Min Stack

The goal here is to design a stack that supports standard operations like push, pop, and top, but also allows for retrieving the minimum element of the stack in constant time (O(1)). Here’s how we can approach it:

Approach

1. Create a Stack: We’ll use two stacks, one for storing the actual elements and the other for keeping track of the minimum values. This will allow us to achieve constant time complexity for each function.
2. Push Operation: When pushing an element onto the stack, also update the minimum stack if the pushed value is less than or equal to the current minimum.
3. Pop Operation: Pop elements from both the actual stack and the minimum stack to ensure that the minimum stack stays in sync with the actual stack.
4. Top Operation: Return the top element from the actual stack.
5. GetMin Operation: Return the top element from the minimum stack, which always stores the minimum element of the actual stack.

Code

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class MinStack:
    def __init__(self):
        self.stack = []
        self.min_stack = []

    def push(self, val: int) -> None:
        self.stack.append(val)
        # If the minimum stack is empty or val is less than or equal to the current minimum
        if not self.min_stack or val <= self.min_stack[-1]:
            self.min_stack.append(val)

    def pop(self) -> None:
        if self.stack[-1] == self.min_stack[-1]:
            self.min_stack.pop()
        self.stack.pop()

    def top(self) -> int:
        return self.stack[-1]

    def getMin(self) -> int:
        return self.min_stack[-1]

Example

The given code snippet and example in the problem statement will produce the output: [null,null,null,null,-3,null,0,-2]

Complexity Analysis

• Time Complexity: All the operations (push, pop, top, and getMin) are done in constant time (O(1)).
• Space Complexity: The space complexity is (O(N)), where (N) is the number of elements in the stack, as we are using two stacks to store the elements and their corresponding minimums.

Identifying Problem Isomorphism

The problem “Min Stack” can be approximately mapped to the problem “Max Stack”.

In “Min Stack”, the task is to design a stack that supports push, pop, top, and retrieving the minimum element. In “Max Stack”, the task is to design a stack that supports push, pop, top, and retrieving the maximum element.

The similarity is that they involve designing a stack with an additional operation: retrieving the minimum element in “Min Stack” and the maximum element in “Max Stack”.

“Min Stack” and “Max Stack” share a similar level of complexity as they both require maintaining an auxiliary stack or data structure to keep track of the minimum/maximum element in the stack during the push operation.

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class MinStack:
    def __init__(self):
        self.stack = []
        self.minStack = []

    def push(self, val: int) -> None:
        self.stack.append(val)
        if self.minStack:
            val = min(self.minStack[-1],val)
        self.minStack.append(val)

    def pop(self) -> None:
        self.stack.pop()
        self.minStack.pop()

    def top(self) -> int:
        return self.stack[-1]

    def getMin(self) -> int:
        return self.minStack[-1]

Problem Classification

Language Agnostic Coding Drills

This code is implementing a special type of stack called MinStack which in addition to standard stack operations (push, pop, top), it has an additional operation called getMin that returns the minimum element in the stack in O(1) time complexity. It achieves this by maintaining a separate stack called minStack which keeps track of the minimum element for each state of the stack.

Let’s break down the key concepts and drills to understand this:

1. Understanding the concept of a stack: A stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out). Mainly the following three basic operations are performed in the stack:

   • Push: Adds an item in the stack. If the stack is full, then it is said to be an Overflow condition.
   • Pop: Removes an item from the stack. The items are popped in the reversed order in which they are pushed. If the stack is empty, then it is said to be an Underflow condition.
   • Top/Peek: Returns top element of stack.

2. Building a basic Stack in Python: Implementing push, pop, and top operations.

3. Understanding minStack: A MinStack is a type of stack that is able to return the smallest element in the stack in constant time. It does this by keeping track of the minimum element for each possible state of the stack. This is usually implemented by having another stack that corresponds to the minimum elements.

4. Building a MinStack in Python: In addition to the basic stack operations, implement a getMin operation that returns the minimum element in constant time.

5. Understanding classes and methods in Python: This includes the special __init__ method for initializing instances of a class, and how to define and call methods on class instances.

Let’s now describe the problem-solving approach:

1. Creating a MinStack: We create a MinStack by initializing two empty lists in the __init__ method, stack and minStack. stack is the actual stack, while minStack is a support stack that holds the minimum values.

2. Push operation: In the push operation, we first add the new element to the stack. Then we check if minStack is empty or not. If it’s not empty, we calculate the minimum of the new element and the top element of minStack, and push it to the minStack. This makes sure that the top element of minStack is always the minimum of all elements in the stack.

3. Pop operation: We simply remove the top element from both the stack and minStack.

4. Top operation: We return the top element of stack.

5. getMin operation: We return the top element of minStack which represents the minimum of all elements currently in the stack.

Targeted Drills in Python

1. Understanding the concept of a stack: Stack is a LIFO data structure – the element added last is the first one to come out. In python, the list data type provides the functionality to be used as a stack. Here’s an example code demonstrating that:

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   # This code demonstrates how to use Python list as a stack. stack = [] # We initialize an empty stack stack.append(1) # We push 1 into the stack stack.append(2) # We push 2 into the stack stack.append(3) # We push 3 into the stack print(stack) # We print the current state of the stack. It should print: [1, 2, 3] top = stack[-1] # We peek into the stack. It should return 3. print(top) popped_element = stack.pop() # We pop from the stack. It should remove 3. print(popped_element) print(stack) # We print the current state of the stack. It should print: [1, 2]

2. Building a basic Stack in Python: Let’s create a Stack class with push, pop, and top methods.

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   class Stack: def __init__(self): self.stack = [] def push(self, val): self.stack.append(val) def pop(self): if len(self.stack) == 0: return None return self.stack.pop() def top(self): if len(self.stack) == 0: return None return self.stack[-1]

3. Understanding minStack: This is more of a conceptual drill, understanding the idea behind minStack should suffice for this.

4. Building a MinStack in Python: Building on the Stack class, let’s add the getMin functionality.

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   class MinStack: def __init__(self): self.stack = [] self.minStack = [] def push(self, val): self.stack.append(val) if self.minStack: val = min(self.minStack[-1], val) self.minStack.append(val) def pop(self): if len(self.stack) == 0: return None self.minStack.pop() return self.stack.pop() def top(self): if len(self.stack) == 0: return None return self.stack[-1] def getMin(self): if len(self.minStack) == 0: return None return self.minStack[-1]

5. Understanding classes and methods in Python: This comes as a part of the previous drills where we defined classes and methods. You may want to read more about this topic for a more in-depth understanding.

Remember, Python is just an example here. The same concepts can be applied to many modern languages such as Java, C++, JavaScript, etc. with syntax differences.

10 Prerequisite LeetCode Problems

The “155. Min Stack” problem on LeetCode involves using a stack data structure and asks for the implementation of a special stack that, in addition to usual stack operations (push and pop), also supports retrieving the minimum element in constant time. This introduces an interesting challenge of maintaining auxiliary information (the minimum element) in an efficient way during stack operations.

Here are 10 problems that can help you prepare for this:

1. 225. Implement Stack using Queues: This problem is about implementing a stack using another data structure, which can help you understand the underlying operations of a stack.

2. 232. Implement Queue using Stacks: In contrast to the previous problem, this one requires implementing a queue using stacks, providing an understanding of the relationship between these two data structures.

3. 716. Max Stack: Similar to Min Stack, this problem involves implementing a stack that supports retrieving the maximum element. It’s the counterpart to the Min Stack problem.

4. 20. Valid Parentheses: This problem requires using a stack to check the validity of parentheses strings, offering practice with stack operations.

5. 946. Validate Stack Sequences: This problem asks to validate stack push and pop sequences, which can deepen your understanding of stack operation sequences.

6. 503. Next Greater Element II: This problem asks for finding the next greater element in a circular array, offering a more complex context where a stack can be useful.

7. 1047. Remove All Adjacent Duplicates In String: This problem can be solved with a stack to remove adjacent duplicates, providing another type of problem where stacks are useful.

8. 682. Baseball Game: This problem requires implementing a stack-like behavior to calculate the total score based on an array of operations, providing another unique context for stack usage.

9. 496. Next Greater Element I: This problem is about finding the next greater element in an array, offering another scenario where a stack can be used.

10. 150. Evaluate Reverse Polish Notation: This problem is about evaluating an expression given in Reverse Polish Notation using a stack, offering practice with stack operations in an expression evaluation context.

These problems will allow you to practice your understanding of the stack data structure and how it can be used in different scenarios.

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Problem Statement:Design a stack that supports push, pop, top, and retrieving the minimum element in constant time.

Implement the MinStack class:

MinStack() initializes the stack object. void push(int val) pushes the element val onto the stack. void pop() removes the element on the top of the stack. int top() gets the top element of the stack. int getMin() retrieves the minimum element in the stack. You must implement a solution with O(1) time complexity for each function.

Example 1:

Input [“MinStack”,“push”,“push”,“push”,“getMin”,“pop”,“top”,“getMin”] [[],[-2],[0],[-3],[],[],[],[]]

Output [null,null,null,null,-3,null,0,-2]

Explanation MinStack minStack = new MinStack(); minStack.push(-2); minStack.push(0); minStack.push(-3); minStack.getMin(); // return -3 minStack.pop(); minStack.top(); // return 0 minStack.getMin(); // return -2

Constraints:

-231 <= val <= 231 - 1 Methods pop, top and getMin operations will always be called on non-empty stacks. At most 3 * 104 calls will be made to push, pop, top, and getMin.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Stepwise Refinement

1. Could you please provide a stepwise refinement of our approach to solving this problem?

2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

3. Could you identify any parts of the problem that can be solved independently?

4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

1. Parameters:

   • What are the inputs to the method?
   • What types are these parameters?
   • What do these parameters represent in the context of the problem?

2. Preconditions:

   • Before this method is called, what must be true about the state of the program or the values of the parameters?
   • Are there any constraints on the input parameters?
   • Is there a specific state that the program or some part of it must be in?

3. Method Functionality:

   • What is this method expected to do?
   • How does it interact with the inputs and the current state of the program?

4. Postconditions:

   • After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
   • What does the return value represent or indicate?
   • What side effects, if any, does the method have?

5. Error Handling:

   • How does the method respond if the preconditions are not met?
   • Does it throw an exception, return a special value, or do something else?

Problem Decomposition

1. Problem Understanding:

   • Can you explain the problem in your own words? What are the key components and requirements?

2. Initial Breakdown:

   • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?

3. Subproblem Refinement:

   • For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?

4. Task Identification:

   • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?

5. Task Abstraction:

   • For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?

6. Method Naming:

   • Can you give each task a simple, descriptive name that makes its purpose clear?

7. Subproblem Interactions:

   • How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

2. Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?

3. If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.

4. If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?

5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

6. Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

1. What are the high-level problem-solving strategies or techniques being used by this code?

2. If you had to explain the purpose of this code to a non-programmer, what would you say?

3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

4. Could you describe the algorithmic approach used by this code in plain English?

5. What are the key steps or operations this code is performing on the input data, and why?

6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Similar Problems

Can you suggest 10 problems from LeetCode that require similar problem-solving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.