## Hilbert R-tree: An improved R-tree using fractals (1994)

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Citations: | 185 - 9 self |

### BibTeX

@INPROCEEDINGS{Kamel94hilbertr-tree:,

author = {Ibrahim Kamel and Christos Faloutsos},

title = {Hilbert R-tree: An improved R-tree using fractals},

booktitle = {},

year = {1994},

pages = {500--509}

}

### Years of Citing Articles

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### Abstract

We propose a new R-tree structure that outperforms all the older ones. The heart of the idea is to facilitate the deferred splitting approach in R-trees. This is done by proposing an ordering on the R-tree nodes. This ordering has to be 'good', in the sense that it should group 'similar' data rectangles together, to minimize the area and perimeter of the resulting minimum bounding rectangles (MBRs). Following [19] we have chosen the so-called '2D-c' method, which sorts rectangles according to the Hilbert value of the center of the rectangles. Given the ordering, every node has a welldefined set of sibling nodes; thus, we can use deferred splitting. By adjusting the split policy, the Hilbert R-tree can achieve as high utilization as desired. To the contrary, the R -tree has no control over the space utilization, typically achieving up to 70%. We designed the manipulation algorithms in detail, and we did a full implementation of the Hilbert R-tree. Our experiments show that the '2-to-...

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- 1975
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- 1990
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- Robinson
- 1981
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- 1982
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Citation Context ...: methods that transform rectangles into points in a higher dimensionality space [16, 8]; methods that use linear quadtrees [9] [1] or, equivalently, the z-ordering [23] or other space filling curves =-=[7]-=- [18]; and finally, methods based on trees (R-tree [15], k-d-trees [4], k-d-B-trees [25], hB-trees [21], cell-trees [13] e.t.c.) One of the most promising approaches in the last class is the R-tree [1... |

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