1. Design and Analysis of Algorithms
Red-Black trees
Haidong Xue
Summer 2012, at GSU

2. Tree height is crucial SEARCH(S, k) MINIMUM(S) MAXIMUM(S)
SUCCESSOR(S, x)
PREDECESSOR(S, x)
INSERT(S, x)
DELETE(S, x)
O(h)
O(h)
O(h)
O(h)
O(h)
O(h)
O(h)
Red-Black tree guarantees that h<=2 lg(n+1)

3. What is a RBT A binary search tree Red-black properties:
Every node is either red or black
The root is black
Every leaf(NIL) is black
If a node is red, then both its children are black
For each node, all simple paths from the node to descendant leaves contain the same number of black nodes

4. What is a RBT A RBT h<=2 lg(n+1) 11 9 12 8 10 11 2 NIL NIL NIL NIL

5. How to maintain a RBT After TREE-INSERT or TREE-DELETE
It is still a binary search tree
But may not still be a RBT
After each insertion or deletion the tree need to be maintained by
RB-INSERT-FIXUP
or RB-DELETE-FIXUP

6. How to maintain a RBT
A component used in RB-INSERT-FIXUP or RB-DELETE-FIXUP is rotation 11 12
Right rotate (x, y)
Link y’s parent to x 9 8 2
Set x’s right y’s left
Set y as x’s right 10
Rotation does not violate binary search tree properties

7. How to maintain a RBT 9 8 2 11 12 10
Left rotate