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

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

Consider the grammar rule $$E \to {E_1} - {E_2}$$ for arithmetic expressions. The code generated is targeted to a CPU having a single user register. The subtraction operation requires the first operand to be in the register. If E₁ and E₂ do not have any common sub-expression, in order to get the shortest possible code.