Bially, T.: Space-filling curves: their generation and their application to band reduction. IEEE Trans. Inform. Theory, IT-15:658--664, (1969)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....curve approximations. Finding the quantised arc length distance s along a Hilbert curve from quantised spatial positions (x, y, z) and vice versa are relatively straightforward operations. In 2D, these conversion algorithms can be represented using a reversible state machine with only four states [10,46]. In 3D, the conversion is slightly more complex [12,49] but it can still be implemented reasonably e#ciently. The advantage of the Hilbert curve encoding of position is that points mapped near each other on the Hilbert curve are guaranteed to be within a certain distance of each other in the ....

T. Bially. Space-Filling Curves: Their Generation and Their Application to Bandwidth Reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, November 1969. 4.1

....is only correct if all homogeneous scale factors are unity. In order for parameter interpolation to be correct under perspective, we have to divide by a common linear perspective correction factor. This factor can be computed by setting up an equation for the reference pervertex parameter values [1, 1, 1]. We will denote this equation by R: R = 1, 1, 1] M 1 . 26) Because we will be dividing by the value given by the evaluation of R, we can use the adjoint A in the above equations rather than M 1 , since common scale factors will cancel. Furthermore, when we have many parameters to ....

....unity. In order for parameter interpolation to be correct under perspective, we have to divide by a common linear perspective correction factor. This factor can be computed by setting up an equation for the reference pervertex parameter values [1, 1, 1] We will denote this equation by R: R = [1, 1, 1] M 1 . 26) Because we will be dividing by the value given by the evaluation of R, we can use the adjoint A in the above equations rather than M 1 , since common scale factors will cancel. Furthermore, when we have many parameters to interpolate, rather than computing a linear equation per ....

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T. Bially. Space-Filling Curves: Their Generation and Their Application to Bandwidth Reduction. IEEE Transactions on Information Theory, 15:658--664, November 1969.

....(0,0) on the H 2 curve has a Hilbert value of 0, while the point (1,1) has a Hilbert value of 2. We call the order of the values on the Hilbert curve the Hilbert order. The Hilbert curve can be generalized for any higher dimensionality. The algorithms to compute the Hilbert values can be found in [2, 3]. 1 1 2 0 0 3 0 1 0 1 2 3 H1 H2 Fig. 4 Hilbert Curves of order 1 and 2 1 3 2 13 12 0 0 1 14 15 2 4 7 8 11 3 5 6 9 10 3.1 Basic Ideas and Properties In the HG tree, all n dimensional values are transformed into 1 dimensional values using Hilbert transformation before they can be used. ....

T. Bially, "Space-Filling Curves: Their Generation and Their Application to Bandwidth Reductin," IEEE Transactions on Information Theory, pp. 658-664, Nov. 1969.

....A set of child pointers and children MBRs are also stored in each node, the child pointer addresses where the child node is stored. The Hilbert R tree [14] uses the index values of the Hilbert space filling curves to represent n dimensional data for indexing. The Hilbert space filling curve [6] [8] 20] basically maps from n dimensional data to a one dimensional value, and vice versa. In [15] Jagadish shows that the Hilbert space filling curve has a better clustering property compared to other space filling curves. In the Hilbert R tree, each child entry of a node is associated with a ....

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Trans. on Information Theory, IT15 (6):658 -- 664, November 1969.

....for parallel dense linear algebra. The use of quad or oct trees (or, in a dual interpretation, space filling curves) is known in parallel computing [1, 22, 23, 37, 39, 44] for improving both load balance and locality. They have also been applied for bandwidth reduction in information theory [2], for graphics applications [16, 30] and for database applications [25] The computations thus parallelized or restructured are reasonably coarse grained, thus making the overheads of maintaining and accessing the data structures insignificant. A recent paper by Frens and Wise [13] champions the ....

.... [20, 36, 38] Instances of this family of layout functions are familiar in parallel computing under the names Morton ordering and Hilbert ordering, and have been used for load balancing purposes [1, 22, 23, 37, 39, 44] They have also been applied for bandwidth reduction in information theory [2], for graphics applications [16, 30] and for database applications [25] Despite the dilation effect described above, canonical layout functions have one major advantage that is heavily exploited for efficiency in address computation. Simple algebraic manipulation of the defining formulas, as ....

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T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

....first order curve) can be regarded as being different states enabling the tree structure to be expressed more compactly as a state diagram. A method for constructing Hilbert Curve state diagrams is described by Lawder [13] who adapts and extends a generic technique originally proposed by Bially [1]. In higher than about 9 dimensions, state diagrams also become too large to accommodate in memory and mappings need to be calculated instead, for example in the manner detailed by Butz [2] and developed by Lawder [12] 4 Application of the Hilbert Curve We now describe how the Hilbert Curve is ....

T. Bially. Space-Filling Curves: Their Generation and Their Application to Bandwidth Reduction. IEEE Transactions on Information Theory, IT-15(6):658-- 664, Nov 1969.

....is based on a C shaped line segment and its three counterparts rotated by 90, 180, and 270 degrees. Figure 2(g) illustrates this layout. The Sfunction for this layout is computationally more complex than any of the others. The fastest method we know of is based on an informal description by Bially [2], which works by driving a finite state machine with pairs of bits from i and j, delivering two bits of S(i;j) at each step. 3.4 Summary We have described five recursive layout functions in terms of spacefilling curves. These layouts grow in complexity from the ones basedon ....

.... Library (ESSL) The application of space filling curves is not new to parallel processing, although most of the applications of the techniques have been tailored to specific application domains [1, 20, 21, 33, 35, 38] They have also been applied for bandwidth reduction in information theory [2], for graphics applications [14, 27] and for database applications [22] Most of these applications have far coarser granularity than our test codes. We have shown that the overheads of these layouts can be reduced enough to make them useful for fine grained computations. Parallel languages and ....

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

....queries which are optimal for the fieldwise xor scheme is a superset of those for the disk modulo scheme [12] The Hilbert curve scheme (HCAM) 3] is based on the idea of space filling curves. A space filling curve visits all points in a d dimensional space exactly once and never crosses itself [1]. It can be used to linearize a set of points (or buckets) in d dimensional space. The buckets are then assigned to disks in a round robin fashion. In other words, the bucket #i 1 ;i 2 ; i d #is assigned to the disk unit number H#i 1 ;i 2 ; i d#mod M where H#i 1 ;i 2 ; i d#is a function ....

T. Bially. Space-filling curves : Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

....extended spatial objects as well. 2.2.1 Space Partitioning Mapping Functions Space Partitioning mapping function based methods provides a mapping function from the domain of data items to the set of processor IDs. For example, a mapping function can be based on the Hilbert Space filling curve [7, 21]. See [10] for a survey of other mapping functions. The Hilbert curve gives a total ordering of points in 2 dimensional space. Polygons can be declustered using the Hilbert method as follows. Let L s be the set of input objects, and let L p be the ordered list of polygons corresponding to the ....

T. Bially. Space-Filling Curves: Their Generation and Their Application to Bandwidth Reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, 1969.

....declustering methods: 1) Space Partitioning with Mapping function based methods provide a mapping function from the domain of data items to the set of processor ids, assuming that all data items and queries are equiprobable. We use a mapping function based on the Hilbert Space filling curve [6, 16]. See [27] for a survey of other mapping functions for declustering. 2) Local load balancing methods [22, 17] consider a sample window of space (similar to the actual range query) at a time, and try to equally distribute the load in that window to all the processors. We use a rectangle as the ....

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, 1969.

....apply transformation of objects in order to overcome the difficulties raised by their spatial extent and dimensionality. The first known algorithm [O86] uses a grid to regularly divide the multidimensional space into small blocks, called pixels, and employs a space filling curve (zordering) B69] to order them. Each object is then approximated by the set of pixels intersected by its MBR, i.e. a set of z values. Since z values are 1 dimensional, the objects can be dynamically indexed using relational index structures, like the B tree, and the spatial join is performed in a sort merge ....

Bially T. Space-Filling Curves: Their Generation and Their Application to Bandwidth Reduction. IEEE Transactions on Information Theory, 15(6): 658-664, 1969.

....space that preserves the physical adjacency of the points. The Morton ordering is achieved by constructing keys for sorting the subdomains by interleaving the bits of the subdomain coordinates. For Hilbert curves, algorithms based on bit manipulation [5] and finite state diagrams [3] exist. An elegant recursive algorithm can be found in [18] We focus on Hilbert reordering because it traverses only contiguous subdomains and thus potentially results in better data locality in the reordered data structure. 3.2 Column and Row Ordering The sorting keys for row or column ....

....int num of dim, double ( coord) The following code segment shows the original data structure for particles in Barnes Hut and the added code for calling the hilbert reordering function. struct definition for a body typedef struct short type; real mass; double pos[3]; body, bodyptr; bodyptr bodytab; program provided coord function double coord(bodyptr btab, int i, int dim) return btab[i] pos[dim] main( serial initialization of particles hilbert reorder(bodytab, sizeof( bodytab) num bodies, 3, coord) parallel ....

T. Bially. Space--filling curves: Their generation and their application to bandwidth reduction. IEEE Trans. Inform. Theory, IT-15:658 -- 664, November 1969.

.... described as being based either on quadtrees [16] or on space filling curves [22, 32, 34] This layout is known in parallel computing as the Morton ordering and has been used for load balancing purposes [7, 25, 26, 33, 36, 40] It has also been applied for bandwidth reduction in information theory [9], for graphics applications [20, 30] and for database applications [27] Figure 4 illustrates this layout. Our interest, however, is in exploiting the benefits of such orderings for multi level memory hierarchies. Morton ordering has the following operational interpretation. Divide the original ....

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

....for parallel dense linear algebra. The use of quad or oct trees (or, in a dual interpretation, space filling curves) is known in parallel computing [44# 1# 23# 22# 37# 39] for improving both load balance and locality. They have also been applied for bandwidth reduction in information theory [2], for graphics applications [16# 30] and for database applications [25] The computations thus parallelized or restructured are reasonably coarse grained, thus making the overheads of maintaining and accessing the data structures insignificant. A recent paper byFrens and Wise [13] champions the ....

.... curves [36# 20# 38] Instances of this family of layout functions are familiar in parallel computing under the names Morton ordering and Hilbert ordering, and havebeenused for load balancing purposes [44# 1# 23# 22# 37# 39] They have also been applied for bandwidth reduction in information theory [2], for graphics applications [16# 30] and for database applications [25] Despite the dilation effect described above, canonical layout functions have one major advantage that is heavily exploited for efficiency in address computation. Simple algebraic manipulation of the defining formulas, as ....

[Article contains additional citation context not shown here]

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

....extended spatial objects as well. 10 2.2.1 Space Partitioning Mapping Functions Space partitioning mapping function based methods provide a mapping function from the domain of data items to the set of processor IDs. For example, a mapping function can be based on the Hilbert Space filling curve [7, 21]. See [10] for a survey of other mapping functions. The Hilbert curve gives a total ordering of points in 2 dimensional space. Polygons can be declustered using the Hilbert method as follows. Let L s be the set of input objects, and let L p be the ordered list of polygons corresponding to the ....

T. Bially. Space-Filling Curves: Their Generation and Their Application to Bandwidth Reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, 1969.

....issues, we limit the context to be a spatial context. The spatial context is defined as follows. For each subband, we scan the pixels inside that subband using a Hilbert space filling curve (see Fig. 10) This self similar curve exploits the coherence in two dimensions, and it guarantees that [2]: i) all pixels in an area are visited before moving on to another area, ii) the sequentially visited pixels are always adjacent in the 2 D image plane. The spatial context is then defined as the n C pixels that appear before the current pixel in the Hilbert scan. In the experiments we use n C = ....

T. Bially. Space filling curves: Their generation and their application to bandwith reduction. IEEE Trans. Inf. Theor., 15,(6):658--664, 1969.

....of splitting a full node. It has been used for image applications, such as QBIC [5] and content based retrieval and navigation in a hypermedia system [9] Kamel et al. 7] developed an improved R tree, the Hilbert R tree, which uses a Hilbert space filling curve for grouping data objects [2] [3] They show that the Hilbert R tree outperforms the R tree. The Hilbert space filling curve has been shown to give a better space localization than other space filling curves [8] Although there has been much research on the Rtree and its variants, there is hardly any development in the k ....

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Trans. on Information Theory, IT-15(6):658 -- 664, November 1969.

.... described as being based either on quadtrees [16] or on space filling curves [22, 32, 34] This layout is known in parallel computing as the Morton ordering and has been used for load balancing purposes [7, 25, 26, 33, 36, 40] It has also been applied for bandwidth reduction in information theory [9], for graphics applications [20, 30] and for database applications [27] Figure 4 illustrates this layout. Our interest, however, is in exploiting the benefits of such orderings for multi level memory hierarchies. Morton ordering has the following operational interpretation. Divide the original ....

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

....and column or row major ordering. benchmark to partition the tree cells. The Morton ordering is achieved by constructing keys for sorting the subdomains by interleaving the bits of the subdomain coordinates. For Hilbert curves, algorithms based on bit manipulation [6] or finite state diagrams [4] were devised previously. An elegant recursive algorithm can be found in [16] Note that the Hilbert ordering traverses only contiguous subdomains and thus can potentially give better data locality in the reordered data structure. To reduce false sharing in Category 1 applications resulted from ....

T. Bially. Space--filling curves: Their generation and their application to bandwidth reduction. IEEE Trans. Inform. Theory, IT-15:658 -- 664, November 1969.

....image as sets of runs [1] 4. Heuristics in computational geometry problems use the mapping. For example, for the travelling salesman problem, the cities are linearly ordered and visited accordingly [2] 5. Locality preserving mappings are used for bandwidth reduction of digitally sampled signals [3] and for graphics display generation [17] 6. In scientific parallel processing, locality preserving linearization techniques are preferred for dynamic unstructured mesh partitioning [14] Sophisticated mapping functions have been proposed in the literature. One, based on interleaving bits from ....

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Trans. on Information Theory, IT-15(6):658--664, November 1969.

....ECC Before we discuss the merits of various declustering methods, we define some terminology used throughout this report. We assume that the file is being declustered over M disks, which are numbered as 0; 1; 2; M Gamma 2; M Gamma 1. a b c d X X X X X X X X X X X X X X bucket[1,1] buket[2,1] bucket[1,2] bucket[2,2] Salary Age a b c d Figure 1: Cartesian product file: Each rectangle corresponds to a bucket A file F is called a Cartesian product file if it satisfies the following definition: Definition 1 Let D i denote the i th attribute domain of a k attribute file and let each D i ....

....we discuss the merits of various declustering methods, we define some terminology used throughout this report. We assume that the file is being declustered over M disks, which are numbered as 0; 1; 2; M Gamma 2; M Gamma 1. a b c d X X X X X X X X X X X X X X bucket[1,1] buket[2,1] bucket[1,2] bucket[2,2] Salary Age a b c d Figure 1: Cartesian product file: Each rectangle corresponds to a bucket A file F is called a Cartesian product file if it satisfies the following definition: Definition 1 Let D i denote the i th attribute domain of a k attribute file and let each D i be ....

[Article contains additional citation context not shown here]

T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Trans. on Information Theory, IT-15(6):658--664, November 1969.

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Bially, T.: Space-filling curves: their generation and their application to band reduction. IEEE Trans. Inform. Theory, IT-15:658--664, (1969)

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T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

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T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

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T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658--664, Nov. 1969.

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