runtime of 2 dependent nested for loops

Question

for (i=1; i<=n ;i=i*2){

  for (j=1; j<=i ;j++){

  basic_step;

  }
}

Regarding the above nested loops, I can't seem to understand why is the following runtime analysis is correct (specifically the first equality), when $k$ is defined as $\log i$, meaning $i=2^k$. What is the process that one should come up with, in order to decide this setting and the sigmas' indices? While I do understand that the outer loop runs for $\Theta(\log n)$ times, I still can't grasp the setting of $k$. A clarification will be very much appreciated!

$$\sum_{k=0}^{\log n}\sum_{j=1}^{2^k}1$$

Answer 1

The value of $i$ at the $k$'th iteration (starting from zero) is $i_k := 2^k$. The inner loop runs $i_k$ times, for a total of $i_0 + i_1 + \cdots + i_K$, where $K$ is the largest number such that $i_K \leq n$, that is $\lfloor \log_2 n \rfloor$. Therefore the running time is $$ \sum_{k=0}^{\lfloor \log_2 n \rfloor} 2^k = 2^{\lfloor \log_2 n \rfloor + 1} - 1 = \Theta(n). $$