CS 758/858: Algorithms http://www.cs.unh.edu/~ruml/cs758 Red-Black - PowerPoint PPT Presentation

CS 758/858: Algorithms http://www.cs.unh.edu/~ruml/cs758 Red-Black Trees Deletion Fixup 1 handout: slides Wheeler Ruml (UNH) Class 7, CS 758 – 1 / 16

Red-Black Trees ■ Red-Black Trees ■ BST Deletion ■ Single Child ■ Immed. Succ. ■ Deep Succ. ■ Break Deletion Fixup Red-Black Trees Wheeler Ruml (UNH) Class 7, CS 758 – 2 / 16

Red-Black Trees node: data, left, right, parent, color Red-Black Trees ■ Red-Black Trees 1. every node is either red or black ■ BST Deletion ■ Single Child 2. the root is black ■ Immed. Succ. 3. (consider nil to be black) ■ Deep Succ. ■ Break 4. both children of a red node are black Deletion Fixup 5. from any node, all paths to leaves have the same ‘black height’ Wheeler Ruml (UNH) Class 7, CS 758 – 3 / 16

Plain Binary Tree Deletion 4 cases of delete( z ): Red-Black Trees ■ Red-Black Trees 1. no left child, or no kids: substitute right subtree at parent. ■ BST Deletion ■ Single Child 2. no right child: substitute left subtree at parent. ■ Immed. Succ. 3. successor y is z ’s right child: ■ Deep Succ. ■ Break (a) substitute y for z Deletion Fixup (b) attach z ’s left subtree as y ’s left subtree 4. successor y is deeper: (a) substitute y ’s right subtree for y (b) attach z ’s right subtree as y ’s right subtree (c) as above, substitute y for z (d) as above, attach z ’s left subtree as y ’s left subtree What if it’s a red-black tree? Wheeler Ruml (UNH) Class 7, CS 758 – 4 / 16

Cases 1 and 2: Single Child 1. every node is either red or black Red-Black Trees 2. the root is black ■ Red-Black Trees ■ BST Deletion 3. (consider nil to be black) ■ Single Child ■ Immed. Succ. 4. both children of a red node are black ■ Deep Succ. 5. from any node, all paths to leaves have the same ‘black ■ Break height’ Deletion Fixup deleting z with single child x 1. x takes z ’s place Wheeler Ruml (UNH) Class 7, CS 758 – 5 / 16

Cases 1 and 2: Single Child 1. every node is either red or black Red-Black Trees 2. the root is black ■ Red-Black Trees ■ BST Deletion 3. (consider nil to be black) ■ Single Child ■ Immed. Succ. 4. both children of a red node are black ■ Deep Succ. 5. from any node, all paths to leaves have the same ‘black ■ Break height’ Deletion Fixup deleting z with single child x 1. x takes z ’s place 2. book uses y for z for short code 3. if y ( = z ) was black, we have ‘extra black’ at x , so call fixup routine at x Wheeler Ruml (UNH) Class 7, CS 758 – 5 / 16

Case 3: Two Children, Successor is Child 1. every node is either red or black Red-Black Trees 2. the root is black ■ Red-Black Trees ■ BST Deletion 3. (consider nil to be black) ■ Single Child ■ Immed. Succ. 4. both children of a red node are black ■ Deep Succ. 5. from any node, all paths to leaves have the same ‘black ■ Break height’ Deletion Fixup deleting z , successor y is right child 1. y takes z ’s place and color 2. attach z ’s left subtree as y ’s left subtree Wheeler Ruml (UNH) Class 7, CS 758 – 6 / 16

Case 3: Two Children, Successor is Child 1. every node is either red or black Red-Black Trees 2. the root is black ■ Red-Black Trees ■ BST Deletion 3. (consider nil to be black) ■ Single Child ■ Immed. Succ. 4. both children of a red node are black ■ Deep Succ. 5. from any node, all paths to leaves have the same ‘black ■ Break height’ Deletion Fixup deleting z , successor y is right child 1. y takes z ’s place and color 2. attach z ’s left subtree as y ’s left subtree 3. if y was black, we need ‘extra black’ at y ’s right child x , so call fixup routine at x Wheeler Ruml (UNH) Class 7, CS 758 – 6 / 16

Case 4: Two Children, Successor is Deeper deleting z , successor y is deep down Red-Black Trees ■ Red-Black Trees 1. substitute y ’s right child x for y ■ BST Deletion ■ Single Child 2. attach z ’s right subtree as y ’s right subtree ■ Immed. Succ. as in simpler case: ■ Deep Succ. ■ Break 3. y takes z ’s place and color Deletion Fixup 4. attach z ’s left subtree as y ’s left subtree 5. if y was black, we need ‘extra black’ at x , so call fixup routine at x Wheeler Ruml (UNH) Class 7, CS 758 – 7 / 16

Break asst 4: write verifier ■ Red-Black Trees ■ Red-Black Trees ■ ■ BST Deletion ■ Single Child ■ Immed. Succ. ■ Deep Succ. ■ Break Deletion Fixup Wheeler Ruml (UNH) Class 7, CS 758 – 8 / 16

Red-Black Trees Deletion Fixup ■ Fix-up Loop ■ Case 1 ■ Case 2 ■ Case 3 ■ Case 4 ■ Complexity ■ EOLQs Red-Black Tree Deletion Fixup Wheeler Ruml (UNH) Class 7, CS 758 – 9 / 16

Deletion Fix-up Loop need to find a red node to make black Red-Black Trees Deletion Fixup ■ Fix-up Loop when x red or root, color black and terminate ■ Case 1 ■ Case 2 ■ Case 3 x is non-root black node. ■ Case 4 assume x is a left child (other cases symmetric). ■ Complexity ■ EOLQs Must have sibling w , since x holds ‘extra blackness’. 4 cases: 1. w is red 2. w and both its children are black 3. w is black, its right child is black, its left child is red 4. w is black, its right child is red fix-up loop invariant: all properties hold if ‘extra black’ at x is considered, heights of fringe (greek) nodes preserved Wheeler Ruml (UNH) Class 7, CS 758 – 10 / 16

Case 1 case 1: w is red. so parent and children must be black. Red-Black Trees Deletion Fixup ■ Fix-up Loop solution: ■ Case 1 ■ Case 2 1. rotate and recolor to get black sibling (moves red ■ Case 3 horizontally) ■ Case 4 ■ Complexity 2. fall through to case 2, 3, or 4 ■ EOLQs Wheeler Ruml (UNH) Class 7, CS 758 – 11 / 16

Case 2 case 2: w and both its children are black Red-Black Trees Deletion Fixup ■ Fix-up Loop solution: ■ Case 1 ■ Case 2 1. color w red. subtree at parent now ‘black-balanced’. ■ Case 3 2. move x ’s blackness (and w ’s) up the tree ■ Case 4 ■ Complexity 3. recur at parent ■ EOLQs if from case 1, x now red, so will terminate Wheeler Ruml (UNH) Class 7, CS 758 – 12 / 16

Case 3 case 3: w is black, its right is black, left is red Red-Black Trees Deletion Fixup ■ Fix-up Loop solution: ■ Case 1 ■ Case 2 1. rotate right and move red over to right child ■ Case 3 2. fall through to case 4 ■ Case 4 ■ Complexity ■ EOLQs Wheeler Ruml (UNH) Class 7, CS 758 – 13 / 16

Case 4 case 4: w is black, its right child is red Red-Black Trees Deletion Fixup ■ Fix-up Loop solution: ■ Case 1 ■ Case 2 1. rotate and recolor to annihilate red with x ’s black ■ Case 3 2. set x to root to force termination ■ Case 4 ■ Complexity ■ EOLQs Wheeler Ruml (UNH) Class 7, CS 758 – 14 / 16

Complexity finding successor is O (lg n ) Red-Black Trees Deletion Fixup ■ Fix-up Loop one fixup iteration is constant time ■ Case 1 ■ Case 2 ■ Case 3 fixup loops only when moving up, so is O (lg n ) ■ Case 4 ■ Complexity ■ EOLQs how many rotations are performed? Wheeler Ruml (UNH) Class 7, CS 758 – 15 / 16

EOLQs For example: Red-Black Trees Deletion Fixup What’s still confusing? ■ ■ Fix-up Loop What question didn’t you get to ask today? ■ ■ Case 1 ■ Case 2 What would you like to hear more about? ■ ■ Case 3 ■ Case 4 Please write down your most pressing question about algorithms ■ Complexity and put it in the box on your way out. ■ EOLQs Thanks! Wheeler Ruml (UNH) Class 7, CS 758 – 16 / 16