nov 2005 |

Externally published | Yes |

### Bibliographical note

Funding Information:The first author’s work was partially supported by NSF grant CCR-0121555 and an NSF Graduate Research Fellowship. Part of that work was done while the author was an intern at Microsoft Research. The work of the second author was done as part of a post-doc fellowship at The Hebrew University, and was supported in part by the Landau Center and by a grant from the Israeli Science Foundation (195/02).

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