CS 261 Data Structures. AVL Trees

Transcription

1 CS 261 Data Structures AVL Trees

2 AVL Implementation struct AVLNode { TYPE val; struct AVLNode *left; struct AVLNode *rght; int hght; /* Height of node*/ ;

3 Compute the Balance Factor (BF) int _height(struct AVLNode *cur) { if(cur == 0) return -1 else return cur->hght; int balancefact (struct AVLnode * current) { return ( _height(current->right) - _height(current->left) );

4 Compute Height void _setheight(struct AVLNode *cur) { int lh = _height(cur->left); int rh = _height(cur->rght); if(lh < rh) cur->hght = 1 + rh; else cur->hght = 1 + lh;

5 Add Adam Alex Abner Angela Abigail Adela Alice

6 Add Insert at the Leaf Level Adam Alex Alex Abner Angela Abner Angela Abigail Adela Alice Abigail Adela Alice Adam

7 Add struct AVLnode * AVLnodeAdd (struct AVLnode* current,type e) { struct AVLnode * newnode; if (current == 0) { /* allocate memory for newnode; */ return newnode;/* why not balance here?*/ else...

8 Add struct AVLnode * AVLnodeAdd (struct AVLnode* current, TYPE e) {... if (current == 0) {... else if( LT(e, current->value) ) current->left = AVLnodeAdd(current->left, e); else current->right = AVLnodeAdd(current->right, e); return balance(current);

9 Add What is the traversal type? struct AVLnode * AVLnodeAdd (struct AVLnode* current, TYPE e){ struct AVLnode * newnode; if (current == 0) { newnode = (struct AVLnode *) malloc(sizeof(struct AVLnode)); assert(newnode!= 0); newnode->value = e; newnode->left = newnode->right = 0; return newnode; else if( LT(e, current->value) ) else current->left = AVLnodeAdd(current->left, e); current->right = AVLnodeAdd(current->right, e); return balance(current);

10 Balance struct AVLnode * balance (struct AVLnode * current) { int cbf = balancefact(current); if (cbf < -1) { /* rotation right */ else if (cbf > 1) { /* rotation left */ _setheight(current); return current;

11 Balance... if (cbf < -1) {/* rotation right */ if (balancefact(current->left) > 0) /* double rotation: rotate left the left child */ current->left = rotateleft(current->left); return rotateright(current); else if (cbf > 1) {......

12 Balance... if (cbf < -1) {... else if (cbf > 1) {/* rotation left */ if (balancefact(current->right) < 0) /* double rotation: rotate right the right child */ current->right = rotateright(current->right); return rotateleft(current);...

13 Balance struct AVLnode * balance (struct AVLnode * current) { int cbf = balancefact(current); if (cbf < -1) {/* rotation right */ if (balancefact(current->left) > 0)) /* double rot. */ current->left = rotateleft(current->left); return rotateright(current); else if (cbf > 1) {/* rotation left */ if (balancefact(current->right) < 0) /* double rot. */ current->right = rotateright(current->right); return rotateleft(current); _setheight(current); return current;

14 Rotation Left struct AVLnode * rotateleft (struct AVLnode * current){ struct AVLnode * newtop = current->right;... current 1(0) 2(3) 4(2) newtop 3(0) 5(1) 6(0)

15 Rotation Left struct AVLnode * rotateleft (struct AVLnode * current){ struct AVLnode * newtop = current->right; current->right = newtop->left; newtop->left = current;... 2(3) 4(2) 1(0) 4(2) 2(3) 5(1) 3(0) 5(1) 1(0) 3(0) 6(0) 6(0)

16 Rotation Left struct AVLnode * rotateleft (struct AVLnode * current){ struct AVLnode * newtop = current->right; current->right = newtop->left; newtop->left = current; _setheight(current); _setheight(newtop); return newtop; 2(3) 4(2) 1(0) 4(2) 2(1) 5(1) 3(0) 5(1) 1(0) 3(0) 6(0) 6(0)

17 Rotation Right struct AVLnode * rotateright (struct AVLnode * current){ struct AVLnode * newtop = current->left; current->left = newtop->right; newtop->right = current; _setheight(current); _setheight(newtop); return newtop; 6(3) 4(2) 4(2) 7(0) 3(1) 6(1) 3(1) 5(0) 2(0) 5(0) 7(0) 2(0)

18 Remove: Who fills the hole in the tree? Answer: the leftmost child of the right child (smallest element in right subtree)

19 Remove void removeavltree(struct AVLTree *tree, TYPE val) { if (containsavltree(tree, val)) { tree->root = _removenode(tree->root, val); tree->cnt--; cur TYPE _leftmost(struct AVLNode *cur) { while(cur->left!= 0) { 6(3) cur = cur->left; 4(2) 7(0) return cur->val; 2(0) 3(1) 5(0)

20 Remove struct AVLNode * _removenode(struct AVLNode *cur, TYPE val) { struct AVLNode *temp; if(eq(val, cur->val)){ /* replace cur with the leftmost child of the right child of cur */ else if(lt(val, cur->val)) cur->left = _removenode(cur->left, val); else cur->rght = _removenode(cur->rght, val); return balance(cur);

21 Remove struct AVLNode * _removenode(struct AVLNode *cur, TYPE val) { struct AVLNode *temp; if(eq(val, cur->val)){ if(cur->rght!= 0){ cur->val = _leftmost(cur->rght); cur->rght = _removeleftmost(cur->rght); return balance(cur); else { temp = cur->left; free(cur); return temp;...

22 Remove struct AVLNode *_removeleftmost(struct AVLNode *cur) { struct AVLNode *temp; if(cur->left!= 0){ cur->left = _removeleftmost(cur->left); return balance(cur); temp = cur->rght; free(cur); return temp;

23 AVL Trees: Sorting An AVL tree can sort a collection of values: 1. Copy data into the AVL tree: O(??) 2. Copy them out using the?? traversal: O(??)

24 AVL Trees: Sorting An AVL tree can sort a collection of values: Copy data into the AVL tree: O(n log 2 n) Copy them out using the in-order traversal: O(n)

25 AVL Trees: Sorting Execution time à O(n log n): Matches that of quick sort Unlike quick sort, no problems if data is already sorted (which degrades quick sort to O(n 2 )) But, requires extra storage for the AVL tree