An implementation of a queue Q, using two stacks S1 and S2, is given below:

void insert(Q, x) {
  push(S1, x);
}
void delete(Q) {
  if (stack - empty(S2)) then
  if (stack - empty(S1)) then {
    print(“Q is empty”);
    return;
  } else
    while (!(stack - empty(S1))) {
      x = pop(S1);
      push(S2, x);
    }
  x = pop(S2);
}

Let n insert and m ($\le$ n) delete operations be performed in an arbitrary order on an empty queue Q. Let x and y be the number of push and pop operations performed respectively in the process. Which one of the following is true for all m and n?

The order in which insert and delete operations are performed matters here. The best case: Insert and delete operations are performed alternatively. In every delete operation, 2 pop and 1 push operations are performed. So, total m+ n push (n push for insert() and m push for delete()) operations and 2m pop operations are performed. The worst case: First n elements are inserted and then m elements are deleted. In first delete operation, n + 1 pop operations and n push operation are performed. Other than first, in all delete operations, 1 pop operation is performed. So, total m + n pop operations and 2n push operations are performed (n push for insert() and n push for delete())