Problem Description

The task is to implement a queue with the FIFO (first in, first out) principle using two stacks. In a typical queue, elements are added, or 'pushed', to the back and removed, or 'popped', from the front. Additionally, you should be able to 'peek' at the element at the front without removing it and test if the queue is 'empty'. This should be done using only the standard operations of a stack: 'push to top', 'peek/pop from top', 'size', and 'is empty'.

Intuition

The key to solving this problem is to use two stacks, stk1 and stk2, to invert the order of elements twice so that they come out in the same order that they went in. Initially, all new elements are pushed onto stk1. However, we can't directly pop from stk1 for the queue's pop operation because stacks follow LIFO order. So, to get the FIFO order of a queue, we reverse stk1 by popping all its elements and pushing them onto stk2. The first element pushed into stk1 (and therefore the first in queue order) is now at the top of stk2, allowing us to perform the FIFO pop correctly.

The move method is our helper that handles this transfer if stk2 is empty. It is lazily called only when necessary (when popping or peeking). This efficiency is important as it minimizes the number of operations. Once all elements are transferred to stk2, they can be popped or peeked in the correct FIFO order.

The empty method simply checks both stacks. If both are empty, the queue is empty.

The elegance of this solution arises from the delayed transfer of elements until necessary (amortized analysis), which minimizes the number of total operations needed.

Learn more about Stack and Queue patterns.

Solution Approach

The implementation consists of the following steps, utilizing two stacks, stk1 and stk2, which are simply represented as Python lists:

1. Constructor (__init__): Two empty stacks are initialized. stk1 is for adding new elements (push operation), and stk2 is used for FIFO retrieval (pop and peek operations).

   self.stk1 = []
   self.stk2 = []

2. Push Operation (push): Elements are added to stk1. Each new element is simply appended to the end of stk1, which is the top of the stack.

   def push(self, x: int) -> None:
       self.stk1.append(x)

3. Pop Operation (pop): To remove an element from the front of the queue, we need to get it from the bottom of stk1. The move method is called to transfer elements from stk1 to stk2, if stk2 is empty, effectively reversing the stack order. The element at the top of stk2 (the front of the queue) is then popped.

   def pop(self) -> int:
       self.move()
       return self.stk2.pop()

4. Peek Operation (peek): Similar to pop, but instead of removing the element at the front of the queue, we only retrieve it. move ensures the element is moved to stk2 so that it can be peeked at.

   def peek(self) -> int:
       self.move()
       return self.stk2[-1]

5. Empty Operation (empty): This operation checks if both stk1 and stk2 are empty. The queue is empty if and only if both stacks are empty.

   def empty(self) -> bool:
       return not self.stk1 and not self.stk2

6. Move Helper Method (move): This is an essential method that transfers elements from stk1 to stk2 when stk2 is empty. It's called only before a pop or peek operation and only when necessary (i.e., when stk2 is empty and the next front of the queue needs to be accessed).

   def move(self):
       if not self.stk2:
           while self.stk1:
               self.stk2.append(self.stk1.pop())

At its core, this approach leverages the fact that the stack data structure (using append and pop in Python lists) can be reversed by transferring elements from one stack to another. By having two stacks, we can ensure elements are in the correct FIFO order for queue operations by handling elements in the 'lazy' manner - that is, by only moving elements when absolutely necessary.

Example Walkthrough

Let's walk through a small example to illustrate how the queue implementation using two stacks—stk1 and stk2—works.

Consider the following sequence of operations:

1. push(1) - Add the element '1' to the queue.
2. push(2) - Add the element '2' to the queue after '1'.
3. peek() - Get the element at the front of the queue without removing it.
4. pop() - Remove the element from the front of the queue.
5. empty() - Check if the queue is empty.

Now let's examine how each operation is handled:

1. push(1): stk1 receives the element as [1]. stk2 remains [].
2. push(2): stk1 grows to [1, 2]. stk2 is still [].
3. peek(): We want to see the front element of the queue, which is '1'. However, since stk1 is LIFO, we need to move elements to stk2 to access '1'. move() is called, transferring all elements from stk1 to stk2, resulting in stk1 as [] and stk2 as [2, 1]. Now we can peek the top of stk2 which is '1', the first element.
4. pop(): We need to pop the front element, which is '1'. Since stk2 already has the correct order, we simply pop from stk2. stk2 becomes [2] after popping '1', and stk1 is still [].
5. empty(): To determine if the queue is empty, we check if both stk1 and stk2 are empty. Since stk2 has an element '2', the method returns False, indicating the queue is not empty.

The sequence of stk1 and stk2 after each operation is shown below:

1. After push(1): stk1: [1], stk2: []
2. After push(2): stk1: [1, 2], stk2: []
3. After peek(): stk1: [], stk2: [2, 1]
4. After pop(): stk1: [], stk2: [2]
5. After empty(): No change in stacks, stk1: [], stk2: [2]

The queue is operational, demonstrating that two stacks used in this manner can effectively implement a FIFO queue.

Solution Implementation

Time and Space Complexity

Time Complexity:

• __init__(): O(1) - Initializing two empty stacks takes constant time.
• push(x): O(1) - Append operation on a list (stack) is an amortized constant time operation.
• pop(): Amortized O(1) - In the worst case, this operation can be O(n), where n is the number of elements in stk1, because it has to move all elements from stk1 to stk2 if stk2 is empty. However, each element is only moved once due to the two-stack arrangement, and, across a series of m operations, this gives an average (or amortized) time complexity of O(1).
• peek(): Amortized O(1) - Similar to pop(), it may involve moving all elements from stk1 to stk2 in the worst case, but due to the amortized analysis, it averages to constant time.
• empty(): O(1) - Checking if two lists are empty is a constant time operation.
• move(): Amortized O(1) - Although it can be O(n) in the worst case when moving all elements from stk1 to stk2, it is part of the pop() and peek() operations and contributes to their amortized time complexity.

Space Complexity:

• Overall space complexity for the MyQueue class is O(n), where n is the number of elements in the queue at a given time. This is because all elements are stored in two stacks (stk1 and stk2). No additional space is used that is proportional to the number of elements in the queue except for these two stacks.

Learn more about how to find time and space complexity quickly using problem constraints.