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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