Longest Univalue Path

You are given the root of a binary tree, and the task is to find the length of the longest path where all the nodes have the same value. This path can start and end at any node, not necessarily the root.

Algorithm

  1. Initialize Variables: Initialize a variable max_length to keep track of the maximum length of the univalue path.

  2. Recursion Function: Create a recursive function dfs(node) that returns the length of the longest univalue path starting from the given node.

    • If the node is None, return 0.
    • Recursively call dfs(node.left) and dfs(node.right) to get the univalue path lengths for the left and right children.
    • If the node’s value is equal to its left child’s value, use the left child’s path length; otherwise, set it to 0.
    • Do the same for the right child.
    • The length of the univalue path for the current node is the sum of the left and right path lengths.
    • Update max_length with the maximum of itself and the current node’s univalue path length.
    • Return the maximum of the left and right path lengths plus 1.

  3. Start Recursion: Call dfs(root) to start the recursion from the root.

  4. Result: Return the value of max_length.

Code

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class Solution:
    def longestUnivaluePath(self, root: Optional[TreeNode]) -> int:
        max_length = 0

        def dfs(node):
            nonlocal max_length
            if not node:
                return 0

            left_length = dfs(node.left)
            right_length = dfs(node.right)

            left_length = left_length + 1 if node.left and node.left.val == node.val else 0
            right_length = right_length + 1 if node.right and node.right.val == node.val else 0

            max_length = max(max_length, left_length + right_length)

            return max(left_length, right_length)

        dfs(root)
        return max_length

Explanation

  • The function dfs calculates the longest univalue path starting from the given node.
  • It recursively traverses the left and right children, considering them in the path only if they have the same value as the current node.
  • max_length keeps track of the maximum length found.

Insights

  • Time Complexity: (O(n)), where (n) is the number of nodes in the tree.
  • Space Complexity: (O(h)), where (h) is the height of the tree (due to recursive call stack).

This solution efficiently calculates the length of the longest univalue path in the given binary tree, satisfying the given constraints.

  1. Understand the problem

  2. Manually work on the given examples and see how you can compute the value

  3. From the perspective of the root, I have to find the max of left subtree’s value and right subtree’s value

  4. Longest path may or may not go through the root

  5. Define terms in the problem statement All the nodes in the path must be of the same value Identify the invariant

  6. Recursive approach. Base cases: N = 0 N = 1

    result = 0

Minimize the number of cases
We can one base case where N = 0
When you have only one node, or reach a leaf node
  You are handling them in the same way

N = 2
The node values are the same
result = 1

The node values are NOT the same
result = 0

N = 3
The node values are the same
result = 2

The node values are NOT the same
result = 0

    What is the unit of work?

    • My left child and right child values are the same as my value so the result = 2

    What is the role of a leaf node? Defines the base case What is the role of an intermediate node?

    What is my responsibility as an intermediate node? Should I return a value?

    How are we dealing with leaf nodes?

    • Special case - we hit the base case
    • left child - 0
    • right child - 0

    Recursive cases: Problem Decomposition

    • left subtree
    • right subtree

    We need do the unit of work after the recursive calls

  7. How do we keep track of different lengths?

  8. How do we keep updating the max values? max variable ?

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# Definition for a binary tree node.
# class TreeNode
#     attr_accessor :val, :left, :right
#     def initialize(val = 0, left = nil, right = nil)
#         @val = val
#         @left = left
#         @right = right
#     end
# end

def univalue_path(root)
#     O node
  if root.nil?
      return 0
  end
#   1 node
  if root.left.nil? and root.right.nil?
    return 0
  end
#   2 or more nodes  
  left = univalue_path(root.left)
  right = univalue_path(root.right)
   
   left_edge = 0 
   right_edge = 0
  #  If I have a left child then I will check if my value is same my left child value
  if root.left and root.val == root.left.val
      left_edge = left + 1
  end
    
  if root.right and root.val == root.right.val
     right_edge = right + 1
  end
    
  @max = [@max, left_edge + right_edge].max
    
  [left_edge, right_edge].max
end

# @param {TreeNode} root
# @return {Integer}
def longest_univalue_path(root)
    @max = 0
  univalue_path(root)
    @max
end

Identifying Problem Isomorphism

“687. Longest Univalue Path” is about finding the length of the longest path within a binary tree such that every node on the path has the same value. This path may or may not pass through the root.

This problem can be seen as isomorphic to “250. Count Univalue Subtrees”. The problem “250. Count Univalue Subtrees” asks to count the number of subtrees in the binary tree that are univalue - which means all the nodes in the subtree have the same value.

The isomorphism lies in the fact that both problems focus on ‘univalue’ properties in a binary tree, though they treat it differently. Both problems can be solved with a depth-first search (DFS) strategy that starts from the root, recursively checks subtrees, and updates the count or the maximum length based on whether the subtree meets certain conditions (being univalue).

While the two problems share the same concept of ‘univalue’, they are not exactly the same. The “687. Longest Univalue Path” problem is about finding the longest path of nodes with the same value, whereas the “250. Count Univalue Subtrees” problem is about counting the number of subtrees where all nodes have the same value. Thus, the problems are structurally isomorphic, but they’re not exactly the same.

10 Prerequisite LeetCode Problems

Identify 10 LeetCode problems of lesser complexity, excluding the problem itself that I should solve as preparation for tackling 687. Longest Univalue Path . Include the name of the given problem in the response before the list. Do not add double quotes for the items in the list. Include the reason why that problem is relevant. The format of the response must be:

For the , the following is a good preparation:

Problem Classification

Problem Statement: 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?

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does it map to on Leetcode for 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 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?

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 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?

Is invariant and loop invariant the same for this problem?

Identify Recursion Invariant

Is there an invariant during recursion in this problem?

Is invariant and invariant during recursion 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

  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 code for the solution of this problem.

  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. Do not include the original problem. The response text is of the following format. First provide this as the first sentence: Here are 10 problems that use similar underlying concepts: