byond brand managment by usman asif, rizwan ahmad,kashif javed and rizwan
B-Trees ( Rizwan Rehman)
description
Transcript of 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.
• 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 by ORACLE, DB2, and INGRES.
• 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.
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.
m-way Search Trees
• Each node has up to m – 1 pairs and m children. • m = 2 => binary search tree.
4-Way Search Tree
10 30 35
k < 10 10 < k < 30 30 < k < 35 k > 35
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.
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.
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 5 is 3-4-5 tree (root may be 2-node though).
• B-tree of order 2 is full binary tree.
Minimum # Of Pairs• n = # of pairs.• # of external nodes = n + 1.• Height = h => external nodes on level h + 1.
1 12 >= 23 >= 2*ceil(m/2)
h + 1 >= 2*ceil(m/2)h-1
n + 1 >= 2*ceil(m/2)h-1, h >= 1
level # of nodes
Minimum # Of Pairs
• m = 200.
n + 1 >= 2*ceil(m/2)h-1, h >= 1
height # of pairs
2 >= 1993 >= 19,9994 >= 2 * 106 – 1
5 >= 2 * 108 – 1
h <= log ceil(m/2) [(n+1)/2] + 1
Insert
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1 3 5 6 30 409
Insertion into a full leaf triggers bottom-up node splitting pass.
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Split An Overfull Node
• ai is a pointer to a subtree.
• pi is a dictionary pair.
m a0 p1 a1 p2 a2 … pm am
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.
Insert
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• Insert a pair with key = 2.
• New pair goes into a 3-node.
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Insert Into A Leaf 3-node• Insert new pair so that the 3 keys are in
ascending order.
• Split overflowed node around middle key.
1 2 3
• Insert middle key and pointer to new node into parent.
1 3
2
Insert
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• Insert a pair with key = 2.
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Insert
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• Insert a pair with key = 2 plus a pointer into parent.
Insert
• Now, insert a pair with key = 18.
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2 4
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Insert Into A Leaf 3-node• Insert new pair so that the 3 keys are in
ascending order.
• Split the overflowed node.
16 17 18
• Insert middle key and pointer to new node into parent.
1816
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Insert
• Insert a pair with key = 18.
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2 4
5 6 30 409 16 173
Insert
• Insert a pair with key = 17 plus a pointer into parent.
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Insert
• Insert a pair with key = 17 plus a pointer into parent.
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Insert
• Now, insert a pair with key = 7.
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2 4
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Insert
• Insert a pair with key = 6 plus a pointer into parent.
30 401
2 4
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Insert
• Insert a pair with key = 4 plus a pointer into parent.
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5 7
Insert
• Insert a pair with key = 8 plus a pointer into parent.
• There is no parent. So, create a new root.
30 401
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Insert
• Height increases by 1.
30 401 93 16
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2062
5 7
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