PPT for B trees in data structure and algorithms.

1. B- Trees

2. Motivation
• Large differences between time access to disk,
cash memory and core memory
• Minimize expensive access
(e.g., disk access)
• B-tree: Dynamic sets that is optimized for
disks

3. AB-treeis an M-way search tree with two properties :
1. It is perfectly balanced: every leaf node is at the same
depth
2. Every internal node other than the root, is at least half-
full, i.e. M/2-1 ≤ #keys ≤ M-1
3. Every internal node with k keys has k+1 non-null
children
For simplicity we consider Meven and we use t=M/2:
2.* Every internal node other than the root is at least half-
full, i.e.t-1≤ #keys ≤2t-1, t≤ #children ≤2t
B-Trees

4. Example: a 4-way B-tree
B-tree 4-way tree
B-tree
1. It is perfectly balanced: every leaf node is at the same depth.
2. Every node, except maybe the root, is at least half-full
t-1≤ #keys ≤2t-1
3. Every internal node with k keys has k+1 non-null children
20 40
0 5 10 25 35 45 55
20 40
0 5 25 35 45 55
10

5. B-treeHeight
Claim: any B-tree with nkeys, height hand minimum degree t
satisfies:
Proof:
• The minimum number of KEYS for a tree with height h is
obtainedwhen:
– The root contains one key
– All other nodes containt-1 keys
2
1
log


n
h t

6. B-Tree: Insert X
1. As in M-waytree find the leaf node to which Xshould be
added
2. Add X to this node in the appropriate place among the
values already there
(there are no subtrees to worry about)
3. Number of values in the node after adding the key:
– Fewer than 2t-1: done
– Equal to 2t: overflowed
4. Fix overflowed node

7. Fix an Overflowed
1. Split the node into three parts, M=2t:
– Left: the first t values, become a left child node
– Middle: the middle value at position t, goes up to parent
– Right: the last t-1 values, become a right child node
2. Continue with the parent:
1. Until no overflow occurs in the parent
2. If the root overflows, split it too, and create a new root node
y
… 56 98 ….
60 65 68 83 86 90
… 56 68 98 ….
60 65 83 86 90
split
J
x x
y z

8. Insert example
20 40 60 80
0 5 10 15 25 35 45 55 87 98
Insert 3:
20 40 60 80
0 3 5 1015 25 35 45 55
3;6  tM
62 66 70 74 78
62 66 70 74 78 87 98

9. 61 62 66 7074 78
20 4060 80
0 3 5 10 15 25 35 45 55
Insert 61:
3;6  tM
62 66 70 74 78 87 98
20 40 60 80
0 3 5 10 15 25 35 45 55 87 98
74 7861 62 66
OVERFLOW
20 40 60 70 80
0 3 5 10 15 25 35 45 55 87 98
SPLIT IT

10. Insert 38:
74 7861 62 66
20 40 60 70 80
0 3 5 1015 25 35 45 55 87 98
74 7861 62 66
20 40 60 70 80
0 3 510 15 25 35 38 45 55 87 98
3;6  tM

11. 5 20 4060 7080
Insert 4:
0 3 4 2535 38 45 55 6162 66 87 9874 7810 15
74 786162 66
20 40 60 70 80
0 3 4 510 15
25 35 38 45 55 87 98
74 7861 6266
20 40 60 70 80
25 35 38 45 55 87 98
OVERFLOW
0 3 5 10 15
SPLIT IT
OVERFLOW
SPLIT IT
3;6  tM

12. 0 3 4 25 35 38 45 55 61 62 66 87 9874 78
60
5 20 40 70 80
10 15
5 20 40 60 70 80
0 3 4 25 35 38 45 55 61 62 66 87 9874 7810 15
OVERFLOW
SPLIT IT
3;6  tM

13. Complexity Insert
• Inserting a key into a B-tree of height h is done in a
single pass down the tree and a single pass up the
tree
Complexity: )(log)( nOhO t

14. B Vs B+ Tree

15. B trees
• Generalization of binary search trees
– Not binary trees
– The B stands for Bayer (or Boeing)
• Designed for searching data stored on block-
oriented devices

16. A very small B tree
Bottom nodes are leaf nodes: all their pointers
are NULL

17. In reality
In
tree
ptr
Key
Data ptr
In
tree
ptr
Key
Data ptr
In
tree
ptr
Key
Data ptr
In
tree
ptr
Key
Data ptr
In
tree
ptr
To
Leaf
7
To
leaf
16
To
Leaf
--
Null
Null
--
Null
Null

18. B+ trees
• Variant of B trees
• Two types of nodes
– Internal nodes have no data pointers
– Leaf nodes have no in-tree pointers
• Were all null!

19. B+ tree nodes
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr

20. B+ trees
• Variant of B trees
• Two types of nodes
– Internal nodes have no data pointers
– Leaf nodes have no in-tree pointers
• Were all null!

21. B+ tree nodes
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
In
tree
ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr
Key
Data ptr

22. Example B+-tree
9
5
1 3 5 6 30 409 16 17
16 30
 index node
 leaf/data node