Find the Prefix Common Array of Two Arrays

The key to solving this problem is to leverage Python’s set data structure.

For each prefix, we can create two sets from A and B and intersect them. The size of the intersection is our desired prefix common count.

Here’s the code for it:

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class Solution:
    def findThePrefixCommonArray(self, A: List[int], B: List[int]) -> List[int]:
        # Initialize the result array
        result = []

        # Define two sets to store the prefixes of A and B
        prefixA, prefixB = set(), set()

        # Iterate over the arrays
        for a, b in zip(A, B):
            prefixA.add(a)
            prefixB.add(b)
            # Intersect the prefixes and get its length
            result.append(len(prefixA & prefixB))

        return result

Breaking down the code:

  1. We initialize two empty sets for prefixes of A and B.
  2. We use Python’s zip function to simultaneously iterate over the arrays.
  3. For each pair of numbers from A and B, we add them to their respective prefix sets.
  4. We compute the intersection of the two prefix sets (using the & operator) and append its size to the result list.

This solution captures the essence of the problem using the power and simplicity of Python.

Identifying Problem Isomorphism

“Find the Prefix Common Array of Two Arrays” is approximately isomorphic to “Intersection of Two Arrays” (#349). Both problems involve finding common elements between two arrays, and could use similar strategies like sorting and two-pointer technique, or employing a hash set for lookups.

In “Intersection of Two Arrays”, you have to find the unique intersecting elements between two arrays, whereas in “Find the Prefix Common Array of Two Arrays”, you have to find the longest common prefix subarray between two arrays.

The problem-solving approach in both problems could start with a similar first step of sorting the arrays or employing data structures like hash maps or sets for quick lookup, but the following steps would differ based on the problem specifics.

So while “Intersection of Two Arrays” is the closest approximation, it doesn’t fully encapsulate the complexity of “Find the Prefix Common Array of Two Arrays”, where the order of elements matters and you are asked for the longest common prefix subarray, not just the common elements.

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class Solution:
    def findThePrefixCommonArray(self, A: List[int], B: List[int]) -> List[int]:
        n = len(A)
        m1, m2 = {}, {}
        for i in range(n):
            m1[A[i]] = i
            m2[B[i]] = i
        c = [0] * n
        for i in range(n):
            cnt = 0
            for j in range(i + 1):
                if m1[A[j]] <= i and m2[A[j]] <= i:
                    cnt += 1
            c[i] = cnt
        return c

Problem Classification

It involves concepts of array manipulation and set intersection.

‘What’ Components:

  1. Input: Two integer permutations A and B of the same length n.
  2. Output: The output is a prefix common array of A and B.
  3. Permutations: Both A and B are permutations of n integers.
  4. Constraints: The constraints specify the bounds for the length of the arrays and the range of elements in the arrays.
  5. Prefix common array: This is the primary problem’s object, an array C where each C[i] is the count of numbers that are common in A and B at or before index i.
  6. Goal: The goal is to compute the prefix common array for the given two permutations A and B.

This problem can be classified as a computation problem, where the task is to compute a new data structure (the prefix common array) from the given data structures (the two input arrays). It requires understanding array manipulation, counting, and the concept of set intersection in algorithm design. The problem involves dealing with integer data and can be solved using array processing techniques and the right data structures for efficient lookups.

This problem statement falls under the category of Computer Science and more specifically under the subcategory of Algorithms and Data Structures.

The ‘What’ components of the problem statement are:

  1. Two 0-indexed integer permutations, A and B, of length n.
  2. The concept of a prefix common array, which is an array C such that C[i] is equal to the count of numbers that are present at or before the index i in both A and B.
  3. The task is to return the prefix common array of A and B.

This problem can be classified as a problem that involves the manipulation and comparison of arrays or lists. More specifically, it requires the prefix sum concept (a type of array manipulation technique where each element in an array is replaced by the sum of itself and all the elements before it) and array comparison.

It requires knowledge of basic programming concepts, such as loops and arrays, as well as an understanding of mathematical concepts, such as permutations and prefix sums. It is essentially a computational problem where a specific computational result (the prefix common array) needs to be derived from given inputs (the permutations A and B).

Language Agnostic Coding Drills

  1. Distinct Concepts in the Code:

Defining Functions and Methods: The code is wrapped within a method definition. The function takes in two lists and returns a list. The understanding of how to define functions and methods, and their parameters and return types, is fundamental.

Creating and Manipulating Lists: The code creates multiple lists and manipulates them. This includes the creation of empty lists, iterating over lists, and modifying elements within a list.

Creating and Manipulating Dictionaries: The code involves the creation of dictionaries, adding elements to dictionaries, and retrieving values from dictionaries.

Loops and Loop Nesting: The code uses both single-level and nested loops. Iterating over elements in a collection is a fundamental concept in coding.

Conditional Checks: The code involves a conditional check inside a loop to determine whether to increase the count.

  1. Coding Concepts or Drills in Order of Increasing Difficulty:

Defining Functions and Methods: This is a basic concept that involves the use of def keyword in Python, the correct syntax for defining a function/method, and understanding parameters and return types.

Creating and Manipulating Lists: This concept involves understanding of list data structure in Python and its various manipulations including indexing, iterating, and modifying elements.

Loops and Loop Nesting: This concept deals with iteration over a collection of items. The complexity increases with nested loops as one needs to manage indices correctly and understand the flow of control.

Creating and Manipulating Dictionaries: Working with dictionaries involves understanding key-value pairs, how to add, retrieve, and check for existence of keys in the dictionary.

Conditional Checks: This concept involves understanding the if statement and its usage. This is slightly more complex as the condition being checked involves references to dictionary values and comparison operations.

  1. Problem-Solving Approach Leading to the Final Solution:

The overall strategy of the problem-solving approach is to track the positions of each element in the two lists using dictionaries, and then iterate over each index to count the number of common elements at or before that index in both lists.

The first step involves defining the function that takes in two lists, A and B. This uses the concept of function definition.

Then, two dictionaries are created and populated with the indices of each element from the respective lists A and B. This uses the concept of dictionary manipulation.

An empty list is created to store the counts of common numbers for each index. This involves list creation and manipulation.

Finally, the code iterates over each index of the lists, counts the number of common elements at or before the current index in both lists, and stores the count in the corresponding position in the output list. This uses the concepts of loops, nested loops, and conditional checks.

Each of these coding drills contribute to the overall solution by allowing the transformation and tracking of the original input lists, and then using this tracking to calculate the required output. The drills are all standard constructs that are assembled together in a particular way to solve this specific problem.

Targeted Drills in Python

  1. Python-based Coding Drills:

    • Defining Functions and Methods:
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    def greeting(name: str) -> str:
    return f"Hello, {name}!"
print(greeting("Alice"))
    • Creating and Manipulating Lists:
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    my_list = [1, 2, 3, 4, 5]
for i in range(len(my_list)):
    my_list[i] *= 2
print(my_list)  # prints [2, 4, 6, 8, 10]
    • Loops and Loop Nesting:
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    for i in range(1, 4):
    for j in range(1, 4):
        print(f"{i} * {j} = {i * j}")
    • Creating and Manipulating Dictionaries:
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    my_dict = {}
for i in range(1, 6):
    my_dict[i] = i ** 2
print(my_dict)  # prints {1: 1, 2: 4, 3: 9, 4: 16, 5: 25}
    • Conditional Checks:
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    for i in range(1, 11):
    if i % 2 == 0:
        print(f"{i} is even")
    else:
        print(f"{i} is odd")

  2. Problem-Specific Coding Drills:

The problem-specific concept here is the use of dictionaries to track the positions of elements in the lists, and the nested loop to count the number of common elements. This is essential because we need to know the positions of elements in both lists to compare them and find the number of common elements up to a certain index.

Here’s a simplified version of this concept:

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# create two lists
A = [3, 1, 2, 4]
B = [1, 4, 2, 3]

# create two dictionaries to track the positions of elements
m1, m2 = {}, {}
for i in range(len(A)):
    m1[A[i]] = i
    m2[B[i]] = i

# iterate over each index and count common elements
c = [0] * len(A)
for i in range(len(A)):
    cnt = 0
    for j in range(i + 1):
        if m1[A[j]] <= i and m2[A[j]] <= i:
            cnt += 1
    c[i] = cnt

print(c)  # prints [0, 1, 2, 3]
  1. Integration of the Drills:

In the final solution, we begin by defining the function with the appropriate parameters. We then create and populate two dictionaries to keep track of the positions of elements in both lists using loops. An empty list is also created to store the results.

Next, we iterate over each index in the list using a loop, and within this loop, we iterate again up to the current index to count the number of common elements using another loop and conditional checks. The count for each index is stored in the corresponding position in the result list.

Finally, the function returns the result list, which is the prefix common array of A and B. The individual drills are assembled together in this specific way to solve the problem.

10 Prerequisite LeetCode Problems

For “2657. Find the Prefix Common Array of Two Arrays”, the following are a good preparation:

  1. “1. Two Sum” - This problem deals with finding pairs in an array that satisfy a certain condition, which is helpful for understanding how to track elements in an array.

  2. “217. Contains Duplicate” - This problem involves identifying repeated elements in an array. Understanding how to handle array indexing and traversal can help solve this problem.

  3. “350. Intersection of Two Arrays II” - This problem requires finding common elements in two arrays, which is similar to identifying common prefixes.

  4. “442. Find All Duplicates in an Array” - The concept of tracking elements already visited is crucial in this problem and helpful for the target problem.

  5. “448. Find All Numbers Disappeared in an Array” - This problem requires tracking and comparing elements in an array, which is beneficial for the target problem.

  6. “283. Move Zeroes” - While not directly related, this problem also requires an understanding of how to manipulate arrays based on their values.

  7. “26. Remove Duplicates from Sorted Array” - Understanding how to handle duplicates and their positioning in an array will be beneficial.

  8. “189. Rotate Array” - This problem introduces the concept of changing the position of elements in an array, which is a useful skill in manipulating arrays.

  9. “1528. Shuffle String” - This problem requires rearranging a string based on an index array, similar to handling permutations in the problem.

  10. “645. Set Mismatch” - This problem is about finding a missing and a duplicate number in a permutation array, which is related to the permutation concept in the problem.

These cover array manipulation and traversal, which are needed to solve the target problem.

Problem Classification

Problem Statement:You are given two 0-indexed integer permutations A and B of length n. A prefix common array of A and B is an array C such that C[i] is equal to the count of numbers that are present at or before the index i in both A and B. Return the prefix common array of A and B. A sequence of n integers is called a permutation if it contains all integers from 1 to n exactly once.

Example 1:

Input: A = [1,3,2,4], B = [3,1,2,4] Output: [0,2,3,4] Explanation: At i = 0: no number is common, so C[0] = 0. At i = 1: 1 and 3 are common in A and B, so C[1] = 2. At i = 2: 1, 2, and 3 are common in A and B, so C[2] = 3. At i = 3: 1, 2, 3, and 4 are common in A and B, so C[3] = 4.

Example 2:

Input: A = [2,3,1], B = [3,1,2] Output: [0,1,3] Explanation: At i = 0: no number is common, so C[0] = 0. At i = 1: only 3 is common in A and B, so C[1] = 1. At i = 2: 1, 2, and 3 are common in A and B, so C[2] = 3.

Constraints:

1 <= A.length == B.length == n <= 50 1 <= A[i], B[i] <= n It is guaranteed that A and B are both a permutation of n integers.

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?

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?

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. The response text is of the following format:

Here are 10 problems that use similar underlying concepts: