1. Week 7 - Friday
CS221

2. Last time
What did we talk about last time?
BST deletion
2-3 trees

3. Questions?

4. Project 2

5. 2-3 tree practice Add the following keys to a 2-3 tree: 62 11 32 7 4524 88 25 28 90

6. Red-black trees
Student Lecture

7. Red-black trees

8. An interlude in a formicary
One hundred ants are walking along a meter long stick
Each ant is traveling either to the left or the right with a constant speed of 1 meter per minute
When two ants meet, they bounce off each other and reverse direction
When an ant reaches an end of the stick, it falls off
Will all the ants fall off?
What is the longest amount of time that you would need to wait to guarantee that all ants have fallen off?

9. "As if"
On the previous slide, we can look at the problem "as if" ants were passing through each other with no effect
This idea of looking at a problem as if it is something else can make solving it easier
Coding up a 2-3 tree is annoying (but possible)
By creating a data structure that is (somehow) equivalent, we can get the job done in an easier way
Sometimes the way we implement an algorithm and the way we analyze it are different

10. Red-black trees A red-black tree is a form of binary search tree
Each node looks like a regular BST node, with one additional piece of information: color
A node can either be red or black
Null values are considered black
The color allows us to simulate a 2-3 tree
We can think of a red node is actually part of a 3 node with its parent

11. Red-black tree definition
A red-black tree is a BST with red and black nodes and the following properties:
Red nodes lean left from their parents
No node has two red children
The tree has perfect black balance
In other words, every path from the root to a null has the same number of black nodes on the way
The length of this path is called the black height of the tree
The book describes the link as having a color (which is probably easier to think about), but the color has to be stored in the node

12. 2-3 tree implementation How do we implement a 2-3 tree?
Answer: We don't.
It is (of course) possible, but it involves having weird 2-nodes and 3-nodes that inherit from some abstract node class or interface
It's a huge pain
Note that 2-3 trees are essentially a special case of a B-tree, and someone does have to implement those
Instead, we use red-black trees which are structurally the same as 2-3 trees (if you squint)

13. 2-3 tree mapping to a red-black treeJ A C H L R P S X M E J A C H L R P S X

14. Building the tree
We can do an insertion with a red-black tree using a series of rotations and recolors
We do a regular BST insert
Then, we work back up the tree as the recursion unwinds
If the right child is red, we rotate the current node left
If the left child is red and the left child of the left child is red, we rotate the current node right
If both children are red, we recolor them black and the current node red
You have to do all these checks, in order!
Multiple rotations can happen
It doesn't make sense to have a red root, so we always color the root black after the insert

15. We perform a left rotation when the right child is redY X B A C
Current Y X B A C
Current
We perform a left rotation when the right child is red

16. Right rotation Z Y B A D
Current X C Z Y B A D
Current X C
We perform a right rotation when the left child is red and its left child is red

17. Recolor Y X B A D
Current Z C Y X B A D
Current Z C
We recolor both children and the current node when both children are red

18. Red-black tree practice
Add the following keys to a red-black tree: 62 11 32 7 45 24 88 25 28 90

19. Analysis of red-black trees
The height of a red-black tree is no more than 2 log n
Find is Θ(height), so find is Θ(log n)
Since we only have to go down that path and back up to insert, insert is Θ(log n)
Delete in red-black trees is messy, but it is also actually Θ(log n)

20. AVL Trees

21. AVL trees Another approach to balancing is the AVL tree
Named for its inventors G.M. Adelson-Velskii and E.M. Landis
Invented in 1962
An AVL tree is better balanced than a red-black tree, but it takes more work to keep such a good balance
It's faster for finds (because of the better balance)
It's slower for inserts and deletes
Like a red-black tree, all operations are Θ(log n)

22. AVL definition An AVL tree is a binary search tree where
The left and right subtrees of the root have heights that differ by at most one
The left and right subtrees are also AVL trees

23. AVL tree? 10 6 14 1 9 17 2 7 15

24. Balance factor
The balance factor of a tree (or subtree) is the height of its right subtree minus the height of its left
In an AVL tree, the balance factor of every subtree is -1, 0, or +1
What’s the balance factor of this tree? 6 1 9 2

25. How do we build an AVL tree?
Carefully.
Every time we do an add, we have to make sure that the tree is still balanced
There are 4 cases
Left Left
Right Right
Left Right
Right Left

26. Left Left 3 2 1 B A C D 3 2 1 B A C D
Right Rotation

27. Right Right 2 1 B A C D 3 3 2 1 B A C D
Left Rotation

28. Left Rotation + Right Rotation
Left Right 3 2 1 B A C D 3 2 1 B A C D
Left Rotation + Right Rotation

29. Right Rotation + Left Rotation
Right Left 2 1 B A C D 3 3 2 1 B A C D
Right Rotation + Left Rotation

30. Let’s make an AVL tree!
Let’s just add these numbers in order:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
What about these?
7, 24, 92, 32, 2, 57, 67, 84, 66, 75

31. Balancing a Tree by Construction

32. How to make a balanced tree
What if we knew ahead of time which numbers we were going to put into a tree?
How could we make sure the tree is balanced?
Answer:
Sort the numbers
Recursively add the numbers such that the tree stays balanced

33. Balanced insertion
Write a recursive method that adds a sorted array of data such that the tree stays balanced
Assume that an add(int key) method exists
Use the usual convention that start is the beginning of a range and end is the location after the last legal element in the range
public void balance(int[] data, int start, int end )

34. DSW algorithm
Takes a potentially unbalanced tree and turns it into a balanced tree
Step 1
Turn the tree into a degenerate tree (backbone or vine) by right rotating any nodes with left children
Step 2
Turn the degenerate tree into a balanced tree by doing sequences of left rotations starting at the root
The analysis is not obvious, but it takes O(n) total rotations

35. Which method is best?
How much time does it take to insert n items with:
Red-black or AVL tree
Balanced insertion method
Unbalanced insertion + DSW rebalance
How much space does it take to insert n items with:

36. Self-Adjusting Trees

37. Self-Adjusting Trees
Balanced binary trees ensure that accessing any element takes O(log n) time
But, maybe only a few elements are accessed repeatedly
In this case, it may make more sense to restructure the tree so that these elements are closer to the root

38. Restructuring strategies
Single rotation
Rotate a child about its parent if an element in a child is accessed, unless it's the root
Moving to the root
Repeat the child-parent rotation until the element being accessed is the root

39. Splaying
A concept called splaying takes the rotate to the root idea further, doing special rotations depending on the relationship of the child R with its parent Q and grandparent P
Cases:
Node R's parent is the root
Do a single rotation to make R the root
Node R is the left child of Q and Q is the left child of P (respectively right and right)
Rotate Q around P
Rotate R around Q
Node R is the left child of Q and Q is the right child of P (respectively right and left)
Rotate R around P

40. Danger, Will Robinson!
We are not going deeply into self-organizing trees
It's an interesting concept, and rigorous mathematical analysis shows that splay trees should perform well, in terms of Big O bounds
In practice, however, they usually do not
An AVL tree or a red-black tree is generally the better choice
If you come upon some special problem where you repeatedly access a particular element most of the time, only then should you consider splay trees

41. Mid-semester Evaluations

42. Upcoming

43. Next time…
Hash tables
Hashing functions

44. Reminders
Read section 3.4
Finish Project 2
Due tonight by midnight