Question

Suppose you have a deque \(D\) containing the numbers (1,2,3,4,5,6,7,8) in this order. Suppose further that you have an initially empty queue \(Q .\) Give a pseudo-code description of a function that uses only \(D\) and \(Q(\text { and no other variables or objects) and results in } D\) storing the elements \((1,2,3,5,4,6,7,8),\) in this order.

Short answer

Use a queue to temporarily hold some values and swap the necessary ones before returning them back to the deque.

Step-by-step solution

- Initialize the algorithm

Start by understanding that we need to rearrange the elements in the deque (D) to match the order (1,2,3,5,4,6,7,8). Use the queue (Q) to assist in temporarily holding values.

- Remove elements before the change point

Remove the first three elements from the front of deque (D) sequentially and enqueue them into (Q) so that we can access them easily later.Pseudo-code: for i in range(1, 4): Q.enqueue(D.pop_front())

- Remove the elements to be swapped

Remove the next two elements which need to be swapped in the desired final sequence.Pseudo-code: x = D.pop_front()y = D.pop_front()

- Place swapped elements back into deque

Push these two elements back into the deque (D) in the reverse order.Pseudo-code: D.push_front(x)D.push_front(y)

- Re-insert the previously dequeued elements

Enqueue the elements stored in queue Q back into deque D in their original order.Pseudo-code: while not Q.is_empty(): D.push_back(Q.dequeue())

- Verify final order in deque D

Ensure that the deque contains elements in the final desired order: (1,2,3,5,4,6,7,8).

Key concepts

data structures

Data structures are fundamental to programming and algorithm design. They organize and store data in ways that enable efficient access and modification. In this exercise, we focus on two data structures: deque (double-ended queue) and queue.
Deques allow insertion and removal of elements from both ends, making them versatile for various algorithmic tasks. They can be used wherever elements need to be dynamically added or removed from either the front or the back. This can be particularly useful in simulations and algorithms like breadth-first search (BFS).
Queues, on the other hand, follow a First-In-First-Out (FIFO) principle. This is useful when the order of processing elements is significant, such as scheduling tasks or processing requests in the order they arrive.

queue manipulation

Queue manipulation is a key skill when working with data structures. In this exercise, we use both deques and queues to rearrange elements efficiently.
To solve the problem, we need to move elements between these two structures without using any additional variables. This involves extracting, storing, and then re-inserting elements precisely.
1. **Extract Elements**: Start by removing three elements from the deque and enqueuing them into the queue.
2. **Swap Elements**: Remove two elements from the deque that need to be swapped and re-insert them in reverse order.
3. **Reinsert Elements**: Finally, dequeue the stored elements from the queue and add them back to the deque.
Understanding these manipulations helps in creating efficient algorithms that manage data effectively.

algorithm design

Algorithm design is the process of creating a step-by-step method to solve a problem. This involves understanding the problem, planning a sequence of actions, and using appropriate data structures.
In our exercise, we designed an algorithm to rearrange elements using only a deque (D) and a queue (Q). The key steps include:
1. **Initialization**: Recognize the initial setup and the desired outcome.
2. **Element Extraction**: Use loops and basic operations to extract elements in the required sequence.
3. **Swapping and Reinsertion**: Carefully swap and reinsert elements to achieve the desired order.
Effective algorithm design ensures that we manage available resources optimally and achieve the correct result with minimal operations.

C++ programming

C++ is a powerful language for implementing complex algorithms and data structures. It provides built-in support for deques and queues through its Standard Template Library (STL).
Here's how you might implement this exercise in C++ using STL containers:
1. **Include necessary headers**: `#include ` and `#include `
2. **Initialize deque and queue**: Use `std::deque D` and `std::queue Q`.
3. **Perform operations**: Use functions like `pop_front()`, `push_front()`, and `push_back()` for deque operations, and `enqueue()`/`dequeue()` for queue operations.
By leveraging C++ and its robust STL, we can efficiently implement sophisticated data manipulations.