Complexity of two algorithms

Question

I am studying algorithms complexity, and I have a question about the difference between the following two algorithms:

Algorithm #1:

sum = 0
i = 1

while (i < n) {

   for j = 1 to i {
       sum = sum + 1
   }

   i = i*2;
}

return sum

Algorithm #2:

sum = 0
i = 1

while (i < n) {

   for j = 1 to n {
       sum = sum + 1
   }

   i = i*2;
}

return sum

The only difference is the 'for' loop, but what is the difference between the time complexity of these algorithms ? When do I have to multiply or add the complexity of nested loops?

Answer 1

Let say for simplisity that n is a power of 2 i.e. 2^k. Then it is obvious that outer loop will proceed k times and on each step inner loop will proceed:

1 to 1 i.e. 2^0
1 to 2 i.e. 2^1
1 to 4 i.e. 2^2
...
1 to        2^k

So we just need to find the sum 2^0 + 2^1 + ... + 2^k

It is knows as 2^(k+1) - 1 = 2^k * 2 - 1 = 2*n + 1

So omitting constants we get O(n)

Second one is simple. Outer loop is log(n), inner n so you get O(n*log(n)).