Uhlmann, J.: 1991b, `Satisfying general proximity/similarity queries with metric trees'. Information Processing Letters 40, 175--179.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the original set into a fixed number of subsets and chooses a representative from each of the subsets. The representative and the maximum distance from the representative to a point of the corresponding subset are also maintained to support nearest neighbor queries. The metric tree of Uhlmann [20] and the vantage point tree (vp tree) of Yanilos [23] are somehow similar to the first technique of [9] as they partition the elements into two groups according to a representative, called a vantage point. In [23] the vp tree has been generalized to a multi way tree. In order to reduce the number ....

....vp tree has been generalized to a multi way tree. In order to reduce the number of distance calculations, Baeza Yates et al. [2] suggested using the same vantage point in all nodes that belong to the same level. Then, a binary tree degenerates into a simple list of vantage points. Another method [20] is the generalized hyper plane tree (gh tree) which partitions the data set into two by picking two points as representatives and assigning the remaining to the closest representative. Bozkaya and Ozsoyoglu [7] 6] proposed an extension of the vp tree called multi vantage point tree (mvp tree) ....

J. K. Uhlmann, "Satisfying General Proximity/Similarity Queries with Metric Trees," Information Processing Letter, Vol. 40, No. 4, November 25, 1991, pp. 175-179.

....diagram cannot be built for arbitrary metric spaces, using only the distance between objects as a distinctive property. This has been proved recently in [12] Consequently one can build just approximations to such diagram, if we intend to use it for similarity index. The generalized hyperplanes [13], are the rst known reference to such a construction. Instead of building the Voronoi diagram for every database element, a hierarchy of divisions is de ned using two elements per level. This is generalized for using more than one element of the Voronoi diagram in the Geometric Near Neighbor ....

J. Uhlmann. Satisfying general proximity /similarity queries with metric trees. Information Processing Letters, 40:175-179, 1991.

....so as to predict its probable future behavior) etc. Since the problem has appeared in unrelated areas, the corresponding algorithms and data structures seem to emerge from a great diversity, and different approaches have been proposed and analyzed separately, often under different assumptions [5, 20, 22, 19, 21, 23, 13,15, 1, 4, 14, 18, 3, 11, 17, 7, 8, 24]. Due to space limitations we refer the reader to a recent survey where all the known approaches for similarity searching are discussed [9] Currently, the only realistic way to compare two different algorithms is to apply them to the same data set. We present a unified complexity model for the ....

....Fig. 3. With two rings we define an equivalence based on being at the same distance to both points. However, the resulting class is partitioned. 6 Pivot Based and Clustering Algorithms A large class of methods to index metric spaces are just variants of what we call pivot based algorithms [5, 20, 22, 21, 23, 13, 15, 1, 14, 18, 3, 11, 7, 8, 24]. The idea is an extension of Example 3, using more pivots in order to decrease the external complexity. Instead of just one pivot, one selects h pivots p 1 Delta Delta Delta p h 2 U, and stores all the distances d(u; p i ) for all u 2 U. This set of distances is the index. Now, given a query ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175--179, 1991.

....are saved in the backtracking process because only one di erent pivot per level exists. However, the tree is taller. A variant called Fixed Height fq tree (fhq tree) is also proposed where all the leaves are at the same depth h, regardless of the bucket size. Vantage Point Trees (vp trees) [17, 19] are designed for continuous distance functions. The root has two equal size subtrees that divide the elements in closer to and farther from the root. This can be extended to m ary trees (mvp trees) 5, 4] Generalized hyperplane trees (gh trees) 17] use two pivots for each tree node and divide ....

....size. Vantage Point Trees (vp trees) 17, 19] are designed for continuous distance functions. The root has two equal size subtrees that divide the elements in closer to and farther from the root. This can be extended to m ary trees (mvp trees) 5, 4] Generalized hyperplane trees (gh trees) [17] use two pivots for each tree node and divide the space according to which of the two pivots is closer to each object. If this is generalized to an m ary partition then a Geometric Near neighbor Access Tree (gna tree) is obtained [5] which makes a Voronoi like partition of the space [1] among the ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. IPL, 40:175-179, 1991.

....at each partition. 4.2 Continuous Distance Functions We present now the data structures designed for the continuous case. They can be used also for discrete spaces with virtually no modifications. VPT The first approach designed for continuous distance functions is called Metric Trees in [54]. A more complete work on the same idea [59] calls them Vantage Point Trees or VPTs. They build a binary tree recursively, taking any element as the root p and taking the median of the set of all distances, M = medianfd(p; u) u 2 Ug. Those elements u such that d(p; u) M are inserted into the ....

....of the forest. The VPF, of O(n) size, is built using O(n 2 Gammaae ) time and answers 14 queries in O(n 1 Gammaae log n) distance evaluations, where 0 ae 1 depends on r . Unfortunately, to achieve ae 0, r has to be too small to be of use in high dimensions. GHT Another proposal of [54] is the Generalized Hyperplane Tree (GHT) This is a binary tree built recursively as follows. At each node, two pivots p 1 and p 2 are selected. The elements closer to p 1 than to p 2 go into the left subtree and those closer to p 2 into the right subtree. To answer queries of type (a) in this ....

[Article contains additional citation context not shown here]

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175--179, 1991.

....are saved in the backtracking process because only one different pivot per level exists. However, the tree is taller. A variant called Fixed Height fq tree (fhq tree) is also proposed where all the leaves are at the same depth h, regardless of the bucket size. Vantage Point Trees (vp trees) [13, 15] are designed for continuous distance functions. The root has two equal size subtrees that divide the elements in closer to and farther from the root. This can be extended to m ary trees (mvp trees) 5, 4] Generalized hyperplane trees (gh trees) 13] use two pivots for each tree node and divide ....

....size. Vantage Point Trees (vp trees) 13, 15] are designed for continuous distance functions. The root has two equal size subtrees that divide the elements in closer to and farther from the root. This can be extended to m ary trees (mvp trees) 5, 4] Generalized hyperplane trees (gh trees) [13] use two pivots for each tree node and divide the space according to which of the two pivots is closer to each object. If this is generalized to an m ary partition then a Geometric Near neighbor Access Tree (gna tree) is obtained [5] which makes a Voronoi like partition of the space [1] among the ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. IPL, 40:175--179, 1991.

....are saved in the backtracking process because only one different pivot per level exists. However, the tree is taller. A variant called Fixed Height fq tree (fhq tree) is also proposed where all the leaves are at the same depth h, regardless of the bucket size. Vantage Point Trees (vp trees) [18, 20] are designed for continuous distance functions. The root has two equal size subtrees that divide the elements in closer to and farther from the root. This can be extended to m ary trees (mvp trees) 4, 3] Finally, algorithms like AESA [19] LAESA [14, 13] and its variants [16, 7] and Fixed ....

....distance evaluations by increasing the number of pivots. Hence, in general these methods use as many pivots as they can, and they are normally well below their optimum. For more details on these results the reader should see [9] 2. 2 Clustering Algorithms Generalized Hyperplane Trees (gh trees) [18] use two centers for each tree node and divide the space according to which of the two centers is closer to each object. At search time the query enters into the subtrees whose zone of influence has a nonempty intersection with the query ball. Bisector Trees [12, 17] are similar but the zones ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. IPL, 40:175--179, 1991.

....Many comparisons are saved in the backtracking process because only one different pivot per level exists. However, the tree is taller. A variant called Fixed Height FQT (FHQT) is also proposed where all the leaves are at the same depth h, regardless of the bucket size. Vantage Point Trees (VPTs) [36, 39] are designed for continuous distance functions. The root has two equal size subtrees that divide the elements in closer to and farther from the root. This can be extended to m ary trees (MVPTs) 10, 9] Finally, algorithms like AESA [37] LAESA [31, 30] and its variants [33, 13] and Fixed ....

....They select a set of centers, which are elements from U, and divide the space so that each center has its zone of influence. Each zone is normally divided recursively. The algorithms differ in how the centers are selected, how the zones are delimited, etc. Generalized Hyperplane Trees (GHTs) [36] use two centers for each tree node and divide the space according to which of the two centers is closer to each object. This is like dividing the space with a hyperplane formed by the points at the same distance from both centers. At search time the query enters into the subtrees whose zone of ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175--179, 1991.

.... Claimed Query Extra CPU Structure Complexity Complexity Complexity query time BKT [19, 58] n pointers O(n log n) O(n ) FQT [5] n: n log n pointers O(n log n) O(n ) FHQT [5, 4, 6] n: nh pointers O(nh) O(log n) O(n ) FQA [24] nhb bits O(nh) O(log n) O(n log n) VPT [61, 67, 25] n pointers O(n log n) O(log n) MVPT [16, 15] n pointers O(n log n) O(log n) VPF [68] n pointers O(n 2 ) O(n 1 log n) BST [43, 51] n pointers O(n log n) not analyzed GHT [61, 18] n pointers O(n log n) not analyzed GNAT [16] nm 2 distances O(nm log m n) not ....

.... O(log n) O(n ) FQA [24] nhb bits O(nh) O(log n) O(n log n) VPT [61, 67, 25] n pointers O(n log n) O(log n) MVPT [16, 15] n pointers O(n log n) O(log n) VPF [68] n pointers O(n 2 ) O(n 1 log n) BST [43, 51] n pointers O(n log n) not analyzed GHT [61, 18] n pointers O(n log n) not analyzed GNAT [16] nm 2 distances O(nm log m n) not analyzed VT [31, 50, 62] n pointers O(n log n) not analyzed MT [26] n pointers O(n(m: m 2 ) log m n) not analyzed SAT [47] n pointers O(n log n= log log n) O(n 1 (1= log log n) AESA [63] n 2 ....

[Article contains additional citation context not shown here]

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175-179, 1991.

....selection of a centroid distributes the distances better. The problem with the algorithm [33] is that it needs O(n 2 ) space and build time. In this sense it is close to [25] This is unacceptably high for all by very small databases. Some approaches designed for continuous distance functions [31, 37, 8, 9, 12, 24] are not covered in this brief review. The reason is that these structures do not use all the information obtained from the comparisons, since this cannot be done in continuous spaces. It can, however, be done (and it is done) in discrete spaces and this fact makes the reviewed structures superior ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175--179, 1991.

....algorithms to build it depend heavily on coordinate information. Nevertheless, the concept itself has inspired several approaches constructing a more or less fine approximation to either the Voronoi graph or its dual, the Delaunay triangulation. In this line we can find generalized hyper planes [17], the GNATS (Geometric Neighbor Access Trees) 7] and more recently the SB algorithm [11] and the SAT (Spatial Approximation Tree) 14] The key idea in all these algorithms is to build a proximity graph allowing to search by approaching spatially to the query, as opposed to the pivot based ....

....predecessors. Overcoming the Curse of Dimensionality 3 If, on the other hand, the distance function is continuous, then additional work has to be done. It is impossible to assign directly one branch for each distance outcome, hence some discretization has to be carried out. In the Metric Trees [17] it is suggested to binarize the distance outcome using as threshold the median of the distance from the pivot to all its associated elements. A more complete work on the same idea is presented in the Vantage Point Trees (VPT) 19] This tree is generalized to use more than one pivot per node and ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175--179, 1991.

....data based only on a triangle inequality obeying distance metric. We show how this, in its own right, gives a fast and effective clustering of data. But more importantly we show how it can produce a well balanced structure similar to a Ball Tree (Omohundro, 1991) or a kind of metric tree (Uhlmann, 1991; Ciaccia, Patella, Zezula, 1997) in a way that is neither topdown nor bottom up but instead middleout . We then show how this structure, decorated with cached sufficient statistics, allows a wide variety of statistical learning algorithms to be accelerated even in thousands of ....

....1977) we can perform clustering, and a very wide class of non parametric statistical techniques on enormous data sources hundreds of times faster than previous algorithms (Moore, 1999) but only up to about 8 10 dimensions. This paper replaces the kd trees with a certain kind of Metric tree (Uhlmann, 1991) and investigates the extent to which this replacement allows acceleration on real valued queries in higher dimensions. To achieve this, in Section 3 we introduce a new tree free ag glomerative method for very quickly computing a spatial hierarchy (called the anchors hierarchy) this is needed ....

Uhlmann, J. K. (1991). Satisfying general proximity /similarity queries with metric trees. Information Processing Letters, 40, 175--179.

....remarks the authors clearly anticipate generalization to continuous settings such as R n . The idea that vantage points near the corners of the space are better than those near the center was described in [28] and much later in [32] More recent papers describing vantage point approaches are [30, 29, 25] and [32] who describe variants of what we refer to as a vantage point tree. Also see [10] for very recent work on search in metric spaces. The well known kd tree of Friedman and Bentley [15, 16, 4, 3] recursively divides a pointset in R d by projecting each element onto a distinguished ....

Uhlmann, J. K. Satisfying general proximity /similarity queries with metric trees. Information Processing Letters (November 1991).

....Way, Princeton NJ 08540, and this work was completed while a visitor at the Princeton University computer science department. email: pny cs.princeton.edu trinsically moderate dimension, recursive projectiondecomposition techniques such as kd trees [17, 18, 4, 3] and vantage point techniques [31, 30, 27, 33] for general metric spaces of intrinsically low dimension, are e ective. As dimension d 1, these tree techniques perform well only if the number of points n grows exponentially in d or in the case of Vornoi diagrams if space increases exponentially. The motivation for this work is the ....

Uhlmann, J. K. Satisfying general proximity /similarity queries with metric trees. Information Processing Letters (November 1991).

....can not be neglected remains to be done as future research. We introduce the mvp tree (multi vantage point tree) as a general solution to the problem of answering similarity based queries efficiently for high dimensional metric spaces. The mvp tree is similar to the vp tree (vantage point tree) [Uhl91] in the sense that both structures use relative distances from a vantage point to partition the domain space. In vp trees, at every node of the tree, a vantage point is chosen among the data points, and the distances of this vantage point from all other points (the points that will be indexed ....

....specifying an interval) by making use of the pre computed distances. The technique of storing and using pre computed distances may be effective for data domains with small cardinality, however, space requirements and search complexity becomes overwhelming for larger domains. Uhlmann introduced [Uhl91] two hierarchical index structures for similarity search. The first one is the vp tree (vantage point tree) The vp tree basically partitions the data space into spherical cuts around a chosen vantage point at each level. This approach, referred as the ball decomposition in the paper, is similar ....

[Article contains additional citation context not shown here]

J. K. Uhlmann, "Satisfying General Proximity/Similarity Queries with Metric Trees", Information Processing Letters, vol 40, pages 175-179, 1991.

....show that such an approach can incur a considerable amount of inaccuracy for n nearest neighbor search. Therefore we study an alternative approach that uses distance based indexing, known as Metric Space Model (MSM) indexing. In particular we examine the Vantage Point tree (vp tree) method [36, 39, 10]. This approach can obviously save the overhead of inferring points in a multidimensional space, and can also avoid the difficulty in preserving distances. Our main contributions are the following. 1 1. We propose an n nearest neighbor search algorithm for the vp tree index method. From our ....

....Structures Quite a number of distance based indexing structures have been proposed. A summary of some of these methods can be found in [6, 7] Previous work includes techniques suggested in [8] which contains some of the basic ideas for later methods, the generalized hyperplane tree (gh tree) [36], the vantage point tree (vptree) 36, 39, 10] the Geometric Near neighbor Access Tree (GNAT) 7] the mvp tree [6] which is a variation of the vp tree, and the M tree [11] In [7] the GNAT is compared to the binary vp tree in a set of experiments and is found to incur more expensive construction ....

[Article contains additional citation context not shown here]

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:4:175--179, November 1991.

....queries in metric spaces has recently attracted the attention of researchers. The work of Burkhard and Keller [3] provided different interesting techniques for partitioning a metric data set in a recursive fashion where the recursive process is materialized as a tree. The metric tree of Uhlmann [4] and the vantage point tree (vp tree) of Yanilos [5] are somehow similar to a technique of [3] as they partition the elements into two groups according to a representative, called a vantage point. In [5] the vp tree has also been generalized to a multi way tree. In order to reduce the number of ....

....been generalized to a multi way tree. In order to reduce the number of distance calculations, Baeza Yates et al. [6] suggested to use the same vantage point in all nodes that belong to the same level. Then, a binary tree degenerates into a simple list of vantage points. Another method of Uhlmann [4] is the generalized hyper plane tree (gh tree) The gh tree partitions the data set into two by picking two points as representatives and assigning the remaining to the closest representative. Bozkaya and Ozsoyoglu [7] proposed an extension of the vp tree called multi vantage point tree (mvp tree) ....

Uhlmann, J.K.: Satisfying General Proximity/SimilarityQueries with Metric Trees. IPL 40(4) (1991) 175-179.

....they claim that it outperforms the SS tree and the SR tree, by comparing their improvement relative to the R tree. One caveat in the above approach is that the design of the X tree is based on the assumption that data and query points are uniformly distributed. The VP tree The VP trees [82, 90, 16] use a simple and intuitive idea. Given a set of points P , we select a point p to be the vantage point. We compute the distance from p to all other points in P , and we use these distances to compare the points. Formally, for some point x 2 P , let Pi p (x) d(p; x) Given two points x and y in ....

J. K. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40(4):175--179, November 1991.

....can not be neglected remains to be done as future research. We introduce the mvp tree (multi vantage point tree) as a general solution to the problem of answering similarity based queries efficiently for high dimensional metric spaces. The mvp tree is similar to the vp tree (vantage point tree) [Uhl91] in the sense that both structures use relative distances from a vantage point to partition the domain space. In vp trees, at every node of the tree, a vantage point is chosen among the data points, and the distances of this vantage point from all other points (the points that will be indexed ....

....(by specifying an interval) by making use of the pre computed distances. The technique of storing and using pre computed distances may be effective for data domains with small cardinality, however, space requirements and search complexity becomes overwhelming for larger domains. Uhlmann introduced [Uhl91] two hierarchical index structures for similarity search. The first one is the vp tree (vantage point tree) The vp tree basically partitions the data space into spherical cuts around a chosen vantage point at each level. This approach, referred as the ball decomposition in the paper, is similar ....

[Article contains additional citation context not shown here]

J. K. Uhlmann, "Satisfying General Proximity/Similarity Queries with Metric Trees", Information Processing Letters, vol 40, pages 175-179, 1991.

....of the remaining points is computed. Based on these distances, the points are separated into two or several different branches. For each branch, the structure is constructed recursively. J. K. Uhlmann outlined the foundation for two different methods, more generally referred to as metric trees [Uhl91] One of these methods, subsequently called vp trees, 3 was implemented by P. N. Yianilos [Yia93] The basic construction of a vp tree is to break the space up using spherical cuts. To build it, pick a point in the data set (this is called the vantage point, hence the name vp tree) Now, ....

J. Uhlmann. Satisfying general proximity / similarity queries with metric trees. Information Processing Letters, 40(4):175--9, November 1991.

....remarks the authors clearly anticipate generalization to continuous settings such as R n . The idea that vantage points near the corners of the space are better than those near the center was described in [27] and much later in [31] More recent papers describing vantage point approaches are [29, 28, 24] and [31] who describe variants of what we refer to as a vantage point tree. Also see [10] for very recent work on search in metric spaces. The well known kd tree of Friedman and Bentley [15, 16, 4, 3] recursively divides a pointset in R d by projecting each element onto a distinguished ....

Uhlmann, J. K. Satisfying general proximity /similarity queries with metric trees. Information Processing Letters (November 1991).

....extraction. In this approach, we can not use conventional spatial index structures, since we do not assume any geometry on the domain of sequences. Instead, our indexing scheme is based on the lengths of sequences and relative distances between sequences. A distance based index structure, vp tree [Uhl91], is used as the underlying index structure in our method. The indexing scheme is used as a major filtering mechanism to eliminate distant sequences in processing a similarity match query. This method is general, in the sense that it can be used for other sequence domains as well. These two ....

....the Euclidean distance function cannot be used directly as a similarity metric because the domain is non numeric and sequences may be of different lengths. In these cases, indexing methods for Euclidean spaces are usually not applicable. Still, there are other distance based index structures [Uhl91, Bri95, BCMW94, CPZ97] that do not assume any geometry of the application domain, but only depend on the fact that the distance function is metric (See section 5 for definition) The simple idea behind these structures is to use some reference object(s) to partition the search space with respect to the distances of ....

[Article contains additional citation context not shown here]

J.K. Uhlmann, "Satisfying General Proximity/Similarity Queries with Metric Trees", Information Processing Letters, Vol 40, pages 175-179, 1991.

....show that the number of traversed nodes is O(n x ) where n is the size of the set and 0 x 1 depends on the metric space. It can also be shown that the fixed height variant traverses a sublinear number of nodes [BYN98b] Some approaches designed for continuous distance functions , e.g. Uhl91, Yia93, Bri95, FL95] are not covered in this brief review. The reason is that these structures do not use all the information obtained from the comparisons, since this cannot be done in continuous spaces. This is, however, done in discrete spaces and this fact makes the reviewed structures ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175--179, 1991.

....index structure. We can not use conventional index structures, since we do not assume any geometry on the domain of sequences. Instead, we designed an indexing scheme which is based on the lengths of sequences and relative distances between sequences. A distance based index structure, vp tree [Uhl91], is used as the underlying index structure in our method. The indexing scheme is used as a major filtering mechanism to eliminate distant sequences in processing a similarity match query. The rest of the paper is organized as follows. In section 2 we provide a brief overview of the previous work ....

....Euclidean distance function cannot be used directly as a similarity metric because the domain is non numeric and sequences may be of different lengths. In these cases, indexing methods for Euclidean spaces are usually not applicable. Still, there are other distance based index structures [Uhl91, Bri95] that do not assume any geometry of the application domain, but only depend on the fact that the distance function is metric (See section 5 for definition) The simple idea behind these structures is to use some reference object(s) to partition the search space with respect to the distances of ....

[Article contains additional citation context not shown here]

J.K. Uhlmann, "Satisfying General Proximity/Similarity Queries with Metric Trees", Information Processing Letters, v40, p175-179,1991.

.... case, it is not possible to produce new objects in the metric space, e.g. to aggregate or divide two objects (in a Euclidean space, bounding rectangles are often used for this purpose) Various methods exist for indexing objects in the metric space model as well as for computing proximity queries [11, 13, 14, 52, 53]. These methods can only make use of the properties of distance functions (non negativity, symmetry, and the triangle inequality) and operate without any knowledge of how objects are represented or how the distances between objects are computed. Such a general approach is usually slower than ....

.... points [48] Another approach is to abandon the goal of indexing the data points based on space occupancy and instead use properties of the distance metric employed (see the discussion of the metric space model in Section 2) If a hierararchical index method based on distance (e.g. [11, 14, 52]) is employed, our algorithm is still applicable. In fact, the k nearest neighbor algorithm presented in [14] is similar to our algorithm in that it uses a priority queue for nodes to guide the traversal of the index. If we use the Euclidean distance metric, the nearest neighbor search region ....

[Article contains additional citation context not shown here]

J. K. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40(4):175--179, November 1991.

....to each of the remaining points is computed. Based on these distances, the points are separated into two or several different branches. For each branch, the structure is constructed recursively. J. K. Uhlmann outlined the foundation for two different methods, generally described as metric trees [Uhl91] One of these methods, subsequently called vp trees 3 , was implemented by P. N. Yianilos [Yia93] The basic construction of a vp tree is to break the space up using spherical cuts. To build it, pick a point in the data set (this is called the vantage point, hence the name vp tree) Now, ....

Uhlmann. Satisfying general proximity / similarity queries with metric trees. Information Processing Letters, 40, 1991.

....was provided by the fact that this chip s notion of string distance is non Euclidian. Here elements of the metric space are strings, e.g. a database of city names and associated postal codes. This early work was described in [4] Independently, Uhlmann has reported the same basic structure [5, 6] calling it a metric tree. There is a sizable Nearest Neighbor Search literature, and the vp tree should properly be viewed as related to and descended from many earlier contributions which we now proceed to summarize. Burkhard and Keller in [7] present three file structures for nearest neighbor ....

J. K. Uhlmann, "Satisfying general proximity/similarity queries with metric trees," Information Processing Letters, November 1991.

....of the performance of FQ trees is presented in [3] which disregarding some complications can be applied to BK trees as well. We show the results in the Appendix. We also give an analysis of fixed height FQ trees which is new. Some approaches designed for continuous distance functions , e.g. [19, 23, 7, 10], are not covered in this brief review. The reason is that these structures do not use all the information obtained from the comparisons, since this cannot be done in continuous spaces. This is, however, done in discrete spaces and this fact makes the reviewed structures superior to those for ....

J. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information Processing Letters, 40:175--179, 1991.

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Uhlmann, J.: 1991b, `Satisfying general proximity/similarity queries with metric trees'. Information Processing Letters 40, 175--179.

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