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Recursion

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Programming

Recursion

- What does the following program do?
#include <iostream>

using namespace std;

int fac(int n){

// Assume n >= 0

int product;

if(n <= 1)

return 1;

product = n * fac(n-1);

return product;

}

void main(){// driver function

int number;

do{

cout << "Enter integer (negative to stop): ";

cin >> number;

if(number >= 0)

cout << fac(number) << endl;

}while(number >= 0);

}

- Assume the number typed is 3.
fac(3) :

3 <= 1 ? No.

product3 = 3 * fac(2)

fac(2) :

2 <= 1 ? No.

product2 = 2 * fac(1)

fac(1) :

1 <= 1 ? Yes.

return 1

product2 = 2 * 1 = 2

return product2

product3 = 3 * 2 = 6

return product3

fac(3)has the value 6

- 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) * (n-2) * ... * 1

or

n! = n * (n-1)! and 1! = 1

- A recursive solution may be simpler to write (once you get used to the idea) than a non-recursive solution.
- But a recursive solution may not be as efficient as a non-recursive solution of the same problem.

// Non-recursive factorial function

// Compute the factorial using a loop

int fac(int n){

// Assume n >= 0

int k, product;

if(n <=1)

return 1;

product = 1;

for(k=1; k<=n; k++)

product*= k;

return product;

}

- Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

where each number is the sum of the preceding two.

- Recursive definition:
- F(0) = 0
- F(1) = 1
- F(n) = F(n-1) + F(n-2)

- Binary search:
- Compare search element with middle element of the array:
- If not equal, then apply binary search to half of the array (if not empty) where the search element would be.

- How to write recursively?
int rec(1-2 parameters){

if(stopping condition)

return stopping value;

// second stopping condition if needed

return value/rec(revised parameters)

+-*/ rec(revised parameters);

}

- How to write exp(int x, int y) recursively?
int exp(int x, int y){

if(y==0)

return 1;

return x * exp(x, y-1);

}

- Write a recursive function that takes a double array and its size as input and returns the sum of the array:
double asum(int a[], int size){

if(size==0)

return 0;

return asum(a, size-1)+a[size-1];

}

- Write a recursive function that takes a double array and its size as input and returns the product of the array:
double aprod(int a[], int size){

if(size==0)

return 1;

return aprod(a, size-1)*a[size-1];

}

- Write a recursive function that counts the number of zero digits in a non-negative integer
- zeros(10200) returns 3
int zeros(int n){

if(n==0)

return 1;

if(n < 10)

return 0;

if(n%10 == 0)

return 1 + zeros(n/10);

else

return zeros(n/10);

}

- Write a recursive function to determine how many factors m are part of n. For example, if n=48 and m=4, then the result is 2 (48=4*4*3).
int factors(int n, int m){

if(n%m != 0)

return 0;

return 1 + factors(n/m, m);

}