K. Clarkson, "An algorithm for approximate closest-point queries," Proc. 10th ACM Symp. on Computational Geometry, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....p # is the true nearest neighbor to q. In other words, p is within relative error # of the true nearest neighbor. The approximate nearest neighbor problem has been heavily studied recently. Examples include algorithms by Bern [Ber93] Arya and Mount [AM93b] Arya, et al. AMN 98] Clarkson [Cla94] Chan [Cha97] Kleinberg [Kle97] Indyk and Motwani [IM98] and Kushilevitz, Ostrovsky and Rabani [KOR98] In this study we restrict attention to data structures of size O(dn) based on hierarchical spatial decompositions, and the kd tree in particular. In large part this is because of the ....

K. L. Clarkson, An algorithm for approximate closest-point queries, Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994, pp. 160--164.

....neighbor searching are based on simple spatial subdivisions, having poor worst case performance, but relatively good average case performance. Recently it has been suggested that another way to achieve good performance in practice is to compute approximate rather than exact nearest neighbors [2, 4, 7]. A very simple algorithm that works well for uniformly distributed data is the bucketing algorithm (sometimes called Elias s algorithm [15] Rivest [13] presented an analysis of the performance of this algorithm for points uniformly distributed on the vertices of a d dimensional hypercube, ....

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proceedings of the 10th Annual ACM Symposium on Computational Geometry, pages 160--164, 1994.

....function of dimension. Thus for reasonably large dimensions, brute force search is often the most e#cient in practice. One approach to reducing the search time is through approximate nearest neighbor search. A number of data structures for approximate nearest neighbor searching have been proposed [1, 3, 11]. The phenomenon of concentration of distance would suggest that approximate nearest neighbor searching is meaningless. Fortunately, the distributions that arise in applications tend to be clustered in lower dimensional subspaces [6] Good search algorithms take advantage of this low dimensional ....

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160--164, 1994.

....to relax the requirement of finding the true nearest neighbor, it is possible to achieve significant improvements in running time and at only a very small loss in the performance of the vector quantizer. Recently this approach has been studied by us and some others from a theoretical perspective [1, 2, 10, 7]. In this work, however, we are more concerned with practical aspects of the search algorithms. We present three algorithms for nearest neighbor searching: 1) the standard k d tree search algorithm [14, 23] which has been enhanced to use incremental distance calculation, 2) a further ....

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160--164, 1994.

....given a set of points in U d , to compute a query for the points lying in a d dimensional box R = a 1 ; b 1 ] a d ; b d ] Known data structures providing sublinear search time have space cost growing exponential with the dimension d. This is known as the curse of dimensionality [9]. Hence, for d of moderate size, a query is often most efficiently computed by a linear scan of the input. A straightforward optimization of this approach using space O(dn) is to keep the points sorted by each of the d coordinates. Then, for a given query, we can restrict the scan to the ....

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proceedings of the 10th Annual Symposium on Computational Geometry, pages 160--164, Stony Brook, NY, USA, June 1994. ACM Press.

....which use practical data structures such as kd trees, R trees, R trees, and Hilbert R trees; see e.g. 82, 99, 123, 121, 74, 94, 161, 174] But these algorithms also su er from curse of dimensionality. This has lead to the development of algorithms for nding approximate nearest neighbors [24, 25, 26, 45, 116, 123] or for special cases, such as when the distribution of query points is known in advance [48, 178] For a given parameter 0 and a query point , an approximate nearest neighbor query ( NN query) asks for returning a point whose distance is at most (1 ) times the distance between and its ....

....the distance between and its nearest neighbor in S. This relaxation is quite meaningful in the context of the applications mentioned above. Arya et al. 25] showed that an NN query can be answered in O( 1= d ) log n) time using O(n) space. The query bound was later improved by Clarkson [45] and Chan [35] to O( 1= d 1) 2 log n) the algorithm by Chan has smaller preprocessing time. Many approximation techniques based on distance preserving random projections of Geometric Optimization September 7, 2000 Proximity Problems 20 points onto lower dimensional subspaces have been ....

K. L. Clarkson, An algorithm for approximate closest-point queries, Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994, pp. 160-164.

....[18, 12, 36, 28, 1, 29] currently the best algorithms can find a nearest neighbor in time poly(d; log n) but they need exponential (n Theta(d) storage. On the other hand, there is little evidence in the form of concrete lower bounds to support the curse of dimensionality conjecture [13]; i.e. the belief that in high dimension the problem is indeed intractable (see below for more details) Our results. We present here significant improvements over recently discovered lower bounds for nearest neighbor search [10] Specifically, our main concern is nearest neighbor search in the ....

....In this version of the problem, the search algorithm is required to find a database point whose distance to the query is within a factor of 1 of the distance to a nearest neighbor, where 0 is a predefined value. The best available (randomized) algorithm, following a long line of work [5, 13, 6, 26, 25, 27], uses (for an arbitrary constant , when stated in terms of the cell probe model) poly(n; d) cells of size O(d) each, and searches probing O(log log d) cells. This randomized upper bound nearly matches a recent deterministic lower bound of Omega Gamma 40 log d= log log log d) 11] which holds ....

K. Clarkson. An algorithm for approximate closest-point queries. In Proc. of 10th SCG, pp. 160--164, 1994.

....in R d , find the point of P closest to an arbitrary query point q under L t . Efficient NN searching for high d is an important problem in many application areas such as pattern classification, information retrieval, multimedia and voice recognition and has been researched extensively. See [1, 2, 5, 10, 13, 16, 17, 18] for a few examples of previous work. Unfortunately for high d most of the proposed methods are of theoretical interest only as they tend to degrade due to the dimensionality curse (a notable exception is [17] As before, we attempt to deflate the curse by seeking an efficient algorithm that ....

K. L. Clarkson. An algorithm for approximate closestpoint queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160--164, 1994.

.... provided the data structure satisfies certain mild assumptions [7] Note that the query time of the above approach is exponential in d, so it is impractical even for moderate values of d (say d 10) This has lead to the development of algorithms for finding approximate nearest neighbors [88, 26, 29, 28, 169, 166] or for special cases, such as when the distribution of query points is known in advance [84, 262] See [132, 169, 167, 119, 148, 222, 236] for a few heuristics for answering nearest neighbor queries. 7.3 Linear programming queries Let S be a set of n halfspaces in R d . We wish to preprocess ....

K. L. Clarkson, An algorithm for approximate closest-point queries, Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994, pp. 160--164.

....an index on disk such that all nearest neighbors to any query point are physically adjacent on disk. We discuss this curse of dimensionality in more detail in Section 2. Fortunately, in many cases it is sufficient to perform an approximate search that returns many but not all nearest neighbors [2, 15, 25, 27, 28]. A feature vector is often an approximate characterization of an object, so we are already dealing with approximations anyway. For instance, in content based image retrieval [17, 39] and document copy detection [9, 11, 18] it is usually acceptable to miss a small fraction of the target ....

K. Clarkson. An algorithm for approximate closest-point queries. Proceedings of the 10th SCG, pages 160--64, 1994.

....R trees, and Hilbert R trees; see e.g. 87, 104, 129, 127, 78, 99, 170, 184] Even these algorithms suffer from the curse of dimensionality. This has lead to the development of algorithms for finding approximate nearest neighbors Geometric Optimization June 6, 2000 Proximity Problems 20 [25, 26, 27, 49, 122, 129] or for special cases, such as when the distribution of query points is known in advance [52, 188] For a given parameter 0 and a query point , an approximate nearest neighbor query ( NN query) asks for returning a point p 2 S so that d(p; 1 )d(p 0 ; for all p 0 2 S. This ....

....mentioned above. Arya et al. 27] showed that an NN query can be answered in O( 1= d ) log n) time using O(n) space. Note that the size of their data structure is independent of , and that can be specified as a part of the query. Although the query bound was later improved by Clarkson [49] and Chan [37] to O( 1= d Gamma1) 2 log n) is fixed for all queries in both the data structures and the size of their data structures depends on . Moreover, the data structure by Arya et al. 27] is practical and works well for dimensions up to 20 Gamma Gamma30. Many approximation ....

K. L. Clarkson, An algorithm for approximate closest-point queries, Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994, pp. 160--164.

....lower bounds for halfspace emptiness queries in both the online and o ine settings. Finally, in Section 9, we o er our conclusions. ## ######### ########### # This curse of dimensionality can sometimes be avoided by requiring only an approximation of the correct output; see, for example, [7, 24, 37]. SPACE TIME TRADEOFFS FOR EMPTINESS QUERIES 5 #### ########### We begin by reviewing the de nition of the semigroup arithmetic model, originally introduced by Fredman to study dynamic range searching problems [33] and later re ned for the static setting by Yao [56] A semigroup (## ) is a set ....

K. L. Clarkson, An algorithm for approximate closest-point queries, in Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994, pp. 160-164.

....lower bounds for halfspace emptiness queries in both the online and o ine settings. Finally, in Section 9, we o er our conclusions. 2. Semigroup Arithmetic. 3 This curse of dimensionality can sometimes be avoided by requiring only an approximation of the correct output; see, for example, [7, 24, 37]. SPACE TIME TRADEOFFS FOR EMPTINESS QUERIES 5 2.1. De nitions. We begin by reviewing the de nition of the semigroup arithmetic model, originally introduced by Fredman to study dynamic range searching problems [33] and later re ned for the static setting by Yao [56] A semigroup (S; is a set ....

K. L. Clarkson, An algorithm for approximate closest-point queries, in Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994, pp. 160-164.

....tree with two important properties. First, the height of the tree is O(log n) Second, the cells associated with the nodes of the tree are fat (i.e. have bounded aspect ratio) Recently, Duncan et al. 13] have given another construction of a partition tree that has these two properties. Clarkson [10] and later Chan [9] showed how to reduce the query time to O( 1=ffl) d Gamma1) 2 log n) but it increased the space requirement to O( 1=ffl) d Gamma1) 2 n log n) Kleinberg [20] provided an algorithm that has query time polynomial in log n, d and ffl but requires O( n log d) 2d ) space. ....

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160--164, 1994.

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K. Clarkson, "An algorithm for approximate closest-point queries," Proc. 10th ACM Symp. on Computational Geometry, 1994.

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K. Clarkson. An algorithm for approximate closest-point queries. In Proc. of 10th SCG, pp. 160--164, 1994.

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K. Clarkson. An algorithm for approximate closest-point queries. In Proc. of 10th SCG, pp. 160--164, 1994.

No context found.

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Symp. Comput. Geom., pages 160--164, 1994.

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K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160-164, 1994.

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K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160--164, 1994.

No context found.

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160--164, 1994.

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K. L Clarkson. An Algorithm for Approximate Closest-Point Queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., 1994, pages 160--164.

No context found.

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 160--164, 1994.

No context found.

K. L. Clarkson. An algorithm for approximate closest-point queries. In Proceedings of the 10th Annual ACM Symposium on Computational Geometry, pages 160-164, 1994.

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Clarkson, K.L. An algorithm for approximate closest-point queries, Proc. 10th Annu. ACM Symp. Comput. Geom. 10 (1994), 160--164.

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