## On Approximate Nearest Neighbors in Non-Euclidean Spaces (1998)

Venue: | In FOCS |

Citations: | 33 - 8 self |

### BibTeX

@INPROCEEDINGS{Indyk98onapproximate,

author = {Piotr Indyk},

title = {On Approximate Nearest Neighbors in Non-Euclidean Spaces},

booktitle = {In FOCS},

year = {1998},

pages = {148--155}

}

### Years of Citing Articles

### OpenURL

### Abstract

The nearest neighbor search (NNS) problem is the following: Given a set of n points P = fp 1 ; : : : ; png in some metric space X, preprocess P so as to efficiently answer queries which require finding a point in P closest to a query point q 2 X. The approximate nearest neighbor search (c-NNS) is a relaxation of NNS which allows to return any point within c times the distance to the nearest neighbor (called c-nearest neighbor). This problem is of major and growing importance to a variety of applications. In this paper, we give an algorithm for (4dlog 1+ae log 4de + 3)-NNS algorithm in l d 1 with O(dn 1+ae log n) storage and O(d log n) query time. In particular, this yields the first algorithm for O(1)-NNS for l 1 with subexponential storage. The preprocessing time is close to linear in the size of the data structure. The algorithm can be also used (after simple modifications) to output the exact nearest neighbor in time bounded by O(d log n) plus the number of (4dlog 1+ae log 4d...

### Citations

775 | An optimal algorithm for approximate nearest neighbor searching in fixed dimensions
- ARYA, MOUNT, et al.
- 1998
(Show Context)
Citation Context ... d O(n) (here and later we denote ffl = c \Gamma 1). The dependence on ffl was later reduced by Clarkson [Cl94] and Chan [Chan97] to ffl \Gamma(d\Gamma1)=2 . Arya, Mount, Netanyahu, Silverman, and Wu =-=[AMNSW94]-=- obtained optimal O(n) preprocessing cost, but with query time growing as O(d d log n). Euclidean/Manhattan norm. Kleinberg [Kl97] gave an algorithm with O(n log d) 2d preprocessing and query time pol... |

711 | Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality
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Citation Context ...in situations when the quality of the result cannot be compromised. The approximate nearest neighbor problem was recently a subject of extensive research (see Section 1.1). The most recent results of =-=[IM98]-=- and [KOR98] give algorithms for approximate nearest neighbor in d-dimensional Euclidean space with polynomial storage and query time polynomial in log n and d. These algorithms are of mainly theoreti... |

431 | Efficient and effective querying by image content - Faloutsos, Barber, et al. |

408 |
Extensions of lipschitz maps into a hilbert space
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Citation Context ...uclidean /Manhattan norms. The algorithm of [Kl97] is based random projections, which are not well defined for l p norms with p ? 2. The first algorithm of [IM98] uses the Johnson-Lindenstrauss Lemma =-=[JL84]-=- to reduce the dimensionality to O(log n); this lemma provably does not hold for other norms (like l 1 [Mat96, Mat]). The algorithm of [KOR98] and the second result of [IM98] solve the problem by embe... |

335 | Combining fuzzy information from multiple systems - Fagin - 1996 |

280 | On Lipschitz embedding of finite metric spaces in Hilbert space - Bourgain - 1985 |

190 | Efficient search for approximate nearest neighbor in high dimensional spaces
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- 1998
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Citation Context ...ns when the quality of the result cannot be compromised. The approximate nearest neighbor problem was recently a subject of extensive research (see Section 1.1). The most recent results of [IM98] and =-=[KOR98]-=- give algorithms for approximate nearest neighbor in d-dimensional Euclidean space with polynomial storage and query time polynomial in log n and d. These algorithms are of mainly theoretical interest... |

170 | Two algorithms for nearest-neighbor search in high dimensions
- KLEINBERG
- 1997
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Citation Context ...ffl \Gamma(d\Gamma1)=2 . Arya, Mount, Netanyahu, Silverman, and Wu [AMNSW94] obtained optimal O(n) preprocessing cost, but with query time growing as O(d d log n). Euclidean/Manhattan norm. Kleinberg =-=[Kl97]-=- gave an algorithm with O(n log d) 2d preprocessing and query time polynomial in d, ffl, and log n, and another algorithm with preprocessing polynomial in d, ffl, and n but with query time O(n + d log... |

122 | Fuzzy Queries In Multimedia Database Systems - Fagin - 1998 |

114 |
Efficient Similarity Searching
- Agrawal, Faloutsos, et al.
- 1993
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Citation Context ...r worst-case scenarios and are likely to be much better for instances occurring in practice. Also, the algorithm is neighborhood sensitive. The l 1 norm is of significant practical interest (e.g. see =-=[ALSS95]-=-). Its advantage is that it enables to combine different similarity measures (color, texture, etc) without any assumptions about their additivity [AGGM98]; it has also other attractive properties whic... |

113 | A randomized algorithm for closest-point queries - CLARKSON - 1988 |

76 | On the Analysis of Indexing Schemes
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- 1997
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Citation Context ...n algorithm for a general class of productsmetrics. Finally, we show that for any c ! 3 the c-NNS problem in l 1 is provably hard for a version of the indexing model introduced by Hellerstein et. al. =-=[HKP97]-=- (our upper bound can be adapted to work in this model). Supported by a Stanford Graduate Fellowship and NSF Award CCR-9357849. 1 Introduction The nearest neighbor search (NNS) problem is the followin... |

68 |
An algorithm for approximate closest-point queries
- Clarkson
- 1994
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Citation Context ...nd Mount [AM93] gave an algorithm with query time O(1=ffl) d log 3 n and preprocessing O(1=ffl) d O(n) (here and later we denote ffl = c \Gamma 1). The dependence on ffl was later reduced by Clarkson =-=[Cl94]-=- and Chan [Chan97] to ffl \Gamma(d\Gamma1)=2 . Arya, Mount, Netanyahu, Silverman, and Wu [AMNSW94] obtained optimal O(n) preprocessing cost, but with query time growing as O(d d log n). Euclidean/Manh... |

55 | Approximate nearest neighbor queries revisited. Symposium on Computational Geometry, 352–358
- Chan
- 1997
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Citation Context ...roughly O(n log d n) storage and O(log d n) query time. Arbitrary norm. The first result for approximate nearest neighbor in ! d was due to Bern [Bern93]. His result (after recent improvement by Chan =-=[Chan97]-=-) guarantees polynomial storage and query time polynomial in d and log n, but with c polynomial in d. Arya and Mount [AM93] gave an algorithm with query time O(1=ffl) d log 3 n and preprocessing O(1=f... |

55 |
Point location in arrangements of hyperplanes
- MEISER
- 1993
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Citation Context ...ncy for the c-NNS problem. 1.1 Related work In this section we review the work on approximate nearest neighbor in metric spaces. We note that for the exact case the best known result is due to Meiser =-=[Me93]-=-, who obtained an algorithm with storage ~ O(n d ) and query time ~ O(d 5 ); this can be improved to O(d 4 log d log n) query time and roughly O((n= log n) bd=2c ) preprocessing ([AE], p. 46). The sto... |

44 |
Approximate closest-point queries in high dimensions
- BERN
- 1993
(Show Context)
Citation Context ...r this case Gabow et al [GBT84] gave an algorithm with roughly O(n log d n) storage and O(log d n) query time. Arbitrary norm. The first result for approximate nearest neighbor in ! d was due to Bern =-=[Bern93]-=-. His result (after recent improvement by Chan [Chan97]) guarantees polynomial storage and query time polynomial in d and log n, but with c polynomial in d. Arya and Mount [AM93] gave an algorithm wit... |

33 |
Approximate nearest neighbor searching
- Arya, Mount
- 1993
(Show Context)
Citation Context ...! d was due to Bern [Bern93]. His result (after recent improvement by Chan [Chan97]) guarantees polynomial storage and query time polynomial in d and log n, but with c polynomial in d. Arya and Mount =-=[AM93]-=- gave an algorithm with query time O(1=ffl) d log 3 n and preprocessing O(1=ffl) d O(n) (here and later we denote ffl = c \Gamma 1). The dependence on ffl was later reduced by Clarkson [Cl94] and Chan... |

27 | Approximate Nearest Neighbor Algorithms for Hausdorff Metrics via Embeddings
- Farach-Colton, Indyk
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Citation Context ...re the individual metric spaces represent various properties of images (color, texture, etc). Finally, Farach-Colton and Indyk recently gave several results on embedding of Hausdorff metrics into l 1 =-=[FI98]-=-. We complement our algorithmic results with a lower bounds for c-NNS in the indexing model introduced by Hellerstein et al [HKP97]. In particular, we show that for any c 2 [1; 3) there exists an inst... |

21 |
Similarity Search
- Gionis, Indyk, et al.
- 1999
(Show Context)
Citation Context ...s of significant practical interest (e.g. see [ALSS95]). Its advantage is that it enables to combine different similarity measures (color, texture, etc) without any assumptions about their additivity =-=[AGGM98]-=-; it has also other attractive properties which make it useful for fuzzy information systems [Fagin96, Fagin98]. Moreover, it has important theoretical properties which enable us to further generalize... |

21 |
Deterministic superimposed coding with applications to pattern matching
- Indyk
- 1997
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Citation Context ...y of the instance is very high (equal to the instance size) so it still leaves a possibility for a much better result for low dimensional spaces. However, by using superimposed codes (as discussed in =-=[I97]-=-) one can reduce (n; n; k)-SQ to (n; k 2 log n; k)- SQ. Therefore, we obtain the following result Corollary 2 For any 1sc ! 3, block size B, redundancy rs1 and 1sasB there exists an instance of size n... |

14 |
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Citation Context ...n l d 1 with finite d and no distortion. Moreover, if we allow a small multiplicative distortion t, then any n-point metric space can be embedded in l d 1 with small dimension d = ~ O(n 1 b(t+1)=2c ) =-=[Mat96]-=-. By exploiting this property we obtain a tc(d; ae)- approximation algorithm for a product of k arbitrary finite metrics of size s with query time ~ O(ks 1 b(t+1)=2c ) and storage ~ O(ks 1 b(t+1)=2c n... |

9 | On embedding expanders into l p spaces - Matousek - 1997 |

8 | Fast algorithms for t-spanners and stretch-t paths - Cohen - 1999 |

2 |
Scaling and related techniques for computational geometry
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- 1984
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Citation Context ...log d log n) query time and roughly O((n= log n) bd=2c ) preprocessing ([AE], p. 46). The storage requirements can be significantly reduced when the underlying norm is l 1 - for this case Gabow et al =-=[GBT84]-=- gave an algorithm with roughly O(n log d n) storage and O(log d n) query time. Arbitrary norm. The first result for approximate nearest neighbor in ! d was due to Bern [Bern93]. His result (after rec... |

1 |
Geometric Range Searching and its Relatives", manuscript
- Agarwal, Eriksson
- 1997
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Citation Context ... due to Meiser [Me93], who obtained an algorithm with storage ~ O(n d ) and query time ~ O(d 5 ); this can be improved to O(d 4 log d log n) query time and roughly O((n= log n) bd=2c ) preprocessing (=-=[AE]-=-, p. 46). The storage requirements can be significantly reduced when the underlying norm is l 1 - for this case Gabow et al [GBT84] gave an algorithm with roughly O(n log d n) storage and O(log d n) q... |