Advanced Tree. Structures. AVL Tree. Outline. AVL Tree Recall, Binary Search Tree (BST) is a special case of. Splay Tree (Ch 13.2.

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1 ttp://1...0/csd/ Data tructure Capter 1 Advanced Tree tructures Dr. atrick Can cool of Computer cience and Engineering out Cina Universit of Tecnolog AVL Tree Recall, Binar earc Tree (BT) is a special case of Binar Tree All elements stored in te left subtree of a node wit value K ave values < K All elements stored in te rigt subtree of a node wit value K ave values >= K 1 Outline AVL Tree (C 1..1) pla Tree (C 1..) AVL Tree roblem of BT? Unbalance Actuall affects te operating time 1 4

2 AVL Tree An AVL (Adelson-Velskii and Landis) Tree is a binar searc tree wit a balanced condition Balanced condition is relaed a little bit Ever node in te tree, te eigt of te left and rigt subtrees differ b at most 1 AVL Tree AVL Tree Node violates te AVL propert AVL tree 6 Basicall follows insertion strateg of BT Ma cause violation of AVL tree propert Rebalance is needed For Eample After adding, 4, 6 Te Tree becomes unbalance Heigt = 0 Heigt = Rebalance is needed 4 6 8

3 Onl nodes tat are on te pat from te insertion point to te root migt ave teir balance altered Rebalance te tree at te deepest suc node guarantees tat te entire tree satisfies te AVL propert Te rebalance metod is called Rotation k Rigt-Rigt z Left-Left Rigt-Left z Left-Rigt RR LL LR LR RL 1 LR RL 1 11 ubtrees: Four situations of violation: Insertion appens in or + Rigt-Rigt Left-Left Rigt-Left Left-Rigt Rotation Two Rotations: ingle Rotation Handle Left-Left and Rigt-Rigt situation Double Rotation Handle Left-Rigt and Left-Rigt situation 1

4 : Rotation ingle Rotation Rigt-Rigt z : Rotation ingle Rotation How about Rigt-Left or Left-Rigt? ingle Rotation cannot solve Rigt-Left and Left-Rigt cases Rigt-Left Left-Rigt 1 : Rotation ingle Rotation Left-Left : Rotation Double Rotation Rigt-Left

5 : Rotation Double Rotation Rigt-Left : Rotation Double Rotation Left-Rigt : Rotation Double Rotation Rigt-Left Rotation Algoritm Insert te node as te same as BT insertion Trace te pat from te inserted node towards te root For eac node encountered, ceck if eigts of left and rigt subtree differ b at most 1 If es, go to net node If no, perform an appropriate rotation tops Wen rotation is performed Or, we ve cecked all nodes in te pat 18

6 Rotation Insert,,, 1, 1, ingle ingle Double 1,, 1, 14, 1, Double 14 Double ingle mall Eercise!!!! Build AVL Tree b inserting,, 1, 14, 1, 6 Compleit Compleit for AVL Insertion is Θ(log n) 1. Insert a value to te tree: Θ(log n) ame as ordinar BT. Trace te pat from te new leaf towards te root, for eac node on te pat: Θ(log n) Ceck eigt difference: Θ(1) If satisfies AVL propert, proceed to net node If not, perform a rotation: Θ(1) Cecking stops Wen a single rotation is performed Or, we ve cecked all nodes in te pat 4

7 AVL Tree: Deletion Algoritm Delete te node as te same as BT deletion Trace te pat from te last node deleted towards te root For eac node encountered, ceck if eigts of left and rigt subtree differ b at most 1 If es, go to net node If no, perform an appropriate rotation Continue to trace te pat until we reac te root Unlike insertion, more tan one node ma need rotation AVL Tree: Deletion Double Rotation O I k X -1 I 1 I O Delete in X I 1 I O I 1 I X I -1 I I 1 I O O I 1 X AVL Tree: Deletion ingle Rotation k1 Delete in X I O +1-1 X I O X I O O I -1 * Te eigt of Inside subtree (I) can be -1 AVL Tree: Deletion

8 mall Eercise!!!! Delete 46,, 6,, in tis AVL Tree pla Tree imilar to AVL Tree, pla Tree contains some rules for improving BT Different from AVL Tree, pla Tree is not guaranteed to be balanced but te total cost of all accesses will remain ceap M operations will take O(m log n) time for a tree of n node wenever m > n It means te average time for operation is O(log n) Remain tat te worst case of BT mabe O(n) 1 mall Eercise!!!! Delete 46,, 6,, ingle Double 0 pla Tree: Algoritm Wen a node is accessed (insert, delete or searc), splaing is performed Objective: Move to te root (if is deleted, it s parent is moved) plaing is a series of rotations ingle Rotation Zigzag Rotation (Double Rotation) Zigzig Rotation (Double Rotation)

9 pla Tree: ingle Rotation Te node is accessed arent a c a b b c As te same as AVL ingle Rotation pla Tree: Zigzig Rotation d a b c c a b a b c d d pla Tree: Zigzag Rotation a randparent arent b c d a b d c Te node is accessed As te same as AVL Double Rotation a b c d 4 pla Tree: Algoritm Use double rotation until reaces eiter te root or te cild of te root Zigzag Rotation Zigzig Rotation If is te cild of te root, use single rotate to make te te root 6

10 pla Tree: Eample Zigzig Rotation 8 18 Assume 8 is searced pla Tree: Eample ingle Rotation pla Tree: Eample Zigzag Rotation 8 mall Eercise!!!! Do te following operations in tis pla Tree: Add Delete

11 41 ZigZig ZigZig ZigZag Add ZigZag ingle Delete