B-Trees

1
B-Trees

• 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
AVL Trees

• n 230 109 (approx).
• 30 lt height lt 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.

3
m-way Search Trees

• Each node has up to m 1 pairs and m children.
• m 2 gt binary search tree.

4
4-Way Search Tree
k gt 35
k lt 10
10 lt k lt 30
30 lt k lt 35
5
Definition Of B-Tree

• Definition assumes external nodes (extended m-way
  search tree).
• B-tree of order m.
• m-way search tree.
• Not empty gt 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.

6
2-3 And 2-3-4 Trees

• B-tree of order m.
• m-way search tree.
• Not empty gt 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.

7
Insert
8
4
15 20
1 3
5 6
30 40
9
16 17
Insertion into a full leaf triggers bottom-up
node splitting pass.
8
Insert
8
4
15 20
1 3
5 6
30 40
9
16 17

• Insert a pair with key 2.
• New pair goes into a 3-node.

9
Insert Into A Leaf 3-node

• Insert new pair so that the 3 keys are in
  ascending order.

• Split overflowed node around middle key.

• Insert middle key and pointer to new node into
  parent.

10
Insert
8
4
15 20
1 3
5 6
30 40
9
16 17

• Insert a pair with key 2.

11
Insert
8
4
2
15 20
3
1
5 6
30 40
9
16 17

• Insert a pair with key 2 plus a pointer into
  parent.

12
Insert

• Now, insert a pair with key 18.

13
Insert Into A Leaf 3-node

• Insert new pair so that the 3 keys are in
  ascending order.

• Split the overflowed node.

• Insert middle key and pointer to new node into
  parent.

14
Insert

• Insert a pair with key 18.

15
Insert
8
17
2 4
15 20
18
1
5 6
30 40
9
3
16

• Insert a pair with key 17 plus a pointer into
  parent.

16
Insert
17
8
2 4
15
20
1
18
5 6
30 40
9
3
16

• Insert a pair with key 17 plus a pointer into
  parent.

17
Insert

• Now, insert a pair with key 7.

18
Insert
8 17
6
2 4
15
20
7
1
18
30 40
9
3
16
5

• Insert a pair with key 6 plus a pointer into
  parent.

19
Insert
8 17
4
6
2
15
20
5
7
18
30 40
9
16
1
3

• Insert a pair with key 4 plus a pointer into
  parent.

20
Insert
8
4
17
6
2
15
20
1
3
5
7
18
30 40
9
16

• Insert a pair with key 8 plus a pointer into
  parent.
• There is no parent. So, create a new root.

21
Insert
8
4
17
6
2
15
20
18
30 40
9
16
1
3
5
7

• Height increases by 1.

22
Delete From A Leaf

• Delete the pair with key 16.
• 3-node becomes 2-node.

23
Delete From A Leaf
8
2 4
15 20
1
5 6
30 40
9
3
17

• Delete the pair with key 17.
• Deletion from a 2-node.
• Check an adjacent sibling and determine if it is
  a 3-node.
• If so borrow a pair and a subtree via parent
  node.

24
Delete From A Leaf
8
2 4
15 30
1
5 6
9
3
40
20

• Delete the pair with key 20.
• Deletion from a 2-node.
• Check an adjacent sibling and determine if it is
  a 3-node.
• If not, combine with sibling and parent pair.

25
Delete From A Leaf
8
2 4
15
1
30 40
5 6
9
3

• Delete the pair with key 30.
• Deletion from a 3-node.
• 3-node becomes 2-node.

26
Delete From A Leaf
8
2 4
15
1
5 6
9
3
40

• Delete the pair with key 3.
• Deletion from a 2-node.
• Check an adjacent sibling and determine if it is
  a 3-node.
• If so borrow a pair and a subtree via parent
  node.

27
Delete From A Leaf
8
2 5
15
1
6
9
4
40

• Delete the pair with key 6.
• Deletion from a 2-node.
• Check an adjacent sibling and determine if it is
  a 3-node.
• If not, combine with sibling and parent pair.

28
Delete From A Leaf
8
2
15
1
4 5
9
40

• Delete the pair with key 40.
• Deletion from a 2-node.
• Check an adjacent sibling and determine if it is
  a 3-node.
• If not, combine with sibling and parent pair.

29
Delete From A Leaf
8
2
1
4 5
9 15

• Parent pair was from a 2-node.
• Check an adjacent sibling and determine if it is
  a 3-node.
• If not, combine with sibling and parent pair.

30
Delete From A Leaf
2 8
1
4 5
9 15

• Parent pair was from a 2-node.
• Check an adjacent sibling and determine if it is
  a 3-node.
• No sibling, so must be the root.
• Discard root. Left child becomes new root.

31
Delete From A Leaf
2 8
1
4 5
9 15

• Height reduces by 1.

32
Delete A Non-Leaf

• Deletion from interior node is transformed into a
  deletion from a leaf node.