each node in the tree, the difference in height of its two subtrees is at the most p. AVL tree is a BST that is height-balanced-1-tree.

Transcription

1 Data Structures CSC212 1 AVL Trees A binary tree is a eigt-balanced-p-tree if for eac node in te tree, te difference in eigt of its two subtrees is at te most p. AVL tree is a BST tat is eigt-balanced-tree

2 Data Structures CSC212 2 AVL Trees A binary tree is an AVL tree if Eac node satisfies BST property key of te node is grater tan te key of eac node in its left subtree and is smaller tan te key of eac node in its rigt subtree Eac node satisfies eigt-balanced-tree property te difference between te eigts of te left subtree and rigt subtree of te node does not exceed one.

3 Data Structures CSC An Imbalanced Tree A Balanced Tree

4 Data Structures CSC212 4 Insert 1, 2, 3, 4 and 5 in te given order BST after insertions AVL Tree after insertions

5 Data Structures CSC212 5 Wy AVL Trees? Wen data elements are inserted in a BST in sorted order: 1, 2, 3, 1 BST becomes a degenerate tree. Searc operation FindKey takes O(n), wic is as inefficient as in a list. 3 It is possible tat after a number of insert and delete operations a binary tree may become imbalanced and increase in eigt. Can we insert and delete elements from BST so tat its eigt is guaranteed to be O(log n)? AVL tree ensures tis. 2 n

6 Data Structures CSC212 6 Specification of AVL Tree ADT Elements: Any data type Structure: A binary tree suc tat if N is any node in te tree ten all nodes in its left subtree are smaller tan N, and all nodes in its rigt subtree are larger tan N. Te eigt difference of te two subtrees of any node is at te most one. Note: : A node N is larger tan te node M if key value of N is larger tan tat of M and vice versa. Domain: Number of elements is bounded d Operations: Operation Specification void empty() Precondition: none. Process: returns true if te AVL as no nodes.

7 Data Structures CSC212 7 Operation Specification void findkey (int k) Precondition: none. Process: searces te AVL for a node wose key = k. If found ten tat node will be set as te current node and returns true. And if searc failed, ten (1) AVL empty ten just return false; or (2) AVL not empty ten current node is te node to wic a node containing k would be attaced as a cild if it were added to AVL and return false. void insert(intint k, Type val) Precondition: AVL not full. Process: (1) if we already ave a node wit key = k ten current retain its old value (te value prior to calling tis operation) and return false; or (2) insert a new node wit te given key and data and setting it as current node, return true. void update(type) Precondition/Requires : AVLis not empty. Process: update te value of data of te current node, key remains uncanged. Type retrieve() Precondition: AVL is not empty. Process: returns data of te current node.. bool remove_key(int k) Precondition: none. Process: removes te node containing te key = k, and in case BST is not empty ten sets te root as te current node. Returns true or false depending on weter te operation was successful or not.

8 Data Structures CSC212 8 Representation of AVL Tree ADT Since AVL is a special binary tree, it can be represented using - Linked List Note : Array is not suitable for AVL

9 Data Structures CSC212 9 Implementation of AVL public class AVLNode <T> { public int key, bal; public T data; public AVLNode<T> left, rigt; } public AVLNode(int k, T val) { key = k; bal = ; data = val; left = rigt = null; } public AVLNode (int k, AVLNode<T> l, AVLNode<T> r) { key = k; bal = ; } left = l; rigt = r;

10 Data Structures CSC212 1 Implementation of AVL public class Flag { boolean value; /** Creates a new instance of Flag */ public Flag() { value = false; } public Flag(boolean v){ value = v; } public boolean get_value (){ return value; } public void set_value(boolean l v){ value = v; } }

11 Data Structures CSC212 Implementation of AVL public class AVL <T> { AVLNode<T> root, current; public BST() private AVLNode<T> findparent (AVLNode<T> p) private AVLNode<T> find_min(avlnode<t> p) private AVLNode<T> remove_aux(int key, AVLNode<T> p, Flag flag) } public boolean empty() public T retrieve () public boolean findkey(int tkey) public boolean insert (int k, T val) public boolean remove_key (int tkey) public boolean update(int key, T data)

12 Data Structures CSC Note Tere is always a unique pat from te root to te new node. It is called te searc pat. A pivot node is a node closest to te new node on te searc pat, wose balance factor is eiter 1 or. Insert

13 Data Structures CSC No Pivot node 3 3 Insert Insert Pivot node 55 65

14 Data Structures CSC If, after an insert operation or a delete operation, an AVL tree becomes imbalanced, adjustments must be made to te tree to cange it back into an AVL tree. Tese adjustments are called rotations. Rotations are eiter single or double rotations.

15 Data Structures CSC Insert Operation Step 1: Insert a node into te AVL tree as it is inserted in a BST Step 2: Examine te searc pat to see if tere is a pivot node. Tree cases may arise. Case 1: Tere is no pivot node. No adjustment required. Case 2: Te pivot node exists and te subtree of te pivot node to wic te new node is added as smaller eigt. No adjustment required Case 3: Te pivot node exists and te subtree to wic te new node is added as te larger eigt. Adjustment required.

16 Data Structures CSC Insert: Case 1 No pivot node Insert Insert

17 Data Structures CSC Insert: Case 2 Pivot node exits New node added to te sorter subtree of te Pivot node. Pivot Node 2 2 Insert Insert Pivot Node 45 7

18 Data Structures CSC Note In tis case If balance factor of te pivot node is, te new node is inserted in te left subtree If balance factor of te pivot node is, te new node is inserted in te rigt subtree

19 Data Structures CSC Insert: Case 3 Pivot node exits New node added to te longer subtree of te Pivot node. 6 Insert 5 Pivot Node Note: AVL Tree is no more an AVL Tree after insertion.

20 Data Structures CSC212 2 Note Case 3 as furter 4 cases: Left-left case: An insertion in te left subtree of te left cild of te pivot node Rigt-left case: An insertion in te rigt subtree of te left cild of pivot node Left-rigt case: An insertion in te left subtree of te rigt cild of pivot node Rigt-rigt case: An insertion in te rigt subtree of te rigt cild of pivot node In te cases left-left and rigt-rigt rigt, single rotation is applied after new node is inserted. In te cases rigt-left and rigt-left left,, double rotation is applied after new node is inserted.

21 Data Structures CSC Insert: Left-left subcase te tree te tree B Pivot Single Rotation A A T3 B T2 T2 T3 New Node New Node Note: Rotation is applied about pivot node.

22 Data Structures CSC B Pivot A A T3 Single Rotation B T2 T2 T3 New Node New Node public void rotateleftcild(avlnode<t> B) { AVLNode<T> A = B.left; B.left = A.rigt; A.rigt = B; B = A; }

23 Data Structures CSC Example 1 6 Insert Single rotation 8 1

24 Data Structures CSC Insert: Rigt-rigt subcase te tree te tree Pivot B Single Rotation A T3 T2 A T3 B T2 New Node New Node

25 Data Structures CSC Pivot B A T3 T2 A Single Rotation T3 B T2 New Node New Node public void rotaterigtcild(avlnode<t> B) { AVLNode<T> A = B.rigt; B.rigt = A.left; A.left = B; B = A; }

26 Data Structures CSC Example 2 6 Insert Single rotation

27 Data Structures CSC Insert: Left-rigt subcase te tree te tree B Pivot Double Rotation C Pivot A T4 C T2 T3 A B T2 T3 T4

28 Data Structures CSC B Pivot C Pivot A T4 C Double Rotation T2 T3 A B T2 T3 T4 public void doublerotateleftcild(avlnode<t> B) { rotaterigtcild(b.left B.left); rotateleftcild(b); }

29 Data Structures CSC Example 3 6 Insert Double rotation 8 1

30 Data Structures CSC212 3 Insert: Left-rigt subcase te tree te tree B Pivot Double Rotation C Pivot A T4 C T2 T3 A B T2 T3 T4

31 Data Structures CSC Example 4 6 Insert Double rotation 8 1

32 Data Structures CSC Insert: Rigt- Left subcase te tree te tree Pivot B Double Rotation Pivot C T4 T3 C A T2 T4 B T3 T2 A

33 Data Structures CSC Pivot B Pivot C T4 T3 C A T2 Double Rotation T4 B T3 T2 A public void doublerotaterigtcild(avlnode<t> B) { rotateleftcild(b.rigt B.rigt); rotatrigtcild(b); }

34 Data Structures CSC Example Insert Double rotation

35 Data Structures CSC Insert: Rigt- Left subcase te tree te tree Pivot B Double Rotation Pivot C T4 A B A C T3 T2 T3 T2 T4

36 Data Structures CSC Example Insert Double rotation

37 Data Structures CSC Delete Operation Step 1: : Delete te node as in BST. Leaf node will always be deleted. Step 2: Ceck eac node p on te pat from te root to te parent node W of te deleted node. Start from te node on te pat closest to W. Tree cases are possible. Case 1: : Node p as balance factor. No rotation needed. Case 2: Node p as balance factor of (or 1) and a node was deleted from rigt sub-tree (or left subtree). No rotation needed. Case 3: Node p as balance factor of (or 1) and a node was deleted from left sub-tree (rigt sub-tree). Rotation needed. Four sub-cases are possible.

38 Data Structures CSC Let q be te cild of p wit larger eigt r be te cild of q wit larger eigt Sub-case (left-left) left) r is left cild of q, and q is left cild of p Apply Single rotation Sub-case -2 (rigt-rigt g ) r is rigt cild of q, and q is rigt cild of p Apply Single rotation Sub-case -3 (left-rigt ) r is left cild of q, and q is rigt cild of p Apply double rotation Sub-case -4 (rigt-left) r is rigt cild of q, and q is left cild of p Apply Single rotation te tree p -2 q T4 r T2 T3

39 Data Structures CSC Case 1 Balance factor of p is te tree te tree p p Node to be deleted.

40 Data Structures CSC212 4 Case 2 Balance factor of p is and node is deleted from left subtree te tree te tree p p Node to be deleted.

41 Data Structures CSC Case 2 Balance factor of p is and node is deleted from rigt subtree te tree te tree p p Node to be deleted.

42 Data Structures CSC Case 3 Subcase 1 (left-left) r is left cild of q, and q is left cild of p te tree te tree -2 p q r T4 Single Rotation r T2 T3 q p T4 T2 T3

43 Data Structures CSC Case 3 Subcase 2 (rigt-rigt) r is rigt cild of q, and q is rigt cild of p te tree te tree +2 p q Single Rotation q p r T4 T2 r T3 T4 T2 T3

44 Data Structures CSC Case 3 Subcase 3 (left-rigt) r is rigt cild of q, and q is left cild of p te tree te tree -2 p Double Rotation r q q p r T4 T2 T3 T4 T2 T3

45 Data Structures CSC Case 3 Subcase 4 (rigt-left) r is left cild of q, and q is rigt cild of p te tree te tree +2 p q Single Rotation r p q T4 r T4 T2 T3 T2 T3

46 Data Structures CSC Deletion: Example c e j m n p Delete p s b d k o r u a g i l t f

47 Data Structures CSC Deletion: Example c e j m n +2 o Delete p s b d k Sub-Case 1 Single Rotation a g i l t f r u

48 Data Structures CSC Deletion: Example c e j -2 m o s u b d k n r t a g i l f Sub Case 8 Double Rotation

49 Data Structures CSC Deletion: Example After Double Rotation c e j k m s a b d g i l o u n f r t

50 Data Structures CSC212 5 Exercise 1 (i) Is te following tree an AVL tree? If not, convert it into an AVL tree (ii) Insert te nodes 7, 125, 65, 72, and 61 in te tree given in part (i), one at a time, and sow te result of eac insertion in a separate diagram. Ensure tat te resulting tree satisfies AVL condition. (iii) Delete node 12 from te AVL tree you get in part (ii) and sow te result in a separate diagram. Make sure te resulting tree is an AVL tree.

51 Data Structures CSC Exercise 1 1. Wat is an AVL tree and wen is it used? 2. Wat is te time complexity for inserting N keys into an AVL tree? 3. Insert te following keys into an empty AVL tree 5, 1, 2, 6, 8, 3, 9 4. Delete 88 and ten

52 Data Structures CSC Delete 38 from te following AVL tree