AVL Trees Knowing More - PowerPoint Presentation

AVL Trees

Knowing MoreHow balanced are we?

Knowing MoreHow balanced are we? How many nodes off each side? 3 7

Knowing MoreHow many nodes? Determine on demand

Knowing MoreHow many nodes? Determine on demand int size(Node curNode ) if curNode is null, return 0 else return 1 + size( leftChild ) + size( rightChild)

Knowing MoreHow many nodes? Determine on demand T(n) = 2T(n/2) + 1 O(n) int size(Node curNode ) if curNode is null, return 0 else return 1 + size( leftChild) + size(rightChild)

Knowing MoreHow many nodes? Store in tree Update on insert / delete O(1)

Knowing More How balanced are we? How many nodes off each side????? 7 7

Knowing MoreHow balanced are we? What is the longest path in each direction? 3 3

Knowing MoreHow balanced are we? What is the longest path in each direction? Tree ops dependent on longest path Consistent height is key 3 3

Knowing MoreHeight of node Determine on demand int height(Node curNode ) if curNode is null, return -1 else return 1 + max( height(leftChild), height( rightChild ))

Knowing MoreHeight of node Determine on demand T(n) = 2T(n/2) + 1 O(n) per node int height(Node curNode ) if curNode is null, return -1 else return 1 + max( height( leftChild), height( rightChild ))

Knowing MoreHeight of node Store in each node Update on insert / delete O(1) per node 0 0 0 1 1 2 3

AVL TreesGeorgii Adelson-Velsky Evgenii Mikhailovich Landis

AVL Rules Every node has balance factor Balance = Height(left child) – Height(right child) Height of Null = -1 0 0 0 1 2 3 -1 -1 -1 -1 -1 -1 -1

AVL RulesNode must have balance factor of -1, 0 or 1 Rotate to fix bad nodes

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1 2

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1 2 4

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1 2 4 7

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1 2 4 7 12

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1 2 4 7 12

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1 2 4 7 12

AVL RulesWorst case: Height 0 1 2 3 4 Nodes 1 2 4 7 12

AVL RulesWorst case: N odes at height h N h = Nh-1 + Nh-2 + 1 Height 0 1 2 3 4 Nodes 1 2 4 7 12

AVL RulesWorst case proof + 1  

AVL RulesWorst case proof N(h - 1) is > N(h -2) + 1 + + 1  

AVL RulesWorst case proof + 1 + + 1  

AVL RulesRecurrence relationship:  

AVL RulesRecurrence relationship:  

AVL RulesRecurrence relationship:  

AVL RulesRecurrence relationship: N(0) = 1 Need h - 2k = 0 Solve for k: k = h/2  

AVL RulesRecurrence relationship: k = h/2  

AVL RulesRecurrence relationship: N(0) = 1  

AVL RulesWorst case proof  

AVL RulesWorst case proof  

AVL RulesWorst case proof  

AVL RulesWorst case proof  

So…. Guaranteed log(n) height O( logn ) Insert/Delete/Remove But… Higher constant factors for insert/remove  

Maintaining BalanceInserting/deleting node changes balance by 1 Affects whole chain

Maintaining Balance-2 or +2 needs to be balanced 2 cases Outside Cases (single rotation) : Longest chain is left-left or right-right of unbalanced Inside Cases (double rotation) : Longest chain is left-right or right-left of unbalanced

Case 1A valid AVL subtree f A D G h h h b h h+1

Case 1Insertion creates problem on left-left or right-right f A D G h h+1 h b h+2

Rotate away from problemCase 1 f A D G h h+1 h b h+1

Case 2Problem created on left-right or right-left path f A G h h h b C E d h - 1 h - 1 h+1

Case 2Problem created on left-right or right-left path f A G h h h+1 b C E d h h - 1 h+2

Case 2Do Zig- Zag Rotation f A G h h h+2 b C E d h h - 1 h+1

Case 2Do Zig- Zag Rotation f A G h h h+2 b C E d h h - 1 h+1 h+1

DeletionsDeletions like rotations – rotate to restore balance

DeletionsDeletions like rotations – rotate to restore balance May unbalance multiple nodes

DeletionsDeletions like rotations – rotate to restore balance May unbalance multiple nodes

DeletionsDeletions like rotations – rotate to restore balance May unbalance multiple nodes