1. 22C:21 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, it’s THAT easy
… and a little personal touch is always better

3. The myStack class // Instance variables private static int[] stack;
private static int top=0;
private static int MAX=100;
// Default constructor
public myStack() {
stack=new int[MAX];
top=-1; }

4. Push(element) public void push(int b) { if(top>=MAX)
System.out.println("Overflow!");
return; }
stack[++top]=b;
this.print();

5. Pop() public int pop() { if(top<0)
System.out.println("Underflow!");
return(-1); }
int temp=top--;
this.print();
return stack[temp];

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);
int moregrabage=SysStack.pop();
return number1.add(number2);

8. Output for n=3 fibonacci(3) has been called... Current Stack: -- 32

9. Output for n=3 contd. fibonacci(1) has been called... Current Stack:-- 1 2 3
fibonacci(1) returning value...

10. Output for n=3 contd. fibonacci(0) has been called... Current Stack:-- 2 3
fibonacci(0) returning value...

11. Output for n=3 contd. fibonacci(2) returning value... Current Stack:-- 3
fibonacci(1) has been called... 1

12. … should be empty at the end
Output for n=3 contd.
fibonacci(1) returning value...
Current Stack: -- 3
fibonacci(3) returning value...
… should be empty at the end

13. What happens for n=5? 5
Fibonacci(5) 4 3 5
System Stack
Empty

14. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 4 5
System Stack

15. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)1 3 4 5
System Stack

16. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)1 2
Fibonacci(2) 3 1 4 5
System Stack

17. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)1 2 1 2
Fibonacci(2) 3 4
Fibonacci(1) 1 5
System Stack

18. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)1 2
Fibonacci(2) 3 4 5
System Stack

19. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)2 1 2
Fibonacci(2) 3 4
Fibonacci(0) 5
System Stack

20. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)1 2
Fibonacci(2) 3 4 5
System Stack

21. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)1 3 4 5
System Stack

22. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)1 1
Fibonacci(1) 3 4 5
System Stack

23. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3)
System Stack

24. What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 2 4 5
System Stack

25. What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 12 4 5
System Stack

26. What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 1
Fibonacci(1) 2 4 5
System Stack

27. What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 22 4 5
System Stack

28. What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2)
Fibonacci(0) 2 4 5
System Stack

29. What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 2
System Stack

30. What happens for n=5? 5
Fibonacci(5) 4 3
Fibonacci(4) 4 5
System Stack

31. What happens for n=5? 5
Fibonacci(5) 3 5
System Stack

32. What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 3 5
System Stack

33. What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 12 3 5
System Stack

34. What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 12 3 5
System Stack

35. What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 22 3 5
System Stack

36. What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2)
Fibonacci(0) 2 3 5
System Stack

37. What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 2
System Stack

38. What happens for n=5? 5
Fibonacci(5)
Fibonacci(3) 3 1 3 5
System Stack

39. What happens for n=5? Fibonacci(5) 5 Fibonacci(3) 3 1 Fibonacci(1) 1 3
System Stack

40. What happens for n=5? 5
Fibonacci(5)
Fibonacci(3) 3 3 5
System Stack

41. What happens for n=5? 5
Fibonacci(5) 5
System Stack

42. What happens for n=5?
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:
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()
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.