Minimum Height Trees

To find the Minimum Height Trees (MHTs), we can start by trimming the leaves of the tree iteratively until we’re left with one or two nodes (roots of MHTs). Here’s the step-by-step approach:

Create an Adjacency List: Construct an adjacency list for the given nodes and edges.
Find Initial Leaves: Identify the nodes that are leaves (having only one neighbor).
Trim Leaves Iteratively: In each iteration, remove the current leaves and update the neighbor counts for adjacent nodes. Add new leaves to the list for the next iteration.
Collect Results: When you are left with one or two nodes, these nodes are the roots of the MHTs.

Here’s the code:

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from collections import defaultdict

class Solution:
    def findMinHeightTrees(self, n: int, edges: List[List[int]]) -> List[int]:
        if n == 1:
            return [0]

        # Step 1: Create an adjacency list
        adj_list = defaultdict(set)
        for a, b in edges:
            adj_list[a].add(b)
            adj_list[b].add(a)

        # Step 2: Find initial leaves
        leaves = [node for node in adj_list if len(adj_list[node]) == 1]

        # Step 3: Trim leaves iteratively
        while n > 2:
            new_leaves = []
            for leaf in leaves:
                neighbor = adj_list[leaf].pop()
                adj_list[neighbor].remove(leaf)
                if len(adj_list[neighbor]) == 1:
                    new_leaves.append(neighbor)
                n -= 1
            leaves = new_leaves

        # Step 4: Collect results
        return leaves

This code first constructs the adjacency list representation of the tree. Then it iteratively trims the leaves, reducing the number of nodes until only one or two are left, which are the roots of the MHTs.

The time complexity of this code is O(n), where n is the number of nodes, and the space complexity is also O(n) because of the use of the adjacency list.

Define the Terms Tree: A tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any connected graph without simple cycles is a tree.

Tree Height: The height of a rooted tree is the number of edges on the longest downward path between the root and a leaf.

Minimum Height Tree: Among all possible rooted trees, those with minimum height (i.e. min(h)) are called minimum height trees

Define the Interface Input: n (number of vertices), edges (array of arrays) Output: array of integers

Analyze the Input and Output

Identify the Invariants

Identify the constraints

1 <= n <= 2 * 104
edges.length == n - 1
0 <= ai, bi < n
ai != bi
All the pairs (ai, bi) are distinct.
The given input is guaranteed to be a tree and there will be no repeated edges.

Search the Problem Space

Classify the Problem

Analyze the Examples Edge case: Example 3. Simple case: Example 2

Solve the Problem by Hand 3 nodes - (0 rooted tree will have a height of: 2) (1 rooted tree will have a height of: 1) (2 rooted tree will have a height of: 2) - 1 mht

4 nodes - 1 mht (0 rooted tree will have a height of: 3) (1 rooted tree will have a height of: 2) (2 rooted tree will have a height of: 2) (3 rooted tree will have a height of: 3)

Do you have to create all possible trees?

No. This is brute force approach. How do we identify the vertices that will result in mht?
Hint: You have to use the degree of the node somehow to figure out the answer. As we are creating the connections between the vertices, we can keep track of the degree of a vertex.

If the degree becomes 0, get back to parent root.

6 nodes - 2 mht

Describe the Pattern

The root is changing
What is not changing? The intermediate node remains the same. What is the property of this intermediate node? Intermediate node is connected to all other nodes. Degree of a vertex Degree of one is 3
The leaves are changing

Tree can have only one root. Tree can have many leaves.

In example 2, node 3 has degree 4 and becomes one of the mht is rooted in 3.

Distill the Observations to Create Insights

Brute Force Approach

Analyze Time and Space Complexity

Time: O( ) Space: O( )

Outline the Solution

If n is 1 return [0] (example 3 - edge case)
Create an adjacency list
Count the degree for every vertex
If n is 2, we can return the edge.
The terminating condition is n > 2
Select all the vertices with degree = 1
Go thorough the leaves and delete them from the tree Remove the edge of the leaf from the tree
Update the degree of the intermediate nodes that the leaves were connnected decrease it by one.
The number of vertices must be reduced by the number of leaves
Return the remaining vertices in the tree.

Key Takeaways

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# @param {Integer} n
# @param {Integer[][]} edges
# @return {Integer[]}
def find_min_height_trees(n, edges)
  return [0] if n == 1
 
    list = Hash.new{|h, v| h[v] = []}
    degree = Hash.new(0)

    edges.each do |a, b|
      list[a] << b
      degree[a] += 1
        list[b] << a
      degree[b] += 1
    end

    leaves = {}

    while n > 2
      leaves = degree.select{|k, v| v == 1}

        leaves.each do |vertex, _|
           degree.delete(vertex)

           list[vertex].each do |c|
              if degree.key?(c)
                 degree[c] -= 1 
              end
           end
        end
        n -= leaves.size
    end

    degree.keys
end

Identifying Problem Isomorphism

“Minimum Height Trees” falls into the class of graph theory problems, specifically those that deal with understanding tree structures and graph traversal.

A simpler problem to this is “LeetCode Find Center of Star Graph”. In this problem, you are given an undirected star graph, and you need to find the center of this star graph. This is simpler because a star graph has a unique center, and finding it involves less complexity compared to computing minimum height trees.

A more complex problem is “Network Delay Time”. This problem requires finding how long it will take for a signal to reach all nodes in a network, starting from a given node. It is more complex because it requires a more advanced graph algorithm, Dijkstra’s algorithm, to find the shortest paths to all other nodes from the start.

The reasoning for these selections:

“Find Center of Star Graph” involves identifying a particular node in a graph, a similar task to “Minimum Height Trees” where you need to find nodes (roots) to minimize tree height. However, “Find Center of Star Graph” is simpler because of the star graph structure.
“Network Delay Time” requires a deeper understanding of graph traversal and pathfinding algorithms. Like “Minimum Height Trees”, it involves selecting a starting point and traversing the graph, but with the added complexity of weights on the edges and finding shortest paths.

So, the order of problems from simpler to more complex, based on understanding of graph structures and graph traversal, would be:

“Find Center of Star Graph”
“Minimum Height Trees”
“Network Delay Time”

This mapping is approximate. While these problems all deal with the broad concept of graph structures and traversal, the specific problem constraints and details can influence the complexity and solution approach.

10 Prerequisite LeetCode Problems

Identify 10 LeetCode problems of lesser complexity, excluding the problem itself that I should solve as preparation for tackling . 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 problems 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?

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?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level 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

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

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 high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, 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 language-agnostic 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 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.

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 problem-specific 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.