Implementing Red Black Trees

Ellen WalkerCPSC 201 Data Structures

Hiram College

Implementing Red-Black Trees

• Several implementations are possible (and many are on the Internet)– Some include parent links in RedBlackNodes - can be iterative

– Textbook’s is recursive, recognizing need for rotations on the way back up the tree (after recursive call)

• These slides follow your textbook, but…– Code here is incomplete– In places, code has been simplified for presentation

• For complete, correct code, see your textbook and/or the book code

About These Slides

• Follow textbook’s presentation but…

• … code is incomplete• … code is slightly simplified for ease of presentation– Example: casts are removed

• Use these notes as a guide for understanding when reading the actual textbook code

Classes (Koffman & Wolfgang)

SearchTree

BinarySearchTree

BSTWithRotate

RBTree

(abstract class)

(Adds RotateLeft and RotateRight)

Single Rotation (RotateRight)

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root

temp = root.left;root.left = temp.right;temp.right = root;return temp;

temp

temp

root

Double Rotation

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root

root.left = rotateLeft(root.left); return RotateRight(root);

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RedBlackNode<E>

• E data– Data contained in this node

• RedBlackNode left, RedBlackNode right– References to children

• boolean isRed– Color of this node (true means Red)– Set to true in constructor, because all nodes (except root) are red when added

Recall: BST Recursive Add

If (item.equals(localRoot.data))addreturn = null; //duplicates not addedReturn localRoot;

If (item.compareTo(localRoot.data)<0) Add to the left return localRoot;Else Add to the right return localRoot;

Recall: BST Recursive Add (left side)

If localRoot.left == null localRoot.left = new Node(item);

return localRoot;Else localRoot.left = add(localRoot.left,item);

return localRoot;

Add Recursion

• Base case at leaf of tree– Create and link a new node

• Recursive case– Recursively add on the correct side– Link result of recursion back into the tree, or the changes will be lost!

– localRoot.left = add(localRoot.left, item); //not just add(localRoot.left, item);

Top Down Insertion Algorithm

• Search the binary tree in the usual way for the insertion algorithm

• If you pass a 4-node (black node with two red children) on the way down, split it

• Insert the node as a red child, and use the split algorithm to adjust via rotation if the parent is red also.

• Force the root to be black after every insertion.

RBT Adding to the Left

If localRoot.left == null localRoot.left = new Node(item); return localRoot;Else //Split me if I’m a 4-node moveBlackDown(localRoot); localRoot.left = add(localRoot.left, item); //Deal with consecutive red child & grandchild //case 1: red left child with red left grandchild

//case 2: red left child with red right grandchild

RotateRight to Fix Left-Left Reds

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localRoot

localRoot.left.isRed = false;localRoot.isRed = true;Return rotateRight(localRoot);

Double Rotation to Fix Left-Right Reds

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localRoot.left = rotateLeft(localRoot.left); localRoot.left.isRed = false; localRoot.isRed = true;return RotateRight(localRoot);

Putting the code all together

• Insert appropriate tests to recognize the various red-red cases

• Repeat all the “left code” for the right side– Swap “left” vs. “right” everywhere

• Write the “starter add method”– Deals with an empty tree– Makes the root black at the end

Starter Method

If (root == null) { root = new RedBlackNode<E> (item); root.isRed = false; //root is always black addReturn = true; //item was added}Else { root = add(root, item); //sets addReturn root.isRed = false; //root is always black

}return addReturn;

Insert 1, 2, 3

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Insert red leaf(2 consecutive red nodes!)

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Left single rotation

Continued: Insert 4, 5

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4-node (2 red children)split on the way downRoot remains black

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Single rotation to avoidconsecutive red nodes

Continued, Insert 9, 6

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4-node (3,4,5) split on the way down,4 is now red (passed up)

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Double rotation

Deletion Algorithm

• Find the node to be deleted (NTBD)– On the way down, if you pass a 2-node upgrade it by borrowing from its neighbor and/or parent

• If the node is not a leaf node, – Find its immediate successor, upgrading all 2-nodes

– Swap value of leaf node with value of NTBD

• Remove the current leaf node, which is now NTBD (because of swap, if it happened)

Red-black “neighbor” of a node

• Let X be a 2-node to be deleted• If X is its parent’s left child, X’s right neighbor can be found by:– Let S = parent’s right child. If S is black, it is the neighbor

– Otherwise, S’s left child is the neighbor.

• If X is parent’s left child, then X’s left neighbor is grandparent’s left child.

Neighbor examples2

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Right neighbor of 1 is 3

Right neighbor of 3 is 5

Left neighbor of 5 is 3

Left neighbor of 3 is 1

Upgrade a 2-node

• Find the 2-node’s neighbor (right if any, otherwise left)

• If neighbor is also a 2-node (2 black children)– Create a 4-node from neighbors and their parent.– If neighbors are actually siblings, this is a color swap.

– Otherwise, it requires a rotation

• If neighbor is a 3-node or 4-node (red child)– Move “inner value” from neighbor to parent, and “dividing value” from parent to 2-node.

– This is a rotation

Deletion Examples (Delete 1)

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Make a 4-node from 1, sibling 3, and “divider value” 2.[Single rotation of 2,4,6]

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Delete 64

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Upgrade 6 by color flip, swap with successor (9)

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Delete 4

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No 2-nodes to upgrade, swap with successor (5)

Delete 22

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Find 2-node enroute to successor (3)Neighbor is 3-node (6,9) Shift to get (3,5) and 9 as children,6 up to parent.

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Single rotation

Delete 2 (cont’d)

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Swap with successor

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Remove leaf