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7rtree - CSE-5120-Fall-2009 An R-tree for data points...

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CSE-5120-Fall-2009 R-Trees: Index structure for Spatial Searching Guttman, SIGMOD 1984 R-tree : a height-balanced tree has some similarity to a B-tree records in its leaf nodes pointing to data objects. An R-tree for spatial objects A B C D E F G H J K I N M L A B C D E F G H I J K L M N . . . . . . x-l x-h y-l y-h . . . pointer to child An R-tree for data points A B C D E F G H J K I N M L A B C D E F G H I J K L M N point x point x is under A only point y point y is under B only Leaf nodes contain ( I, object identifier ) I is an n -dimensional rectangle which is the bounding box of the spatial object I = ( I 0 , I 1 , ..., I n - 1 ) I i is an interval [a,b] Non-leaf nodes contain ( I, child pointer ) where I covers all rectangles in the lower node’s entries. Node size: maximum M , minimum m l M 2 m 47
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Some properties of an R -tree: 1. In a leaf node, I is the smallest rectangle that contains the n -dimensional data ob- ject. 2. For each entry in a non-leaf node, I is the smallest rectangle that contains the rectan- gles in the child node. 3. The root node has at least two children un- less it is a leaf, and at most M children. 4. All leaves appear on the same level. The height of an R-tree containing N index records is at most | log m N | - 1 The maximum number of nodes is » N m + » N m 2 + ... + 1 Worst case space utilization for all nodes ex- cept the root is m M . Range Search A B C D E F G H J K I N M L A B C D E F G H I J K L M N Query Range Search Similar to B -tree in some way. However, more than one subtree under a node may be visited, not possible to guarantee good perfor- mance . Algorithm Search : Given: an R-tree, Query: a search rectangle Q. 1. search subtrees check each entry E to see if E.I overlaps Q . For all overlapping entries , Search the subtree pointed to by its pointer. 2. search leaf node check all entries E to see if E.I overlaps Q .
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