Outline
•Introduction
•Definitions
•Height-Balanced Trees
•Picking Out AVL Trees
•Why AVL Trees?
•Added Complexity
•Violations
•The Balance Factor
•What is a rotation?
•Single Rotations
•Double Rotations
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Height-Balanced Trees - Data Structures - Lecture Slides, Slides of Data Structures and Algorithms
Some concept of Data Structures are Abstract, Balance Factor, Complete Binary Tree, Dynamically, Storage, Implementation, Sequential Search, Advanced Data Structures, Graph Coloring Two, Insertion Sort. Main points of this lecture are: Height-Balanced Trees, Definitions, Introduction, Added Complexity, Violations, Balance Factor, Rotation, Single Rotations, Double Rotations, First Balanced Binary
Typology: Slides
2012/2013
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Outline
- Introduction
- Definitions
- Height-Balanced Trees
- Picking Out AVL Trees
- Why AVL Trees?
- Added Complexity
- Violations
- The Balance Factor
- What is a rotation?
- Single Rotations
- Double Rotations
Introduction
• AVL tree is the first balanced binary search
tree (name after its discovers, Adelson-Velskii
and Landis).
• The AVL tree is a balanced search tree that has
an additional balance condition.
• The balance condition must be easy to
maintain and ensures that the depth of the
tree is O(logN)
Height-Balanced Trees
- Height of a tree is the length of the longest path from a root to a leaf.
- A binary search tree T is a height-balanced k-tree or HB[k]- tree, if each node in the tree has the HB[k] property.
- A node has the HB[k] property if the height of the left and right sub-trees of the node differ in height by at most k.
- A HB[1] tree is called an AVL tree.
Picking Out AVL Trees
- Notation: NODE = Left-Ht, Right-Ht
Yes! 5 = 2, 1 4 = 1, 0 3 = 0, 0 7 = 0, 0
No! 10 = 3, 1
Why AVL Trees?
- The worst case for Binary Search Trees:
Insert: O(n).
Delete: O(n).
Search :O(n).
The worst case for AVL trees:
Insert: O(log n).
Delete: O(log n).
Search:O(log n).
Recall when comparing the time complexity
O(log n) < O(n)
Why AVL Trees? (Cont’d)
• AVL trees allow for logarithmic Inserts and
Deletes, and they allow for easy traversals,
such as the “In-order” traversal if we want to
sort our elements.
Violations
No!
No! 5 = 3, 4 = 2,
Notation: NODE = Left-Ht, Right-Ht
The Balance Factor
• In order to detect when a “violation” of a AVL
tree occurs, we need to have each node keep
track of the difference in height between its
right and left sub-trees.
• Notation for the balance factor:
bf (node) = Left-Ht – Right- Ht
To be a valid AVL Tree, -1 <= bf(x) <= 1 for all
nodes.
The Balance Factor (Cont’d)
Notation for the balance factor:
bf (node) = Left-Ht – Right- Ht
bf(8) = 2 – 0 = 2
bf(4) = 1 – 0 = 1
Bf(3) = 0 – 0 = 0
Invalid Tree!!
What is a rotation?
• A rotation is an operation to rearrange nodes
to maintain balance of an AVL tree after
adding and removing a node
• There are two case to perform the rotation
1) Single Rotation
2) Double Rotation
When to perform rotations?
we need to perform a rotation anytime a node’s
balance factor is 2 or -
Single Rotations
After inserting (a) 60; (b) 50; (c) 20 into an
initially empty binary search tree, the tree is
not balanced; d) a corresponding AVL tree
rotates its nodes to restore balance.
Single Rotations (Cont’d)
- Algorithm for left rotation
Algorithm rotateLeft(nodeN) nodeC = right child of nodeN Set nodeN ’s right child to nodeC ’s left child Set nodeC ’s left child to nodeN