rtrees14 - Advanced Tree Structures RTrees(14 Introduction...

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Introduction In some of the previous sets of notes we examined many different variants of search trees. In this set of notes we will examine a special variant of the B-tree which is suited to representing spatial data. R-Trees Spatial data are the types of objects which are utilized frequently in many different application areas. Computer-assisted design (CAD), geographical data, and VLSI design layout are examples of application areas in which spatial data is created, searched, and deleted. This type of data requires special data structures to process in an efficient manner. For example, we might request that all counties in an area specified by geographical coordinates be printed or that all buildings within walking distance from a particular police station be identified. There have been many different data structures that have been developed to represent this type of data, R-trees were one of the first structures developed to handle such data and are still commonly used. There have been, as with most of the advanced tree structures that we have examined, several variants of R-trees including R + -trees and R * -trees. A R-tree is a height-balanced tree which is an extended variant of the B-tree. Objects are represented in an R-tree by their minimum bounding rectangle (MBR). R-tree are characterized by the following properties: 1. Every leaf node contains between m and M index records (where m M/ 2), unless it is the root. 2. For each index record ( I , tuple-identifier ) in a leaf node, I is the minimum bounding rectangle that spatially contains the m -dimensional data object represented by the indicated tuple. 3. Every internal node has between m and M children, unless it is the root. 4. For each entry ( I, child-pointer ) in an internal node, I is the minimum bounding rectangle that spatially contains the rectangles in the child node. 5. The root node has at least two children unless it is a leaf node. 6. All the leaf nodes appear on the same level. 7. All MBRs have sides parallel to the axis of a global coordinate system as shown in Figure 1. R-Trees - 1 Advanced Tree Structures – R - Trees (14)
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c b d e g f Figure 1 – A collection of spatial objects. Each node in the tree corresponds to a disk page. A leaf node consists of a number of entries with the form: ( I, tuple-id ), where I is an MBR, and a tuple-id is the unique identifier for the tuple in the database holding the object corresponding to that MBR. I is represented as I = (I 0 , …, I m-1 ), where I i is a closed, bounded interval [a, b] along direction i . Nonfinite intervals can also be considered, by having a , b , or both equal to infinity. Internal nodes are composed of a number of entries of the form: ( I, child-ptr ) where I is the MBR for all rectangles in the lower node (the child node) pointed to by child-ptr . Each node in the tree can have a maximum of M entries and a minimum of m (where m M/2 ) entries, unless it is the root. The root node has at least two children, unless it is a leaf.
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rtrees14 - Advanced Tree Structures RTrees(14 Introduction...

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