C. Faloutsos, "Gray Codes for Partial Match and Range Queries," IEEE Trans. Software Eng., vol. 14, no. 10, pp. 1,381-1,392, Oct. 1988. Early version available as UMIACS-TR-87-4, also CS-TR1796.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....at the Technion. The authors are with the Computer Science Department, Technion Israel Institute of Technology, Haifa 32000, Israel (e mail: moosh, etzion cs. technion.ac.il) Communicated by T. E. Fuja, Associate Editor at Large. Publisher Item Identifier S 0018 9448(99)07312 5. hashing [10], puzzles, such as the Chinese Rings and Tower of Hanoi [13] ordering of documents on shelves [19] signal encoding [20] data compression [22] and circuit testing [23] Finally, for an excellent survey on Gray codes the interested reader is referred to [24] The classic example of a Gray code ....

C. Faloutsos, "Gray codes for partial match and range queries," IEEE Trans. Software Eng., vol. 14, pp. 1381--1393, 1988.

....these numbers are then used to obtain a representative number for the spatial objects. Techniques on ordering multi dimensional objects using single dimensional values have been proposed. These include the Peano curve [18] locatiohal keys [1] Zordering [23] Hilbert curve [11] and gray ordering [10]. The mapping functions used in mapping must preserve the proximity between data well enough in order to yield reasonably good spatial search. A spatial object with extent is represented by a set of numbers or one dimensional objects. These one dimensional points are indexed using B trees. One ....

C. Faloutsos. Gray-codes for partial match and range queries. IEEE Transactions on Software Engineering, 14(10):1381-1393, 1988.

....on the data points and on the directory regions. A space filling curve is a mapping that maps the unit interval onto the n dimensional unit hyperrectangle continuously. While there are other space filling curves such as the Peano curve (also known as the Z curve) 23] and the Gray code curve [5], it was shown that the Hilbert curve achieves better clustering than the others [7, 13] The desirable features of the Hilbert curve are that the points close on the Hilbert curve are close in the domain space, and the points close in the domain space are likely to be close on the Hilbert curve. ....

C. Faloutsos, "Gray codes for partial match and range queries," IEEE Transactions on Software Engineering, Vol. 14, No. 10, pp. 1381-1393, 1988.

....ones by using Hilbert space filling curve [18] A space filling curve is a mapping that maps the unit interval onto the n dimensional unit hyper cube continuously. While there are other space filling curves such as the Peano curve (also known as the Z curve) 19] and the Gray code curve [20], it was shown that the Hilbert curve achieves better clustering than the others [21, 22] The desirable features of the Hilbert curve are that the points close on the Hilbert curve are close in the domain space, and the points close in the domain space are likely to be close on the Hilbert ....

C. Faloutsos, "Gray codes for partial match and range queries," IEEE Transactions on Software Engineering, 14(10), 1988, 1381-1393.

....and aspirations for extracting valuable information become more sophisticated. Mapping multi dimensional data to one dimension, enabling simple and well understood one dimensional access methods to be exploited, has been suggested as a solution in the literature, for example by Faloutsos [3, 4]. One way of effecting such a mapping is to utilize space filling curves which pass through every point in a space once so giving a one one correspondence between the coordinates of the points and the one dimensional sequence numbers of the points on the curve. For the most part, interest in ....

C. Faloutsos. Gray Codes for Partial Match and Range Queries. IEEE Transactions on Software Engineering, 14(10):1381--1393, Oct 1988.

....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: `Gray Codes for Partial Match and Range Queries', IEEE Transactions on 15 Software Engineering, Vol.14, No.10, 1381-1393, 1988.

....storing points based on the bit interleaving ordering is referred to as an N tree [251] or a zkd tree [206] in the literature. See [224] for a more detailed discussion on the applications of bit interleaving in spatial data structures. Geometric Range Searching and Its Relatives 13 Faloutsos [117] suggested using Gray codes to define a linear order on points. In general, space filling curves can be used to define a linear ordering on input points ; Hilbert and Morton curves, shown in Figure 2, are the some of the widely used space filling curves in the context of range searching. See [1, ....

C. Faloutsos, Gray codes for partial match and range queries, IEEE Trans. on Software Eng., 44 (1988), 1381--1393.

....as the page assignments of each buckets on that disk, assuming that there are four disks in the system. The disk assignment in both gures is based on the DM declustering [8] Several other single disk allocation mechanisms have been proposed to linearize a multidimensional space on a single disk [9]. In Figure 1(a) we assign the pages in the row major order for each disk. For example, bucket (3,0) is assigned to page 3 of disk 3 whereas bucket (0,3) is assigned to page 0 of disk 3. In Figure 1(b) the page allocation is done in a column major round robin way. Di erent other techniques can ....

C. Faloutsos. Gray codes for partial match and range queries. IEEE Transactions on Software Engineering, 14(10):1381-1393, October 1988.

.... [CSW89] Gray codes have found applications in such diverse areas as circuit testing [RC81] signal encoding [Lud81] ordering of documents on shelves [Los92] data compression [Ric86] statistics [DH94] graphics and image processing [ASD90] processor allocation in the hypercube [CS90] hashing [Fal88], computing the permanent [NW78] information storage and retrieval [CCC92] and puzzles, such as the Chinese Rings and Towers of Hanoi [Gar72] In recent variations on combinatorial Gray codes, generation problems have been considered in which the difference between successive objects, although ....

C. Faloutsos. Gray codes for partial match and range queries. IEEE Transactions on Software Engineering, 14(10):1381--1393, 1988.

....by the database server, which is not negligible, if the number of intervals is very high (e.g. in the 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 ....

Faloutsos C.: `Gray Codes for Partial Match and Range Queries', IEEE Transactions on Software Engineering (TSE) , Vol. 14, No. 10, 1988, pp. 1381-1393.

....k dimensional objects to line segments, using the socalled z transform. This transformation 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 transformed space. Improved distance preserving transformations have been proposed [FALO88], which achieve better clustering of nearby points, by using Gray codes. The original z transform induces an ordering of the k d points, which is the very same one that a (k dimensional) quad tree uses to scan pixels in a k dimensional space. The transformation of a rectangle is a set of line ....

C. Faloutsos, "Gray Codes for Partial Match and Range Queries," IEEE Trans. on Software Engineering Vol. 14(10), October 1988, pp. 1381-1393.

....which work well with one dimensional data are not suitable for spatial data, and hence new methods are required. There are two kinds of spatial access methods: tree or directory based (such as R trees [10] KDB trees [18] grid files [11, 15] and order based (such as z order [16, 17] gray order [6], Hilbert order [7] etc. This paper focuses upon the latter. Order based methods, also referred to as space filling curves, map points from a multidimensional data space into the one dimensional disk space. These curves span a data space of grid points, without visiting any point more than ....

....data space into four quadrants, and further dividing each quadrant into four more quadrants and so on. Examples of such methods are: z order, gray order, hilbert order and nu order. The binary z ordering (BZ) method was proposed and studied in [16, 17] The gray z2 ordering (GZ) was proposed in [6] and shown to outperform BZ in terms of spatial clustering. Subsequently, Hilbert (H) ordering was described in [7] and shown to be superior to both BZ and GZ ordering. In [13] another technique called Gray nu ordering (GN) was proposed, but the study showed that Hilbert was the best among these ....

Faloutsos, C., "Gray Codes for Partial Match and Range Queries," IEEE Transactions on Software Engineering, Vol. 14, No. 10, October 1988, pp. 1381-1393.

....xm Gamma1 y m Gamma1 xm Gamma2 : y 0 . A B tree storing points based on the bit interleaving ordering is referred to as an N tree [285] or a zkd tree [228] in the literature. See [251] for a more detailed discussion on the applications of bit interleaving in spatial data structures. Faloutsos [121] suggested using Gray codes to define a linear order on points. In general, space filling curves 5 can be used to define a linear ordering on input points; Hilbert and Morton curves, shown in Figure 2, are the some of the space filling curves commonly used for this purpose. See [1, 30, 126, 175] ....

C. Faloutsos, Gray codes for partial match and range queries, IEEE Trans. on Software Eng., 44 (1988), 1381--1393.

....cases where an optimal allocation method exists, we provide simple allocation methods that can be easily implemented. Previous research concentrated on efficient retrieval of images for circular queries [CR93] and, more generally, on the disk allocation of relational databases for range queries [Fal88,LSR92,CC92] as well as the organization of raster graphics memory [CLRS86] We discuss their relationship to our work as well as future extensions in the conclusion of the paper. 2 Problem Statement Consider the domain of discourse, an n 1 Theta n 2 array whose elements (i; j) where 0 i n 1 ....

.... (the two attributes are the x and y axes, and the rest of the attributes correspond to the data associated with each unique combination of x and y) Optimality results for partial match queries have been derived [AE93] However, for range queries, only a few allocation methods have been proposed [Fal88,GD90,GDQ92,CC92,LSR92,CR93] Ghandeharizadeh, DeWitt and Qureshi [GD90,GDQ92] demonstrated, using simulation, that declustering data on multiple disks results in significant improve ments in response time for range queries. In [Fal88] Faloutsos uses gray code techniques and in [FB93] ....

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C. Faloutsos. Gray codes for partial match and range queries. IEEE Transactions on Software Engineering, 14(10):1381--1393, October 1988.

....results have shown similar results independent of any data distribution assumptions. Nonetheless, it is certainly the case that some orderings are (substantially) worse than others. Several linearizations have been proposed, including the Zcurve (or bit interleaving) 17] the Gray code curve [3, 4], the Hilbert curve [6, 12] and variants (cf. 15] There have been several analyses of the expected performance of many of these linearization curves, including [7, 13] In [5, 12] it was shown that the Hilbert curve was the best linear ordering for a wide variety of applications. Our results ....

C. Faloutsos. Gray codes for partial match and range queries. IEEE Transactions on Software Engineering, 1987.

....the bits representing the x and y coordinates, starting from the x coordinate. For example, 3; 5) 011; 101) translates to 011011 = 27. All the space filling curves can also be described in such a fashion using a suitable bit interleaving function. Graycode Curve. The graycode curve [4] uses the same bit interleaving function as the Z curve. While the Z curve visits points in the numerical order of the results of applying the interleaving function to the Graycode curve Hilbert curve Figure 2: Graycode and Hilbert curves for grids of size 8 Theta 8. points, the graycode curve ....

C. Faloutsos, Gray codes for partial match and range queries, IEEE Transactions on Software Engineering, 14(10) (1988) 1381-1393.

No context found.

C. Faloutsos, "Gray Codes for Partial Match and Range Queries," IEEE Trans. Software Eng., vol. 14, no. 10, pp. 1,381-1,392, Oct. 1988. Early version available as UMIACS-TR-87-4, also CS-TR1796.

.... into (which will be proportional to the response time for this query) From the theoretical point of view, it proposes a methodology which we believe will be useful in the analysis of other quadtree related methods (e.g. methods using space filling curves, such as the z ordering [17] Gray codes [7], or the Hilbert curve [10] The methodology consists of two steps: Step 1 Solve the problem for the magic rectangles (which is easy) Step 2 Show that the formula for an arbitrary rectan gle can be derived by linear interpolation from suitable magic rectangles. Future work includes the ....

C. Faloutsos, "Gray Codes for Partial Match and Range Queries," IEEE Trans. Software Eng., vol. 14, no. 10, pp. 1,381-1,392, Oct. 1988. Early version available as UMIACS-TR-87-4, also CS-TR1796.

.... into (which will be proportional to the response time for this query) From the theoretical point of view, it proposes a methodology which we believe will be useful in the analysis of other quadtree related methods (e.g. methods using space filling curves, such as the z ordering [17] Gray codes [7], or the Hilbert curve [10] The methodology consists of two steps: Step 1 solve the problem for the magic rectangles (which is easy) Step 2 show that the formula for an arbitrary rectangle can be derived by linear interpolation from suitable magic rectangles. Future work includes the ....

C. Faloutsos. Gray codes for partial match and range queries. IEEE Trans. on Software Engineering, 14(10):1381--1393, October 1988. early version available as UMIACS-TR-87-4, also CS-TR-1796.

....in geometric complexity problems use distancepreserving mappings: e.g. to solve the traveling salesman problem, the cities are ordered in a linear ordering, and visited in this order. 1] The three space filling curves we compare are the Peano curve, the reflected binary graycode (RBG) curve [7] and the Hilbert curve. The first two have been used before as distance preserving mappings for partial match retrieval. The hypothesis is that the proposed Hilbert curve yields a better distance preserving mapping, because it avoids long jumps between points. Among the different types of queries ....

....binary string. Note: The the z value of the point (1,6) in Fig. 4 is 22) Higher dimensions of the Peano curve are calculated the same way. For k dimensional curves, read and interleave k numbers instead of two numbers. The algorithm to compute r values for the RBG is similar to the Peano curve [7]. The only difference is that the bit strings are considered as gray codes instead of binary codes. Specifically: 9 Algorithm R 1. Read in the (integer) x and y coordinates. 2. Convert them to the equivalent gray codes (i.e. bit strings) 3. Interleave the bits of the two bit strings 4. ....

Faloutsos, C., "Gray Codes for Partial Match and Range Queries," IEEE Trans. on Software Engineering, vol. 14, no. 10, pp. 1381-1393, Oct. 1988. early version available as UMIACS-TR-87-4, also CS-TR-1796

....the Hilbert curve to impose a linear ordering on the data rectangles. A space filling curve visits all the points in a k dimensional grid exactly once and never crosses itself. The Z order (or Morton key order, or bit interleaving, or Peano curve) the Hilbert curve, and the Gray code curve [6] are examples of space filling curves. In [7] it was shown experimentally that the Hilbert curve achieves the best clustering among the three above methods. Next we provide a brief introduction to the Hilbert curve: The basic Hilbert curve on a 2x2 grid, denoted by H 1 , is shown in Figure 1. To ....

C. Faloutsos. Gray codes for partial match and range queries. IEEE Trans. on Software Engineering, 14(10):1381--1393, October 1988. early version available as UMIACS-TR-87-4, also CS-TR-1796.

....k dimensional objects to line segments, using the so called z transform. This transformation 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 transformed space. Improved distancepreserving transformations have been proposed [4], which achieve better clustering of nearby points, by using Gray codes. The original ztransform induces an ordering of the k d points, which is the very same one that a (kdimensional) quad tree uses to scan pixels in a k dimensional space. The transformation of a rectangle is a set of line ....

C. Faloutsos, "Gray Codes for Partial Match and Range Queries," IEEE Trans. on Software Engineering , 1988. (to appear)

....use space filling curves (or fractals) and specifically, the Hilbert curve. A space filling curve visits all the points in a k dimensional grid exactly once and never crosses itself. The Z order (or Morton key order, or bit interleaving, or Peano curve) the Hilbert curve, and the Gray code curve [6] are examples of space filling curves. In [7] it was shown experimentally that the Hilbert curve achieves the best clustering among the three above methods. Next we provide a brief introduction to the Hilbert curve: The basic Hilbert curve on a 2x2 grid, denoted by H 1 , is shown in Figure 5. To ....

C. Faloutsos. Gray codes for partial match and range queries. IEEE Trans. on Software Engineering, 14(10):1381--1393, October 1988. early version available as UMIACS-TR-87-4, also CS-TR-1796.

No context found.

Faloutsos, C. (1988). Gray codes for partial match and range queries. IEEE Transactions on Software Engineering, 14(10):1381--1393.

No context found.

C. Faloutsos. Gray codes for partial match and range queries. IEEE Trans. on software Engineering, 14:1381--1393, 1988.

No context found.

C. Faloutsos. Gray Codes for Partial Match and Range Queries. IEEE Transactions on Software Engineering, 14(10):1381-1393, 1988.

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