Count the Number of Complete Components

To solve this problem, we need to identify the complete connected components in the graph. A connected component is said to be complete if there exists an edge between every pair of its vertices.

The following steps outline the approach to find the number of complete connected components:

  1. Initialize the Graph: Create a graph using an adjacency list or adjacency matrix to represent the vertices and edges.

  2. Identify Connected Components: Perform a Depth-First Search (DFS) or Breadth-First Search (BFS) to identify all the connected components.

  3. Check for Completeness: For each connected component, check if it is complete. A connected component with k vertices is complete if and only if it has k * (k - 1) / 2 edges.

  4. Count Complete Components: Count the number of complete connected components and return the count.

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class Solution:
    def countCompleteComponents(self, n: int, edges: List[List[int]]) -> int:
        # Initialize the adjacency list
        graph = [[] for _ in range(n)]
        for a, b in edges:
            graph[a].append(b)
            graph[b].append(a)

        # Perform Depth-First Search to find connected components
        def dfs(node, visited, component):
            visited[node] = True
            component.append(node)
            for neighbor in graph[node]:
                if not visited[neighbor]:
                    dfs(neighbor, visited, component)

        visited = [False] * n
        complete_count = 0

        for i in range(n):
            if not visited[i]:
                component = []
                dfs(i, visited, component)

                # Check if the connected component is complete
                edge_count = sum(len(graph[v]) for v in component) // 2
                if edge_count == len(component) * (len(component) - 1) // 2:
                    complete_count += 1

        return complete_count

The time complexity of this code is O(n^2), as we need to traverse all the vertices and edges, and the space complexity is O(n + e), where n is the number of vertices and e is the number of edges.

Identifying Problem Isomorphism

“Count the Number of Complete Components” can be mapped to “Number of Connected Components in an Undirected Graph”.

Reasoning:

Both problems involve identifying distinct connected components within a given structure. In “Count the Number of Complete Components”, you have to find all “complete” components in a graph or another data structure. In “Number of Connected Components in an Undirected Graph”, you are required to find all the connected components in a graph.

These problems are isomorphic as both rely on concepts of graph theory and need the exploration of the graph to count the distinct components. Although the conditions for being a “complete” component may differ from what defines a “connected” component, the overall procedure of traversing the graph and marking visited nodes is similar.

“Number of Connected Components in an Undirected Graph” is simpler, as it does not require checking for the completeness of a component, but just its connectivity. “Count the Number of Complete Components” might involve additional checks or conditions, making it more complex.

10 Prerequisite LeetCode Problems

For this, the following are a good preparation:

  1. “323. Number of Connected Components in an Undirected Graph” - The problem teaches how to count connected components in an undirected graph, which is the main concept involved in the target problem.

  2. “547. Number of Provinces” - This problem is about counting the connected components in a graph and is a good introduction to the concept.

  3. “200. Number of Islands” - This problem is about counting connected components in a 2D grid, which can help you to understand how to traverse a graph.

  4. “684. Redundant Connection” - This problem helps understand how to manage and manipulate graph connections.

  5. “785. Is Graph Bipartite?” - This problem will help in understanding how to divide a graph into components based on certain conditions.

  6. “261. Graph Valid Tree” - This problem will help in understanding how to validate the connectivity of a graph.

  7. “207. Course Schedule” - This problem is about managing dependencies in a graph, which can be helpful for understanding complex component interactions.

  8. “210. Course Schedule II” - This problem is an extension of “207. Course Schedule” and provides further practice on managing graph dependencies.

  9. “133. Clone Graph” - This problem will help in understanding how to clone or replicate a graph, which is useful for various graph operations.

  10. “1202. Smallest String With Swaps” - This problem will help in understanding how to manipulate the nodes of a graph based on certain conditions.

These focus on handling and operating on graphs, and specifically on understanding connected components, which is crucial for solving the main problem.

Problem Classification

Problem Statement:You are given an integer n. There is an undirected graph with n vertices, numbered from 0 to n - 1. You are given a 2D integer array edges where edges[i] = [ai, bi] denotes that there exists an undirected edge connecting vertices ai and bi. Return the number of complete connected components of the graph. A connected component is a subgraph of a graph in which there exists a path between any two vertices, and no vertex of the subgraph shares an edge with a vertex outside of the subgraph. A connected component is said to be complete if there exists an edge between every pair of its vertices.

Example 1:

Input: n = 6, edges = [[0,1],[0,2],[1,2],[3,4]] Output: 3 Explanation: From the picture above, one can see that all of the components of this graph are complete.

Example 2:

Input: n = 6, edges = [[0,1],[0,2],[1,2],[3,4],[3,5]] Output: 1 Explanation: The component containing vertices 0, 1, and 2 is complete since there is an edge between every pair of two vertices. On the other hand, the component containing vertices 3, 4, and 5 is not complete since there is no edge between vertices 4 and 5. Thus, the number of complete components in this graph is 1.

Constraints:

1 <= n <= 50 0 <= edges.length <= n * (n - 1) / 2 edges[i].length == 2 0 <= ai, bi <= n - 1 ai != bi There are no repeated edges.

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:

The problem “Count the Number of Complete Components” can be mapped to “Number of Connected Components in an Undirected Graph”.

Reasoning:

Both problems involve determining the number of complete or connected components in a given graph. The primary task is to traverse the graph and increment the count whenever a separate component is discovered.

Although the naming and framing of these problems might be different, the core computational task remains largely the same. However, it’s worth mentioning that depending on the specific conditions or additional requirements in the problem “Count the Number of Complete Components”, it could potentially be more complex than “Number of Connected Components in an Undirected Graph”.

“Count the Number of Complete Components” is related to graph theory and concepts such as connected components and depth-first search (DFS) or breadth-first search (BFS). Here are 10 problems to prepare for it:

  1. 200. Number of Islands: This problem is a basic introduction to the concept of finding connected components in a 2D grid.

  2. 323. Number of Connected Components in an Undirected Graph: This problem will help you practice finding connected components in a general graph.

  3. 547. Number of Provinces: This problem is another example of finding connected components in a graph, with a slightly different representation.

  4. 1042. Flower Planting With No Adjacent: This problem is about graph coloring, which is a similar concept to finding complete components.

  5. 695. Max Area of Island: This problem is related to finding connected components in a 2D grid, with an additional requirement of finding the maximum area.

  6. 133. Clone Graph: This problem will give you practice in traversing and copying a graph, which is a useful skill in solving graph problems.

  7. 207. Course Schedule: This problem introduces the concept of directed graphs and topological sorting.

  8. 210. Course Schedule II: This problem expands on the previous one by requiring you to return a valid order of courses, which introduces ordering to connected components.

  9. 261. Graph Valid Tree: This problem will test your understanding of the properties of trees as a specific kind of graph.

  10. 399. Evaluate Division: This problem involves building a graph from given equations and then querying it, a concept that can be related to complete components.

By practicing these problems, you will gain a strong understanding of graph theory and connected components, which will be invaluable for solving the “Count the Number of Complete Components” problem.

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class DSU:
    def __init__(self, n):
        self.p = list(range(n))
        self.rank = [0] * n
        self.count = [1] * n
    def find(self, x):
        if x != self.p[x]:
            self.p[x] = self.find(self.p[x])
        return self.p[x]
    def union(self, x, y):
        xx, yy = self.find(x), self.find(y)
        if xx == yy: return
        self.count[xx] = self.count[yy] = self.count[xx] + self.count[yy]
        if self.rank[xx] < self.rank[yy]: self.p[xx] = yy
        else: self.p[yy] = xx
        if self.rank[xx] == self.rank[yy]: self.rank[xx] += 1
    def size_of_group_that_x_is_a_part_of(self, x):
        return self.count[self.find(x)]

class Solution:
    def countCompleteComponents(self, n, edges):
        uf, counter = DSU(n), Counter()
        for x, y in edges: 
            uf.union(x, y)
            counter[x] += 1
            counter[y] += 1

        groups = set(uf.find(i) for i in range(n))

        for i in range(n):
            if uf.size_of_group_that_x_is_a_part_of(i) != counter[i] + 1:
                groups.discard(uf.find(i))

        return len(groups)

Problem Classification

Domain Categorization:

The provided problem statement falls into the domain of Graph Theory and Graph Algorithms. It involves the manipulation and analysis of a graph to determine the number of complete connected components.

What Components:

  1. An integer n is given which represents the number of vertices in an undirected graph.
  2. A 2D integer array edges is given where edges[i] = [ai, bi] denotes that there exists an undirected edge connecting vertices ai and bi.
  3. The task is to return the number of complete connected components of the graph.
  4. A connected component is defined as a subgraph in which there exists a path between any two vertices, and no vertex of the subgraph shares an edge with a vertex outside of the subgraph.
  5. A connected component is said to be complete if there exists an edge between every pair of its vertices.

Problem Classification:

This problem can be classified as a Graph Analysis Problem. It involves understanding the structure of a graph and identifying specific components within the graph (in this case, complete connected components).

Such problems often require the use of classic graph algorithms such as depth-first search (DFS) or breadth-first search (BFS) to explore the graph, along with additional logic to identify and count the specific components of interest. Understanding the properties of graphs and the principles of graph traversal is crucial to solving these types of problems.

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 this problem map to the corresponding isomorphic problem on Leetcode ?

Language Agnostic Coding Drills

  1. Identify Coding Concepts:

    a. Class Definition: The code defines two classes, DSU and Solution, each with its own methods. This involves understanding of object-oriented programming.

    b. Initialization Method: The __init__ function in the DSU class serves to initialize the object’s state. This concept is fundamental to object-oriented programming.

    c. Recursive Method: The find method in the DSU class uses recursion to find the parent of a given node.

    d. Union Operation: This is a specific operation from Disjoint Set Union (DSU) data structure, which merges two subsets.

    e. Set Data Structure: The code uses a set to store the unique groups identified.

    f. Counting using Dictionary: The code uses the Counter from collections to keep track of the counts of elements.

  2. Ordered List of Concepts with Difficulty Level:

    a. Class Definition (Basic): This is a basic concept of object-oriented programming.

    b. Initialization Method (Basic): This is also a basic concept in object-oriented programming.

    c. Set Data Structure (Basic): This is a fundamental data structure used in many programming problems.

    d. Counting using Dictionary (Intermediate): While dictionaries are a fundamental data structure, using them for counting requires a slightly higher level of understanding.

    e. Recursive Method (Intermediate): Recursion can be a challenging concept for beginners, as it involves thinking about problems in a self-similar way.

    f. Union Operation (Advanced): The union operation is specific to the Disjoint Set Union (DSU) data structure. This requires a more advanced understanding of data structures and algorithms.

  3. Problem-Solving Approach:

    a. Creating DSU Class: The DSU class is used to represent the graph. The DSU class provides methods for performing operations like find and union.

    b. Creating Solution Class: The Solution class is used to encapsulate the main algorithm. It includes a method countCompleteComponents that utilizes the DSU class to find and count the complete components.

    c. Applying Union Operation: The union operation is used to merge the vertices that are connected by an edge. This operation helps in identifying the groups or components within the graph.

    d. Counting Occurrences: Counting the number of occurrences of each vertex in the edges helps us know the degree of each vertex.

    e. Identifying Groups: After applying union operations, the parent node of each group is identified. This information is used to determine the complete components of the graph.

    f. Filtering Complete Components: Finally, we filter out the groups that are not complete components by comparing the size of each group with the degrees of the vertices in it. The groups that pass the condition are the complete components.

By integrating these concepts and steps, we can build a comprehensive solution for counting the number of complete components in a graph.

Targeted Drills in Python

Drill 1 - Class Definition

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class MyClass:
    def my_method(self):
        print("Hello, world!")

In this drill, we’ve defined a basic class and a simple method within it.

Drill 2 - Initialization Method

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class MyClass:
    def __init__(self, name):
        self.name = name

This drill introduces the concept of the __init__ method, which is used to initialize object states.

Drill 3 - Set Data Structure

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my_set = set()
my_set.add(1)
my_set.add(2)
print(my_set)

This drill introduces the set data structure and how to add elements to a set.

Drill 4 - Counting using Dictionary

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from collections import Counter

numbers = [1, 2, 2, 3, 3, 3]
counter = Counter(numbers)
print(counter)

This drill introduces the Counter dictionary, which is used for counting the frequency of elements in a collection.

Drill 5 - Recursive Method

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def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n - 1)

This drill introduces the concept of recursion. Here, we’ve used recursion to calculate the factorial of a number.

Drill 6 - Union Operation (Specific to DSU)

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class DSU:
    def __init__(self, n):
        self.parent = list(range(n))

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

    def union(self, x, y):
        self.parent[self.find(x)] = self.find(y)

dsu = DSU(5)
dsu.union(1, 2)

This drill introduces the concept of the union operation in DSU, where we are merging two subsets.

Order of Integration

  1. We would start by defining the DSU class as in Drill 6, with a few extra methods for keeping track of ranks and sizes of the groups.

  2. Next, we would define the Solution class as in Drill 1, with the countCompleteComponents method.

  3. Inside the countCompleteComponents method, we would implement the functionality of Drills 2, 3 and 4 to initialize the DSU, create a set for storing groups, and a counter for counting occurrences of each vertex.

  4. Then, we’d iterate over the edges of the graph and apply the union operation as in Drill 6.

  5. Finally, we’d implement the functionality of identifying and filtering the complete components. This involves checking the size of each group against the degree of the vertices in it and discarding the groups that do not meet the condition. The remaining groups are the complete components. This would involve combining elements from all drills in a cohesive manner.