CS1102 Tut 7 AVL Trees

1
CS1102 Tut 7 AVL Trees

• Max Tan
• tanhuiyi_at_comp.nus.edu.sg
• COM1-01-09 Tel[phone]http//www.comp.nus.edu.s
  g/tanhuiyi

2
Group Assignments

• Group 1 Q2
• Group 2 Q3
• Group 3 Q4
• Group 4 Q1

3
First 15 minutes

• Write an AVL ADT
• What are the public and private methods you need
  ?
• Public operations Add, Delete, Get, Size
• What do you need to do after you add?
• Balance
• When balancing, you need to rotateLeft and
  rotateRight
• To balance, you need to know the height of the
  left and right subtrees

4
First 15 minutes

• Algorithm for Balance, consider the cases
• Case1 Left.height gt right.height

5
First 15 minutes

• Algorithm for Balance, consider the cases
• Case2 Right.height gt left.height

6
First 15 minutes

• So, break it up into two cases and consider them
  seperately
• Write a rotateLeft and a rotateRight method to
  further simplify the problem!

7
First 15 minutes

• In this case we simply rotate right about x
• rotateRight(x)

8
First 15 minutes

• In this case just rotating right wont solve the
  problem! (why? See next slide)
• rotateLeft(y)
• rotateRight(x)

9
First 15 minutes
10
First 15 minutes

• Simply rotate about x
• rotateLeft(x)

11
First 15 minutes

• Again simply rotating about x wont solve this
• rotateRight(z)
• rotateLeft(x)

12
Question 1

• Draw the AVL tree after performing each of the
  following operation consecutively on an initially
  empty binary search tree insert 8, insert 6,
  insert 12, insert 3, insert 10, insert 9, delete
  12, delete 8, insert 7, and insert 8.

13
Question 1

• After we insert 8, 6, 12, 3, 10, 9, the tree is
  as below

Violation!
14
Question 1

• Do a rotate right at node 12, and obtain the AVL
  tree as below

15
Question 1

• delete 12

16
Question 1

• delete 8

17
Question 1

• Insert 7 and 8

18
Question 1

• Violation

Violation
CASE INSERT INSIDE
1
3
19
Question 1

• Violation Case Insert Inside (2 rotations)

Violation
20
Question 1

• Violation Case Insert Inside (2 rotations)

Violation
21
Question 2
22
Question 2

• Step 1 83 is removed, Violation at 95, so rotate
  left at 95

23
Question 2

• Step 1 83 is removed, Violation at 95, so rotate
  left at 95 (result)

24
Question 2

• Step 2 Equivalent to insert inside, so we need
  to rotate left about 34 first

25
Question 2

• Step 2 Equivalent to insert inside, so we need
  to rotate left about 34 first (result)

26
Question 2

• Step 3 Violation at 67, so rotate right about 67

27
Question 2

• Step 3 Violation at 67, so rotate right about 67
  (result)

28
Question 2

• An AVL tree may no longer be balanced when we
  delete an item from it. How many sub-trees will
  be re-balanced in the worst case when an item is
  deleted from an AVL tree? Give an example to
  support your argument.

29
Question 2

• Consider cases where rotation is needed

30
Question 2
Delete 3
31
Question 2
32
Question 3

• Construct minimal AVL trees of height equals to
  1, 2, 3, 4, and 5. What is the number of nodes in
  a minimal AVL tree of height 6? How many
  different shapes of a minimal AVL tree of height
  h can have?

33
Question 3

• Can you see the recursion?

H 1
H 2
H 3
H 4
H 5
H6 has1 12 7 20 nodes
34
Question 3

• Can you see the recursion?

MinNodes(h) 1 MinNodes(h-1)
MinNodes(h-2)
35
Question 3

• How about the number of shapes?

Two ways to permuate 2 shapes
36
Question 3

• How about the number of shapes?

What about the permutations of the items in the
subtrees ??? Recursion!
37
Question 3

• How about the number of shapes?

NumShapes(Node n) 2 NumShapes(n.left)
NumShapes(n.right)
38
Question 4

• Given an AVL tree with n integer items and two
  integers a and b, where a and b can be any
  integers with a lt b,
• (a) Implement an algorithm to count the number of
  nodes within the range a, b
• (b) What is the time complexity of your
  algorithm. Explain your answer.

39
Question 4
67
rangeCount(curr, 48, 95)
48
96
34
55
95
98
25
42
46
12
40
Question 4
Within the range, search both subtrees
67
rangeCount(curr, 48, 95)
48
96
34
55
95
98
25
42
46
12
41
Question 4
67
Within the range, search both subtrees
48
96
rangeCount(curr, 48, 95)
34
55
95
98
25
42
46
12
42
Question 4
67
Within the range, search both subtrees
48
96
rangeCount(curr, 48, 95)
34
55
95
98
25
42
46
12
43
Question 4
67
Smaller then 48, search right subtree
48
96
rangeCount(curr, 48, 95)
34
55
95
98
25
42
46
12
44
Question 4
67
Smaller then 48, search right subtree
48
96
rangeCount(curr, 48, 95)
34
55
95
98
25
42
46
12
45
Question 4
67
Within range, return size
48
96
rangeCount(curr, 48, 95)
34
55
95
98
25
42
46
12
46
Question 4
67
Within range, return size
48
96
rangeCount(curr, 48, 95)
34
55
95
98
25
42
46
12
47
Question 4
Larger then 95, search left subtree
67
rangeCount(curr, 48, 95)
48
96
34
55
95
98
25
42
46
12
48
Question 4
Larger then 95, search left subtree
67
rangeCount(curr, 48, 95)
48
96
34
55
95
98
25
42
46
12