### Abstract

Given a metric space (X, d_{X}), c ≥ 1, r > 0, and p,q ∈ [0,1], a distribution over mappings ℋ: X → ℕ is called a (r,cr,p,g)-sensitive hash family if any two points in X at distance at most r are mapped by ℋ to the same value with probability at least p, and any two points at distance greater than cr are mapped by ℋ to the same value with probability at most q. This notion was introduced by Indyk and Motwani in 1998 as the basis for an efficient approximate nearest neighbor search algorithm, and has since been used extensively for this purpose. The performance of these algorithms is governed by the parameter ρ = log(1/p)/log(1/q), and constructing hash families with small ρ automatically yields improved nearest neighbor algorithms. Here we show that for X = ℓ_{1} it is impossible to achieve ρ ≤ 1/2c. This almost matches the construction of Indyk and Motwani which achieves ρ ≤ 1/c.

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

Title of host publication | Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06 |

Publisher | Association for Computing Machinery |

Pages | 154-157 |

Number of pages | 4 |

ISBN (Print) | 1595933409, 9781595933409 |

DOIs | |

State | Published - 2006 |

Externally published | Yes |

Event | 22nd Annual Symposium on Computational Geometry 2006, SCG'06 - Sedona, AZ, United States Duration: Jun 5 2006 → Jun 7 2006 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
---|---|

Volume | 2006 |

### Other

Other | 22nd Annual Symposium on Computational Geometry 2006, SCG'06 |
---|---|

Country | United States |

City | Sedona, AZ |

Period | 6/5/06 → 6/7/06 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Keywords

- Locality Sensitive Hashing
- Lower Bounds
- Nearest Neighbor Search

## Fingerprint Dive into the research topics of 'Lower bounds on locality sensitive hashing'. Together they form a unique fingerprint.

## Cite this

*Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06*(pp. 154-157). (Proceedings of the Annual Symposium on Computational Geometry; Vol. 2006). Association for Computing Machinery. https://doi.org/10.1145/1137856.1137881