Question

See solution below, I would like some constructive feedback.

what is the running time in O(n) below.

int a = 0;
int k = n*n*n; //n^3
while(k > 1) 
{
  for (int j=0; j<k; j++) //runs from 0->k
  { a--; }
k = k/2; //divides by 2 each iteration
}

each time the for loop runs, it gives a constant x k.

= 0xn^3 + 1x (n^3/2) + 2x(n^3/4) +…+ nx(n^3/2^n)
= n^3 (0 + 1/2 + 2/4 +…+ n/2^n) <- anyone know how I can simplify this down further?

Edit: i am assuming we would use the arithmetic series somehow….

Answer 1

Let’s use k first

in first while loop, the for loop runs k times

in second while loop, the for loop runs k/2 times

in third while loop, the for loop runs k/4 times

…

so in total it runs ( k + k/2 + k/4 + k/8 + … + 1) times

after extract the k, it’s k * ( 1 + 1/2 + 1/4 + 1/8 + … + 1/k)

as k increases, the part in parentheses becomes 2, which we can ignore

change k to n^3, the result is O(n^3)