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1.9 b tree
- 1. Chapter 15 B External Methods – B-Trees
- 2. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-2 B-Trees • To organize the index file as an external search tree – Use block numbers for child pointers • A child pointer value of –1 is used as the null pointer Figure 15.10a –Figure 15.10a – Blocks organized into a 2-3 tree
- 3. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-3 B-Trees • If the index file is organized into a 2-3 tree – Each node would contain • Either one or two index records, each of the form <key, pointer> • Three child pointers Figure 15.10b –Figure 15.10b – A single node of the 2-3 tree
- 4. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-4 B-Trees • An external 2-3 tree is adequate, but an improvement is possible • To improve efficiency – Allow each node to have as many children as possible • In an external environment, the advantage of keeping a search tree short far outweighs the disadvantage of performing extra work at each node • Block size should be the only limiting factor for the number of children
- 5. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-5 B-Trees • Binary search tree – If a node N has two children, it must contain one key value • 2-3 tree – If a node N has three children, it must contain two key values • General search tree – If a node N has m children, it must contain m – 1 key values
- 6. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-6 B-Trees Figure 15.11Figure 15.11 a) A node with two children; b) a node with three children; c) a node with m children
- 7. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-7 B-Trees • B-tree of degree m – All leaves are at the same level – Nodes • Each node contains between m – 1 and m/2 records • Each internal node has one more child than it has records • Exception: The root can contain as few as one record and can have as few as two children – Example • A 2-3 tree is a B-tree of degree 3 – Each node contains between (3-1) = 2 and 3/2 = 1 records.
- 8. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-8 B-Trees Figure 15.13Figure 15.13 A B-tree of degree 5
- 9. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-9 B-Trees • Retrieval – Generalized search tree retrieval • Insertion into a B-tree – Step 1: Insert the data record into the data file – Step 2: Insert a corresponding index record into the index file
- 10. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-10 B-Trees Figure 15.14a and bFigure 15.14a and b The steps for inserting 55
- 11. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-11 B-Trees Figure 15.14c-eFigure 15.14c-e The steps for inserting 55
- 12. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-12 B-Trees • Deletion from a B-tree – Step 1: Locate the index record in the index file and delete it from the index file – Step 2: Delete the data record from the data file
- 13. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-13 B-Trees Figure 15.15a and bFigure 15.15a and b The steps for deleting 73
- 14. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-14 B-Trees Figure 15.15cFigure 15.15c The steps for deleting 73
- 15. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-15 B-Trees Figure 15.15dFigure 15.15d The steps for deleting 73
- 16. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-16 B-Trees Figure 14.15e and fFigure 14.15e and f The steps for deleting 73
- 17. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-17 Traversals • Accessing only the search key of each record, not the data file – Not efficiently supported by the hashing implementation – Efficiently supported by the B-tree implementation • The search keys can be visited in sorted order by using an inorder traversal of the B-tree • Accessing the entire data record – Not efficiently supported by the B-tree implementation
- 18. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-18 Multiple Indexing • Advantage – Allows multiple data organizations • Disadvantage – More storage space – Additional overhead for updating each index whenever the data file is modified
- 19. © 2004 Pearson Addison-Wesley. All rights reserved 15 B-19 Multiple Indexing Figure 15.16Figure 15.16 Multiple index files
- 20. Chi-Cheng Lin, Winona State University 20 Variations of B-Trees • The fewer nodes are in a B-tree, the better. (Why?) • Problems of B-tree: – It could be only half full • More nodes required • Space wasted – Inorder traversal “jumps” around nodes • B* -Trees: introduced by Donald Knuth • B+ -Trees: introduced by H. Wedekind
- 21. Chi-Cheng Lin, Winona State University 21 B* -Trees • B* -tree of degree m – All leaves are at the same level – Nodes • Each node contains between m – 1 and (2m – 1)/3 records • Each internal node has one more child than it has records • Exception: The root can contain as few as one record and as many as (2m-2) children – Example • B* -tree of degree 9 – Each node contains between (9 – 1) = 8 and (2×9 – 1)/3 = 5 records.
- 22. Chi-Cheng Lin, Winona State University 22 B* -Trees • Splitting nodes – When a node overflows, it is not split right away – A split is delayed by redistributing the keys between a node and its sibling – When both a node and its sibling are full, an insertion to the node will split the two nodes into three nodes • A B*-Tree is always two-third full instead of half full number of nodes in the tree is reduced
- 23. Chi-Cheng Lin, Winona State University 23 B* -Trees
- 24. Chi-Cheng Lin, Winona State University 24 B+ -Trees • References to data are only made from the leaves – All index records can be found from the leaves • Two sets of nodes – Index set • Internal nodes • Provides fast access of data – Sequence set • Leaves, linked sequentially • Provides efficient inorder traversal
- 25. Chi-Cheng Lin, Winona State University 25 B+ -Trees