1. CprE 185: Intro to Problem Solving (using C)
Instructor: Alexander Stoytchev

2. Administrative Stuff
HW8 is out (due this 8pm)

3. Recursion (part 2) CprE 185: Intro to Problem Solving
Iowa State University, Ames, IA
Copyright © Alexander Stoytchev

4. Chapter 10 (Recursion)

5. Examples of Recursion

6. [

7. The von Koch Curve and Snowflake[

8. Divide it into three equal parts[

9. Replace the inner third of it with an equilateral triangle[

10. Repeat the first two steps on all lines of the new figure[

11. [

12. Quick Review of the Last Lecture (with more in depth examples)

13. Recursion
Recursion is a fundamental programming technique that can provide an elegant solution certain kinds of problems.
© 2004 Pearson Addison-Wesley. All rights reserved

14. Recursive Thinking
A recursive definition is one which uses the word or concept being defined in the definition itself
When defining an English word, a recursive definition is often not helpful
But in other situations, a recursive definition can be an appropriate way to express a concept
Before applying recursion to programming, it is best to practice thinking recursively
© 2004 Pearson Addison-Wesley. All rights reserved

15. Circular Definitions Debugger – a tool that is used for debugging
© 2004 Pearson Addison-Wesley. All rights reserved

16. Recursive Definitions
Consider the following list of numbers:
24, 88, 40, 37
Such a list can be defined as follows:
A LIST is a: number
or a: number comma LIST
That is, a LIST is defined to be a single number, or a number followed by a comma followed by a LIST
The concept of a LIST is used to define itself
© 2004 Pearson Addison-Wesley. All rights reserved

17. Recursive Definitions
The recursive part of the LIST definition is used several times, terminating with the non-recursive part:
number comma LIST
, 88, 40, 37
, 40, 37
, 37
number 37
© 2004 Pearson Addison-Wesley. All rights reserved

18. Infinite Recursion
All recursive definitions have to have a non- recursive part
If they didn't, there would be no way to terminate the recursive path
Such a definition would cause infinite recursion
This problem is similar to an infinite loop, but the non-terminating "loop" is part of the definition itself
The non-recursive part is often called the base case
© 2004 Pearson Addison-Wesley. All rights reserved

19. Recursive Definitions
N!, for any positive integer N, is defined to be the product of all integers between 1 and N inclusive
This definition can be expressed recursively as:
1! = 1
N! = N * (N-1)!
A factorial is defined in terms of another factorial
Eventually, the base case of 1! is reached
© 2004 Pearson Addison-Wesley. All rights reserved

20. Recursive Definitions5!
5 * 4!
4 * 3!
3 * 2!
2 * 1! 1 2 6 24
120
© 2004 Pearson Addison-Wesley. All rights reserved

21. Example: Factorial_Iterative.c
int factorial(int a) {
int i,result;
result=1;
for(i=1;i<=a;i++)
result=result*i;
return result; }

22. Example: Factorial_Recursive.c
int factorial(int a) {
if(a==1)
return 1;
else
return a*factorial(a-1); }

23. Recursive Execution 6! (6 * 5!) (6 * (5 * 4!)) (6 * (5 * (4 * 3!)))
(6 * (5 * (4 * (3 * 2!))))
(6 * (5 * (4 * (3 * (2 * 1!)))))
(6 * (5 * (4 * (3 * (2 * (1 * 0!))))))
(6 * (5 * (4 * (3 * (2 * (1 * 1))))))
(6 * (5 * (4 * (3 * (2 * 1)))))
(6 * (5 * (4 * (3 * 2))))
(6 * (5 * (4 * 6)))
(6 * (5 * 24))
(6 * 120)
720

24. Recursive Programming
A function in C can invoke itself; if set up that way, it is called a recursive function
The code of a recursive function must be structured to handle both the base case and the recursive case
Each call to the function sets up a new execution environment, with new parameters and local variables
As with any function call, when the function completes, control returns to the function that invoked it (which may be an earlier invocation of itself)

25. Recursive Programming
Consider the problem of computing the sum of all the numbers between 1 and any positive integer N
This problem can be recursively defined as:
© 2004 Pearson Addison-Wesley. All rights reserved

26. Example: Sum_Iterative.c
int sum(int a) {
int i,result;
result=0;
for(i=1;i<=a;i++)
result=result+i;
return result; }

27. Example: Sum_Recursive.c
int sum(int a) {
if(a==1)
return 1;
else
return a+sum(a-1); }

28. Recursive Programming
// This function returns the sum of 1 to num
int sum (int num) {
int result;
if (num == 1)
result = 1;
else
result = num + sum (n-1);
return result; }
© 2004 Pearson Addison-Wesley. All rights reserved

29. Recursive Control Flow
In Recursive calls methods can call themselves, but typically with different arguments each time
sum(1);
sum(1)
sum(2)
return 1;

30. Recursive Control Flow
In Recursive calls methods can call themselves, but typically with different arguments each time
main()
sum(3)
sum(2)
sum(1)
sum(1);
sum(2);
sum(3);
return 1;

31. Recursive Programming
main
sum
sum(3)
sum(1)
sum(2)
result = 1
result = 3
result = 6
© 2004 Pearson Addison-Wesley. All rights reserved

32. Recursion: Fibonacci Numbers
The sequence: {0,1,1,2,3,5,8,13,...}

33. Fibonacci Sequence #include<stdio.h> #include<stdlib.h>
int fib(int n) {
int ret;
if (n < 2) {
ret = 1; }
else{
ret= fib(n-1) + fib(n-2);
return ret;
int main()
int i;
for(i=0;i<10;i++)
printf("%d ", fib(i));
system("pause");

34. Mathematical notation v.s. C code
int fib(int n)
{ if(n <= 1)
return n; //base case else
return fib(n-1) + fib(n-2); }

35. Execution Trace

36. Mystery Recursion void mystery1(int a, int b)
{ if (a <= b) { int m = (a + b) / 2; printf(“%d “, m); mystery1(a, m-1); mystery1(m+1, b); }
} int main() { mystery1(0, 5); system(“pause”); }

37. Think of recursion as a tree …

38. … an upside down tree

40. (a=0, b=5) (a=0, b=1) (a=3, b=5) (a=0, b=-1) (a=1, b=1) (a=3, b=3)

41. 2 (a=0, b=5) m=(a+b)/2 => print 2 (a=0, b=1) (a=3, b=5) (a=0, b=-1)

42. 2 4 1 3 5 (a=0, b=5) (a=0, b=1) (a=3, b=5) (a=0, b=-1) (a=1, b=1)4
(a=0, b=-1)
(a=1, b=1)
(a=3, b=3)
(a=5, b=5) 1 3 5
(a=1, b=0)
(a=2, b=1)
(a=3, b=2)
(a=4, b=3)
(a=5, b=4)
(a=6, b=5)

43. Traversal Order 2 4 1 3 5 (a=0, b=5) (a=0, b=1) (a=3, b=5) (a=0, b=-1)4
(a=0, b=-1)
(a=1, b=1)
(a=3, b=3)
(a=5, b=5) 1 3 5
(a=1, b=0)
(a=2, b=1)
(a=3, b=2)
(a=4, b=3)
(a=5, b=4)
(a=6, b=5)

44. Questions?

45. Stack Animation
animations/tsang/html/STACK/stack1024.html

46. Memory Organization [

47. The stack during a recursive call to factorial

48. The stack during a recursive call to gcd[

49. Recursive Programming
Note that just because we can use recursion to solve a problem, doesn't mean we should
For instance, we usually would not use recursion to solve the sum of 1 to N problem, because the iterative version is easier to understand
However, for some problems, recursion provides an elegant solution, often cleaner than an iterative version
You must carefully decide whether recursion is the correct technique for any problem

50. Maze Traversal We can use recursion to find a path through a maze
From each location, we can search in each direction
Recursion keeps track of the path through the maze
The base case is an invalid move or reaching the final destination

51. Traversing a maze

56. Towers of Hanoi
The Towers of Hanoi is a puzzle made up of three vertical pegs and several disks that slide on the pegs
The disks are of varying size, initially placed on one peg with the largest disk on the bottom with increasingly smaller ones on top
The goal is to move all of the disks from one peg to another under the following rules:
We can move only one disk at a time
We cannot move a larger disk on top of a smaller one

57. Original Configuration
Towers of Hanoi
Original Configuration
Move 1
Move 2
Move 3

58. Towers of Hanoi
Move 4
Move 5
Move 6
Move 7 (done)

59. Animation of the Towers of Hanoi
WangApr01/RootWang.html

60. Towers of Hanoi
An iterative solution to the Towers of Hanoi is quite complex
A recursive solution is much shorter and more elegant

61. Indirect Recursion
A method invoking itself is considered to be direct recursion
A method could invoke another method, which invokes another, etc., until eventually the original method is invoked again
For example, method m1 could invoke m2, which invokes m3, which in turn invokes m1 again
This is called indirect recursion, and requires all the same care as direct recursion
It is often more difficult to trace and debug

62. Indirect Recursion m1 m2 m3

63. THE END