MindSpace

Left-leaning Red-Black Trees

I think this is the easiest algorithm to learn and code. Robert Sedgewick’s Java code is always succinct. I have learnt how to code Tries based on his code. Now his Red-black tree code is easy to remember.

http://www.cs.princeton.edu/~rs/talks/LLRB/LLRB.pdf

Though the properties of this type of tree are different from the original Red-black tree I think it is sufficient to remember this condensed code.

These are the three lines and some small methods that we should remember mainly.

        if ( isRed(h.right) && !isRed(h.left)) h = rotateLeft(h);
        if ( isRed(h.left) && isRed(h.left.left)) h = rotateRight(h);
        if ( isRed(h.left) && isRed(h.right)) colorFlip(h);

Original binary tree before rotation

Final tree

I think the fundamental rules are these apart from the other Red-black tree rules.

1. Disallow right-leaning red nodes.
2. Disallow more than two consecutive red nodes.

Red-Black Tree

I will refine this post as I code more methods. This is an attempt to learn this data structure and explain what I think I understand using pictures drawn using TEX. I use the tikz editor.

The algorithm code is based on pseudo-code from “Introduction to Algorithms” by Thomas H. Cormen et al. I use Java language features like enums etc. wherever possible.

Sample TEX code

[->,>=stealth',level/.style={sibling distance = 5cm/#1,
  level distance = 1.5cm}] 
\node [arn_n] {3}
    child{ node [arn_r] {2} 
 				child{ node [arn_nil] {NIL}}
    }
    child{ node [arn_r] {4}
							child{ node [arn_nil] {NIL}}
	}
;

Insert

public class RedBlack {

    private final Node SENTINEL= new Node();
    private Node root= new Node();

    public RedBlack() {
        root = SENTINEL;
    }

    enum Color {
        BLACK,
        RED};

    class Node{

        public void setColor(Color color) {
            this.color = color;
        }

        private Color color = Color.BLACK;

        Node left,right,p;
        int key;
    }

    public void insert( RedBlack tree,
                        Node z){
        
        Node y =  SENTINEL;
        Node x =  tree.root;
        
        while( x != SENTINEL){
            y = x;
            if ( z.key < x.key){
                x = x.left;
            }else
                x = x.right;
        }
        
        z.p = y;
        
        if( y == SENTINEL){
            tree.root = z;
        }else if ( z.key < y.key){
                z = y.left;
        }else{
                z = y.right;
        }
        z.left = z.right = SENTINEL;
        z.setColor(Color.RED);
    }
}

I check the colors like this.

    public void inOrder(Node x){

        if( x != SENTINEL){
            inOrder(x.left);
            System.out.println(x.key + "[" + x.color + "]");
            inOrder(x.right);
        }
    }

Now this tree has to be fixed so that it conforms to the rules that govern a Red-Black tree. So I code a method to fix it. This method is based on the pseudo-code from the same book. But I think there is a bug which I haven’t found. This tree after it is fixed looks like this.

The black height property is 2.

The black height property for the left branch is 2 but it is 3 for one of the paths on the right.

I may have to read some other book because finding this bug is going to be hard.

Actually the tree should be like this.