Red Black Tree Essentials

1. Red Black Tree Essentials Notes from “Introduction to Algorithms”, Cormenet al.

2. Definition Red Black Trees A red-black tree is a binary search tree with an extra bit of storage per node. The extra bit represents the color of the node. It's either red or black. Each node contains the fields: color, key, left, right, and p. Any nil pointers are regarded as pointers to external nodes (leaves) and key bearing nodes are considered as internal nodes of the tree. See the video at: http://www.youtube.com/watch?v=vDHFF4wjWYU

3. Properties Red-black tree properties: 1. Every node is either red or black. 2. The root is black. 3. Every leaf (nil) is black. 4. If a node is red then both of its children are black. 5. For each node, all paths from the node to descendant leaves contain the same number of black nodes.

4. Implications • From these properties, it can be shown (with a proof by induction) that the tree has a height no more than 2 * Lg(n + 1). • Thus, worst case lookUp, insert, delete are all Θ(Log n).

5. Three essential methods • Rotation • Insertion • Insert-fixUp

6. LeftRotate(T,x) pre: right[x] != nil[T] pre: root's parent is nill[T] Left-Rotate(T,x) y = right[x] right[x] = left[y] p[left[y]] = x p[y] = p[x] if p[x] == nil[T] then root[T] = y else if x == left[p[x]] then left[p[x]] = y else right[p[x]] = y left[y] = x p[x] = y

7. Red Black Insert RB-Insert(T,z) y = nil[T] x = root[T] while x != nil[T] y = x if key[z] < key[x] then x = left[x] else x = right[x] p[z] = y if y = nil[T] root[T] = z else if key[z] < key[y] then left[y] = z else right[y] = z left[z] = nil[T] right[z] = nil[T] color[z] = RED RB-Insert-fixup(T,z)

8. RB-Insert-fixup(T,z) RB-Insert-fixup(T,z) { while(z's parent is Red) { set y to be z's uncle if uncle y is Red { color parent and uncle black color grandparent red set z to grandparent } else { // the uncleisblack if (zigzag) { // make it a zigzig set z to parent rotatetozigzig } // rotate the zigzigand finish color parent of z black color grandparent of z red rotate grand parent of z } } // end while color root black }