C. Faloutsos and Y. Rong, "Dot: A Spatial Access Method Using Fractals," Proc. IEEE Conf. Data Eng., pp. 152-159, Kobe, Japan, Apr. 1991. Early version available as UMIACS-TR-89-31, CS-TR2214.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....compatible with the Zorder curve. Sorting and Grouping the Photons Block Hashing sorts photons and inserts them into a B tree [15,39] using the Hilbert curve encoding of the position of each photon as the key. This method of spatially grouping points was first proposed by Faloutsos and Rong [19] for a di#erent purpose. Since a B tree stores photon records only at leaves, therefore with careful construction the leaf nodes of the B tree can serve as the photon blocks used in the later stages of Block Hashing. One advantage of using a B tree for sorting is that insertion cost is ....

C. Faloutsos and Y. Rong. DOT: A Spatial Access Method Using Fractals. In Proc. 7th Int. Conf. on Data Engineering,, pages 152--159, Kobe, Japan, 1991. 4.1

....Thus, it is critical to select an appropriate space filling curve, that will keep k small most of the time. The most popular candidates are the Peano curve (also known as z order) proposed initially by Orenstein and Merret [OM84] and the Hilbert curve, proposed originally by Faloutsos and Rong [FR91] These techniques can easily be extended to dimensions higher than 2. 2.2.3 Storing spatial objects using point access methods Both the techniques based on the grid file, and those using space filling curves, are natively capable of storing only multidimensional points. However, in many ....

....methods, is to transform a d dimensional (hyper) rectangle into a point in 2d space. Under this transformation, a rectangular intersection query is transformed into a 2ddimensional dominance query. This transformation was employed in the DOT (Double Transformation) technique of Faloutsos and Rong [FR91] in the work of Kanellakis et al. KRV 93] for d = 1) and in a number of other works. For objects with complicated shape, it may be desirable to decompose them into multiple rectangles, if using the object s MBR provides a crude approximation. Gaede [Gae95] studies empirically the e#ects ....

C. Faloutsos and Y. Rong. DOT: A spatial access method using fractals. In Proc. IEEE Intl. Conf. Data Engineering, pages 152-- 159, 1991.

....indexes has to be recomputed for join processing [9] One problem of Z order as space filling curve is the long diagonal jumps, where the consecutive Z value points connecting these jumps are far apart in X Y space. The spatial clustering of Z ordering can be improved by using the Hilbert curve [1, 5]. However, algorithms to process range query and spatial join using Hilbert curve need to be worked out. Recursive definition of Hilbert curve may pose some challenges here. 1.2 Transformed space Z order Orenstein evaluated the spatial query within native space and transformed space using zkd ....

....Second, the distribution of points in dual space may be highly non uniform. Third, the distance relationship between two close objects may be arbitrarily far apart from each other in dual space [9] Special transformation techniques and split strategies have to used to overcome these problems [1, 4, 8]. 2.2 Is it sufficient to implement Z order B trees only, Or are R trees truly required What optimizer smarts are required for Z order B trees What overall performance loss will we need to tolerate Z order B tree can help in all spatial queries, i.e. R trees are not necessary. Needed ....

Christos Faloutsos and Yi Rong. Dot: A spatial access method using fractals. In Proceedings of the Seventh International Conference on Data Engineering, April 8-12,

....has to be recomputed for join processing [13] Another problem of Z order as space filling curve is the long diagonal jumps, where the consecutive Z value points connecting these jumps are far apart in X Y space. The spatial clustering of Z ordering can be improved by using the Hilbert curve [2, 6]. However, algorithms to process range query and spatial join using Hilbert curve need to be worked out. Recursive definition of Hilbert curve may pose some challenges here. Distance between two points in original space are often different from those space filling curve for both Z order and ....

....the original space are not reflected in the transformed space. The distance between two close objects in the original space may be arbitrarily far apart from each other in the transformed space [13] Special transformation techniques and split strategies have to be used to overcome these problems [2, 5, 10]. 2 Discussion 2.1 Do we need transformations on the dimensions Mapping to higher dimensional space is not needed if one is willing to decompose or replicate extended objects into a collection of cells compatible with the resolution of Z order grid. See [3, 4] for details. R tree, R tree, and ....

Christos Faloutsos and Yi Rong. Dot: A spatial access method using fractals. In Proceedings of the Seventh International Conference on Data Engineering, April 8-12, 1991, Kobe, Japan, pages 152--159.

....tries to preserve the distance, that is, points that are close in the k d space are likely to be close in the 1 d space. The set of ordered MBRs can be organized by any one dimensional access method using the ordering number as a key. Several SAMs using the ordering technique have been proposed [Abel83,84, Oren86, Falo88, Falo91]. The major advantage of this technique is that the SAM inherits all the good properties of the underlying one dimensional access method (e.g. B tree, hashing techniques) However, the introduction of artificial one dimensional ordering may require further enlargement of the MBR, which ....

C. Faloutsos, Y. Rong: `DOT: A Spatial Access Method Using Fractals', Proceedings of the 7th Data Engineering Conference, Kobe, Japan, 152-159, 1991.

....a query. A rectangle Q d i=1 [a i ; b i ] in R d can be mapped to the point (a 1 ; a 2 ; a d ; b 1 ; b 2 ; b d ) in R 2d , and a rectangleintersection query can be reduced to orthogonal range searching. Many heuristic data structures based on this scheme are proposed; see [120, 213, 230] for a sample of such results. The second approach is to construct a data structure on S directly in R d . The most popular data structure based on this approach is the R tree by Guttman [146] R1 R2 R3 R4 R5 A B C D E F G H I R1 R2 R3 R4 R5 R6 A B C D E F G H I Figure 5. An ....

C. Faloutsos and Y. Rong, DOT: A spatial access method using fractals, Proc. 7th IEEE Internat. Conf. on Data Engineering, 1991, pp. 152--158.

....the points in multiple dimensions by a space filling curve, with a specific resolution of the space, and use a one dimensional access method with this ordering. They 2 perform transformations on higher order keys to impose total ordering. Example methods include Z ordering [32] and Hilbert Curves [2, 11, 22]. Multidimensional B trees [35] and K dB trees [33] establish a correspondence between the levels of the index and dimensions. These approaches limit the opportunities for clustering according to connectivity. Other spatial access methods capture the isotropic nature of proximity by recursively ....

C. Faloutsos and Y. Rong. "DOT: A Spatial Access Method Using Fractals". In Proc. of the 7th Intl Conference on Data Engineering. IEEE, 1991.

....the Hilbert Curve which was suggested by Faloutsos [4] although the idea was not fully developed. We are not aware of any actual implementation based on this approach prior to our own, but there has been some useful theoretical work supported by simulation on the clustering properties of the curve [5], 11] 13] and [10] These studies generally agree on the superior clustering properties of the Hilbert Curve. Section 4 introduces the concept of a space filling curve using the Hilbert Curve as illustration. Sections 5 to 8 then discuss our own work on the Hilbert Curve in some detail, ....

Christos Faloutsos and Yi Rong. DOT: A Spatial Access Method Using Fractals. In: Proceedings of the Seventh International Conference on Data Engineering, April 8-12, 1991, Kobe, Japan, pages 152-159. IEEE Computer Society.

....include row wise ordering, the Gray ordering [21] the Hilbert curve [22] the z ordering (section 4.7) also called the Peano curve) Performance studies [2] show that the last two curves are the most suitable for spatial data. Section 4.7 discusses z ordering in more detail. Faloutsos and Rong [20] proposed an indexing method that combines the two transformation techniques. First, the minimum bounding box of an object in the d dimensional space is mapped into an nk dimensional point, and then this point is mapped into a one dimensional point using a spacefilling curve. 4.1 The ....

C. Faloutsos and Y. Rong. DOT: A spatial access method using fractals. In International Conference on Data Engineering, pages 152--159, Los Alamitos, Ca., USA, April 1991. IEEE Computer Society Press.

....thousands) 2. 6 Alternative Techniques Several improvements of the Z ordering concept are well known (cf. figure 3) Some authors propose the use of different curves such as Gray Codes [Fal 86, Fal 88] the Hilbert Curve [FR 89, Jag 90] or other variations [Kum 94] Many studies [Oos 90, Jag 90, FR 91] prefer the Hilbert curve among the proposals, due to its best distance preservation properties (also called spatial clustering properties) Klu 98] proposes a great variety of space filling curves and makes a comprehensive performance study using a relational implementation. As this performance ....

Faloutsos C., Rong Y.: `DOT: A Spatial Access Method Using Fractals', Proc. 7th Int. Conf. on Data Engineering, Kobe, Japan, 1991, pp. 152-159.

....query. A rectangle Q d i=1 [a i ; b i ] in R d can be mapped to the point (a 1 ; a 2 ; a d ; b 1 ; b 2 ; b d ) in R 2d , and a rectangleintersection query can be reduced to orthogonal range searching. Many heuristic data structures based on this scheme have been proposed; see [125, 235, 257] for a sample of such results. The second approach is to construct a data structure on S directly in R d . The most popular data structure based on this approach is the R tree, originally introduced by Guttman [157] R1 R2 R3 R4 R 5 A B C D E F G H I R1 R2 R3 R4 R5 R6 A B C D E F G H I Figure ....

C. Faloutsos and Y. Rong, DOT: A spatial access method using fractals, Proc. 7th IEEE Internat. Conf. on Data Engineering, 1991, pp. 152--158.

....and as such, we will be using it to compare our approach later in the paper. Other structures approaches have been proposed for indexing and querying spatial data. Next we briefly review some of them. The interested reader is referred to Samet s book [11] which surveys several others. In [2] the authors present an approach (named DOT) based on fractal functions which is also capable of transforming MBRs in two or higher dimensions to points in onedimensional space. Thus DOT (as well as our approach) uses a B tree to index MBRs. Although DOT is shown to outperform the classical ....

....of transforming MBRs in two or higher dimensions to points in onedimensional space. Thus DOT (as well as our approach) uses a B tree to index MBRs. Although DOT is shown to outperform the classical R tree [3] it is doubtful whether it can outperform the R tree (given the results in [2]) The issue of parallelizing DOT s approach was not investigated. The use of B trees to index MBRs has also been suggested in [13] However, in that paper, the authors propose the use of four B trees, each one indexing one of the corner coordinates that determine the MBRs. We propose ....

[Article contains additional citation context not shown here]

Faloutsos, C., and Rong, Y. DOT: A spatial access method using fractals. In 7th ICDE (Kobe, Japan, April 1991), pp. 152--159.

....R trees would be needed to provide performance comparable to a single (non parallel) R tree. As we shall see later in the paper our approach requires only two disks to outperform the R tree. To our knowledge no research has been done on parallelizing the R tree or the R tree. In [FR91] the authors present an approach (named DOT) based on fractal functions which is also capable of transforming MBRs in two or higher dimensions to points in onedimensional space. Thus DOT (as well as our approach) uses a B tree to index MBRs. Although DOT is shown to outperform the classical ....

....of transforming MBRs in two or higher dimensions to points in onedimensional space. Thus DOT (as well as our approach) uses a B tree to index MBRs. Although DOT is shown to outperform the classical R tree [Gut84] it is doubtful whether it can outperform the R tree (given the results in [FR91]) The issue of parallelizing DOT s approach was not investigated. The use of B trees to index MBRs has also been suggested in [TP95] However, in that paper, the authors propose the use of four B trees, each one indexing one of the corner coordinates that determine the MBRs. We propose ....

C. Faloutsos and Y. Rong. DOT: A spatial access method using fractals. In Proceedings of the Seventh International Conference on Data Engineering, pages 152--159, Kobe, Japan, April 1991.

....structures based on the R tree index aim primarily at indexing spatial data via N dimensional minimum bounding rectangles, they can also be used to index one dimensional ranges, e.g. valid time ranges. There are other techniques for indexing spatial data, such as the Z ordering [Ore86] and DOT [FR91] We also discuss those in more detail later in Section 5. The contribution of this paper is to address the problem of indexing VTDBs by using the proposed MAP21 approach. MAP21 makes use of a standard B tree (for an introduction refer to [EN94, Chapter 5] among many other texts) which is ....

....R trees and R trees [BKSS90, TP95, KSCL95] We thus chose to use the R tree There are other techniques for indexing spatial data. Among those we can cite those based on mapping N dimensional objects to points in the N dimensional space, such as the Z ordering [Ore86] and the DOT approach [FR91] The Z ordering transforms a spatial object, not necessarily a rectangle, to one or more, possibly disjoint, segments in the one dimensional space. Similarly to the R tree, this results in the possible replication of pointers to objects, which is a feature we would rather avoid. A good ....

[Article contains additional citation context not shown here]

C. Faloutsos and Y. Rong. DOT: A spatial access method using fractals. In Proceedings of the 7th IEEE International Conference on Data Engineering, pages 152--159, Kobe, Japan, April 1991.

No context found.

C. Faloutsos and Y. Rong, "Dot: A Spatial Access Method Using Fractals," Proc. IEEE Conf. Data Eng., pp. 152-159, Kobe, Japan, Apr. 1991. Early version available as UMIACS-TR-89-31, CS-TR2214.

....arrays, e.g. tuples of the form (x, y, z, t, temperature) which can be stored in some multiresolution, hierarchical fashion, clustering related (i.e. nearby) points together. Whenever a transformation is used (e.g. a twodimensional rectangle corresponds to a fourdimensional point [9], 12] a polyhedron is mapped to a high dimensionality point [15] We focus on rectilinear hyperrectangles, that is, n d rectangles with sides aligned with the axis. The problem we examine here is the following: 1041 4347 97 10.00 1997 IEEE . ....

C. Faloutsos and Y. Rong, "Dot: A Spatial Access Method Using Fractals," Proc. IEEE Conf. Data Eng., pp. 152-159, Kobe, Japan, Apr. 1991. Early version available as UMIACS-TR-89-31, CS-TR2214.

....arrays, e.g. tuples of the form (x; y; z; t; temperature) which can be stored in some multi resolution, hierarchical fashion, clustering related (i.e. nearby) points together. ffl Whenever a transformation is used (e.g. a 2 dimensional rectangle corresponds to a 4 dimensional point [9, 12]; a polyhedron is mapped to a high dimensionality point [14] The problem we examine here is the following: Given a hyper rectangle of size s 1 Theta s 2 Theta : s n , Find the number of blocks that it will span on the average. Previous attempts have been restricted to 2 dimensional ....

C. Faloutsos and Y. Rong. Dot: a spatial access method using fractals. In IEEE Data Engineering Conference, pages 152--159, Kobe, Japan, April 1991. early version available as UMIACS-TR-89-31, CS-TR-2214.

No context found.

C. Faloutsos and Y. Rong, "DOT: A Spatial Access Method Using Fractals," Proc. 7th IEEE Int. Conf. on Data Engineering, 152--159, 1991.

No context found.

C. Faloutsos and Y. Rong, "DOT: A Spatial Access Method Using Fractals," Proc. 7th IEEE Int. Conf. on Data Engineering, 152--159, 1991.

No context found.

C. Faloutsos and Y. Rong. DOT: A spatial access method using fractals. In Proceedings of the 7th International Conference on Data Engineering. IEEE Computer Society Press, 1991.

No context found.

C. Faloutsos and Y. Rong. DOT: A Spatial Access Method Using Fractals. In Proc. 7th Int. Conf. on Data Engineering,, pages 152--159, Kobe, Japan, 1991. 4

No context found.

C. Faloutsos and Y. Rong, "Dot: A spatial access method using fractals," in Proceedings of International Conference on Data Engineering, pp. 152--159, 1991.

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