LECTURE 23: Recursion (Chapter 10)

Definition: A function that calls itself is said to be recursive (also, if function f1 calls function f2 that calls f1, it is called recursive).

Often, recursion is an alternative to looping

Recursion is computationally less efficient, but it allows writing natural-looking programs

Example: Factorial

Definition:

n! = n*(n-1)*…*2*1 (4! = 4*3*2*1)

0! = 1

Factorial can also be defined (recursively) as:

n! = n*(n-1)! if n > 0, n! = 1 if n=0

The recursive formula can be implemented in C as:

int factorial(int n)

{

int ans;

if (n==0)

ans = 1;

else

ans = n * factorial(n-1);

return(ans);

}

Let us compare this solution to the non-recursive one:

int factorial(int n)

{

int ans;

int i;

ans = n;

for (i=n-1; i >= 1; i++)

ans = ans * i;

return (ans);

}

Analysis of function factorial – “Tracing a Recursive Function”:

Read 10.2, p. 507-511

Example: Greatest Common Divisor – gcd(m,n)

Definition: gcd is the largest integer that divides both m and n (we will assume m > n)

gcd(6,4) = 2, gcd(12,6) = 6, gcd(7,6) = 1;

gcd can be found recursively as:

gcd(m,n) = n if m%n = 0, gcd(m,n) = gcd(n, m%n) otherwise

To implement this solution is C:

int gcd(int m, int n)

{

int ans;

if(m%n == 0)

ans = n;

else

ans = gcd(n, m%n);

return(ans);

}

Most common form of recursion:

if “simple case”

solve it

else

redefine the problem using recursion