CSC 213 – Large Scale Programming

CSC 213 –Large Scale Programming Lecture 18: Zen & the Art of O(log n) Search

Today’s Goal • Discuss AVL Trees • Relationship to BinaryTree and BST • What an AVL Tree looks like and how they work • Effects on big-Oh notation • AVL Tree Limitations

Dictionary ADT • Dictionary ADT maps key to 1 or more values • Used wherever search is (e.g., everywhere) • Implementation using Sequence takes O(n) time • With good batch of hash, could get O(1) time • But could still end up with O(n) time • If data are random, BST takes O(log n) time • But for ordered data, BST still takes O(n) time • There must be a better way!

Binary Search Trees • May not be complete • But faster when they are • Use different ordering • Lower keys in left subtree • Higher keys in right subtree • Equal keys not specified, but must be consistent 6 9 2 10 1 4

AVL Tree Definition • Another type of BST • Algorithm guarantees O(log n) time • Does not allow tree to become linked list • Keeps tree in balance • AVL Tree balanced when each Node’s children differ in height by at most 1 • Maintains balance through trinode restructuring 4 6 3 2 2 9 2 1 1 8 1 4 1 5 Node heights are shown in red

Trinode Restructuring • Insertion & removal can unbalance tree • Trinode restructuring restores tree’s tao • Used when node’s children’s height differ • Takes node, taller child, & taller grandchild • Grandchild must be taller of taller child’s children • Move median node to root of subtree • Other 2 nodes in restructuring become its children

Trinode Restructuring • Case 1: Single rotation (e.g., 3 in a row) • Move child of 3 to root of subtree • Subtrees become grandchildren of node • No change in the order of subtrees b a T0 b a c T2 T3 T1 T0 c T1 T2 T3

Trinode Restructuring • Case 2: Double rotation • Taller child & grandchild go opposite ways • Move grandchild up to the root • Subtrees become grandchildren of node, but maintain order a b T0 c a c T3 T0 T2 T1 b T3 T1 T2

Insertion in an AVL Tree • Begins like normal BST insertion • Example: insert(5) 4 7 3 2 2 9 2 1 1 8 1 4 2 1 6 1 5

Insertion in an AVL Tree • Travel up tree checking nodes for balance • Must increase tao of unbalanced nodes 7 2 9 8 1 4 6 5

Insertion in an AVL Tree • Must walk up tree starting at inserted Node • Stop once we hit the root node • Update height for each Node along path • Check if Node’s children are balanced • Perform rotations when balance needs restoration • Not all insertions require rebalancing • Will need at most 1 insertion per Node • But some insertions need multiple rebalancings

Removal in an AVL Tree • Start with normal BST removal • But removal may cause drop in the TAO • Example: remove(7) 4 7 8 3 2 1 2 9 2 1 1 8 1 5 1 1 4 6

Removal in an AVL Tree • Start with normal BST removal • But removal may cause drop in the TAO • Example: remove(7) 4 7 8 3 1 2 9 2 1 1 5 1 1 4 6

Removal in an AVL Tree • Start with normal BST removal • But removal may cause drop in the TAO • Example: remove(7) 7 5 2 8 6 1 4 9

Removal in an AVL Tree • Again walk up tree checking for balance • Update heights as we go • Only examines Nodes along path • Height of nodes not on path cannot be changed • May need multiple restructuring operations • Uses same restructuring as insert • Use unbalanced node’s taller child & taller grandchild • Does not matter if child & grandchild on path

Restructuring for Dummies • Store the 7+1 Nodes in local variables • Plus one records parent Node of this subtree • Set all left, right, & parent from pattern • Median node is root; subtrees maintain order a a b b T0 c T0 b a c a c c T3 T3 T0 T2 T0 T2 T1 T1 b T3 T1 T1 T2 T2 T3

In the Next Lecture • Discuss even faster ways of searching • No difference in big-Oh notation, however • But provide significant speedup in real-life • Learn new ways of amortizing costs • Discover proper use of the term “splay” • When and why we splay a tree • What it means to splay • Any other excuses I can think of to say splay

CSC 213 – Large Scale Programming