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A storage structure called an R-tree is useful for indexing of rectangles and other polygons An R-tree is a balanced tree structure with the indexed polygons stored in leaf nodes, much like a B+ -tree However, instead of a range of values, a rectangular bounding box is associated with each tree node The bounding box of a leaf node is the smallest rectangle parallel to the axes that contains all objects stored in the leaf node The bounding box of internal nodes is, similarly, the smallest rectangle parallel to the axes that contains the bounding boxes of its child nodes The bounding box of a polygon is de ned, similarly, as the smallest rectangle parallel to the axes that contains the polygon Each internal node stores the bounding boxes of the child nodes along with the pointers to the child nodes Each leaf node stores the indexed polygons, and may optionally store the bounding boxes of the polygons; the bounding boxes help speed up checks for overlaps of the rectangle with the indexed polygons if a query rectangle does not overlap with the bounding box of a polygon, it cannot overlap with the polygon either (If the indexed polygons are rectangles, there is of course no need to store bounding boxes since they are identical to the rectangles) Figure 236 shows an example of a set of rectangles (drawn with a solid line) and the bounding boxes (drawn with a dashed line) of the nodes of an R-tree for the set of rectangles Note that the bounding boxes are shown with extra space inside them, to make them stand out pictorially In reality, the boxes would be smaller and t tightly on the objects that they contain; that is, each side of a bounding box B would touch at least one of the objects or bounding boxes that are contained in B The R-tree itself is at the right side of Figure 236 The gure refers to the coordinates of bounding box i as BBi in the gure
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We shall now see how to implement search, insert, and delete operations on an R-tree Search: As the gure shows, the bounding boxes associated with sibling nodes may overlap; in B+ -trees, k-d trees, and quadtrees, in contrast, the ranges do not overlap A search for polygons containing a point therefore has to follow all child nodes whose associated bounding boxes contain the point; as a result, multiple paths may have to be searched Similarly, a query to nd all polygons that intersect a given polygon has to go down every node where the associated rectangle intersects the polygon Insert: When we insert a polygon into an R-tree, we select a leaf node to hold the polygon Ideally we should pick a leaf node that has space to hold a new entry, and whose bounding box contains the bounding box of the polygon However, such a node may not exist; even if it did, nding the node may be very expensive, since it is not possible to nd it by a single traversal down from the root At each internal node we may nd multiple children whose bounding boxes contain the bounding box of the polygon, and each of these children needs to be explored Therefore, as a heuristic, in a traversal from the root, if any of the child nodes has a bounding box containing the bounding box of the polygon, the R-tree algorithm chooses one of them arbitrarily If none of the children satisfy this condition, the algorithm chooses a child node whose bounding box has the maximum overlap with the bounding box of the polygon for continuing the traversal Once the leaf node has been reached, if the node is already full, the algorithm performs node splitting (and propagates splitting upward if required) in a manner very similar to B+ -tree insertion Just as with B+ -tree insertion, the R-tree insertion algorithm ensures that the tree remains balanced Additionally, it ensures that the bounding boxes of leaf nodes, as well as internal nodes, remain consistent; that is, bounding boxes of leaves contain all the bounding boxes of the polygons stored at the leaf, while the bounding boxes for internal nodes contain all the bounding boxes of the children nodes The main difference of the insertion procedure from the B+ -tree insertion procedure lies in how the node is split In a B+ -tree, it is possible to nd a value such that half the entries are less than the midpoint and half are greater than the value This property does not generalize beyond one dimension; that is, for more than one dimension, it is not always possible to split the entries into two sets so that their bounding boxes do not overlap Instead, as a heuristic, the set of entries S can be split into two disjoint sets S1 and S2 so that the bounding boxes of S1 and S2 have the minimum total area; another heuristic would be to split the entries into two sets S1 and S2 in such a way that S1 and S2 have minimum overlap The two nodes resulting from the split would contain the entries in S1 and S2 respectively The cost of nding splits with minimum total area or overlap can itself be large, so cheaper heuristics, such as the quadratic split heuristic are used (The heuristic gets is name from the fact that it takes time quadratic in the number of entries)
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