1. Announcements Midterm will be given on 10/21/99
Extra help session at today & tomorrow at SN014 by TA
Homework #4 is due next Tuesday
Quiz #3 is given today
Continue with lectures on R-B Trees
UNC Chapel Hill
M. C. Lin

2. Insertion in R-B Trees It can be done in O(lg n) time. Basic steps:
Use Tree-Insert from BST to insert a node x into T
Color the node x red.
Fix up the modified tree by re-coloring nodes and performing rotation to preserve R-B tree property.
UNC Chapel Hill
M. C. Lin

3.  This routine runs in O(lg n) time.
UNC Chapel Hill
M. C. Lin

4. RB-Insertion
UNC Chapel Hill
M. C. Lin

5. Example of RB-Insert Operation
UNC Chapel Hill
M. C. Lin

6. RB-Insert
UNC Chapel Hill
M. C. Lin

7. RB-Insert For p[x] is a left child:
Case 1 (the child x, its parent and its uncle y are red): Push the pointers up the tree and change the colors, till x and y are of different colors.
Case 2 (x and its parent are red, but y is black): if problem node is a right child, left rotate.
Case 3 (x and its parent are red, but y is black): if problem node is a left child, right rotate.
Symmetric case for p[x] as a right child
UNC Chapel Hill
M. C. Lin

8. Algorithm Analysis
While loop only repeats if case 1 is executed, then the pointer x moves up the tree. The total number of times the while loop can be executed is O(h).
The height of a R-B tree on n nodes is O(lg n), so Tree-Insert takes O(lg n) time.
 TOTAL: O(lg n)
UNC Chapel Hill
M. C. Lin

9. Review on Tree-Delete (T, x)
if x has no children  case 0
then remove x
if x has one child  case 1
then make p[x] point to child
if x has two children  case 2
then swap x with its successor
perform case 0 or case 1 to delete it
UNC Chapel Hill
M. C. Lin

10. RB-Delete
Implementation: use sentinels to treat a NIL child of a node x as an ordinary node whose parent is x. (Sentinel is a dummy object that help simplify boundary conditions.)
A minor modification of Tree-Delete: after splicing out a node, it calls RB-Delete-Fixup to change colors and performs rotations to restore red-black properties.
Replace “NIL” by “nil[T]”
line 7: p[x] p[y] is performed unconditionally
RB-Delete-Fixup(T, x)
UNC Chapel Hill
M. C. Lin

11. UNC Chapel Hill
M. C. Lin

12. Example of RB-Delete-Fixup
UNC Chapel Hill
M. C. Lin

13. UNC Chapel Hill
M. C. Lin

14. Analysis on RB-Delete
Total procedure without RB-Delete-Fixup takes O(lg n) time
RB-Delete-Fixup
cases 1, 3 & 4 takes constant time
case 2 takes at most O(lg n) time
 RB-Delete takes O(lg n) time
 All RB-tree operations take O(lg n) time
UNC Chapel Hill
M. C. Lin