What does the following algorithm approximate?
(Assume m > 1, $$ \in > 0$$)

x = m;
y = 1;
while(x - y > ε){
 x = (x + y) / 2;
 y = m/x;
}
print(x);

The time complexity of the following C function is (assume n > 0)

int recursive(int n){
 if(n == 1){
   return (1);
 }
 return (recursive(n - 1) + recursive(n - 1));
}

Let A[1,...,n] be an array storing a bit (1 or 0) at each location, and f(m) is a function whose time complexity is O(m). Consider the following program fragment written in a C like language:

counter = 0;
for(i = 1; i <= n; i++){
 if(A[i]==1){
   counter++;
 }else{
   f(counter); counter = 0;
 }
}

The complexity of this program fragment is

The tightest lower bound on the number of comparisons, in the worst case, for comparison-based sorting is of the order of