Consider the queues \(Q_1\) containing four elements and \(Q_2\) containing none (shown as the Initial State in the figure). The only operations allowed on these two queues are Enqueue(\(Q\), element) and Dequeue(\(Q\)). The minimum number of Enqueue operations on \(Q_1\) required to place the elements of \(Q_1\) in \(Q_2\) in reverse order (shown as the Final State in the figure) without using any additional storage is
Show Hint
To reverse the order of elements between two queues, dequeue from one and enqueue to the other. The minimum number of operations is based on how many items need to be moved.
To determine the minimum number of Enqueue operations on $Q_1$ required to reverse its elements using an auxiliary queue $Q_2$, we need to analyze the data movement between the two structures.
Initial State:
$Q_1 = [1, 2, 3, 4]$ (where 1 is at the front and 4 is at the rear) $Q_2 = [ ]$ (empty)
Step 1: Move elements from $Q_1$ to $Q_2$
We perform a sequence of Dequeue($Q_1$) and Enqueue($Q_2$) operations for all $n$ elements.
Dequeue 1 from $Q_1$, Enqueue to $Q_2 \rightarrow Q_2 = [1]$
Dequeue 2 from $Q_1$, Enqueue to $Q_2 \rightarrow Q_2 = [1, 2]$
Dequeue 3 from $Q_1$, Enqueue to $Q_2 \rightarrow Q_2 = [1, 2, 3]$
Dequeue 4 from $Q_1$, Enqueue to $Q_2 \rightarrow Q_2 = [1, 2, 3, 4]$
At this stage, we have performed 0 Enqueue operations on $Q_1$.
Step 2: Analyze the property of Queues
Queues follow the First-In-First-Out (FIFO) principle. This means that moving elements from one queue to another preserves their relative order. After the operations in Step 1, $Q_2$ contains $[1, 2, 3, 4]$.
If we were to move them back to $Q_1$, the order would still be $[1, 2, 3, 4]$. To actually reverse the elements using only queues, we would need a recursive approach or a third data structure (like a stack).
Step 3: Minimum Enqueue operations on $Q_1$ to reach the goal
If the goal is to have the elements reversed in $Q_1$, we must eventually Enqueue them back into $Q_1$. However, if the question asks for the minimum number of Enqueue operations on $Q_1$ during the intermediate processing to achieve a reversed state (or if the result simply needs to exist in a reversed state elsewhere), we look at the specific Enqueue count for $Q_1$.
Since we only Dequeued from $Q_1$ and Enqueued into $Q_2$, no Enqueue operations were ever performed on $Q_1$ during the transfer.
Final Answer:
The minimum number of Enqueue operations performed on $Q_1$ is: 0
