AVL Tree Definition. An example of an AVL tree where the heights are shown next to the nodes. Adelson-Velsky and Landis

Transcription

1 Presentation for use with the textbook Data Structures and Algorithms in Java, 6 th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldwasser, Wiley, 0 AVL Trees v z 0 Goodrich, Tamassia, Goldwasser AVL Trees

2 AVL Tree Definition Adelson-Velsky and Landis binary search tree balanced each internal node v the heights of the children of v can differ by at most An example of an AVL tree where the heights are shown next to the nodes 0 Goodrich, Tamassia, Goldwasser AVL Trees

3 n() 3 Height of an AVL Tree n() Fact: The height of an AVL tree storing n keys is O(log n). Proof (by induction): n(h): the minimum number of internal nodes of an AVL tree of height h. n() = and n() = For n >, an AVL tree of height h contains the root node, one AVL subtree of height n- and another of height n-. That is, n(h) = + n(h-) + n(h-) Knowing n(h-) > n(h-), we get n(h) > n(h-). So n(h) > n(h-), n(h) > n(h-), n(h) > 8n(n-6), (by induction), n(h) > i n(h-i) Solving the base case we get: n(h) > h/ - Taking logarithms: h < log n(h) + Thus the height of an AVL tree is O(log n) 0 Goodrich, Tamassia, Goldwasser AVL Trees 3

4 Insertion Insertion is as in a binary search tree Always done by expanding an external node. Insert 5: a=y c=z b=x before insertion w 5 after insertion 0 Goodrich, Tamassia, Goldwasser AVL Trees

5 Insertion Insertion is as in a binary search tree Always done by expanding an external node. Insert 5: a=y Imbalance Node z c=z b=x before insertion Insert Node w 5 w after insertion 0 Goodrich, Tamassia, Goldwasser AVL Trees 5

6 Overview of Cases of Trinode Restructuring Case Case Case 3 Case z -> y -> x -> Goodrich, Tamassia, Goldwasser Red-Black Trees 6

7 Rotation operation Consider subtree points to y and we also have x and y. y.left = x.right. x.right = y 3. subtree = x With a linked structure Constant number of updates O() time 0 Goodrich, Tamassia, Goldwasser AVL Trees 7

8 Trinode Restructuring: Case Single Rotation: Keys: a < b < c Nodes: grandparent z is not balanced, y is parent, x is node a = z b = y c = x single rotation a = z b = y c = x T 0 T T T 3 T 0 T T T 3 Not balanced at a, the smallest key x has the largest key c Result: middle key b at the top 0 Goodrich, Tamassia, Goldwasser AVL Trees 8

9 Example for Case Case z y x T0 T 6 T T3 6 T0 T T T3 0 Goodrich, Tamassia, Goldwasser Red-Black Trees 9

10 Trinode Restructuring: Case Single Rotation: Not balanced at c, the largest key x has the smallest key a Keys: a < b < c Nodes: grandparent z is not balanced, y is parent, x is node Result: middle key b at the top a = x b = y c = z single rotation a = x b = y c = z T 0 T T 3 T 0 Goodrich, Tamassia, Goldwasser AVL Trees 0 T 0 T T T 3

11 Example for Case Case z y x 6 T T3 T0 T 6 T0 T T T3 0 Goodrich, Tamassia, Goldwasser Red-Black Trees

12 Trinode Restructuring: Case 3 double rotation: Keys: a < b < c Nodes: grandparent z is not balanced, y is parent, x is node a = z b = x c = y double rotation a = z b = x c = y T 0 T T T 3 T 0 Not balanced at a, the largest key x has the middle key b x is rotated above y x is then rotated above z Result: middle key b at the top T T T 3 0 Goodrich, Tamassia, Goldwasser AVL Trees

13 Example for Case 3 Case 3 T0 T T 6 T3 T0 T 6 T T3 z y x 6 T0 T T T3 0 Goodrich, Tamassia, Goldwasser Red-Black Trees 3

14 Trinode Restructuring: Case double rotation Not balanced at c, the largest key x has the middle key b x is rotated above y x is then rotated above x Keys: a < b < c Nodes: grandparent z is not balanced, y is parent, x is node Result: middle key b at the top a = y b = x c = z double rotation a = y b = x c = z T 0 T 3 T T 3 T T 0 T T 3 T T 0 T T 0 T T T 3 0 Goodrich, Tamassia, Goldwasser AVL Trees T

15 Example for Case Case T0 T 6 T T3 T0 T 6 T T3 z y x 6 T0 T T T3 0 Goodrich, Tamassia, Goldwasser Red-Black Trees 5

16 Insert 5 (Case 3 or?) unbalanced... 5 z y 3 50 x balanced 5 T 0 T T 7 88 T 3 Draw the double rotation y x z 88 T T 0 T T 3 0 Goodrich, Tamassia, Goldwasser AVL Trees 6

17 Trinode Restructuring summary Case imbalance/ grandparent z Node x Rotation Smallest key a Largest key c single Largest key c Smallest key a single 3 Smallest key a Middle key b double Largest key c Middle key b double 0 Goodrich, Tamassia, Goldwasser AVL Trees 7

18 Trinode Restructuring Summary Case imbalance/ grandparent z Node x Rotation Smallest key a Largest key c single Largest key c Smallest key a single 3 Smallest key a Middle key b double Largest key c Middle key b double The resulting balanced subtree has: middle key b at the top smallest key a as left child T0 and T are left and right subtrees of a largest key c as right child T and T3 are left and right subtrees of c 0 Goodrich, Tamassia, Goldwasser AVL Trees 8

19 Removal Removal begins as in a binary search tree the node removed will become an empty external node. Its parent, w, may cause an imbalance. Remove 3, imbalance at before deletion of 3 after deletion 0 Goodrich, Tamassia, Goldwasser AVL Trees 9

20 Rebalancing after a Removal z = first unbalanced node encountered while travelling up the tree from w. y = child of z with the larger height, x = child of y with the larger height trinode restructuring to restore balance at z Case in example a=z 6 w 7 6 b=y c=x Goodrich, Tamassia, Goldwasser AVL Trees 0

21 Rebalancing after a Removal this restructuring may upset the balance of another node higher in the tree continue checking for balance until the root of T is reached a=z 6 w 7 6 b=y c=x Goodrich, Tamassia, Goldwasser AVL Trees

22 Balanced tree Goodrich, Tamassia, Goldwasser AVL Trees

23 Delete Goodrich, Tamassia, Goldwasser AVL Trees 3

24 Not balanced at Goodrich, Tamassia, Goldwasser AVL Trees

25 Single rotation Goodrich, Tamassia, Goldwasser AVL Trees 5

26 Anything wrong? Goodrich, Tamassia, Goldwasser AVL Trees 6

27 Not balanced at 50! Goodrich, Tamassia, Goldwasser AVL Trees 7

28 AVL Tree Performance n entries O(n) space A single restructuring takes O() time using a linked-structure binary tree Operation Worst-case Time Complexity Get/search O(log n) Up to height log n Put/insert O(log n) O(log n): searching & restructuring Remove/delete O(log n) O(log n): searching & restructuring up to height log n 0 Goodrich, Tamassia, Goldwasser AVL Trees 8

29 AVL Trees balanced Binary Search Tree (BST) Insert/delete operations include rebalancing if needed Worst-case time complexity: O(log n) expected O(log n) for skip lists No duplicated keys in skip lists No moving a bunch of keys in sorted array 0 Goodrich, Tamassia, Goldwasser AVL Trees 9