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AVL Tree

AVL Tree. Outline. AVL Tree Definition Properties Operations. AVL Trees. Unbalanced Binary Search Trees are bad. Worst case: operations take O(n). AVL (Adelson-Velskii & Landis) trees maintain balance.

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AVL Tree

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  1. AVL Tree

  2. Outline • AVL Tree • Definition • Properties • Operations

  3. AVL Trees • Unbalanced Binary Search Trees are bad. Worst case: operations take O(n). • AVL (Adelson-Velskii & Landis) trees maintain balance. • For each node in tree, height of left subtree and height of right subtree differ by a maximum of 1. X X H H-2 H-1

  4. 10 10 5 20 5 20 43 3 3 1 2 1 3 AVL Trees 5

  5. AVL Trees 12 8 16 4 10 14 2 6

  6. Insertion for AVL Tree • After insert 1 12 8 16 4 10 14 2 6 1

  7. Insertion for AVL Tree • To ensure balance condition for AVL-tree, after insertion of a new node, we back up the path from the inserted node to root and check the balance condition for each node. • If after insertion, the balance condition does not hold in a certain node, we do one of the following rotations: • Single rotation • Double rotation

  8. Insertions in AVL Trees Let the node that needs rebalancing be . There are 4 cases: Outside Cases (require single rotation) : 1. Insertion into left subtree of left child of . 2. Insertion into right subtree of right child of . Inside Cases (require double rotation) : 3. Insertion into right subtree of left child of . 4. Insertion into left subtree of right child of . The rebalancing is performed through four separate rotation algorithms.

  9. Insertions Causing Imbalance k2 k1 HP=HQ=HR • An insertion into the subtree: • P (outside) - case 1 • Q (inside) - case 2 • An insertion into the subtree: • Q (inside) - case 3 • R (outside) - case 4 k1 k2 R P P Q Q R

  10. Single Rotation (case 1) HA=HB+1 HB=HC k2 k1 k1 k2 A C A B B C

  11. Single Rotation (case 4) HA=HB HC=HB+1 k2 k1 k1 k2 C A A B B C

  12. Problem with Single Rotation • Single rotation does not work for case 2 and 3 (inside case) k2 k1 k1 k2 R P P Q Q R HQ=HP+1 HP=HR

  13. Double Rotation: Step k3 k3 k1 k2 D D k2 k1 A C B C B A HA=HB=HC=HD

  14. Double Rotation: Step k2 k1 k3 A D B C

  15. Double Rotation k3 k2 k1 k1 k3 D k2 A B C D A B C HA=HB=HC=HD

  16. Double Rotation k2 k1 k3 k1 k3 A k2 A D B C D B C HA=HB=HC=HD

  17. 11 11 8 20 4 20 4 16 27 3 16 8 27 3 Example • Insert 3 into the AVL tree 8

  18. 11 11 8 20 5 20 4 16 27 4 16 8 27 5 Example • Insert 5 into the AVL tree 8

  19. AVL Trees: Exercise • Insertion order: • 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55

  20. Remove Operation in AVL Tree • Removing a node from an AVL Tree is the same as removing from a binary search tree. However, it may unbalance the tree. • Similar to insertion, starting from the removed node we check all the nodes in the path up to the root for the first unbalance node. • Use the appropriate single or double rotation to balance the tree. • May need to continue searching for unbalanced nodes all the way to the root.

  21. Deletion X in AVL Trees • Deletion: • Case 1: if X is a leaf, delete X • Case 2: if X has 1 child, use it to replace X • Case 3: if X has 2 children, replace X with its inorder predecessor (and recursively delete it) • Rebalancing

  22. Delete 55 (case 1) 60 20 70 10 40 65 85 80 90 5 15 30 50 55

  23. Delete 55 (case 1) 60 20 70 10 40 65 85 80 90 5 15 30 50 55

  24. Delete 50 (case 2) 60 20 70 10 40 65 85 80 90 5 15 30 50 55

  25. Delete 50 (case 2) 60 20 70 10 40 65 85 50 80 90 5 15 30 55

  26. prev Delete 60 (case 3) 60 20 70 10 40 65 85 80 90 5 15 30 50 55

  27. Delete 60 (case 3) 55 20 70 10 40 65 85 80 90 5 15 30 50

  28. prev Delete 55 (case 3) 55 20 70 10 40 65 85 80 90 5 15 30 50

  29. Delete 55 (case 3) 50 20 70 10 40 65 85 80 90 5 15 30

  30. prev Delete 50 (case 3) 50 20 70 10 40 65 85 80 90 5 15 30

  31. Delete 50 (case 3) 40 20 70 10 30 65 85 80 90 5 15

  32. prev Delete 40 (case 3) 40 20 70 10 30 65 85 80 90 5 15

  33. Delete 40 : Rebalancing 30 20 70 10 Case ? 65 85 80 90 5 15

  34. Delete 40: after rebalancing 30 10 70 20 65 85 5 80 90 15 Single rotation is preferred!

  35. Minimum Element in AVL Tree • An AVL Tree of height H has at least FH+3-1 nodes, where Fi is the i-th fibonacci number • S0 = 1 • S1 = 2 • SH = SH-1 + SH-2 + 1 SH-1 SH-2 H H-2 H-1

  36. AVL Tree: analysis (1)

  37. AVL Tree: analysis (2) • The depth of AVL Trees is at most logarithmic. • So, all of the operations on AVL trees are also logarithmic. • The worst-case height is at most 44 percent more than the minimum possible for binary trees.

  38. Summary • Find element, insert element, and remove element operations all have complexity O(log n) for worst case • Insert operation: top-down insertion and bottom up balancing

  39. Further Study • http://telaga.cs.ui.ac.id/WebKuliah/IKI10100/resources/animation/data-structure/avl/avltree.html • Section 19.4 of Weiss book

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