### Abstract

Let (X,d
_{X}) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f: X → T such that for every x ∈ X, → where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.

Original language | English (US) |
---|---|

Pages (from-to) | 581-615 |

Number of pages | 35 |

Journal | Combinatorica |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2010 |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*30*(5), 581-615. https://doi.org/10.1007/s00493-010-2302-z

**Maximum gradient embeddings and monotone clustering.** / Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 30, no. 5, pp. 581-615. https://doi.org/10.1007/s00493-010-2302-z

}

TY - JOUR

T1 - Maximum gradient embeddings and monotone clustering

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2010/9

Y1 - 2010/9

N2 - Let (X,d X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f: X → T such that for every x ∈ X, → where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.

AB - Let (X,d X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f: X → T such that for every x ∈ X, → where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.

UR - http://www.scopus.com/inward/record.url?scp=79952549793&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952549793&partnerID=8YFLogxK

U2 - 10.1007/s00493-010-2302-z

DO - 10.1007/s00493-010-2302-z

M3 - Article

VL - 30

SP - 581

EP - 615

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 5

ER -