To understand how stacks work, it’s important to know the basics of this simple type of data structure.

A stack uses a rule called Last In, First Out (LIFO). This means that the last item you add to the stack is the first one you take out. This is different from another type of structure called a queue, where the first item added is the first one removed. Let’s look at the main actions related to stacks.

Push Operation

The push operation is how you add something to the stack.

When you push an item, you place it at the top of the stack, which means it’s the easiest to get to.

You can think of it like stacking boxes; each new box goes on top of the last one.

Here’s how it works:

1. Increase the Top Index: Before adding a new item, we increase the index that shows what’s at the top.
2. Store the Item: The new item is saved at the new top index.
3. Time Complexity: This operation takes a constant amount of time, noted as $O(1)$, no matter how many items are in the stack.

This process keeps each new item at the top, following the LIFO rule.

Pop Operation

The pop operation is what you use to take the top item off the stack.

Here are the steps for this action:

1. Check for Underflow: We first check if the stack is empty; if it is, we can’t remove anything.
2. Get the Top Item: We look at the top item and usually store or return it.
3. Decrease the Top Index: We then lower the index for the top of the stack.
4. Time Complexity: Like push, this operation also takes constant time, $O(1)$, since it only removes one item from the top.

This action follows the LIFO principle by removing only the most recently added item.

Peek Operation

The peek operation lets you see what is on top of the stack without taking it off.

It’s useful when you want to check the last item added without changing anything in the stack.

Here’s how peek works:

1. Check for Empty Stack: Just like pop, we first make sure the stack isn’t empty to avoid errors.
2. Return the Top Item: We get the value of the top item without removing it.
3. Time Complexity: Peeking also takes constant time, $O(1)$, since it doesn’t modify the stack.

Peek is great when you want to see the top item but don’t want to lose it.

Size Operation

The size operation helps you find out how many items are currently in the stack. Here’s how it works:

1. Check Size Variable: Most stacks keep a counter to track how many items are present.
2. Return Size: We can return the current count when asked.
3. Time Complexity: Getting the size generally takes constant time, $O(1)$.

Knowing the size of the stack helps avoid problems like underflow—trying to remove from an empty stack—or overflow—trying to add to a full stack.

IsEmpty Operation

We also need a way to check if the stack is empty. This is important to avoid problems when pushing or popping items.

The isEmpty operation includes:

1. Checking Size or Top Value: Most stacks just check if the size is zero or if the top index is below zero.
2. Return Boolean Result: It will return true if the stack is empty, and false if not.
3. Time Complexity: This operation is quick, taking constant time, $O(1)$.

This helps make sure we don’t try to remove something from an empty stack.

Summary of Key Operations

The main actions that define how stacks work are:

• Push: Adds an item to the top ($O(1)$)
• Pop: Removes and returns the top item ($O(1)$)
• Peek: Shows the top item without removing it ($O(1)$)
• Size: Gives the current count of items ($O(1)$)
• IsEmpty: Checks if the stack is empty ($O(1)$)

These actions are what make stacks useful, and they help keep the LIFO rule.

How Stacks Are Used

Stacks can be built in different ways, mainly using arrays or linked lists.

Array-based Implementation

In an array-based stack, we keep a fixed-size list, along with an index to show what’s at the top. This has some benefits:

1. Easier Memory Management: All items are stored next to each other in memory, which can make things faster.
2. Constant Time Access: We can quickly access any item in the array.

However, there are drawbacks:

• Fixed Size: We have to set a maximum size for the stack, which could lead to overflow if we hit the limit unless we can resize it.

Linked List Implementation

In a linked list stack, each item points to the next one. Here’s what that looks like:

1. Dynamic Size: The stack can grow and shrink based on how many items are there, so we don’t have to worry about overflow.
2. Flexible Memory Use: Memory is only used for items when we need to add them.

But, there are some downsides:

• Needs Extra Memory: Each item takes up more space since it needs to hold a pointer to the next item.
• Slower Access: Accessing items can be less efficient since they’re not stored right next to each other.

Uses of Stacks

Stacks are very helpful in computer science and programming. Here are some common ways they are used:

1. Managing Function Calls: When a function calls another function, the call is pushed onto the stack. When it finishes, it gets popped off.

2. Evaluating Expressions: Stacks help in math problems by managing the order of operations, like in converting and calculating expressions.

3. Backtracking Algorithms: Many problems, like solving a maze, use stacks to remember previous choices before going back.

4. Memory Management: Stacks help allocate memory for local variables in certain programming languages.

5. Undo Features: Many programs use stacks for undo actions, allowing users to undo the last thing done by popping it off the stack.

In conclusion, stacks are an important part of data structures, characterized by actions like push, pop, peek, size, and isEmpty, that work together to maintain their LIFO rule. Understanding these operations and how to use stacks can improve your problem-solving skills in programming.