Balanced Binary Search Tree

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Balanced Binary Search Tree

• Fall 2004
• CSE, POSTECH

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Balanced Binary Search Trees

• height is O(log n), where n is the number of
  elements in the tree
• AVL (Adelson-Velsky and Landis) trees
• red-black trees
• get, put, and remove take O(log n) time

3
Balanced Binary Search Trees

• Indexed AVL trees
• Indexed red-black trees
• Indexed operations also take
• O(log n) time

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Balanced Search Trees

• weight balanced binary search trees
• 2-3 2-3-4 trees
• AA trees
• B-trees
• BBST
• etc.

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AVL Tree

• Definition
• Binary tree.
• If T is a nonempty binary tree with TL and TR as
  its left and right subtrees, then T is an AVL
  tree iff
• TL and TR are AVL trees, and
• hL hR ? 1 where hL and hR are the heights of
  TL and TR, respectively

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Balance Factors
-1
1
1
-1
0
1
0
0
-1
0
0
0
0

• This is an AVL tree.

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Height

• The height of an AVL tree that has n nodes is at
  most 1.44 log2 (n2).
• The height of every n node binary tree is at
  least log2 (n1).

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AVL Search Tree
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put(9)
-1
10
0
1
1
7
40
-1
0
1
-1
0
45
8
3
30
0
-1
0
0
0
0
60
35
9
1
20
5
0
25
10
put(29)
-1
10
1
1
7
40
-1
0
1
0
45
8
3
30
0
0
-1
0
0
-2
60
35
1
20
5
0
-1
RR imbalance gt new node is in right subtree of
right subtree of blue node (node with bf -2)
25
0
29
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put(29)
-1
10
1
1
7
40
-1
0
1
0
45
8
3
30
0
0
0
0
0
60
35
1
25
5
0
0
20
29
RR rotation.
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AVL Rotations

• RR
• LL
• RL
• LR

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Imbalance Types

• After an insertion, when the balance factor of
  node A is 2 or 2, the node A is one of the
  following four imbalance types
• LL new node is in the left subtree of the left
  subtree of A
• LR new node is in the right subtree of the left
  subtree of A
• RR new node is in the right subtree of the right
  subtree of A
• RL new node is in the left subtree of the right
  subtree of A

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Rotation

• Definition
• To switch children and parents among two or three
  adjacent nodes to restore balance of a tree.
• A rotation may change the depth of some nodes,
  but does not change their relative ordering.

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Left Rotation

• Definition
• In a binary search tree, pushing a node A down
  and to the left to balance the tree.
• A's right child replaces A, and the right child's
  left child becomes A's right child.

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Right Rotation

• Definition
• In a binary search tree, pushing a node A down
  and to the right to balance the tree.
• A's left child replaces A, and the left child's
  right child becomes A's left child.
• Animated rotation example http//www.cs.queensu.c
  a/home/jstewart/applets/bst/bst-rotation.html

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AVL Rotations

• To balance an unbalanced AVL tree (after an
  insertion), we may need to perform one of the
  following rotations LL, RR, LR, RL

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An LL Rotation
19
An LR Rotation
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Inserting into an AVL Search Tree
Insert(29)

• Where is 29 going to be inserted into?

• After the insertion, is the tree still an AVL
  search tree? (i.e., still balanced?)

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Inserting into an AVL Search Tree
Insert(29)

• What are the balance factors for 20, 25, 29?

• RR imbalance ? new node is in the right subtree
  of right subtree of node 20 (node with bf -2)

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After RR Rotation

• What would the left subtree of 30 look like
  after an RR rotation?

• After the RR rotation, is the resulting tree an
  AVL search tree now?

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Single and Double Rotations

• Single rotations the transformations done to
  correct LL and RR imbalances
• Double rotations the transformations done to
  correct LR and RL imbalances
• The transformation for an LR imbalance can be
  viewed as an RR rotation followed by an LL
  rotation
• The transformation for an RL imbalance can be
  viewed as an LL rotation followed by an RR
  rotation (see Exercise 15)
• See Figure 11.10 for the AVL-search-tree-insertion
  algorithm

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Deletion from an AVL Search Tree

• Deletion of a node may also produce an imbalance
• Imbalance incurred by deletion is classified
  intothe types R0, R1, R-1, L0, L1, and L-1
• Rotation is also needed for rebalancing
• Read the observations after deleting a node from
  an AVL search tree
• Read Section 11.2.6 for Deletion from an AVL
  search tree

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An R0 Rotation
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An R1 Rotation
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An R-1 Rotation
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Complexity of Dictionary OperationsSearch,
Insert, and Delete
Data Structure
Worst Case
Expected
Hash Table
O(n)
O(1)
Binary SearchTree
O(n)
O(log n)
BalancedBinary Search Tree
O(log n)
O(log n)

• n is the number of elements in dictionary.

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Complexity of Other OperationsAscend,
Search(rank), and Delete(rank)
Data Structure
ascend
get and remove
Hash Table
O(D n log n)
O(D n log n)
O(n)
Indexed BST
O(n)
IndexedBalanced BST
O(n)
O(log n)

• D is the number of buckets.

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Animated AVL Trees

• http//www.seanet.com/users/arsen/avltree.html
• http//www.cs.jhu.edu/goodrich/dsa/trees/avltree.
  html

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Red Black Trees

• Colored Nodes Definition
• Binary search tree.
• Each node is colored red or black.
• Root and all external nodes are black.
• No root-to-external-node path has two consecutive
  red nodes.
• All root-to-external-node paths have the same
  number of black nodes

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Example Red Black Tree
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Red Black Trees

• Colored Edges Definition
• Binary search tree.
• Child pointers are colored red or black.
• Pointer to an external node is black.
• No root to external node path has two consecutive
  red pointers.
• Every root to external node path has the same
  number of black pointers.

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Example Red Black Tree
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Red Black Tree

• The height of a red black tree that has n
  (internal) nodes is between log2(n1) and
  2log2(n1).
• java.util.TreeMap gt red black tree