COMP2521
Data Structures & Algorithms
Week 4.2
AVL Trees
Author: Hayden Smith 2021
In this lecture
Why?
- Rebalancing methods often have an O(n) worse case, and we want to find something with O(log(n)) worst case.
What?
- AVL Tree Overview
- Examples
- Pseudocode
AVL Tree
AVL trees fix imbalances as soon as they occur. It aims to have the tree reasonably maintain balance with an O(log(n)) cost.
General Method:
- Store information on each node (amortisation) about it's height/balance
- Do repairs on tree balance locally, not on overall structure
Invented by Georgy Adelson-Velsky and Evgenii Landis (1962)
AVL Tree Method
A tree is unbalanced (height) when:
This tree can be repaired by rotation:
- If left subtree is too deep, rotate right
- If right subtree is too deep, rotate left
This process would normally be O(n) as you have to traverse all nodes to figure out the height of any particular sub-tree.
However, we will store the balance data on each node (height or balance) which will make things faster, but also mean more to maintain.
|height(left) - height(right)| > 1
AVL Tree Structure(practical)
Red numbers are height. Typically stored on node.
Green numbers are balance. Stored or calculated based on immediate children.
Rule: Rotate whenever tree |balance| > 1
AVL Tree Structure(practical)
Insertion of 6 will trigger a rotate right on 8.
AVL Insertion Algorithm
Pseudocode
Pseudocode
insertAVL(tree, item):
if tree is empty:
return new node containing item
else if item = data(tree):
return tree
if item < data(tree):
left(tree) = insertAVL(left(tree),item)
else if item > data(tree):
right(tree) = insertAVL(right(tree),item)
LHeight = height(left(tree))
RHeight = height(right(tree))
if (LHeight - RHeight) > 1:
if item > data(left(tree)):
left(tree) = rotateLeft(left(tree))
tree = rotateRight(tree)
else if (RHeight - LHeight) > 1:
if item < data(right(tree)) then
right(tree) = rotateRight(right(tree))
tree=rotateLeft(tree)
return tree
AVL Tree Performance
- Average/worst-case search performance of O(log(n))
- Finite time spent on maintaining height data in each node
- Finite time spent rebalancing locally when there is imbalance
- Trees are always height-balanced. Subtree depths differ by +/- 1
- May not be weight-balanced, as subtrees may differ in size (even if height only varies by 1)
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COMP2521 21T2 - 4.2 - AVL Trees
By haydensmith
