具有因变量和对数增量的嵌套循环的时间复杂度

Question

what is the time complexity of this loop, O(N) or O(N (logN ))

also can you explain how you deduced

for (int i = 1; i <= n;  i *= 2) {   
    for (int j = 0; j < i; j++) {
        // Statement(s) that take(s) constant time
   }
}

i have an explanation but it feels wrong

Answer 1

I think you are confused because of the statement O(n + log(n)) , as you thought that the outer loop runs logN times and inner loop runs N times so the answer should be O(NlogN) . You are wrong here because the inner loop doesn't run N times, it only runs i times as explained. Now when you sum all the i over outer loop, you will get that 2*2^k - 1 statement. This will come out to be order of N as given in the explanation.