1. Topic 6 Recursion

2. Program Development and Design Using C++, Third Edition
Objectives
You should be able to:
Mentally execute recursive functions
Write and understand programs which contain recursive functions
Convert a while loop into a call to a recursive function
Program Development and Design Using C++, Third Edition

3. Recursion (Chapter 6 in textbook-section 6.6)
A function that is defined in terms of itself is called recursive
e.g. f(x)=
if x=0
2f(x-1) + x2 if x>0
Program Development and Design Using C++, Third Edition

4. Program Development and Design Using C++, Third Edition
Example of fun()
2*fun(2)+3*3
int fun(int x) {
if (x==0)
return 0;
else
return 2*fun(x-1)+x*x; }
2*(2*fun(1)+2*2)+3*3
2*(2*(2*fun(0)+1*1)+2*2)+3*3
Recursive call for fun(3):
=2*(2*(2*0+1)+4)+9
Returned value of fun(3): 21
Program Development and Design Using C++, Third Edition

5. Program Development and Design Using C++, Third Edition
Recursion
Recursive functions
Functions that call themselves
Can only solve a base case
If not base case
Break problem into smaller problem(s)
Launch new copy of function to work on the smaller problem (recursive call/recursive step)
Slowly converges towards base case
Function makes call to itself inside the return statement
Eventually base case gets solved
Answer works way back up, solves entire problem
Program Development and Design Using C++, Third Edition

6. Recursion: How the Computation Is Performed
C++ allocates new memory locations for function parameters and local variables as each function is called
Allocation made dynamically in memory stack
Memory stack: Area of memory used for rapidly storing and retrieving data
Last-in/first-out
Program Development and Design Using C++, Third Edition

7. Program Development and Design Using C++, Third Edition
Rules of Recursion
Two fundamental rules of recursion:
Base cases. You must always have some base cases which can be solved without recursion.
Making Progress. For the cases that are to be solved recursively, the recursive call must always be to a case that makes progress toward a base case.
Program Development and Design Using C++, Third Edition

8. An incorrect recursive function
int sum(int n) {
return n+sum(n-1); }
This function does not have a base case
Program Development and Design Using C++, Third Edition

9. A correct recursive function
int sum(int n) {
if (n==0)
return 0;
else
return n+sum(n-1); }
void main()
cout<<sum(3);
Program Development and Design Using C++, Third Edition

10. How the computation is performed: The stack
int sum(int n) {
if (n==0)
return 0;
else
return n+sum(n-1); }
void main()
cout<<sum(3);
sum(0)
n=0
sum(1)
n=1
1+0=1
sum(2)
n=2
2+1=3
sum(3)
n=3
3+3=6
main()
Output: 6
Program Development and Design Using C++, Third Edition

11. A non-terminating recursive program
int Bad(int n) {
if (n==0)
return 0;
else
return Bad(n/3+1)+n-1; }
Program Development and Design Using C++, Third Edition

12. Program Development and Design Using C++, Third Edition
Factorial example
factorial
n! = n * ( n – 1 ) * ( n – 2 ) * … * 1
Recursive relationship ( n! = n * ( n – 1 )! )
5! = 5 * 4!
4! = 4 * 3!…
Base case (1! = 0! = 1)
Program Development and Design Using C++, Third Edition

13. Recursive factorial function
int factorial(int n) {
if ( n==0 ) // base case
return 1;
else // recursive step
return n*factorial(n - 1); }
Program Development and Design Using C++, Third Edition

14. Testing the factorial function
void main() {
int i;
// Loop 5 times. During each iteration, calculate
// factorial( i ) and display result.
for ( i = 0; i <= 5; i++ )
cout << i << "! = "<< factorial( i ) << endl; }
Output:
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
Program Development and Design Using C++, Third Edition

15. Drawing a triangle recursively
void Triangle(int n) {
//draws a right-angled triangle
if (n!=0)
line(n,'*'); //function Line from previous chapter
cout<<endl;
Triangle(n-1); }
Program Development and Design Using C++, Third Edition

16. Testing the function Triangle
#include <iostream>
using namespace std;
void line(int x, char ch) ; //function prototypes
void Triangle(int n);
void line(int x, char ch) //function definitions {
int i;
for(i=1; i<=x; i++)
cout << ch ; }
void Triangle(int n)
if (n!=0)
line(n, '*');
cout<<endl;
Triangle(n-1);
Program Development and Design Using C++, Third Edition

17. Testing the function Triangle(ctd)
void main() {
int x;
cout<<"Please enter the height of the triangle:";
cin>>x;
while (x<=0)
cout<<"Please enter a positive height of the triangle:"; }
Triangle(x);
Program Development and Design Using C++, Third Edition

18. The connection between recursion and looping
We can always re-write a while loop as a recursive function:
while (<expression>)
<statement> //This is any sequence of statements in curly brackets
This while loop can be replaced by a call to the recursive function f:
void f() {
if (<expression>)
<statement>;
f(); }
Program Development and Design Using C++, Third Edition

19. A non-recursive Triangle function
void Triangle(int n) {
while (n!=0)
line(n,'*');
cout<<endl;
n--; }
Program Development and Design Using C++, Third Edition

20. Program Development and Design Using C++, Third Edition
Example Using Recursion: Fibonacci Sequence (exercises 6.6, pg. 355, ex. 1)
Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8...
Each number sum of two previous ones (except for the first two numbers).
Write a recursive function that returns the nth number in the Fibonacci sequence, when n is passed to the function as argument.
For example: when n=8, the function should return the eighth number in the sequence, which is 21 (starting from 0).
Program Development and Design Using C++, Third Edition

21. The recursive Fibonacci function
long fibonacci( long n ) {
if ( n == 0 || n == 1 ) // base case
return n;
else
return fibonacci( n - 1 ) + fibonacci( n – 2 ); }
Program Development and Design Using C++, Third Edition

22. The recursive Fibonacci function
return 1
return 0
return +
Program Development and Design Using C++, Third Edition

23. What does this function do?
void Rev() {
int i;
cin>>i;
if (i!=-1)
Rev();
cout<<i<<endl; }
Program Development and Design Using C++, Third Edition

24. Program Development and Design Using C++, Third Edition
Answer:
Reads a sequence of integers, terminated by -1 and outputs them in reverse order (excluding the -1).
Example run:
User enters: 2 5 7 8 -1
Output: 8 2
Program Development and Design Using C++, Third Edition

25. How to think recursively?
Recursive functions are implemented using an if-else (or switch) statement that leads to different cases.
Break the problem into subproblems. Every recursive call reduces the original problem by bringing it closer and closer to the base case until it becomes that case.
Identify the base case. This is the simplest case (could be more than one base cases) that will be used to stop recursion.
Program Development and Design Using C++, Third Edition

26. Example of thinking recursively: Palindrome problem
Considering writing a recursive palindrome function.
A string is a palindrome if it reads the same from the left and from the right.
This problem can be divided into two subproblems:
Check whether the first and last characters of the string are equal.
Ignore these two characters and check whether the rest of the substring is a palindrome. (The second problem is the same as the original problem with a smaller size).
There are two base cases:
The two end characters are not the same, in which case the string not a palindrome
The string size is 0 or 1, in which case the string is a palindrome
Program Development and Design Using C++, Third Edition

27. Recursive palindrome function
Classroom/lab exercise….
Program Development and Design Using C++, Third Edition

28. Program Development and Design Using C++, Third Edition
Exercises:
The recursive solution to the problem of finding the Fibonacci number we gave earlier, is very inefficient (why?). Write a non-recursive Fibonacci function.
Program Development and Design Using C++, Third Edition

29. Exercises (continued):
The greatest common divisor (gcd) of two integers p and q is the largest integer that divides both (e.g. gcd of 18 and 12 is 6). An algorithm to compute the greatest common divisor of two integers p and q is the following:
Let r be the remainder of p divided by q. If r is 0, then q is the greatest common divisor. Otherwise, set p equal to q, then q equal to r, and repeat the process.
Write a recursive function that implements the above algorithm.
Program Development and Design Using C++, Third Edition

30. Exercises (continued):
Write a recursive function called locate that takes a string str, a character ch and an integer i and returns the index position of the first occurrence of character ch in string str. (The function assumes that there is an occurrence of the character at or after the index i in the string str).
Example:
cout<<locate("hello", 'l',0);
should print 2.
Program Development and Design Using C++, Third Edition