### 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 |

Pages | 154-157 |

Number of pages | 4 |

Volume | 2006 |

State | Published - 2006 |

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

### Fingerprint

### Keywords

- Locality Sensitive Hashing
- Lower Bounds
- Nearest Neighbor Search

### ASJC Scopus subject areas

- Software
- Geometry and Topology
- Safety, Risk, Reliability and Quality
- Chemical Health and Safety

### Cite this

*Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06*(Vol. 2006, pp. 154-157)

**Lower bounds on locality sensitive hashing.** / Motwani, Rajeev; Naor, Assaf; Panigrahy, Rina.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06.*vol. 2006, pp. 154-157, 22nd Annual Symposium on Computational Geometry 2006, SCG'06, Sedona, AZ, United States, 6/5/06.

}

TY - GEN

T1 - Lower bounds on locality sensitive hashing

AU - Motwani, Rajeev

AU - Naor, Assaf

AU - Panigrahy, Rina

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Locality Sensitive Hashing

KW - Lower Bounds

KW - Nearest Neighbor Search

UR - http://www.scopus.com/inward/record.url?scp=33748088520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748088520&partnerID=8YFLogxK

M3 - Conference contribution

SN - 1595933409

SN - 9781595933409

VL - 2006

SP - 154

EP - 157

BT - Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06

ER -