Fileprocessing lec-7
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This document discusses B-trees and algorithms for inserting and deleting keys from a B-tree. It explains that B-TREE-INSERT uses B-TREE-SPLIT-CHILD to guarantee recursion never descends to a full node. When the root becomes full during an insertion, it is split and a new root node is created, increasing the tree height by one. For deletion, it ensures a node does not become too small by potentially redistributing keys between nodes. Examples are provided to illustrate deleting keys from B-trees.
![Inserting a key into a B-tree in a single
pass down the tree
• The B-TREE-INSERT procedure uses B-TREE-SPLIT-CHILD to
guarantee that the recursion never descends to a full node.
B-TREE-INSERT(T, k)
1 r ← root[T]
2 if n[r] = 2t - 1
3 then s ← ALLOCATE-NODE()
4 root[T] ← s
5 leaf[s] ← FALSE
6 n[s] ← 0
7 c1[s] ← r
8 B-TREE-SPLIT-CHILD(s, 1, r)
9 B-TREE-INSERT-NONFULL(s, k)
10 else B-TREE-INSERT-NONFULL(r, k)](https://image.slidesharecdn.com/fileprocessing-lec-7-161118210554/85/Fileprocessing-lec-7-2-320.jpg)
![B-TREE-INSERT-NONFULL
• B-TREE-INSERT-NONFULL(x, k)
1 i ← n[x]
2 if leaf[x]
3 then while i ≥ 1 and k < keyi[x]
4 do keyi+1[x] ← keyi[x]
5 i ← i - 1
6 keyi+1[x] ← k
7 n[x] ← n[x] + 1
8 DISK-WRITE(x)
9 else while i ≥ 1 and k < keyi[x]](https://image.slidesharecdn.com/fileprocessing-lec-7-161118210554/85/Fileprocessing-lec-7-5-320.jpg)
![10 do i ← i - 1
11 i ← i + 1
12 DISK-READ(ci[x])
13 if n[ci[x]] = 2t - 1
14 then B-TREE-SPLIT-CHILD(x, i, ci[x])
15 if k> keyi[x]
16 then i ← i + 1
17 B-TREE-INSERT-NONFULL(ci[x], k)](https://image.slidesharecdn.com/fileprocessing-lec-7-161118210554/85/Fileprocessing-lec-7-6-320.jpg)
Fileprocessing lec-7
- 1. File Processing & Organization Course No: 5901227-3 Lecture 7 B-Trees (Continued)
- 2. Inserting a key into a B-tree in a single pass down the tree • The B-TREE-INSERT procedure uses B-TREE-SPLIT-CHILD to guarantee that the recursion never descends to a full node. B-TREE-INSERT(T, k) 1 r ← root[T] 2 if n[r] = 2t - 1 3 then s ← ALLOCATE-NODE() 4 root[T] ← s 5 leaf[s] ← FALSE 6 n[s] ← 0 7 c1[s] ← r 8 B-TREE-SPLIT-CHILD(s, 1, r) 9 B-TREE-INSERT-NONFULL(s, k) 10 else B-TREE-INSERT-NONFULL(r, k)
- 3. • Lines 3-9 handle the case in which the root node r is full: the root is split and a new node s (having two children) becomes the root. Splitting the root is the only way to increase the height of a B-tree. Figure 1 illustrates this case. • The procedure finishes by calling BTREE-INSERT- NONFULL to perform the insertion of key k in the tree rooted at the nonfull root node.
- 4. Figure 1 • Splitting the root with t = 4. Root node r is split in two, and a new root node s is created. The new root contains the median key of r and has the two halves of r as children. • The B-tree grows in height by one when the root is split.
- 5. B-TREE-INSERT-NONFULL • B-TREE-INSERT-NONFULL(x, k) 1 i ← n[x] 2 if leaf[x] 3 then while i ≥ 1 and k < keyi[x] 4 do keyi+1[x] ← keyi[x] 5 i ← i - 1 6 keyi+1[x] ← k 7 n[x] ← n[x] + 1 8 DISK-WRITE(x) 9 else while i ≥ 1 and k < keyi[x]
- 6. 10 do i ← i - 1 11 i ← i + 1 12 DISK-READ(ci[x]) 13 if n[ci[x]] = 2t - 1 14 then B-TREE-SPLIT-CHILD(x, i, ci[x]) 15 if k> keyi[x] 16 then i ← i + 1 17 B-TREE-INSERT-NONFULL(ci[x], k)
- 7. • Lines 3-8 handle the case in which x is a leaf node by inserting key k into x. If x is not a leaf node, then we must insert k into the appropriate leaf node in the subtree rooted at internal node x. • In this case, lines 9-11 determine the child of x to which the recursion descends. • Lines 13-16 guarantee that the procedure never recurses to a full node. • Line 17 then recurses to insert k into the appropriate subtree.
- 8. Deleting a key from a B-tree • Just as we had to ensure that a node didn't get too big due to insertion, we must ensure that a node doesn't get too small during deletion
- 9. Deleting a key from a B-tree • Delete key B from a B-Tree with t=3. D G A B C E F H I
- 10. Deleting a key from a B-tree We want to delete key F from a B-Tree with t =3. D G A B C E F H I
- 11. Deleting a key from a B-tree We want to delete key F from a B-Tree with t =3. G A B C D E F H I
- 12. Deleting a key from a B-tree We want to delete key F from a B-Tree with t =3. C G A B D E F H I
- 13. Deleting a key from a B-tree We want to delete key S from a B-Tree with t=3. Q T OP R S U X
- 14. Deleting a key from a B-tree We want to delete key S from a B-Tree with t=3. Q T OP R S T U X
- 15. Deleting a key from a B-tree Q U O P R S T W V
- 16. Deleting a key from a B-tree We want to delete key U from a B-Tree with t=3. Q T O P R S W V
- 17. Deleting a key from a B-tree • We want to delete I from a B-Tree with t =3. I H G M J K L O P
- 18. Deleting a key from a B-tree • We want to delete I from a B-Tree with t =3. J H G M K L O P
- 19. Deleting a key from a B-tree • We want to delete U from a B-Tree with t =3. R U X P Q ST V W Y Z
- 20. Deleting a key from a B-tree • We want to delete U from a B-Tree with t =3. R X P Q S T V W Y Z