22C:21 Discrete Structures

1
22C21 Discrete Structures

• Problem 7.2. System Stacks on Recursion

2
Stacks just to refresh your memory

• List
• FIFO push/pop
• There is a built-in Stack ADT in java.
• However, I used my own implementation
• because I can
• yes, its THAT easy
• and a little personal touch is always better

3
The myStack class

• // Instance variables
• private static int stack
• private static int top0
• private static int MAX100
• // Default constructor
• public myStack()
• stacknew intMAX
• top-1

4
Push(element)

• public void push(int b)
• if(topgtMAX)
• System.out.println("Overflow!")
• return
• stacktopb
• this.print()

5
Pop()

• public int pop()
• if(toplt0)
• System.out.println("Underflow!")
• return(-1)
• int temptop--
• this.print()
• return stacktemp

6
Fibonacci Numbers

• F0 1
• F1 1
• Fn Fn-1 Fn-2 for n 2
• So
• F2 2, F3 3, F4 5, F5 8
• and so on

7
Modified Fibonacci Code

• public static BigInteger fibonacci(int n)
• System.out.println("fibonacci("n") has been
  called...")
• SysStack.push(n)
• if((n 0) (n 1))
• System.out.println("fibonacci("n")
  returning value...")
• int garbage SysStack.pop()
• return new BigInteger ("1")
• else
• BigInteger number1 fibonacci(n-1)
• BigInteger number2 fibonacci(n-2)
• System.out.println("fibonacci("n") returning
  value...")
• int moregrabageSysStack.pop()
• return number1.add(number2)

8
Output for n3

• fibonacci(3) has been called...
• Current Stack
• --
• 3
• --
• fibonacci(2) has been called...
• Current Stack
• --
• 2
• 3
• --

9
Output for n3 contd.

• fibonacci(1) has been called...
• Current Stack
• --
• 1
• 2
• 3
• --
• fibonacci(1) returning value...
• Current Stack
• --
• 2
• 3
• --

10
Output for n3 contd.

• fibonacci(0) has been called...
• Current Stack
• --
• 0
• 2
• 3
• --
• fibonacci(0) returning value...
• Current Stack
• --
• 2
• 3
• --

11
Output for n3 contd.

• fibonacci(2) returning value...
• Current Stack
• --
• 3
• --
• fibonacci(1) has been called...
• Current Stack
• --
• 1
• 3
• --

12
Output for n3 contd.

• fibonacci(1) returning value...
• Current Stack
• --
• 3
• --
• fibonacci(3) returning value...
• Current Stack
• --
• --

should be empty at the end
13
What happens for n5?
Fibonacci(5)
5
4
3
5
System Stack
Empty
14
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
4
5
System Stack
15
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
3
4
5
System Stack
16
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
2
Fibonacci(2)
3
4
5
System Stack
17
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
1
2
Fibonacci(2)
3
4
1
0
Fibonacci(1)
5
System Stack
18
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
2
Fibonacci(2)
3
4
0
5
System Stack
19
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
0
2
Fibonacci(2)
3
4
0
Fibonacci(0)
5
System Stack
20
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
2
Fibonacci(2)
3
4
5
System Stack
21
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
1
3
4
5
System Stack
22
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
1
1
Fibonacci(1)
3
4
5
System Stack
23
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
Fibonacci(3)
3
4
5
System Stack
24
What happens for n5?
Fibonacci(5)
5
4
3
Fibonacci(4)
2
4
5
System Stack
25
What happens for n5?
Fibonacci(5)
5
Fibonacci(4)
2
Fibonacci(2)
2
4
5
System Stack
26
What happens for n5?
Fibonacci(5)
5
Fibonacci(4)
2
Fibonacci(2)
1
1
0
Fibonacci(1)
2
4
5
System Stack
27
What happens for n5?
Fibonacci(5)
5
Fibonacci(4)
2
Fibonacci(2)
0
2
4
5
System Stack
28
What happens for n5?
Fibonacci(5)
5
Fibonacci(4)
2
Fibonacci(2)
0
0
Fibonacci(0)
2
4
5
System Stack
29
What happens for n5?
Fibonacci(5)
5
Fibonacci(4)
2
Fibonacci(2)
2
4
5
System Stack
30
What happens for n5?
Fibonacci(5)
5
Fibonacci(4)
4
5
System Stack
31
What happens for n5?
Fibonacci(5)
5
3
5
System Stack
32
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
3
5
System Stack
33
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
Fibonacci(2)
2
3
5
System Stack
34
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
Fibonacci(2)
1
1
0
Fibonacci(1)
2
3
5
System Stack
35
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
Fibonacci(2)
0
2
3
5
System Stack
36
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
Fibonacci(2)
0
0
Fibonacci(0)
2
3
5
System Stack
37
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
Fibonacci(2)
2
3
5
System Stack
38
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
1
3
5
System Stack
39
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
1
Fibonacci(1)
1
3
5
System Stack
40
What happens for n5?
Fibonacci(5)
5
Fibonacci(3)
3
3
5
System Stack
41
What happens for n5?
Fibonacci(5)
5
5
System Stack
42
What happens for n5?
System Stack
Empty
43
Note

• Actual system stack holds returned values
• Just the indices has been shown
• Compare with FastFibonacci()

44
Now you know!
45
Quiz 7

• You are given a collection with 6 items. The
  priorities of these items are shown below
• 9 30 25 22 7 50
• 1. Draw all possible max-heaps (as binary trees)
  for this collection of items.
• 2. Start with the max-heap (30, 50, 9, 25, 7,
  22). Perform the following sequence of operations
  on this max-heap
• Insert(45)
• Insert(29)
• Delete()
• Delete()
• Show the max-heap after each operation.
• 3. Answer in a sentence or two Given a max-heap
  can you find the item with 2nd highest priority
  in O(1) time? Explain your answer.
• 4. Answer in a sentence or two Given a max-heap
  can you find the item with smallest priority in
  O(1) time? Explain your answer.