1 / 27

B-Trees ( Rizwan Rehman)

B-Trees ( Rizwan Rehman). Large degree B-trees used to represent very large dictionaries that reside on disk. Smaller degree B-trees used for internal-memory dictionaries to overcome cache-miss penalties.

ethel
Download Presentation

B-Trees ( Rizwan Rehman)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. B-Trees(Rizwan Rehman) • Large degree B-trees used to represent very large dictionaries that reside on disk. • Smaller degree B-trees used for internal-memory dictionaries to overcome cache-miss penalties.

  2. A B-tree is a keyed index structure, comparable to a number of memory resident keyed lookup structures such as balanced binary tree, the AVL tree, and the 2-3 tree. • The difference is that a B-tree is meant to reside on disk, being made partially memory-resident only when entries int the structure are accessed. • The B-tree structure is the most common used index type in databases today. • It is provided byORACLE, DB2, and INGRES.

  3. For a binary search tree, the search time is O(h), where h is the height of the tree. • The height cannot be less than roughly log2n for a tree with n nodes. • If the tree is too large for internal memory, each access of a node is an I/O. • For a tree with 10,000 nodes: • log210000 = 100 disk accesses • We need a structure that would require only 3 or 4 disk accesses.

  4. AVL Trees • n = 230 = 109(approx). • 30 <= height <= 43. • When the AVL tree resides on a disk, up to 43 disk access are made for a search. • This takes up to (approx) 4 seconds. • Not acceptable.

  5. m-way Search Trees • Each node has up to m – 1 pairs and m children. • m = 2 => binary search tree.

  6. 10 30 35 4-Way Search Tree k > 35 k < 10 10 < k < 30 30 < k < 35

  7. Definition Of B-Tree • Definition assumes external nodes (extended m-way search tree). • B-tree of order m. • m-way search tree. • Not empty => root has at least 2 children. • Remaining internal nodes (if any) have at least ceil(m/2) children. • External (or failure) nodes on same level.

  8. 2-3 And 2-3-4 Trees • B-tree of order m. • m-way search tree. • Not empty => root has at least 2 children. • Remaining internal nodes (if any) have at least ceil(m/2) children. • External (or failure) nodes on same level. • 2-3 tree is B-tree of order 3. • 2-3-4 tree is B-tree of order 4.

  9. B-Trees Of Order 5 And 2 • B-tree of order m. • m-way search tree. • Not empty => root has at least 2 children. • Remaining internal nodes (if any) have at least ceil(m/2) children. • External (or failure) nodes on same level. • B-tree of order 5is3-4-5tree (root may be2-node though). • B-tree of order 2 is full binary tree.

  10. level # of nodes Minimum # Of Pairs • n = # of pairs. • # of external nodes = n + 1. • Height = h=> external nodes on level h + 1. 1 1 >= 2 2 >= 2*ceil(m/2) 3 >= 2*ceil(m/2)h-1 h + 1 n + 1 >= 2*ceil(m/2)h-1, h >= 1

  11. height # of pairs Minimum # Of Pairs n + 1 >= 2*ceil(m/2)h-1, h >= 1 • m = 200. >= 199 2 >= 19,999 3 >= 2 * 106 –1 4 >= 2 * 108 –1 5 h <= log ceil(m/2)[(n+1)/2] + 1

  12. Insert 8 4 15 20 1 3 5 6 9 30 40 16 17 Insertion into a full leaf triggers bottom-up node splitting pass.

  13. Split An Overfull Node m a0 p1 a1 p2 a2 …pm am • aiis a pointer to a subtree. • piis a dictionary pair. ceil(m/2)-1 a0 p1 a1 p2 a2 …pceil(m/2)-1 aceil(m/2)-1 m-ceil(m/2) aceil(m/2) pceil(m/2)+1 aceil(m/2)+1 …pm am • pceil(m/2)plus pointer to new node is inserted in parent.

  14. Insert 8 4 15 20 1 3 5 6 9 30 40 16 17 • Insert a pair with key = 2. • New pair goes into a 3-node.

  15. 1 2 3 2 1 3 Insert Into A Leaf 3-node • Insert new pair so that the 3keys are in ascending order. • Split overflowed node around middle key. • Insert middle key and pointer to new node into parent.

  16. Insert 8 4 15 20 1 3 5 6 9 30 40 16 17 • Insert a pair with key = 2.

  17. Insert 8 4 2 15 20 3 1 5 6 9 30 40 16 17 • Insert a pair with key = 2plus a pointer into parent.

  18. 8 2 4 15 20 1 3 5 6 9 30 40 16 17 Insert • Now, insert a pair with key = 18.

  19. 16 17 18 17 16 18 Insert Into A Leaf 3-node • Insert new pair so that the 3keys are in ascending order. • Split the overflowed node. • Insert middle key and pointer to new node into parent.

  20. 8 2 4 15 20 1 3 5 6 9 30 40 16 17 Insert • Insert a pair with key = 18.

  21. Insert 8 17 2 4 15 20 18 1 3 5 6 9 16 30 40 • Insert a pair with key = 17plus a pointer into parent.

  22. Insert 17 8 2 4 15 20 1 18 3 5 6 9 16 30 40 • Insert a pair with key = 17plus a pointer into parent.

  23. 8 17 2 4 15 20 1 18 3 5 6 9 16 30 40 Insert • Now, insert a pair with key = 7.

  24. Insert 8 17 6 2 4 15 20 7 1 18 3 5 9 16 30 40 • Insert a pair with key = 6plus a pointer into parent.

  25. Insert 8 17 4 6 2 15 20 5 7 18 9 16 30 40 1 3 • Insert a pair with key = 4plus a pointer into parent.

  26. Insert 8 4 17 6 2 15 20 1 3 5 7 18 9 16 30 40 • Insert a pair with key = 8plus a pointer into parent. • There is no parent. So, create a new root.

  27. Insert 8 4 17 6 2 15 20 18 9 16 30 40 1 3 5 7 • Height increases by 1.

More Related