Binary Search Tree: Balanced. Data Structures and Algorithms Emory University Jinho D. Choi

Transcription

1 Binary Search Tree: Balanced Data Structures and Algorithms Emory University Jinho D. Choi

2 Binary Search Tree Worst-case Complexity Search Insert Delete Unbalanced O(n) O(n) O(n) + α Balanced O(log n) O(log n) + β O(log n) + α AVL (Adelson-Velskii and Landis) Red-Black

3 Balanced BST - Add & Remove public N add(t key) N node = super.add(key); balance(node); return node; protected abstract void balance(n node); public N remove(t key) N node = findnode(root, key); if (node!= null) N lowest = node.hasbothchildren()? removehibbard(node) : removeself(node); if (lowest!= null && lowest!= node) balance(lowest); return node;

4 Balanced BST - Rotate What about the subtrees? 4

5 Balanced BST - Rotate Left N child = node.getrightchild(); node.setrightchild(child.getleftchild()); node.getparent().replacechild(node, child); child.setleftchild(node); 5

6 Balanced BST - Rotate Left protected void rotateleft(n node) N child = node.getrightchild(); node.setrightchild(child.getleftchild()); if (node.hasparent()) node.getparent().replacechild(node, child); else setroot(child); child.setleftchild(node); 6

7 Binary Search Tree: AVL Data Structures and Algorithms Emory University Jinho D. Choi

8 AVL Node public class AVLNode<T extends Comparable<T>> extends AbstractBinaryNode<T,AVLNode<T>> private int height; // height of this node in the tree public AVLNode(T key) super(key); height = ; public int getheight() return height; public void setheight(int height) height = height; 8

9 AVL Node - Reset Height private void resetheightsaux(avlnode<t> node) if (node!= null) int lh = node.hasleftchild()? node.getleftchild().getheight() : 0; int rh = node.hasrightchild()? node.getrightchild().getheight() : 0; int height = (lh > rh)? lh + : rh + ; if (height!= node.getheight()) node.setheight(height); resetheightsaux(node.getparent()); public void resetheight() resetheightaux(this); 5 8?

10 AVL Node - Balance Factor public int getbalancefactor() if (hasbothchildren()) return left_child.getheight() - right_child.getheight(); else if (hasleftchild()) return left_child.getheight(); else if (hasrightchild()) return -right_child.getheight(); else return 0; 5 max_height(left subtree) - max_height(right subtree)

11 AVL Tree public class AVLTree<T extends Comparable<T>> extends public AVLNode<T> createnode(t key) return new protected void rotateleft(avlnode<t> node) super.rotateleft(node); protected void rotateright(avlnode<t> node) super.rotateright(node); node.resetheights();

12 54867 AVL Tree - Add 5 Balance factor >? Rotate right Rotate left Rotate right

13 AVL Tree - Balance protected void balance(avlnode<t> node) if (node == null) return; int bf = node.getbalancefactor(); if (bf == ) AVLNode<T> child = node.getleftchild(); if (child.getbalancefactor() == -) rotateleft(child); rotateright(node); else if (bf == -) AVLNode<T> child = node.getrightchild(); if (child.getbalancefactor() == ) rotateright(child); rotateleft(node); else balance(node.getparent());

14 Binary Search Tree: Red-Black Data Structures and Algorithms Emory University Jinho D. Choi

15 Red-Black Tree Properties - Every node is either red or black. - The root and all leaves (null) are black. - Every red node must have two black child nodes. - Every path from a given node to any of is descendant leaves must contain the same number of black nodes. 5

16 Red-Black Tree Red node must have two black children. 4 4 Every path must have the same # of black nodes. 6

17 Red-Black Tree Red parent Black uncle 4 4 7

18 Red-Black Tree Red parent Zig Zag. 4 4 Go to the previous step. 8

19 Red-Black - Add 5 node.getparent().settoblack(); uncle.settoblack(); RedBlackNode<T> grandparent = node.getgrandparent(); grandparent.settored(); balance(grandparent); 5 RedBlackNode<T> parent = node.getparent(); 4 if (isroot(node)) node.settoblack(); else if (node.getparent().isred()) RedBlackNode<T> uncle = node.getuncle(); if (uncle!= null && uncle.isred()) balancewithreduncle(node, uncle); else balancewithblackuncle(node); if (grandparent.isleftchild(parent) && parent.isrightchild(node)) rotateleft(parent); node = parent; else if (grandparent.isrightchild(parent) && parent.isleftchild(node)) rotateright(parent); node = parent; node.getparent().settoblack(); grandparent.settored(); if (node.getparent().isleftchild(node)) rotateright(grandparent); else rotateleft(grandparent); 9

20 Agenda Exercise - Reading - 0