GATE CS 2016 Set 1 — Question 51

Let $n$ be the number of elements in the queue. The algorithm attempts to move elements from the queue $Q$ to the stack $S$ such that the elements in $S$ are in non-decreasing order from top to bottom (which means they will be sorted when popped). If the current element at the head of $Q$ is greater than or equal to the top of $S$, it is pushed onto $S$. If it is smaller, the top element of $S$ is popped and enqueued back to $Q$. This behavior is similar to a sorting process where elements are cycled until they can be placed in the correct order.The worst-case scenario for the number of iterations occurs when the input queue is sorted in decreasing order (e.g., $n, n-1, \dots, 1$). In this case, the algorithm performs $n^2$ iterations.Let's trace for small $n$:

• For $n=1$ (Input: $[1]$): 1 iteration.
• For $n=2$ (Input: $[2, 1]$):

1. Dequeue 2, Push 2 ($S=[2], Q=[1]$)
2. $Top(S)=2 > Head(Q)=1$: Pop 2, Enqueue 2 ($S=[], Q=[1, 2]$)
3. Dequeue 1, Push 1 ($S=[1], Q=[2]$)
4. $Top(S)=1 \le Head(Q)=2$: Dequeue 2, Push 2 ($S=[1, 2], Q=[]$)
Total: 4 iterations ($2^2$).

• For $n=3$ (Input: $[3, 2, 1]$): Total 9 iterations ($3^2$).

The pattern for the maximum number of iterations is $n^2$.
Given $n = 16$, the maximum number of iterations is:$$16^2 = 256$$