Red-Black tree

A Red-Black tree is a self-balancing binary search tree. The main difference to a normal binary tree is an extra bit that stores the parity (the color) of each node. This bit can be either be 0 (black) or 1 (red). An example of a red-black tree is shown in Figure 1 (Hardcoded)

The following invariant must be satisfied every time an operation is being made on node of a Red-Black tree:

1. Every node is either red or black
2. The root is always black
3. Every leaf is black
4. If a node is red, then both its children are black
5. All paths from a node to descendant nodes contain the same number of black nodes

The node of a Red-Black tree can be described by the following data type:

from enum import Enum

class RBNode:
    
    def __init__(self, color, key, left, right, parent):
        self.left = left
        self.right = right
        self.parent = parent
        self.key = key
        self.color = color

    def __str__(self):
        return str("(%s)%s" % (self.color, self.key))
    
class NodeColors(Enum):
    BLACK = 0;
    RED = 1;

The data structure of the tree is nothing more than a collection of nodes with the root node as the starting point. The constructor initializes a root node to a provided value

class RBTree:

    def __init__(self, key):
        self.nil = RBNode(NodeColors.BLACK, None, None, None, None)
        self.root = RBNode(NodeColors.BLACK, key, self.nil, self.nil, self.nil)

Your task for this assignment is to construct algorithms that manipulate Red-Black trees.

Constructing a Red-Black Tree

We start constructing a Red-Black tree by calling the constructor

tree = RBTree(5)

Inserting a node consists of three steps:

1. Finding the appropriate branch to insert the node in the tree
2. Inserting the node and creating links to the tree
3. Rebalancing the tree

For Step 1, the procedure is not much different from a normal binary search tree. Specifically, the algorithm first finds the appropriate position for the new node by searching through the tree. In the following snippet, z is the value we would like to insert. However, the snippet has a mistake.

T1 (10 points): Spot and fix the mistake in the search method

def search(tree, z):
    y = tree.nil
    x = tree.root
    while x != tree.nil:
        y = x
        if z.key < x.key:
            x = x.left
        else:
            y = x.right
    return y

Using the (hopefully correct by now!) search method, we can locate the node to insert our new node as a child. This happens by creating the node and linking it to the appropriate super/sub nodes. Note that for new nodes, the starting color is always RED.

T2 (20 points): Write code to construct and link a new node. Hint: Use the search method inside the insert method. Hint: Make the method return the new node, this is usefull for later

def insert(tree, z):
    pass

#Test you implementation here

After inserting the node, the tree potentially needs to be rebalanced. An example of why this is so can be seen in the following plot. What we see here is that the insertion of node 4 violates invariant number 4.

The rebalancing operation can be quite complicated; the pseudocode below takes care of all the cases.

proceduce rb_balance(T, z)
'' T is an RBTree
'' z is the newly inserted node

while z.parent.color == RED
    if z.parent == z.parent.parent.left
        y <- z.parent.parent.right

        if y.color == RED
            z.parent.color <- BLACK
            y.color <- BLACK
            z.parent.parent.color <- RED
            z <- z.parent.parent
        else 
            if z == z.parent.right
                z <- z.parent
                left_rotate(T, z)
            z.parent.color <- BLACK
            z.parent.parent.color <- RED
            right_rotate(T, z.parent.parent)
    else
      y <- z.parent.parent.left
       if y.color == RED
            z.parent.color <- BLACK
            y.color <- BLACK
            z.parent.parent.color <- RED
            z <- z.parent.parent
        else 
            if z == z.parent.left
                z <- z.parent
                right_rotate(T, z)
            z.parent.color <- BLACK
            z.parent.parent.color <- RED
            left_rotate(T, z.parent.parent)
T.root.color <- BLACK

You might be wondering with left_rotate and right_rotate are. They are functions that represent common operations on trees. To understand how they work have a look at the plot below.

Below you can find the Python code for the left_rotate method.

Note: The node you give to the rotate methods is always the parent node of the 2 nodes that are rotating

def left_rotate(tree, x):
    y = x.right
    x.right = y.left

    if y.left != tree.nil:
        y.left.parent = x
    y.parent = x.parent
    
    if x.parent == tree.nil:
        tree.root = y
    elif x == x.parent.left:
        x.parent.left = y
    else:
        x.parent.right = y
    y.left = x
    x.parent = y

T3 (20 points): Implement the code for right_rotate

def right_rotate(tree, x):
    pass

#Test your implementation here

T4 (25 points): Implement the rebalancing pseudocode in Python

def balance(tree, z):
    pass

#Test you implemention here

Validating the Red-Black tree invariants

T5 (25 points): Create a method that will exhaustively search whether the five invariants described above are maintained in the tree.

def check():
    "@return: True, only if tree satisfies all the Red-Black tree invariants"
    pass

#Test all of the five invariants here

Proving performance properties

T6 (10 points): Prove that the average and worse case complexity for the insertion in Red-Black trees is $O(log(n))$.

References

1. A Red-Black tree in Java
2. A Red-Black tree in C
3. Wikipedia on Red-Black Trees