这个函数的时间复杂度是否为O（N）？

Question

What is the time complexity of this function? The function returns minimum value in the array. I think it's O(N) but I can't prove it. Any help would be appreciated!

int[] foo(int arr[], int N)
{
   int k = 1;
   while(k < N)
   {
      for(int i = 0; i+k<N; i+=2*k) 
      {
         if(arr[k] > arr[i+k])
         {
            swap(arr[i], arr[k+i]); //swap values in arr[i] and arr[k+1]
         }
      }
      k = k*2;
   }

   return arr[0];
 }

Answer 1

It's O(N) .

It may seem like O(NlogN) at a first glance, but:

• 1st inner for loop: i = 0, 2, 4, 6, 8, ... ie N/2 operations
• 2nd inner for loop: i = 0, 4, 8, 12, 16, ... ie N/4 operations
• 3rd inner for loop: i = 0, 8, 16, 24, 32, ... ie N/8 operations
• 4th inner for loop: i = 0, 16, 32, 48, 64, ... ie N/16 operations
• etc.

N/2 + N/4 + N/8 + N/16 + ... = N(1/2 + 1/4 + 1/8 + 1/16 ...) = N

https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF