Design a Queue Using Only One Stack

Queue is a FIFO (first-in-first-out) strategy data structure, while Stack is a FILO (first-in-last-out) data structure. The visual description of these data structures is shown in the figure:

The basics of these two data structures are actually implemented by arrays or linked lists, but the API limits their behavior. So let's take a look at how to use "Stack" to implement a "Queue" and how to use "Queue" to implement a "Stack".

1. Using Stack to implement Queue

First, the API of Queue are as follows:

/** Push element x to the back of queue. */

/** Removes the element from in front of queue and returns that element. */

/** Get the front element. */

/** Returns whether the queue is empty. */

We can use two stacks s1,s2 to implement the function of a queue (it may be easier to understand to place the stack horizontally):

private Stack < Integer > s1 , s2 ;

When calling push to enqueue an element, we only need to push the element into s1 . For example, if we push 3 elements into 1,2,3, then the detailed structure is like this:

/** Push element x to the back of queue. */

public void push ( int x ) {

So what if using peek to get the front element of the queue at this time? Logically speaking, the front element should be 1, but in s1 , 1 is pushed to the bottom of the stack. Now we can use s2 to transit elements. When s2 is empty, all elements of s1 can be moved to s2 , at this time the elements in s2 are FIFO (first-in-first-out) order .

/** Get the front element. */

// move elements from s1 to s2

Similarly, for the pop operation, we only need to operate s2 .

/** Removes the element from in front of queue and returns that element. */

// First call peek to ensure that s2 is not empty.

Finally, how to determine the queue is empty? If both stacks are empty, the queue is empty:

/** Returns whether the queue is empty. */

return s1 . isEmpty () && s2 . isEmpty ();

So far, we implement a queue with stacks. The core idea is to use two stacks to cooperate with each other.

It is worth mentioning, what is the time complexity of these operations? What's interesting is that the peek operation may trigger a while loop when it is called. In this case, the time complexity is $O(N)$, but in most cases, the while loop will not be triggered and the time complexity is $O(1)$. Since the pop operation calls peek , its time complexity is the same as peek.

In this case, it can be said that their worst time complexity is $O(N)$, because, including a while loop, it may be necessary to move elements from s1 to s2 .

However, their average time complexity is $O(1)$. We can understand it in this way: For an element, it can only be moved at most once, which means that the average time complexity of each element of the peek operation is $O(1)$.

2. Using Queue to implement Stack

If it is tricky to use 2 stacks to implement a queue, then using queue to implement stack is much more straightforward, requiring only one queue as the basic data structure.

First, the API of stack are as follows:

/** Push element x onto stack. */

/** Removes the element on top of the stack and returns that element. */

/** Get the top element. */

/** Returns whether the stack is empty. */

Let's talk about the push API first. We can add elements directly to the queue, and record the rear element at the same time. Since the rear element is equivalent to the top element of the stack, if we want to use top to get the top element of the stack, we can directly return it:

Queue < Integer > q = new LinkedList <> ();

/** Push element x onto stack. */

public void push ( int x ) {

// x is the rear element of queue, it also the top element of stack.

/** Return the top element. */

Our basic data structure is a FIFO queue, and each pop can only take elements in front of the queue; But the stack is FILO, which means the pop API takes elements from the rear of the queue.

The solution is simple. We can take out the front element of the queue and add it to the rear of the queue, and let the previous rear element line up to the head of the queue, and then we can take it out:

/** Removes the element on top of the stack and returns that element. */

// the previous rear element reaches the head of the queue

There is still a small problem with this implementation. The original rear element has reached to the head of the queue and has been deleted, but the top_elem variable was not updated. We need a little modification:

/** Removes the element on top of the stack and returns that element. */

// Leave the last 2 elements

// Record the new rear element

// Remove the previous rear element

In the end, the empty API is easy to implement, just check if the underlying queue is empty:

/** Returns whether the stack is empty. */

Obviously, if we implement a stack with a queue, the time complexity of the pop operation is $O(N)$, and all other operations are $O(1)$.

I think that implement a stack with a queue is a trivial problem, but implement a queue with a dual stack is worth learning .

After moving elements from stack s1 to s2 , the elements become the FIFO order of the queue in s2. This feature is similar to "Two negatives make an affirmative.," which is not easy to think.

I hope this article is helpful to you.

"Stick to original high-quality articles, and work hard to make algorithmic problems clear."

Design a Queue Using Only One Stack

Source: https://labuladong.gitbook.io/algo-en/ii.-data-structure/implementqueueusingstacksimplementstackusingqueues

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