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); ` `} ` |

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Let n insert and m (<=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?

**(A)** n+m <= x < 2n and 2m <= y <= n+m

**(B)** n+m <= x < 2n and 2m<= y <= 2n

**(C)** 2m <= x < 2n and 2m <= y <= n+m

**(D)** 2m <= x <2n and 2m <= y <= 2n

**Answer:** **(A)** **Explanation:** 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())

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