Largest Component Size by Common Factor

To find the largest connected component in the graph where an edge exists between numbers sharing a common factor greater than 1, we can use disjoint-set data structure (union-find).

We will first create a union-find class that will help us in joining numbers that have a common factor. Then, we will iterate through the numbers, find their factors and join them.

Here’s the code:

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class UnionFind:
    def __init__(self, size):
        self.parent = [i for i in range(size)]
        self.rank = [0] * size

    def find(self, x):
        if x != self.parent[x]:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]

    def union(self, x, y):
        rootX = self.find(x)
        rootY = self.find(y)

        if rootX != rootY:
            if self.rank[rootX] > self.rank[rootY]:
                self.parent[rootY] = rootX
            else:
                self.parent[rootX] = rootY
                if self.rank[rootX] == self.rank[rootY]:
                    self.rank[rootY] += 1

class Solution:
    def largestComponentSize(self, nums: List[int]) -> int:
        max_num = max(nums)
        uf = UnionFind(max_num + 1)
        for num in nums:
            for factor in range(2, int(num**0.5) + 1):
                if num % factor == 0:
                    uf.union(num, factor)
                    uf.union(num, num // factor)

        component_size = Counter(uf.find(num) for num in nums)
        return max(component_size.values())

Here, the UnionFind class takes care of joining the elements that have a common factor, and then we simply use a Counter from collections to find the count of each connected component in the numbers and return the maximum size.

Identifying Problem Isomorphism

“Number of Islands” is approximately isomorphic to “Largest Component Size by Common Factor”.

In “Number of Islands”, you’re given a grid of cells where each cell represents land (1) or water (0), and an island is defined as a group of adjacent land cells (horizontally or vertically). Your task is to find the number of islands.

The mapping between the two problems:

Each cell in the grid in the “Number of Islands” problem maps to an integer in the “Largest Component Size by Common Factor” problem.
Adjacency in the “Number of Islands” problem maps to having a common factor in the “Largest Component Size by Common Factor” problem.
An island in the “Number of Islands” problem maps to a component in the “Largest Component Size by Common Factor” problem.

The core idea of both problems is to find connected components based on some criteria (adjacency in “Number of Islands”, common factor in “Largest Component Size by Common Factor”) and either count them (“Number of Islands”) or find the largest one (“Largest Component Size by Common Factor”).

“Number of Islands” is simpler as it involves a more straightforward concept of adjacency in a grid. The “Largest Component Size by Common Factor” problem is more complex as it requires knowledge of number theory and understanding how to identify and group integers based on common factors.

10 Prerequisite LeetCode Problems

“Largest Component Size by Common Factor” is a graph theory problem where the nodes are the numbers and an edge is drawn between two numbers if they have a common factor greater than 1. You have to find the largest connected component in the graph.

The problem involves graph theory (particularly connected components), union-find data structure, and prime factorization. Here are 10 problems to build these foundational skills:

Problem 200. Number of Islands This problem is about finding connected components in a binary grid. It is a simpler version of the given problem where the graph is explicit, while in the given problem, the graph is implicit.
Problem 547. Number of Provinces Similar to Problem 200, this problem requires finding connected components in a graph. It will help in understanding the depth-first search (DFS) approach to solve such problems.
Problem 684. Redundant Connection This problem introduces the union-find data structure, which is instrumental in solving problems related to connectivity in a graph.
Problem 721. Accounts Merge This problem requires finding connected components in a graph using DFS or union-find.
Problem 785. Is Graph Bipartite? This problem helps with understanding how to deal with more complex graph structures and can improve your understanding of DFS and breadth-first search (BFS).
Problem 1319. Number of Operations to Make Network Connected This problem also involves finding connected components in a graph. It adds a bit more complexity, as it involves making certain operations to connect the components.
Problem 204. Count Primes This problem helps you understand prime numbers, which are key to finding common factors between numbers.
Problem 263. Ugly Number This problem requires finding prime factors of a number, a concept which is used in the given problem to find common factors.
Problem 952. Largest Component Size by Common Factor This problem is very similar to the one you’re preparing for, and it can help you understand how to approach it.
Problem 1192. Critical Connections in a Network This problem introduces the concept of graph theory and DFS. It will help in understanding the importance of connectivity and components in a graph.

First get a good grip on how to find connected components in a graph, understand the union-find data structure, and then how to find prime factors of a number. These concepts are key to solving the given 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?

Problem Classification

Problem Statement:You are given an integer array of unique positive integers nums. Consider the following graph:

There are nums.length nodes, labeled nums[0] to nums[nums.length - 1], There is an undirected edge between nums[i] and nums[j] if nums[i] and nums[j] share a common factor greater than 1. Return the size of the largest connected component in the graph.

Example 1:

Input: nums = [4,6,15,35] Output: 4

Example 2:

Input: nums = [20,50,9,63] Output: 2

Example 3:

Input: nums = [2,3,6,7,4,12,21,39] Output: 8

Constraints:

1 <= nums.length <= 2 * 104 1 <= nums[i] <= 105 All the values of nums are unique.

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 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?

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 following piece of complex software code.

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.

Q&A

What are the reasons for making these mistakes in the given code?