Complexity of a double for loop

Question

I am trying to figure out the complexity of a for loop using Big O notation. I have done this before in my other classes, but this one is more rigorous than the others because it is on the actual algorithm. The code is as follows:

for(cnt = 0, i=1; i<=n; i++) //for any size n
{
    for(j = 1; j <= i; j++)
    {
       cnt++;
    }
}

AND

for(cnt = 0, i=1; i<=n; i*=2) //for any size n
{
    for(j = 1; j <= i; j++)
    {
       cnt++;
    }
}

I have arrived that the first loop is of O(n) complexity because it is going through the list n times. As for the second loop I am a little lost. I believe that it is going through the loop i times for each n that is tested. I have (incorrectly) assumed that this means that the loop is O(n*i) for each time it is evaluated. Is there anything that I'm missing in my assumption. I know that cnt++ is constant time.

Thank you for the help in the analysis. Each loop is in its own space, they are not together.

Answer 1

The outer loop of the first example executes n times. For each iteration of the outer loop, the inner loop gets executed i times, so the overall complexity can be calculated as follows: one for the first iteration plus two for the second iteration plus three for the third iteration and so on, plus n for the n-th iteration.

1+2+3+4+5+...+n = (n*(n-1))/2 --> O(n^2)

The second example is trickier: since i doubles every iteration, the outer loop executes only Log2(n) times. Assuming that n is a power of 2, the total for the inner loop is

1+2+4+8+16+...+n

which is 2^Log2(n)-1 = n-1 for the complexity O(n).

For ns that are not powers of two the exact number of iterations is (2^(Log2(n)+1))-1, which is still O(n):

1      -> 1
2..3   -> 3
4..7   -> 7
8..15  -> 15
16..31 -> 31
32..63 -> 63

and so on.