Principles of Computer Science

Transcription

1 Principles of Computer Science AVL Trees 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 1

2 Today s Topics Review Sorted Binary Tree Traversal Insert AVL Tree Sorted balanced binary tree Problems of insertion Cases of inbalance A solution 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 2

3 Sorted Binary Tree Binary Tree Each parent node has at most two children nodes. Sorted Binary Tree all left descendents are smaller than or equal to the node; and all right descendents are greater than the node. The ordering condition of binary trees 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 3

4 Sorted Binary Tree Sorted Binary Tree Not Sorted Binary Tree 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 4

5 Trees in Java class TreeNode<T> { public T content; public TreeNode<T> left; public TreeNode<T> right; } left root right class BinaryTree<T> { public TreeNode<T> root; } left right left right 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 5

6 Traversing through a binary tree public int sumofallnodes(treenode<integer> node) { int sum = 0; if(node == null) return 0; Base case } sum = node.content + sumofallnodes(node.left) + sumofallnodes(node.right); Recursion 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 6

7 Inserting a New Value into a Sorted Binary Tree Need to keep the ordering condition. Can only insert into leaf nodes Traverse through the tree for an insertion point 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 7

8 Inserting a New Value into a Sorted Binary Tree public void insert(t value, TreeNode<T> node) { if(node == null) return; // This is an error condition if(value <= node.content) { /* INSERT INTO THE LEFT SUBTREE */ if(node.left == null) node.createleftchild(value); else insert(value, node.left) } } if(value > node.content) { /* INSERT INTO THE RIGHT SUBTREE */ <... Symmetric treatment... > } 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 8

9 Problems with Insertion Insertion does not respect the balance of the binary tree. If not supervised, insertions can create long and narrow binary trees. 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 9

10 Problem with lazy insertion: setroot (10) insert (6, root) insert (5, root) insert (4, root) insert (3, root) insert (2, root) insert (1, root) Worst case: binary tree looks like a linked list 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 10

11 Why are narrow trees bad? Data insertion Data search Arrays O(n) O(n) Linked List O(1) O(n) (Balanced) Binary Trees O(log n) O(log n) Narrow Binary Trees O(n) O(n) 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 11

12 The AVL Tree Adelson-Velskii and Landis (AVL) Tree, 1962 AVL trees are balanced AVL condition for every node n, the height of n.left and the height of n.right cannot differ by more than 1. 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 12

13 AVL Condition Ok Violated 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 13

14 Inserting a New Value into an AVL Tree Insert must satisfies Insertion in log(n) Ordering condition of the sorted binary condition AVL condition Our current insertion routine satisfies Insertion in log(n) Ordering condition of the sorted binary condition Our current insertion routine does not satisfy AVL condition; it can make the tree unbalanced 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 14

15 Four cases of imbalance introduced by insertion 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 15

16 Four cases of imbalance introduced by insertion AVL condition guarantees: height(t LL ) height(t LR ) < 2 height(t RL ) height(t RR ) < 2 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 16

17 Four cases of imbalance introduced by insertion Case 1: 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 17

18 Four cases of imbalance introduced by insertion Case 1: 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 18

19 Remedy to Case 1 Single rotation rotatewithleft(k1) 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 19

20 Four cases of imbalance introduced by insertion Case 2: 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 20

21 Four cases of imbalance introduced by insertion Case 2: 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 21

22 Remedy to Case 2 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 22

23 Remedy to Case 2 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 23

24 Remedy to Case 2: Double rotation doublerotatewithleft(k3) 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 24

25 Case 3, Case 4: Mirror images of Case 1 and Case 2 Requires rotatewithright() 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 25

26 Case 3, Case 4: Mirror images of Case 1 and Case 2 Requires doublerotatewithright() 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 26

27 Inserting into AVL Tree Java Implementation 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 27

28 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 28

29 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 29

30 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 30

31 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 31

32 Summary Review of sorted binary trees Inserting new values into sorted binary trees Implications of naïve insertion AVL trees 08/11/2013 CSCI AVL Trees - F.Z. Qureshi 32

33 Readings Ch /11/2013 CSCI AVL Trees - F.Z. Qureshi 33