### 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 (US) |
---|---|

Pages (from-to) | 1180-1190 |

Number of pages | 11 |

Journal | European Journal of Combinatorics |

Volume | 26 |

Issue number | 8 |

DOIs | |

State | Published - Nov 2005 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*European Journal of Combinatorics*,

*26*(8), 1180-1190. https://doi.org/10.1016/j.ejc.2004.07.002

**Metric structures in L1 : Dimension, snowflakes, and average distortion.** / Lee, James R.; Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*European Journal of Combinatorics*, vol. 26, no. 8, pp. 1180-1190. https://doi.org/10.1016/j.ejc.2004.07.002

}

TY - JOUR

T1 - Metric structures in L1

T2 - Dimension, snowflakes, and average distortion

AU - Lee, James R.

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2005/11

Y1 - 2005/11

N2 - We study the metric properties of finite subsets of L1. 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 L1. We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 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 Lp for p ∈. (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the ℓ1 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 ℓ1, 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.

AB - We study the metric properties of finite subsets of L1. 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 L1. We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 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 Lp for p ∈. (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the ℓ1 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 ℓ1, 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.

UR - http://www.scopus.com/inward/record.url?scp=23944474318&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23944474318&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2004.07.002

DO - 10.1016/j.ejc.2004.07.002

M3 - Article

VL - 26

SP - 1180

EP - 1190

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 8

ER -