1. CS200: Algorithms Analysis

2. RED BLACK TREES
Red black trees maintain a balanced tree structure so that height of tree = θ(log n).
Idea is to add a color(r, b) to a BST.
RED-BLACK PROPERTIES
1. every node is r or b.
2. every leaf (nil) is black.
3. if a node is red then both its children are black => there can’t be two successive red nodes in a tree.
4. every simple path from a node to a leaf contains the same # of b nodes.
5. root is black (simplifying assumption for inserts).

3. Black-Height of a node : bh(x) = # of b nodes on path to leaf not counting x.
Show an example tree, label with r or b, and h

4. Example RB Tree
Label with bh (4. every simple path from a node to a leaf contains the same # of b nodes and bh(x) = # of b nodes on path to leaf not counting x.)
A height-h node has bh >= h/2. Why?

15. Nodes

16. Inductive Proof of Lemma 13.1 (Go over on own)
Prove: a red-black tree with n internal nodes has h <=2log(n+1). Proof is based on claim that a sub-tree rooted at node x contains >= 2bh(x)-1 internal nodes.
Proof is by induction on height h of node x.
basis: h=0 => x is nil; black leaf
=> 2bh(x)-1 = 20-1 = 0
inductive step: x is an internal node with 2 children.

17. A sub-tree rooted at x has at least 2bh(x)-1 internal nodes.
bh of children of x have at least bh(x) -1 (if child is black).
 subtree has at least
(2bh(x)-1 -1) + (2bh(x)-1 -1) +1 internal nodes.
(2bh(x)-1 -1) + (2bh(x)-1 -1) +1 = 2(2bh(x)-1 -1) +1= 2bh(x)-1 internal nodes.
Let h be the height of tree then n >=2bh(root) -1 >= 2h/2 -1 (using property 3 of red-black trees).
n >= 2h/2-1
n+1>= 2h/2
log(n+1)>=h/2  2log(n+1)>=h

18. Query Operations
Corollary. The queries SEARCH, MIN, MAX, SUCCESSOR, and PREDECESSOR all run in O(lg n) time on a red-black tree with n nodes.

19. Modifying Operations
The operations INSERT and DELETE cause modifications to the red-black tree: • the operation itself, • color changes, • restructuring the links of the tree via “rotations”.

20. Rotations

21. Insertions

22. Insertions

23. Insertions

24. Insertions

25. Insertions

27. Pseudo Code

28. Graphical Notation

29. Case 1

31. Case 2

32. Case 3

34. Analysis :
Given line 1., what property of RB trees can be violated by above code?

35. Summary
RB Tree properties
Balancing algorithms (rotations)
analysis