Question

The full question is

Supposed we have a function

void foo(int n){
  int i = n;
  while(i > 0){
    //do an O(n) operation
    //do some O(1) operations
    i = sqrt(i) - 1;
  }
}

I just have to find the asymptotic bounds, but I can’t do that until I figure out how many times the loop actually runs. Which I’m guessing is another summation involving a square root, but I’m not sure how to start.

Answer 1

You want to find you how many times the loop will execute.

If i < 2, then the loop will execute at most twice.

Therefore if i < 4 the loop will execute at most 3 times.

Therefore if i < 16 the loop will execute at most 4 times.

Therefore if i < 256 the loop will execute at most 5 times.

…

etc.

You see that if i < 2^(2^m), then the loop will execute at most (m+2) times.

This means that the order of the number of times it will execute is log(log(n)),

since i starts at n.

Thus the overall complexty is O(n*log(log(n)).

(That’s if the number of O(1) operations in each iteration is constant.)