Chapter 1

1. Chapter 1 RECURSION

2. Chapter 1 • Subprogram implementation • Recursion • Designing Recursive Algorithms • Towers of Hanoi • Backtracking • Eight Queens problem

3. Function implementation • Code segment (static part) • Activation record (dynamic part) • Parameters • Function result • Local variables • Return address

4. Function implementation

5. Function implementation #include <iostream.h> int maximum( int x, int y, int z) { int max = x; if ( y > max ) max = y; if ( z > max ) max = z; return max; } int main() { int a, b, c, d1, d2; A1 cout << "Enter three integers: "; A2 cin >> a >> b >> c; A3 d1 = maximum (a, b, c); A4 cout << "Maximum is: " << d1 << endl; A5 d2 = maximum (7, 9, 8); A5 cout << "Maximum is: " << d2 << endl; A7 return 0; }

6. Function implementation d1 = maximum (a, b, c); // a = 7; b = 9; c = 8 A4 Return Address Return value int maximum( int x, int y, int z) { int max = x; if ( y > max ) max = y; if ( z > max ) max = z; return max; } 7 x 9 y 8 z max

7. Function implementation d1 = maximum (a, b, c); // a = 7; b = 9; c = 8 A4 Return Address Return value int maximum( int x, int y, int z) { int max = x; if ( y > max ) max = y; if ( z > max ) max = z; return max; } 7 x 9 y 8 z 7 max

8. Function implementation d1 = maximum (a, b, c); // a = 7; b = 9; c = 8 A4 Return Address Return value int maximum( int x, int y, int z) { int max = x; if ( y > max ) max = y; if ( z > max ) max = z; return max; } 7 x 9 y 8 z 9 max

9. Function implementation d1 = maximum (a, b, c); // a = 7; b = 9; c = 8 A4 Return Address Return value int maximum( int x, int y, int z) { int max = x; if ( y > max ) max = y; if ( z > max ) max = z; return max; } 7 x 9 y 8 z 9 max

10. Function implementation d1 = maximum (a, b, c); // a = 7; b = 9; c = 8 A4 Return Address 9 Return value int maximum( int x, int y, int z) { int max = x; if ( y > max ) max = y; if ( z > max ) max = z; return max; } 7 x 9 y 8 z 9 max

11. Function implementation

12. Function implementation

13. Function implementation • Stack frames: • Each vertical column shows the contents of the stack at a given time • There is no difference between two cases: • when the temporary storage areas pushed on the stack come from different functions, and • when the temporary storage areas pushed on the stack come from repeated occurrences of the same function.

14. Recursion • An object contains itself

15. Recursion

16. Recursion • Recursion is the name for the case when: • A function invokes itself, or • A function invokes a sequence of other functions, one of which eventually invokes the first function again. • In regard to stack frames for function calls, recursion is no different from any other function call. • Stack frames illustrate the storage requirements for recursion. • Separate copies of the variables declared in the function are created for each recursive call.

17. Recursion

18. Recursion

19. Recursion

20. Recursion • In C++, it’s possible for a function to call itself. Functions that do so are called seft-referential or recursive functions. • In some problems, it may be natural to define the problem in terms of the problem itself. • Recursion is useful for problems that can be represented by a simpler version of the same problem. • Example: Factorial 1! = 1; 2! = 2*1 = 2*1! • 3! = 3*2*1=3*2! • …. n! = n*(n-1)! The factorial function is only defined for positive integers. n!=1 if n is equal to 1 n!=n*(n-1)! if n >1

21. Example : #include <iostream.h> #include <iomanip.h> unsigned long Factorial( unsigned long ); int main(){ for ( int i = 0; i <= 10; i++ ) cout << setw( 2 ) << i << "! = " << Factorial( i ) << endl; return 0; } // Recursive definition of function factorial unsigned long Factorial( unsigned long number ){ if (number < 1) // base case return 1; else // recursive case return number * Factorial( number - 1 ); } The output : 0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800

22. Factorial(3) unsigned long Factorial( unsigned long number ){ A1 if (number < 1) // base case A2 return 1; A3 else // recursive case A4 return number * Factorial( number - 1 ); A5} A4 0 1 A4 A4 A4 1 1 1 1 1 A4 A4 A4 A4 A4 2 2 2 2 2 2 A0 A0 A0 A0 A0 A0 A0 3 3 3 3 3 3 3 6

24. Recursion We must always make sure that the recursion bottoms out: • A recursive function must contain at least one non-recursive branch. • The recursive calls must eventually lead to a non-recursive branch. • Recursion is one way to decompose a task into smaller subtasks. • At least one of the subtasks is a smaller example of the same task. • The smallest example of the same task has a non-recursive solution. Example: The factorial function n! = n * (n-1)! and 1! = 1

26. Recursion - Print List

27. Recursion - Print List pTemp = pHead;

28. Recursion - Print List • A list is • empty, or • consists of an element and a sublist, where sublistis a list. pHead pHead

29. Recursion - Print List Algorithm Print(val head<pointer>) Prints Singly Linked List. Pre headpoints to the first element of the list needs to be printed. Post Elements in the list have been printed. Uses recursive function Print. A1.if(head= NULL) // stopping case A1.1.return A2.write (head->data) A3.Print(head->link) // recursive case A4.End Print

30. Recursion - Print List Create List Print(pHead)

31. output 6

32. output 6 10

33. output 6 10 14

34. output 6 10 14 20

35. output 6 10 14 20

36. output 6 10 14 20

37. output 6 10 14 20

38. output 6 10 14 20

39. output 6 10 14 20

40. output 6 10 14 20

41. Recursion - Print List

42. Recursion - Print List

43. Designing Recursive Algorithms

44. Designing Recursive Algorithms

45. Designing Recursive Algorithms

46. Designing Recursive Algorithms

47. Designing Recursive Algorithms

48. Designing Recursive Algorithms

50. Designing Recursive Algorithms