1. Programming Recursion “To Iterate is Human, to Recurse, Divine” -- L. Peter Deutsch

2. Functions – Reminder A group of variables and statements that is assigned a name A sub-program  A function may call other functions. return-type name(arg_type1 arg_name1, arg_type2 arg_name2, …) { function body; return value; }

3. Return Statement - Reminder Return causes the execution of the function to terminate and returns a value to the caller.

4. Scope of Variables - Reminder A variable declared within a function is unrelated to variables declared elsewhere A function cannot directly access variables that are declared in other functions

5. Function Declaration - Reminder We can call a function from the point in the file in which the function has been declared, until the end of the file. return-type name(arg_type1 arg_name1, arg_type2 arg_name2, …);

6. Recursive Function A function defined in terms of itself is called a recursive function. return_value rec_func(type arg1, type arg2, …) { … rec_func(…); … }

7. As we saw, n! = 1*2*3*… *(n-1)*n Thus, we can also define factorial the following way:  0! = 1  n! = n*(n-1)! for n>0 (n-1)!* n Factorial

8. A Recursive Definition C functions can call themselves!  However, not with the same parameters (why?) Some functions can be defined using smaller occurrences of themselves. Such a definition is called a “recursive definition” of a function. Every recursive function must have a “boundary condition”. The function stops calling itself when it is satisfied.  Why is this necessary?

9. Example - Factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } int fact_itr(int n) { int fact = 1; while (n >= 1) { fact *= n; --n; } return fact; }

10. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← ?

11. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ?

12. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← ?

13. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ?

14. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← ?

15. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← 2 * ?

16. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← 2 * ? n ← 1 return ← ?

17. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← 2 * ? n ← 1 return ← 1 * ?

18. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← 2 * ? n ← 1 return ← 1 * ? n ← 0 return ← ?

19. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← 2 * ? n ← 1 return ← 1 * ? n ← 0 return ← 1

20. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← 2 * ? n ← 1 return ← 1 * 1

21. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * ? n ← 2 return ← 2 * 1

22. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * ? n ← 3 return ← 3 * 2

23. Recursive factorial int fact_rec(int n) { if (n == 0) return 1; return n * fact_rec(n-1); } n ← 4 return ← 4 * 6

24. Another Example void print_nums(int num) { if (num == 0) return; printf("%d ", num); print_nums(num - 1); printf("%d ", num); } print call func print 3 2 1 return 0 3 2 1 1 2 3

25. What Does It Do? void foo() { int num; scanf("%d", &num); if (num < 0) return; foo(); printf("%d ", num); }

26. Reverse Print void reverse_print() { int num; scanf("%d", &num); if (num < 0) return; reverse_print(); printf("%d ", num); } Reads an unbounded series of numbers from the user and prints them in reverse order.

27. Power x y = x*x*…*x Recursive definitions (assume non-negative y): 1,y = 0 x y = x*x y-1,y odd (x y/2 ) 2,y even y times

28. rec_pow int rec_pow(int x, int y) { if (y == 0) return 1; if (y % 2 != 0) return x * rec_pow(x, y - 1); else return square(rec_pow(x, y / 2)); }

29. The Three Rules of Recursion 1.Know when to stop. 2.Decide how to take one step. 3.Break the problem down into that step plus a smaller problem.

30. Exercise Write a program that receives two non- negative integers and computes their product recursively. Hint: Notice that the product a*b is actually a+a+…+a (b times).

31. Solution int rec_mul(int a, int b) { /* base condition – a * 0 = 0 */ if (b == 0) return 0; /* save a and call recursively */ return a + rec_mul(a, b-1); }

32. Exercise Given the following iterative version of sum- of-digits calculation Find the recursive definition of this function (don’t forget the base case!) int sum_digits(int n) { int sum = 0; while (n > 0) { sum += n%10; n = n/10; } return sum; }

33. Solution int sum_digit_rec(int n) { if (n == 0) return 0; return n % 10 + sum_digit_rec(n / 10); }

34. More uses Recursion is a general approach to programming functions Its uses are not confined to calculating mathematical expressions! For example – max_rec.c

35. Step By Step int max_rec(int *arr, int size) { int rest_max; if (size == 1) return *arr; rest_max = max_rec(arr + 1, size - 1); if (*arr > rest_max) return *arr; else return rest_max; } 1472 arr: max_rec(, 4) rest_max = max_rec(, 3) rest_max = max_rec(, 2) rest_max = max_rec(, 1) rest_max = 2 7 7

36. What Does It Do? char *rec_func(char *str, char c) { if (*str == '\0') return NULL; if (*str == c) return str; return rec_func(++str, c); }

37. Solution A recursive implementation of strchr

38. Exercise Give a recursive implementation to int strcmp(const char *s, const char *t); The function compares two strings and return:  0 – if s == t  negative – if s < t  positive – if s > t Write a program that accepts two strings from the user and compare them using the above function

39. Solution int strcmp(cons char *s, const char *t) { if (*s != *t) return *s - *t; if (*s == '\0') return 0; return strcmp(++s, ++t); }

40. Towers of Hanoi Initial Setup:  Tower A contains n disks of different sizes.  Disks can only go on top of smaller disks (or directly on the board). ACB

41. Towers of Hanoi Objective:  Move all n disks from tower A to tower C. ACB

42. Towers of Hanoi Moves:  Take the top disk of a tower and move the disk to another tower.  No disk may be put on top of a smaller disk. ACB

43. Towers of Hanoi Move tower with n disks is equivalent to: 1.Move tower with n-1 disks ACB

44. Towers of Hanoi Move tower with n disks is equivalent to: 2. Move a single disk ACB

45. Towers of Hanoi Move tower with n disks is equivalent to: 3. Move tower with n-1 disks ACB

46. Towers of Hanoi - C void move_disk(char from, char to) { printf("move disk from %c to %c\n", from, to); } void move_tower(int height, char from, char to, char temp) { if (height == 1) { move_disk(from, to); } else { move_tower(height - 1, from, temp, to); move_disk(from, to); move_tower(height - 1, temp, to, from); }