Red-Black Trees | Top-Down Insertion
In Bottom-Up insertion of Red-Black Trees, “simple” Binary Search Tree insertion is used, followed by correction of the RB-Tree Violations on the way back up to the root. This can be done easily with the help of recursion. While in Top-Down Insertion, the corrections are done while traversing down the tree to the insertion point. When the actual insertion is done, no further corrections are needed, so no need to traverse back up the tree.
Therefore, the goal of Top-Down insertion is to traverse from the root to the insertion point in such a way that RB properties are maintained. This iterative approach thus makes Top-Down insertion faster than Bottom-Up insertion.
The two basic operations to perform for fixing violations and balancing are-
Following is the detailed Algorithm
The main goal of this algorithm is to create an insertion point at which the parent of the new node is Black, or the uncle of the new node is black.
Let N be the new node to be inserted.
- If Y and Z are Black:
- If X’s Parent is Black:
- X’s Parent P is Red, Grandparent is Black and X and P are both left OR right children of Grandparent G:
- Recolor X, Y, Z
- Rotate P around G
- Color P black
- Color G red
- X’s Parent is Red, Grandparent is Black and X and P are opposite children of Grandparent G
- Recolor X, Y, Z
- Rotate X around P
- Rotate X around G
- Recolor X and G
perform Simple BST insertion. Insert new node N as the left/right child of Y OR Z and make the colour of the newly inserted node as red.
Then Recolor X, Y, Z and continue Down the tree.
If violations exist continue with further cases.
Insert New Node N at the required Position.
Example :
Insert Node 3 in the RB-Tree below –
Below the implementation of the following approach:
Java
// Java implementation for Top-Down // Red-Black Tree Insertion creating // a red black tree and storing an // English sentence into it using Top // down insertion approach import static java.lang.Integer.max; // Class for performing // RBTree operations public class RbTree { TreeNode Root = null; // Function to calculate // the height of the tree int HeightT(TreeNode Root) { int lefth, righth; if (Root == null || (Root.children == null && Root.children[1] == null)) { return 0; } lefth = HeightT(Root.children[0]); righth = HeightT(Root.children[1]); return (max(lefth, righth) + 1); } // Function to check if // dir is equal to 0 int check(int dir) { return dir == 0 ? 1 : 0; } // Function to check if a // node's color is red or not boolean isRed(TreeNode Node) { return Node != null && Node.color.equals("R"); } // Function to perform // single rotation TreeNode SingleRotate(TreeNode Node, int dir) { TreeNode temp = Node.children[check(dir)]; Node.children[check(dir)] = temp.children[dir]; temp.children[dir] = Node; Root.color = "R"; temp.color = "B"; return temp; } // Function to perform double rotation TreeNode DoubleRotate(TreeNode Node, int dir) { Node.children[check(dir)] = SingleRotate(Node.children[check(dir)], check(dir)); return SingleRotate(Node, dir); } // Function to insert a new // node with given data TreeNode Insert(RbTree tree, String data) { if (tree.Root == null) { tree.Root = new TreeNode(data); if (tree.Root == null) return null; } else { // A temporary root TreeNode temp = new TreeNode(""); // Grandparent and Parent TreeNode g, t; TreeNode p, q; int dir = 0, last = 0; t = temp; g = p = null; t.children[1] = tree.Root; q = t.children[1]; while (true) { if (q == null) { // Inserting root node q = new TreeNode(data); p.children[dir] = q; } // Sibling is red else if (isRed(q.children[0]) && isRed(q.children[1])) { // Recoloring if both // children are red q.color = "R"; q.children[0].color = "B"; q.children[1].color = "B"; } if (isRed(q) && isRed(p)) { // Resolving red-red // violation int dir2; if (t.children[1] == g) { dir2 = 1; } else { dir2 = 0; } // If children and parent // are left-left or // right-right of grand-parent if (q == p.children[last]) { t.children[dir2] = SingleRotate(g, last == 0 ? 1 : 0); } // If they are opposite // childs i.e left-right // or right-left else { t.children[dir2] = DoubleRotate(g, last == 0 ? 1 : 0); } } // Checking for correct // position of node if (q.data.equals(data)) { break; } last = dir; // Finding the path to // traverse [Either left // or right ] dir = q.data.compareTo(data) < 0 ? 1 : 0; if (g != null) { t = g; } // Rearranging pointers g = p; p = q; q = q.children[dir]; } tree.Root = temp.children[1]; } // Assign black color // to the root node tree.Root.color = "B"; return tree.Root; } // Print nodes at each // level in level order // traversal void PrintLevel(TreeNode root, int i) { if (root == null) { return; } if (i == 1) { System.out.print("| " + root.data + " | " + root.color + " |"); if (root.children[0] != null) { System.out.print(" " + root.children[0].data + " |"); } else { System.out.print(" " + "NULL" + " |"); } if (root.children[1] != null) { System.out.print(" " + root.children[1].data + " |"); } else { System.out.print(" " + "NULL" + " |"); } System.out.print(" "); return; } PrintLevel(root.children[0], i - 1); PrintLevel(root.children[1], i - 1); } // Utility Function to // perform level order // traversal void LevelOrder(TreeNode root) { int i; for (i = 1; i < HeightT(root) + 1; i++) { PrintLevel(root, i); System.out.print("\n\n"); } } } // Class for representing // a node of the tree class TreeNode { // Class variables String data, color; TreeNode children[]; public TreeNode(String data) { // Color R- Red // and B - Black this.data = data; this.color = "R"; children = new TreeNode[2]; children[0] = null; children[1] = null; } } // Driver Code class Driver { public static void main(String[] args) { // Tree Node Representation // ------------------------------------------- // DATA | COLOR | LEFT CHILD | RIGHT CHILD | // ------------------------------------------- RbTree Tree = new RbTree(); String Sentence, Word; Sentence = "old is gold"; String Word_Array[] = Sentence.split(" "); for (int i = 0; i < Word_Array.length; i++) { Tree.Root = Tree.Insert(Tree, Word_Array[i]); } // Print Level Order Traversal System.out.println("The Level" + "Order Traversal" + "of the tree is:"); Tree.LevelOrder(Tree.Root); System.out.println("\nInserting a" + " word in the tree:"); Word = "forever"; Tree.Root = Tree.Insert(Tree, Word); System.out.println(""); Tree.LevelOrder(Tree.Root); } } |
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C#
// C# implementation for Top-Down // Red-Black Tree Insertion creating // a red black tree and storing an // English sentence into it using Top // down insertion approach using System; // Class for performing // RBTree operations class RbTree { public TreeNode Root = null; // Function to calculate // the height of the tree public int HeightT(TreeNode Root) { int lefth, righth; if (Root == null || (Root.children == null && Root.children[1] == null)) { return 0; } lefth = HeightT(Root.children[0]); righth = HeightT(Root.children[1]); return (Math.Max(lefth, righth) + 1); } // Function to check if // dir is equal to 0 public int check(int dir) { return dir == 0 ? 1 : 0; } // Function to check if a // node's color is red or not public bool isRed(TreeNode Node) { return Node != null && Node.color.Equals("R"); } // Function to perform // single rotation public TreeNode SingleRotate(TreeNode Node, int dir) { TreeNode temp = Node.children[check(dir)]; Node.children[check(dir)] = temp.children[dir]; temp.children[dir] = Node; Root.color = "R"; temp.color = "B"; return temp; } // Function to perform double rotation public TreeNode DoubleRotate(TreeNode Node, int dir) { Node.children[check(dir)] = SingleRotate(Node.children[check(dir)], check(dir)); return SingleRotate(Node, dir); } // Function to insert a new // node with given data public TreeNode Insert(RbTree tree, String data) { if (tree.Root == null) { tree.Root = new TreeNode(data); if (tree.Root == null) return null; } else { // A temporary root TreeNode temp = new TreeNode(""); // Grandparent and Parent TreeNode g, t; TreeNode p, q; int dir = 0, last = 0; t = temp; g = p = null; t.children[1] = tree.Root; q = t.children[1]; while (true) { if (q == null) { // Inserting root node q = new TreeNode(data); p.children[dir] = q; } // Sibling is red else if (isRed(q.children[0]) && isRed(q.children[1])) { // Recoloring if both // children are red q.color = "R"; q.children[0].color = "B"; q.children[1].color = "B"; } if (isRed(q) && isRed(p)) { // Resolving red-red // violation int dir2; if (t.children[1] == g) { dir2 = 1; } else { dir2 = 0; } // If children and parent // are left-left or // right-right of grand-parent if (q == p.children[last]) { t.children[dir2] = SingleRotate(g, last == 0 ? 1 : 0); } // If they are opposite // childs i.e left-right // or right-left else { t.children[dir2] = DoubleRotate(g, last == 0 ? 1 : 0); } } // Checking for correct // position of node if (q.data.Equals(data)) { break; } last = dir; // Finding the path to // traverse [Either left // or right ] dir = q.data.CompareTo(data) < 0 ? 1 : 0; if (g != null) { t = g; } // Rearranging pointers g = p; p = q; q = q.children[dir]; } tree.Root = temp.children[1]; } // Assign black color // to the root node tree.Root.color = "B"; return tree.Root; } // Print nodes at each // level in level order // traversal public void PrintLevel(TreeNode root, int i) { if (root == null) { return; } if (i == 1) { Console.Write("| " + root.data + " | " + root.color + " |"); if (root.children[0] != null) { Console.Write(" " + root.children[0].data + " |"); } else { Console.Write(" " + "NULL" + " |"); } if (root.children[1] != null) { Console.Write(" " + root.children[1].data + " |"); } else { Console.Write(" " + "NULL" + " |"); } Console.Write(" "); return; } PrintLevel(root.children[0], i - 1); PrintLevel(root.children[1], i - 1); } // Utility Function to perform // level order traversal public void LevelOrder(TreeNode root) { int i; for (i = 1; i < HeightT(root) + 1; i++) { PrintLevel(root, i); Console.Write("\n\n"); } } } // Class for representing // a node of the tree public class TreeNode { // Class variables public String data, color; public TreeNode []children; public TreeNode(String data) { // Color R- Red // and B - Black this.data = data; this.color = "R"; children = new TreeNode[2]; children[0] = null; children[1] = null; } } // Driver Code public class Driver { public static void Main(String[] args) { // Tree Node Representation // ------------------------------------------- // DATA | COLOR | LEFT CHILD | RIGHT CHILD | // ------------------------------------------- RbTree Tree = new RbTree(); String Sentence, Word; Sentence = "old is gold"; char[] spearator = { ' ', ' ' }; String []Word_Array = Sentence.Split(spearator, StringSplitOptions.RemoveEmptyEntries); for (int i = 0; i < Word_Array.Length; i++) { Tree.Root = Tree.Insert(Tree, Word_Array[i]); } // Print Level Order Traversal Console.WriteLine("The Level" + "Order Traversal" + "of the tree is:"); Tree.LevelOrder(Tree.Root); Console.WriteLine("\nInserting a" + " word in the tree:"); Word = "forever"; Tree.Root = Tree.Insert(Tree, Word); Console.WriteLine(""); Tree.LevelOrder(Tree.Root); } } // This code is contributed by Rajput-Ji |
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Output:
The LevelOrder Traversalof the tree is: | is | B | gold | old | | gold | R | NULL | NULL | | old | R | NULL | NULL | Inserting a word in the tree: | is | B | gold | old | | gold | B | forever | NULL | | old | B | NULL | NULL | | forever | R | NULL | NULL |
References:
Red Black Trees – UMBC CSEE
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