J. G. Griffiths, An algorithm for displaying a class of spacefilling curves, SoftwarePractice & Experience, v.16 n.5, p.403-411, May 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....especially in the road atlas applications for the mobile devices. Line segments (or polylines) can be used to represent streets, rivers, etc. Other related studies have also used line segment datasets [HS91, HS92] In all the structures, the line segments are sorted based on the Hilbert order [Gri86] of their centroids and kept in an array. The leaf nodes of the structures have pointers (index into the array) to the actual data items. As was mentioned in Section 1, we do not consider dynamic structures in this study, and assume that all the data items are pre loaded into the memory resident ....

....techniques rather than insert item by item to build the data structure. Roussopoulos and Leifker [RL85] use packed R trees for such static databases to lower response times. Further, Kamel and Faloutsos [KF93] suggest using Hilbert value (a linearization technique for multidimensional space [Gri86] for sorting the data items before constructing the bulk loaded R tree. This is the structure that is evaluated in this paper. Typically, such R trees are built in a bottom up fashion, level by level. After the line segments are sorted, for each line segment, starting from the first and going ....

J. G. Griffiths. An Algorithm for Displaying a Class of Space-filling Curves . Software - Practice and Experience (SPE), 16(5):403--411, 1986.

....The Hilbert curve requires in addition to shrinkings, rotations through 90 o of the first and ;90 o of the last square before they are reflected and translated into their respective positions . Hilbert curvedrawing algorithms are generally efficient, recursive procedures [2] 10] 15] [16], 29] On the other hand, algorithms for point coding, the sorts of procedures needed when actually applying the Hilbert curve, tend to be more complex and ad hoc, generally unrelated to each other or to the drawing routines [6] 7] 8] 9] 11] 17] 18] 26] 27] This state of affairs ....

....arc notation broadly indicates how space is ordered within each cell, and is useful in developing the vertex labeling approach. The Sierpinski spacefilling curve is considered fairly easy to work with, due in large part to its high degree of symmetry. For drawing algorithms, see: 2] 10] 15] [16], 22] 29] 30] Most of these are similar, well constructed recursive procedures. These algorithms depend on curve regularity. For 17 Figure 3.2: Sierpinski curve, alternate view based on the same decomposition entry exit = entry exit 0. a b c Figure 3.3: Correspondence between vertex ....

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J. G. Griffiths. An algorithm for displaying a class of space-filling curves. Software---Practice and Experience, 16(5):403--411, May 1986.

....datasets, especially in the road atlas applications for the mobile devices. Line segments (or polylines) can be used to represent streets, rivers, etc. Other related studies have also used line segment datasets [9, 10] In all the structures, the line segments are sorted based on the Hilbert order [7] of their centroids and kept in an array. The leaf nodes of the structures have pointers (index into the array) to the actual data items. As was mentioned in Section 1, we do not consider dynamic structures in this study, and assume that all the data items are pre loaded into the memory resident ....

J. G. Griffiths. An Algorithm for Displaying a Class of Space-filling Curves . Software - Practice and Experience (SPE), 16(5):403--411, 1986.

....than all other R tree variants both in terms of speed (i.e. number of disk accesses required during a search) and density of storage. 54 Figure 4.4: Corner stitching 4.5. 1 The Hilbert curve and Hilbert values The Hilbert curve is a self similar member of the class of space filling curves [Gri86] A space filling curve C satisfies the property 8p8q(p = q) p 2 C; q 2 Z Theta Z where Z Z , as in definition 2.4. The Hilbert curve may be imagined as a tessellation of square tiles on which different orientations of the same basic design S 1 are drawn, and joined up to form a ....

J. G. Griffiths. An algorithm for displaying a class of space-filling curves. Software--- Practice and Experience, 16(5):403--411, May 1986.

....the Hilbert curves of order 2 and 3. When the order of the curve tends to infinity, the resulting curve is a fractal, with a fractal dimension of 2 [15] The Hilbert curve can be generalized for higher dimensionalities. Algorithms to draw the two dimensional curve of a given order, can be found in [11], 13] An algorithm for higher dimensionalities is in [2] The path of a space filling curve imposes a linear ordering, which may be calculated by starting at one end of the curve and following the path to the other end. Figure 5 shows one such ordering for grid size of 4 Theta 4 (see curve H 2 ....

J.G. Griffiths. An algorithm for displaying a class of space-filling curves. Software-Practice and Experience, 16(5):403--411, May 1986.

....Figure 1 shows the curves with order 1, 2 and 3. Figure 1: Hilbert Curve with order 1, 2 and 3 Now each pixel in the image stays in some place of the curve. Finally, follow along the curve, the Hilbert Order of each pixel can be uniquely decided. More detail about the Hilbert Curve can be found in [7, 1, 3]. There is a good property about the Hilbert Curve that can be easily seen from the recursive generation of the curve: Property 2.1 (locality) Suppose H i , H j are two Hilbert Curve with order i, j, and i j. The path of H i follows the path of H j . For example, let j = 1, H i (i 1) must fill ....

J. Griffiths. An algorithm for displaying a class of space-filling curves, Software-Practice and Experience, 16(5):403-411, 1986.

....rotate and reflect the curve at vertex 0 and at vertex 3. The curve can keep growing recursively by following the same rotation and reflection pattern at each vertex of the basic curve. Fig. 2.3 also shows the Hilbert curves of order 2 and 3. An algorithm to draw this curve is given in Griffiths [8] and Wirth [20] 0 1 2 3 H 1 H 2 H 3 Fig. 2.3 Hilbert curves of order 1, 2, and 3 The path of a space filling curve imposes a linear ordering, which may be calculated by starting at one end of the curve and following the path to the other end. Orenstein [14] used the 4 term z ordering to ....

Griffiths, J.G., "An Algorithm for Displaying a Class of Space-filling Curves," Software-Practice and Experience, vol. 16, no. 5, pp. 403-411, May 1986.

....the Hilbert curves of order 2 and 3. When the order of the curve tends to infinity, the resulting curve is a fractal, with a fractal dimension of 2 [22] The Hilbert curve can be generalized for higher dimensionalities. Algorithms to draw the two dimensional curve of a given order, can be found in [12], 18] An algorithm for higher dimensionalities is in [5] The path of a space filling curve imposes a linear ordering on the grid points. Figure 1 shows 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 H H H 1 2 3 Figure 1: Hilbert Curves of order 1, 2 and 3 one such ordering for a 4 Theta 4 ....

J.G. Griffiths. An algorithm for displaying a class of space-filling curves. Software-Practice and Experience, 16(5):403--411, May 1986.

....they group physically nearer values into the same bucket and thus achieve a good approximation of the value domain. A well known technique in spatial databases for capturing the proximity of multi dimensional values in a linear order is to use a spacefilling curve, such as the Hilbert curve [3, 7, 8]. We propose using the Hilbert numbering of attribute value combinations as a sort parameter (denoted by H) to order the data and thus once again reduce the problem to a single dimension. This scheme (called HILBERT) is illustrated in Figure 3, which shows a MaxDiff(H,F) partitioning of Figure 1 ....

J. G. Griffiths. An algorithm for displaying a class of space-filling curves. Software - Practice and Experience, 16(5):403--411, May 1984.

....11 12 13 14 15 H H H 1 2 3 Figure 5: Hilbert Curves of order 1, 2 and 3 infinity, the resulting curve is a fractal, with a fractal dimension of 2 [20] The Hilbert curve can be generalized for higher dimensionalities. Algorithms to draw the two dimensional curve of a given order, can be found in [10], 17] An algorithm for higher dimensionalities is in [5] The path of a space filling curve imposes a linear ordering on the grid points, which may be calculated by starting at one end of the curve and following the path to the other end. Figure 5 shows one such ordering for a 4 Theta 4 grid ....

J.G. Griffiths. An algorithm for displaying a class of space-filling curves. Software-Practice and Experience, 16(5):403--411, May 1986.

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J. G. Griffiths, An algorithm for displaying a class of spacefilling curves, SoftwarePractice & Experience, v.16 n.5, p.403-411, May 1986.

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J. G. Griffiths, An algorithm for displaying a class of spacefilling curves, SoftwarePractice & Experience, v.16 n.5, p.403-411, May 1986.

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J. G. Griffiths, An algorithm for displaying a class of spacefilling curves, SoftwarePractice & Experience, v.16 n.5, p.403-411, May 1986.

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