### Abstract

Given a metric space (X, dx), c ≥ 1, r > 0, and p,q ∈ [0, 1], a distribution over mappings ℋ : X → N is called a (r, cr, p, o)-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 ℋ 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 p 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) |
---|---|

Pages (from-to) | 930-935 |

Number of pages | 6 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 21 |

Issue number | 4 |

DOIs | |

State | Published - 2007 |

### Fingerprint

### Keywords

- Locality sensitive hashing
- Lower bounds
- Nearest neighbor search

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*21*(4), 930-935. https://doi.org/10.1137/050646858

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

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 21, no. 4, pp. 930-935. https://doi.org/10.1137/050646858

}

TY - JOUR

T1 - Lower bounds on locality sensitive hashing

AU - Motwani, Rajeev

AU - Naor, Assaf

AU - Panigrahy, Rina

PY - 2007

Y1 - 2007

N2 - Given a metric space (X, dx), c ≥ 1, r > 0, and p,q ∈ [0, 1], a distribution over mappings ℋ : X → N is called a (r, cr, p, o)-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 ℋ 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 p 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, dx), c ≥ 1, r > 0, and p,q ∈ [0, 1], a distribution over mappings ℋ : X → N is called a (r, cr, p, o)-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 ℋ 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 p 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=56649117650&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56649117650&partnerID=8YFLogxK

U2 - 10.1137/050646858

DO - 10.1137/050646858

M3 - Article

VL - 21

SP - 930

EP - 935

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -