Find the Minimum and Maximum Number of Nodes Between Critical Points
To find the critical points in a linked list and determine the minimum and maximum distances between them, we’ll follow these steps:
Identify Critical Points: Iterate through the linked list, and for each node, check whether it’s a local maxima or local minima. To be a local maxima, the current node’s value must be greater than both the previous and the next node’s values. To be a local minima, the current node’s value must be smaller than both the previous and the next node’s values.
Record the Positions: For every critical point found, record its position in the list (e.g., using a list to store the indices).
Calculate the Distances: Once all the critical points are found, calculate the minimum and maximum distances between the indices.
Return the Result: Return the calculated minimum and maximum distances, or
[1, 1]
if there are fewer than two critical points.
Here’s the code:


 For
head = [5,3,1,2,5,1,2]
, the output will be[1,3]
.  For
head = [1,3,2,2,3,2,2,2,7]
, the output will be[3,3]
.
The time complexity of this solution is ( O(n) ), where ( n ) is the number of nodes in the linked list, and the space complexity is ( O(m) ), where ( m ) is the number of critical points.
The error is occurring because prev1
is assigned the integer value of current.val
, but later in the code, I tried to access prev1.val
. Since prev1
is an integer and not a node, this leads to the AttributeError.
The correct approach is to compare the current node’s value with the previous and next node’s values directly. Here’s the corrected code:


By replacing prev1.val
with prev_val
and modifying the conditions accordingly, we eliminate the AttributeError. This corrected code should now work as intended.
Identifying Problem Isomorphism
“Find the Minimum and Maximum Number of Nodes Between Critical Points” can be mapped approximately to “Longest Increasing Subsequence”.
Here’s the reasoning:
In “Find the Minimum and Maximum Number of Nodes Between Critical Points”, you have to identify critical points in a binary tree and then determine the minimum and maximum distances between these critical points.
In “Longest Increasing Subsequence”, you have to find the length of the longest subsequence of a given sequence in which all elements are sorted in increasing order.
Even though one problem is based on a binary tree and the other on an array, both involve identifying critical sequences (longest increasing subsequence, nodes between critical points) within a data structure.
In both, you’re scanning through the data structure (tree traversal in one, array traversal in the other), keeping track of information as you go (whether it’s the current longest increasing subsequence or the distance between critical points), and updating this information as necessary.
“Longest Increasing Subsequence” is simpler because it deals with a linear data structure (arrays), while “Find the Minimum and Maximum Number of Nodes Between Critical Points” deals with a more complex data structure (binary trees). However, both problems require an understanding of dynamic programming to solve efficiently.
10 Prerequisite LeetCode Problems
For “2058. Find the Minimum and Maximum Number of Nodes Between Critical Points”, the following is a good preparation:
“21. Merge Two Sorted Lists”: This is a fundamental problem that deals with the concept of manipulating linked lists and comparing node values, which is a basic requirement for the main problem.
“83. Remove Duplicates from Sorted List”: This problem introduces the concept of comparing a node with its next node, which is helpful for identifying the local minima and maxima in the main problem.
“141. Linked List Cycle”: This problem helps in understanding how to traverse a linked list and handle edge cases, which can be useful for the main problem.
“206. Reverse Linked List”: It’s a basic problem to get a hang of linked list manipulation. Understanding this will help in traversing the linked list from both directions, a possible necessity when identifying critical points.
“876. Middle of the Linked List”: This problem enhances understanding of counting nodes in a linked list which can be used to calculate distances between critical points.
“19. Remove Nth Node From End of List”: It offers practice in handling pointers at different positions in a linked list, a skill that’s useful for identifying critical points.
“237. Delete Node in a Linked List”: This problem teaches how to handle and remove nodes from a linked list. Useful for edge case scenarios in the main problem.
“234. Palindrome Linked List”: This problem isn’t directly related, but manipulating the linked list for a palindrome check can be a good practice for linked list traversal.
“160. Intersection of Two Linked Lists”: The techniques used for finding the intersection can be used for finding the critical points and calculating the distance between them.
“203. Remove Linked List Elements”: This problem provides practice on scanning a linked list and conditionally modifying its structure, which is key to locating and analyzing critical points.
These cover linked list manipulation, comparison of node values, and handling edge cases, which are crucial for the main problem.
Problem Classification
Problem Statement:A critical point in a linked list is defined as either a local maxima or a local minima. A node is a local maxima if the current node has a value strictly greater than the previous node and the next node. A node is a local minima if the current node has a value strictly smaller than the previous node and the next node. Note that a node can only be a local maxima/minima if there exists both a previous node and a next node.
Given a linked list head, return an array of length 2 containing [minDistance, maxDistance] where minDistance is the minimum distance between any two distinct critical points and maxDistance is the maximum distance between any two distinct critical points. If there are fewer than two critical points, return [1, 1].
Example 1:
Input: head = [3,1] Output: [1,1] Explanation: There are no critical points in [3,1].
Example 2:
Input: head = [5,3,1,2,5,1,2] Output: [1,3] Explanation: There are three critical points:
 [5,3,1,2,5,1,2]: The third node is a local minima because 1 is less than 3 and 2.
 [5,3,1,2,5,1,2]: The fifth node is a local maxima because 5 is greater than 2 and 1.
 [5,3,1,2,5,1,2]: The sixth node is a local minima because 1 is less than 5 and 2. The minimum distance is between the fifth and the sixth node. minDistance = 6  5 = 1. The maximum distance is between the third and the sixth node. maxDistance = 6  3 = 3.
Example 3:
Input: head = [1,3,2,2,3,2,2,2,7] Output: [3,3] Explanation: There are two critical points:
 [1,3,2,2,3,2,2,2,7]: The second node is a local maxima because 3 is greater than 1 and 2.
 [1,3,2,2,3,2,2,2,7]: The fifth node is a local maxima because 3 is greater than 2 and 2. Both the minimum and maximum distances are between the second and the fifth node. Thus, minDistance and maxDistance is 5  2 = 3. Note that the last node is not considered a local maxima because it does not have a next node.
Constraints:
The number of nodes in the list is in the range [2, 105]. 1 <= Node.val <= 105
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.
Clarification Questions
What are the clarification questions we can ask about this problem?
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?
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 realworld 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?
How to visualize these 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 nontechnical 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 ProblemSolving 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
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
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?
Is invariant and loop invariant the same for 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
Parameters:
 What are the inputs to the method?
 What types are these parameters?
 What do these parameters represent in the context of the problem?
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?
Method Functionality:
 What is this method expected to do?
 How does it interact with the inputs and the current state of the program?
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?
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
Problem Understanding:
 Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
 Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
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?
Task Identification:
 Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
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?
Method Naming:
 Can you give each task a simple, descriptive name that makes its purpose clear?
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
Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
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?
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.
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?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
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 code for the solution of this problem.
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
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.
Dissect the code and identify each distinct concept it contains. Remember, this process should be languageagnostic and generally applicable to most modern programming languages.
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.
Next, describe the problemsolving 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 stepbystep 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 Pythonbased coding drills for each of those concepts.
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.
In addition to the general concepts, identify and write coding drills for any problemspecific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
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 problemsolving 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. The response text is of the following format:
Here are 10 problems that use similar underlying concepts: