Stacks are a basic way to store data that works on a Last-In-First-Out (LIFO) system. This means that the last item added is the first one to be taken out. To see how stacks are different from other data storage methods like arrays and linked lists, we need to look at how each of these methods works, how well they perform, and where they are usually used.

How Stacks Work

1. What You Can Do with Stacks:

   • Push: This means adding something to the stack. It takes a constant amount of time, noted as $O(1)$. This is because you just put the item on top without needing to look anywhere else.
   • Pop: This means removing the top item. This also takes $O(1)$ time, as it’s just about taking the item from the top.
   • Peek: This is when you want to see the top item without taking it off. This also takes $O(1)$ time.

2. How Arrays Work:

   • Access: You can get an item at any position in the array quickly, which takes $O(1)$ time.
   • Insert: Adding a new item can take longer, on average $O(n)$ time, since items might have to be moved.
   • Delete: Similar to inserting, removing an item takes about $O(n)$ time due to the need to shift other items.

3. How Linked Lists Work:

   • Access: Getting to an item takes longer, $O(n)$, since you might have to go through several items to find it.
   • Insert: Adding an item at the start is quick ($O(1)$), but to add in the middle or at the end can take longer, anywhere from $O(1)$ to $O(n)$ depending on where it goes.
   • Delete: Removing from the start is quick ($O(1)$), but from somewhere else it can take $O(n)$.

Memory Use

• Stacks:

  • Stacks can be built using arrays or linked lists. They usually don't use much extra memory. If a stack is made with an array, it has a set maximum size. If it grows too big, it can overflow, which means it runs out of space. A linked list doesn’t have this problem but uses extra memory for pointers (which link items together).

• Arrays:

  • Arrays use memory efficiently if their size is known in advance. They let you access items quickly but can waste space if you guess the size too large. If you change the size later, it takes time and resources to copy the items.

• Linked Lists:

  • Linked lists don’t waste space by reserving too much or too little. However, they do use more memory because of the pointers, which can be significant for smaller sets of data.

Where They are Used

• Stacks:

  • Stacks are used a lot in programming for managing function calls (called the call stack). When a function is called, it goes on the stack, and when it’s done, it comes off. This helps keep the program running smoothly.
  • They are also used for undo actions, like in text editors. Each action is added to the stack, and you can undo the latest by removing it.

• Arrays:

  • Arrays are great for storing fixed collections, like a list of students in a class. They allow for quick access and work well when you know how many items you will have.

• Linked Lists:

  • Linked lists are good for situations where you need to add or remove items often. For example, they are used in queues, which manage items in the order they came in.

Performance Comparison

Now that we know how these data structures work, let’s compare them:

• Speed:

  • Stacks allow for fast operations to add and remove items, making them faster than arrays when items are added or deleted often.

• Memory:

  • Stacks can be created using arrays or linked lists, but they generally use memory better than linked lists. Arrays can have limits, though.

• Building Complexity:

  • Stacks are easy to set up, whether you use arrays or linked lists. But arrays can get a bit trickier if you need to change their size.

Space Use Analysis

1. Stack:

   • If a stack is made with an array, it uses as much memory as the number of items you have, $O(n)$. For linked lists, the memory use is $O(n)$ too, but with extra memory needed for pointers.

2. Array:

   • The memory use for an array is $O(n)$. If you don’t fill it up completely, it can waste some space.

3. Linked List:

   • A linked list also uses $O(n)$, but the extra memory for pointers can make it less efficient for smaller sets of data.

Conclusion

In summary, stacks are a powerful data structure that is simple to use and very efficient. They follow the LIFO rule, making them very helpful in certain programming tasks.

When we look at stacks next to arrays and linked lists, we see that while they use space similarly, their speed in managing data makes them unique. Stacks work best when you need quick access and handling of data in a certain order, like in processes called recursion or keeping track of states in applications.

Arrays offer quick access but slow down with lots of changes, while linked lists are flexible but come with extra memory usage. Knowing about these data structures helps programmers choose the right one for their projects and data management needs.