Download Lecture13: Tree III Insertion Order matters Complexity of Operations

Insertion Order matters
• Difference between these binary search trees?
 Insert: 4, 2, 1, 3, 6, 5, 7
 Insert: 1, 2, 3, 4, 5, 6, 7
CSED233: Data Structures (2015F)
Lecture13: Tree III
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Bohyung Han
CSE, POSTECH
[email protected]
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Complexity of Operations
CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
Self‐balancing Trees
• Lookup Time
• Variants of binary search tree
• Self‐balancing
 Balanced tree: log
 Unbalanced tree:  Additional rules for how to structure tree
 Needs operations to move nodes around
• Insertion time
 Balanced tree: log
 Unbalanced tree: • Constraints on height differences of leaves
• Ensures that tree does not degenerate to list.
• Balance does matter for efficient trees.
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
AVL Tree
Insertion
• What is AVL tree?
• Insertion is as in a binary search tree.
 A self‐balancing binary search tree
 Adelson‐Velskii and Landis' tree, named after the inventors
 For every node of tree , the heights of the children of differ by at most 1.
 Height: log
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AVL tree
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Before insertion
After insertion
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
Insertion
CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
Restructuring
• Tree height after insertion
• Single Rotation:
 All nodes along the path increase their height by 1.
 It may violate the AVL tree property.
• We rebalance the subtree of using a restructuring method.

: the first violation node from the bottom along the path
 : ’s child with the higher height
 : ’s child with the higher height

, , : inorder traversal of node , , and 44
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h‐3
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a=z
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h‐2
b=y
h‐2
b=y
single rotation
a=z
h‐3
c=x
c=x
h‐3
T0
T3
T1
a=y
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h‐2
h‐1
h‐3
c=z
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h‐1
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T1
T2
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b=x
c=z
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b=y
single rotation
b=y
a=x
c=z
a=x
T3
T0
T3
T2
T2
T1
T0
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
Restructuring
Insertion Example
• Double rotations:
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h‐1
h‐1
h‐3
double rotation
a=z
h‐2
c = y h‐3
≤h‐3 b = x ≤h‐3
T0
T0
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b=x
a=y
a=y
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T0
T2
T3
T1
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b=x
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a=y
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After the first rotation
After insertion unbalanced
T0
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T2
c=z
b=x
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c=z
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b=x
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c=z
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c=z
a=y
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double rotation
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c = y h‐3
≤h‐3
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h‐2
b=x
a=z
h‐3
T3
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h‐2
T1
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
Insertion Example
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After the second rotation
After the first rotation
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
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Removal
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CSED233: Data Structures
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After deletion of 32
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
Reconstructing after a Removal
Reconstructing after a Removal
• Node definition
• Recursive reconstructing
 Restructuring may upset the balance of another node higher in the tree.
 We must continue checking for balance until the root is reached.
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: the first unbalanced node encountered while travelling up the tree from the violating node  : the child of with the larger height
 : the child of with the larger height
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
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CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
AVL Tree Performance
• Restructuring
 A single restructure takes 1 time.
 Using a linked‐structure binary tree
• Search
 A search takes log
 The height of tree is time.
log , no restructuring is needed.
• Insertion
 An insertion takes log time.
 Restructuring up the tree, maintaining heights is
log
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log
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• Deletion
 A deletion takes log time.
 Restructuring up the tree, maintaining heights is 15
CSED233: Data Structures
by Prof. Bohyung Han, Fall 2015
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