Neighboring Bitwise XOR

The problem is about determining if there exists a valid original binary array that could have formed the given derived array by computing the bitwise XOR (⊕) of adjacent values.

Let’s implement the solution:

  1. Initialize: Start by assuming the first value of the original array (original[0]). It can be either 0 or 1.
  2. Build the Original Array: Build the original array using the derived array and the assumed value of original[0].
  3. Check Validity: Check if the constructed original array is valid according to the rules mentioned in the problem statement.
  4. Return Result: If a valid original array is found, return True. If not, return False.

Here’s the code:

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class Solution:
    def doesValidArrayExist(self, derived: List[int]) -> bool:
        n = len(derived)

        for assumed_value in [0, 1]:
            original = [assumed_value]
            
            for i in range(n - 1):
                original.append(derived[i] ^ original[-1])

            if original[-1] == derived[-1] ^ original[0]:
                return True

        return False

Explanation:

  • The loop for assumed_value in [0, 1] is used to try both possible values for original[0].
  • Inside this loop, we build the original array using the derived array and the current assumed value for original[0].
  • We then check if the last element of the original array is consistent with the rules mentioned in the problem statement.
  • If a valid original array is found, the function returns True.
  • If no valid original array is found after trying both possible values for original[0], the function returns False.

The time complexity of this code is (O(n)), where (n) is the length of the derived array.

Identifying Problem Isomorphism

“Neighboring Bitwise XOR” involves calculating the bitwise XOR of neighboring elements in a given data structure. The XOR operation is a common bitwise operation used in many problems.

An analogous problem is “136. Single Number”. This problem asks you to find a number in an array where every element appears twice except for one which appears only once. The solution uses the property of the XOR operation that a number XORed with itself results in 0 and any number XORed with 0 equals the number itself.

Though the specifics of the problems differ, they both require a good understanding of the XOR bitwise operation and how to apply it in various contexts. So while it’s not an exact match, the approach to solve them has a common theme and thus forms an approximate mapping.

Gaining insights into these problems can help you understand the usage of XOR operation and how it can be applied to solve different types of problems.

10 Prerequisite LeetCode Problems

For “2683. Neighboring Bitwise XOR”, the following are a good preparation:

  1. “136. Single Number” - This problem practices using the XOR operator to find the single number in an array.

  2. “389. Find the Difference” - This problem also uses the XOR operator to find the difference between two strings, reinforcing understanding of XOR operations.

  3. “201. Bitwise AND of Numbers Range” - This problem practices bitwise operations and their properties, which are essential for the main problem.

  4. “191. Number of 1 Bits” - It helps understanding bitwise manipulation techniques, especially for binary values.

  5. “231. Power of Two” - This problem involves bitwise manipulations and provides useful practice in handling binary numbers.

  6. “461. Hamming Distance” - This problem requires the XOR operation to find the Hamming distance, which helps understand XOR properties.

  7. “338. Counting Bits” - This problem practices manipulating binary numbers and understanding their properties.

  8. “190. Reverse Bits” - This problem practices manipulating bits, a crucial aspect of the main problem.

  9. “371. Sum of Two Integers” - This problem provides practice with bitwise manipulation to simulate arithmetic operations.

  10. “476. Number Complement” - This problem involves flipping binary bits which is a good practice for bitwise operations.

These cover XOR operation and its properties, bitwise manipulation, and working with binary numbers, which are key to solving “2683. Neighboring Bitwise XOR”.

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class Solution:
    def doesValidArrayExist(self, derived: List[int]) -> bool:
        for first in [0,1]:
            original=first
            for result in derived[:-1]:
                original=original^result

            if original^first==derived[-1]:
                return True
        return False

Problem Classification

This problem statement falls into the domain of Computer Science, specifically within the sub-domain of bit manipulation and array manipulation.

The ‘What’ components of the problem are:

  1. An input array derived of length n, where n is a non-negative integer.
  2. A rule to generate derived from original where each element at index i in derived is the bitwise XOR (⊕) of the elements at the corresponding index and the next index in original, and for the last element, it is the bitwise XOR of the last element and the first element in original.
  3. The problem requires to determine if there exists a valid binary array original that could have formed derived.

The problem can be further classified as a binary array reconstruction problem. It involves the reconstruction of the original array given the derived array and the rules of transformation from original to derived. Understanding how bitwise operations work is essential to solve this problem. The problem-solving process would require backtracking or iterative methods to check all possible original arrays that could generate the given derived array. It’s also likely that a working knowledge of linear algebra or specific properties of the XOR operation could lead to a more efficient solution.

Language Agnostic Coding Drills

  1. Dissecting the code:

Here are the distinct concepts that this code contains:

  • Basic Python Syntax: The fundamental understanding of Python’s syntax, such as class, method, for loop, list, etc.

  • Bitwise XOR operator: The understanding of how XOR works. XOR is a binary operation that is true if exactly one of the bits is true.

  • Control Flow: The use of if-else statements and loops to control the flow of the program based on certain conditions.

  • List Manipulation: Working with list elements, such as accessing list items using indices and slicing.

  1. List of concepts ordered by increasing difficulty:

    a) Basic Python Syntax: This is the easiest concept as it forms the foundation of coding in Python. Without understanding this, it would be hard to progress to more advanced concepts.

    b) List Manipulation: While this concept is a bit more advanced than the basic Python syntax, it’s still one of the basic skills required when dealing with data in Python.

    c) Control Flow: This is moderately difficult as it involves logical thinking and determining the correct conditions for the program to function as expected.

    d) Bitwise XOR operator: This is relatively difficult as it requires an understanding of binary numbers and bitwise operations, which are more advanced topics in computer science.

  2. Problem-solving approach:

The problem-solving approach in the code involves iterating over the possible first elements (0 or 1) in the original array. For each possible first element, it performs a cumulative XOR operation over the derived array (except the last element), thereby simulating the construction of the original array. Then it checks if the XOR of the simulated original array’s last element and the first element equals the last element in the derived array, as per the rule stated in the problem. If it does, then it means a valid original array can exist and the function returns True. If it does not satisfy the condition for both possible first elements, it means a valid original array does not exist and the function returns False.

Each of the coding drills, i.e., basic Python syntax, list manipulation, control flow, and bitwise XOR operator, are fundamental to implement this approach. For example, list manipulation is used to access and iterate over elements in the derived array, control flow is used to check the condition to determine if a valid original array can exist, and the bitwise XOR operator is used to simulate the construction of the original array and check the conditions. All these drills combined allow the implementation of the solution to this problem.

Targeted Drills in Python

  1. Python-based coding drills for each identified concept:

    a) Basic Python Syntax:

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class MyClass:
    def myMethod(self, myParameter):
        print(myParameter)

# Create an instance of MyClass
myObject = MyClass()
# Call the method myMethod
myObject.myMethod("Hello, world!")

b) List Manipulation:

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# Creating a list
myList = [1, 2, 3, 4, 5]

# Accessing elements in the list
print(myList[0])  # Output: 1

# Slicing a list
print(myList[:-1])  # Output: [1, 2, 3, 4]

c) Control Flow:

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# Using if-else statements
x = 10
if x > 5:
    print("x is greater than 5")
else:
    print("x is not greater than 5")

# Using for loop
for i in range(5):
    print(i)

d) Bitwise XOR operator:

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# Bitwise XOR operation
x = 5  # Binary: 101
y = 3  # Binary: 011
result = x ^ y  # Result: 110, which is 6 in decimal
print(result)  # Output: 6

  1. Problem-specific concepts:

    a) Iterating over two possibilities (0 or 1) - as the first element of the original array can only be 0 or 1, we need to iterate over these two possibilities.

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for first in [0, 1]:
    print(first)

b) Cumulative XOR operation over the list elements - this is essential in our problem as we need to perform a cumulative XOR operation to simulate the construction of the original array.

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myList = [5, 3, 6]  # Binary: 101, 011, 110
original = myList[0]
for result in myList[1:]:
    original = original ^ result
print(original)  # Output: 4 (100 in binary)
  1. Integration of drills:

The problem solution involves the integration of these drills in the following order:

  • First, create a for loop to iterate over the two possibilities of the first element in the original array (Basic Python Syntax, Control Flow, Problem-specific concept a).

  • Then, inside the for loop, perform a cumulative XOR operation over the elements in the derived array, excluding the last element (List Manipulation, Bitwise XOR operator, Problem-specific concept b).

  • Finally, use an if-else statement to check if the XOR of the simulated original array’s last element and the first element equals the last element in the derived array. If it does, return True, otherwise, return False after the for loop ends (Basic Python Syntax, Control Flow).

Problem Classification

Problem Statement:A 0-indexed array derived with length n is derived by computing the bitwise XOR (⊕) of adjacent values in a binary array original of length n. Specifically, for each index i in the range [0, n - 1]: If i = n - 1, then derived[i] = original[i] ⊕ original[0]. Otherwise, derived[i] = original[i] ⊕ original[i + 1]. Given an array derived, your task is to determine whether there exists a valid binary array original that could have formed derived. Return true if such an array exists or false otherwise. A binary array is an array containing only 0’s and 1’s

Example 1:

Input: derived = [1,1,0] Output: true Explanation: A valid original array that gives derived is [0,1,0]. derived[0] = original[0] ⊕ original[1] = 0 ⊕ 1 = 1 derived[1] = original[1] ⊕ original[2] = 1 ⊕ 0 = 1 derived[2] = original[2] ⊕ original[0] = 0 ⊕ 0 = 0

Example 2:

Input: derived = [1,1] Output: true Explanation: A valid original array that gives derived is [0,1]. derived[0] = original[0] ⊕ original[1] = 1 derived[1] = original[1] ⊕ original[0] = 1

Example 3:

Input: derived = [1,0] Output: false Explanation: There is no valid original array that gives derived.

Constraints:

n == derived.length 1 <= n <= 105 The values in derived are either 0’s or 1’s

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: