Question

Describe how you would design a stack that, in addition to the traditional push and pop operations, also supports an operation called min, which returns the element having the minimum value in the stack. Push, pop, and min should all operate in \(\mathrm{O}(1)\) time.

Short answer

Design a stack with a primary stack, a min stack for the minimum values, and ensure synchronized push and pop operations.

Step-by-step solution

Define the Stack Design

To design a stack with a min operation, we will use two stacks: a primary stack to store all the elements as usual, and a secondary min stack to track the minimum values. The min stack will store the minimum value every time an element is pushed onto the primary stack.

Implement the Push Operation

For the push operation, whenever we push a new element onto the primary stack, we compare it with the current top element of the min stack. If the min stack is empty or the new element is smaller or equal, we also push it onto the min stack. This ensures the min stack always has the smallest value at the top.

Implement the Pop Operation

For the pop operation, we remove the top element from the primary stack. Additionally, if the element popping from the primary stack is equal to the top element of the min stack, it means the current minimum is being removed, so we also pop the top element from the min stack.

Implement the Min Operation

For the min operation, simply return the top element of the min stack, since it always contains the minimum value in the primary stack. Accessing the top of the min stack is an O(1) operation.

Key concepts

Stack

A **stack** is a data structure that follows the Last-In-First-Out (LIFO) principle. This means that the last element added to the stack will be the first one to be removed. Think of a stack like a pile of plates; you add or remove the top plate.

Stacks are essential in various real-world applications, such as managing function calls in programming languages and handling undo functionalities in software. Here are some core operations you can perform with stacks:

• **Push**: Add an element to the top of the stack.
• **Pop**: Remove the top element from the stack.
• **Peek/Top**: View the top element without removing it.
• **IsEmpty/Size**: Check if the stack is empty or find the number of elements in it.

Enhancing a stack with additional features, like a **min** operation, can optimize tasks requiring frequent minimum value retrievals. The solution involves using two stacks to maintain the primary stack's integrity and quickly access the minimum value. This structure ensures efficient operations even as the stack grows.

Algorithm Design

**Algorithm design** involves creating a process or set of rules to solve a specific problem. When designing a stack with a minimum value feature, the goal is to maintain efficient time complexity for all operations, ensuring they perform swiftly.

The key insight is using two stacks; one for storing all elements and another for tracking minimums. Here's a breakdown:

• **Primary stack**: Holds every element added to the stack.
• **Min stack**: Keeps track of the current minimum. Each time an element is pushed, it's compared with the min stack's top, ensuring it retains the smallest value without extra computation.

With this design, operations such as push, pop, and accessing the minimum, all achieve constant time ( O(1) ) complexity. This effectiveness stems from minimizing redundant calculations and maintaining simplicity in the underlying data structure, central tenets of good algorithm design.

Complexity Analysis

In **complexity analysis**, we evaluate how efficiently an algorithm uses computational resources, commonly focusing on time and space complexities.

For a stack with push, pop, and min operations, achieving an ( O(1) ) time complexity for each is crucial. This constant time complexity means that as the stack grows in size, the execution speed remains unaffected.

**Time Complexity**

• **Push Operation**: Only involves adding an element to the primary stack and potentially the min stack if it's a new minimum, making it ( O(1) ).
• **Pop Operation**: Similar to push, removing the element from the top of two stacks is straightforward, ensuring ( O(1) ) execution time.
• **Min Operation**: Accessing the top of the min stack to get the minimum value is a simple operation, thus ( O(1) ).

**Space Complexity** relies on maintaining two stacks. Although it slightly increases required space, primarily when storing duplicate elements in the min stack, it's a trade-off for fast operation speed. The additional space usage is justified by the substantial performance benefits gained when operations are performed on the stack.