Question

What is the time complexity to compute this function for n.

int rec(int n)
{
    if (n<=1) {
        return n ;
    }
    int i;
    int sum=0;
    for (i=1; i<n; i++) {
       sum=sum+rec(i); 
    }
    return sum ;
}

Answer 1

well let’s break this out

 (1)        f(n) = f(n-1) + f(n-2) + ... f(1) + 1

however,

 (2)        f(n-1) = f(n-2) + ... f(1) + 1

so plugging (2) into (1) gives

 (3)        f(n) = f(n-1) + f(n-1) = 2 f(n-1)

and f(2) = 1, so this is clearly 2ⁿ (for details: Can not figure out complexity of this recurrence). well, actually 2^n-1 but in big O, it doesn’t quite matter because the ^-1 is the same as /2.