Implement Queue using Stacks
Implementing a queue using two stacks can be done by using one stack to manage the elements being pushed and another to manage the elements being popped or peeked. Here’s a way to achieve that:
 Push Stack (
pushStack
): You’ll use this stack to manage elements being pushed into the queue.  Pop Stack (
popStack
): This stack will help manage elements being popped or peeked.
Whenever you want to pop or peek, you’ll need to make sure that the popStack
has elements. If not, you need to transfer all the elements from pushStack
to popStack
, reversing their order.
Here’s the implementation:


 Push: Append the element to
pushStack
.  Pop: If
popStack
is empty, transfer all elements frompushStack
, then pop frompopStack
.  Peek: Similar to pop, but return the top element from
popStack
without popping.  Empty: Check if both stacks are empty.
This ensures that the first element pushed into the queue is always the first element to be popped or peeked.
Identifying Problem Isomorphism
“Implement Queue using Stacks” is an isomorph to “Implement Stack using Queues”. Both problems require the implementation of one type of data structure using another data structure, with similar operations.
In the “Implement Queue using Stacks” problem, you have to implement a queue, which follows the FIFO (FirstInFirstOut) principle, using stacks that follow the LIFO (LastInFirstOut) principle. You need to support operations like push (adding an element to the end of the queue), pop (removing an element from the front of the queue), and peek (returning the front element of the queue).
Similarly, in the “Implement Stack using Queues” problem, you have to implement a stack using queues. You need to support operations like push (adding an element on top of the stack), pop (removing the top element from the stack), and top (returning the top element of the stack).
These problems are isomorphic as they both require understanding of basic data structure principles and require using the operations of one data structure to implement another. The only difference is the actual data structures involved. If you understand how to solve one problem, you can apply the same type of thinking to solve the other problem, making them isomorphic.
Both problems is of similar difficulty as both require similar levels of understanding and manipulation of basic data structures. They both involve dealing with the operations of one data structure while maintaining the characteristics of another.
“Implement Stack using Queues” is a comparable one to “Implement Queue using Stacks”. These problems are essentially mirror images of each other, with each requiring the implementation of one data structure (queue or stack) using only instances of the other.
In “Implement Stack using Queues”, you are tasked with implementing a stack using only instances of queue data structure and the operations allowed on a queue. This includes implementing functions such as push (insert an element on to the stack), pop (remove the top element from the stack), and top (retrieve the top element on the stack).
While “Implement Stack using Queues” and “Implement Queue using Stacks” are both intermediate in difficulty, if you’re more familiar with queue operations, you may find “Implement Stack using Queues” to be simpler and a good starting point.
After getting comfortable with “Implement Stack using Queues”, the insights gained can then be applied to tackle “Implement Queue using Stacks” more effectively, where the challenge is to implement a queue using stack operations.


Problem Classification
The problem is about designing and implementing a data structure  specifically a queue using two stacks. Thus, the problem can be classified as a Data Structures problem, with a focus on the Queue and Stack structures. It involves understanding of Data Structure Manipulation techniques and Data Structure Design.
Given the need to implement specific operations, such as enqueue (push), dequeue (pop), peek, and empty, the problem also falls under the category of Interface Design and Method Implementation.
In terms of algorithmic complexity, it touches upon Amortized Analysis since some operations (like dequeue and peek) may be expensive in the worstcase scenario (when all elements have to be moved from one stack to another), but if averaged over a sequence of operations, they are efficient.
This problem also highlights the use of Adaptation of Existing Data Structures to solve problems in innovative ways, demonstrating the flexibility and versatility of basic data structures like stacks and queues.
This problem can be classified under:
 Data Structures (Queue, Stack)
 Data Structure Manipulation
 Data Structure Design
 Interface Design
 Method Implementation
 Amortized Analysis
 Adaptation of Existing Data Structures.
Language Agnostic Coding Drills
Here’s the breakdown of the key concepts involved, in the order of complexity:
Understanding basic data structures: Lists (Stacks)
At the foundation, we need to understand how lists (used as stacks here) work, including how to add (append) and remove (pop) elements.
Working with two stacks
The crux of the problem lies in understanding how two stacks can interact with each other and can be used to implement a queue structure. This includes knowing how to move elements from one stack to another.
Using a Class in Python
This involves understanding how to use classes in Python, including creating instance variables, and defining and calling methods.
Implementing a queue using two stacks
This involves understanding the queue data structure and its operations (enqueue, dequeue, peek, and isEmpty), and how to implement these operations using two stacks. This is the most complex part as it requires a good understanding of both queue and stack data structures and their properties, and how to translate these properties and operations from one data structure to another.
From the problem statement to the final solution, the following steps are taken:
The problem asks for an implementation of a queue using only stacks. A queue is a FirstInFirstOut (FIFO) data structure while a stack is a LastInFirstOut (LIFO) data structure. The challenge here is to implement a FIFO structure using only LIFO structures.
To do this, two stacks are used. One stack is used to “reverse” the order of elements (in_stk), while the other is used to keep the “queue” order (out_stk).
When an element is enqueued (push), it is simply pushed onto in_stk.
When an element is dequeued (pop), all elements are transferred from in_stk to out_stk, effectively reversing their order to become FIFO. The top element of out_stk (the oldest element) is then popped.
For the peek operation, the same transfer is done as in the pop operation, but without popping the element. This gives access to the “front” element of the queue.
The queue is empty when both in_stk and out_stk are empty.
This algorithm is based on the amortized complexity, where the expensive pop and peek operations (which require transferring elements) are rare compared to the cheap push operations.
Targeted Drills in Python
We’ll start from basic concepts and build up to more complex ones. We’ll also create some problemspecific drills for the final solution.
Drill 1: Understanding Basic Data Structures: Lists
Let’s start by creating a simple stack and performing some basic operations:


Drill 2: Working with two stacks
This drill will help in understanding how two stacks can interact with each other:


Drill 3: Using a Class in Python
This drill will help you understand how to use classes in Python:


Drill 4: Implementing a queue using two stacks
Now, let’s combine all the previous concepts to implement our queue using two stacks:


You can run these drills one by one, understand the output and then move on to the next one. Once you’re comfortable with all the drills, you can integrate all the concepts to solve the final problem.