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Find The Original Array of Prefix Xor

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ArraysBit Manipulation

You are given an integer array pref of size n. Find and return the array arr of size n that satisfies:

pref[i] = arr[0] ^ arr[1] ^ ... ^ arr[i].

Note that ^ denotes the bitwise-xor operation.

It can be proven that the answer is unique.

Example 1:

Input: pref = [5,2,0,3,1]
Output: [5,7,2,3,2]
Explanation: From the array [5,7,2,3,2] we have the following:
- pref[0] = 5.
- pref[1] = 5 ^ 7 = 2.
- pref[2] = 5 ^ 7 ^ 2 = 0.
- pref[3] = 5 ^ 7 ^ 2 ^ 3 = 3.
- pref[4] = 5 ^ 7 ^ 2 ^ 3 ^ 2 = 1.

Example 2:

Input: pref = [13]
Output: [13]
Explanation: We have pref[0] = arr[0] = 13.

Constraints:

1 <= pref.length <= 10⁵
0 <= pref[i] <= 10⁶

Try coding it in LeetCode

Find The Original Array of Prefix Xor

Medium

Asked by:

24 views

Topics:

ArraysBit Manipulation

You are given an integer array pref of size n. Find and return the array arr of size n that satisfies:

pref[i] = arr[0] ^ arr[1] ^ ... ^ arr[i].

Note that ^ denotes the bitwise-xor operation.

It can be proven that the answer is unique.

Example 1:

Input: pref = [5,2,0,3,1]
Output: [5,7,2,3,2]
Explanation: From the array [5,7,2,3,2] we have the following:
- pref[0] = 5.
- pref[1] = 5 ^ 7 = 2.
- pref[2] = 5 ^ 7 ^ 2 = 0.
- pref[3] = 5 ^ 7 ^ 2 ^ 3 = 3.
- pref[4] = 5 ^ 7 ^ 2 ^ 3 ^ 2 = 1.

Example 2:

Input: pref = [13]
Output: [13]
Explanation: We have pref[0] = arr[0] = 13.

Constraints:

1 <= pref.length <= 10⁵
0 <= pref[i] <= 10⁶

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the range of values in the `pref` array, and is it possible to have negative numbers or zero?
What is the expected return if the input `pref` array is null or empty?
Is the input `pref` array guaranteed to represent a valid prefix XOR array, or do I need to handle cases where no original array can be derived?
What is the maximum possible length of the `pref` array?
Are there any memory constraints I should be aware of, considering the original array will have the same length as `pref`?

Brute Force Solution

Approach

We are given a series of encoded messages and we want to find the original messages. The brute force approach means we'll try every possible combination of original messages until we find the right one. This involves guessing, checking if the guess is correct, and repeating until we are successful.

Here's how the algorithm would work step-by-step:

Start by guessing what the first original message could be. Try all possible numbers.
Based on our first guess, we can figure out what the second original message must be to produce the second encoded message.
Then, using our first two guesses, figure out what the third original message must be to produce the third encoded message.
Keep doing this, figuring out each original message one at a time based on the previous guesses and the encoded messages.
Once we have a complete set of original messages, check if they produce the given encoded messages when combined in the specified way.
If they do, we've found the solution! If not, go back to the first message and try a different guess.
Repeat this whole process of guessing and checking until we find the set of original messages that works.

Code Implementation

def find_original_array_brute_force(encoded_array):
    array_length = len(encoded_array)
    if array_length == 0:
        return []

    for first_element_guess in range(201):
        original_array_guess = [first_element_guess]
        is_valid = True

        # Iterate through the encoded array to construct the original array.

        for i in range(1, array_length):
            next_original_element = encoded_array[i] ^ original_array_guess[i - 1]
            original_array_guess.append(next_original_element)

        # Verify if the generated array matches the encoded array.
        prefix_xor = 0
        for i in range(array_length):
            prefix_xor ^= original_array_guess[i]
            if prefix_xor != encoded_array[i]:
                is_valid = False
                break

        # If the generated array is valid, return it.

        if is_valid:
            return original_array_guess

    return []

Big(O) Analysis

Time Complexity

O(2^n) – The described brute-force approach explores all possible combinations for the original array. Assuming each element in the original array can take on a range of values (in the worst case, consider a boolean array with 2 possibilities), finding the correct combination involves trying all possible original arrays. Therefore, we essentially iterate through all possible arrays of size n, leading to 2^n possibilities in the boolean scenario. Thus the time complexity is O(2^n).

Space Complexity

O(N) – The algorithm constructs a potential original array of size N, where N is the number of encoded messages. Additionally, it may need to store temporary arrays of size N to verify the correctness of each guessed original array. The space used for these auxiliary arrays scales linearly with the input size N, leading to a space complexity of O(N).

Optimal Solution

Approach

The problem gives us a series of values where each value is the cumulative XOR of the original array. To find the original array, we can reverse this process by using the properties of XOR. The key idea is that XORing a number with itself results in zero, and XOR is its own inverse operation.

Here's how the algorithm would work step-by-step:

The first element of the original array is the same as the first element of the XORed series.
For the remaining elements, we XOR each element of the XORed series with the element that comes before it. The result of this operation will be the corresponding element in the original array.
This works because the XOR operation effectively cancels out the previous XORs, leaving us with just the single original value.

Code Implementation

def find_original_array(prefix_xor_array):
    original_array = [0] * len(prefix_xor_array)

    # The first element is the same
    original_array[0] = prefix_xor_array[0]

    for index in range(1, len(prefix_xor_array)):
        # XOR with the previous element cancels out
        # the cumulative XOR, leaving the single value.
        original_array[index] = prefix_xor_array[index] ^ prefix_xor_array[index - 1]

    return original_array

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the input array of size n (where n is the number of prefix XOR values) once. In each iteration, it performs a single XOR operation. Thus, the number of operations scales linearly with the input size n, resulting in a time complexity of O(n).

Space Complexity

O(N) – The algorithm creates a new array of the same size as the input array to store the original array. Where N is the number of elements in the input array, the auxiliary space used is directly proportional to the size of the input array because the algorithm needs to store each element of the original array, which has N elements. Therefore, the space complexity is O(N).

Edge Cases

Case	How to Handle
Null or empty input array `pref`	Return an empty array if `pref` is null or empty, as there's no original array to derive.
Input array `pref` with a single element	Return an array containing only the first element of `pref`, as the original array would consist of just that element.
Integer overflow when calculating XOR	The XOR operation can potentially lead to integer overflow if the values in `pref` are very large; consider using larger data types or modular arithmetic if necessary depending on constraints.
Large input array `pref` to assess time and space complexity.	Ensure the solution has linear time and space complexity (O(n)), avoiding nested loops or data structures that grow excessively with input size.
All elements in `pref` are the same.	The original array will contain the first element of `pref` and the rest will be zeros, handled correctly by the iterative XOR difference.
Zero value in `pref` at any index	The XOR operation handles zeros correctly, so the logic should naturally account for it by producing the prior array value.
Negative numbers as values in the XOR result (if applicable/allowed in the language)	Check the programming language's handling of negative numbers in XOR operation and ensure correct output based on language specification.
Pref array is monotonically increasing or decreasing	Prefix XOR doesn't mandate increasing or decreasing properties, the iterative XOR-difference approach will still reconstruct the array correctly.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the range of values in the `pref` array, and is it possible to have negative numbers or zero?
What is the expected return if the input `pref` array is null or empty?
Is the input `pref` array guaranteed to represent a valid prefix XOR array, or do I need to handle cases where no original array can be derived?
What is the maximum possible length of the `pref` array?
Are there any memory constraints I should be aware of, considering the original array will have the same length as `pref`?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

Start by guessing what the first original message could be. Try all possible numbers.
Based on our first guess, we can figure out what the second original message must be to produce the second encoded message.
Then, using our first two guesses, figure out what the third original message must be to produce the third encoded message.
Keep doing this, figuring out each original message one at a time based on the previous guesses and the encoded messages.
Once we have a complete set of original messages, check if they produce the given encoded messages when combined in the specified way.
If they do, we've found the solution! If not, go back to the first message and try a different guess.
Repeat this whole process of guessing and checking until we find the set of original messages that works.

Code Implementation

def find_original_array_brute_force(encoded_array):
    array_length = len(encoded_array)
    if array_length == 0:
        return []

    for first_element_guess in range(201):
        original_array_guess = [first_element_guess]
        is_valid = True

        # Iterate through the encoded array to construct the original array.

        for i in range(1, array_length):
            next_original_element = encoded_array[i] ^ original_array_guess[i - 1]
            original_array_guess.append(next_original_element)

        # Verify if the generated array matches the encoded array.
        prefix_xor = 0
        for i in range(array_length):
            prefix_xor ^= original_array_guess[i]
            if prefix_xor != encoded_array[i]:
                is_valid = False
                break

        # If the generated array is valid, return it.

        if is_valid:
            return original_array_guess

    return []

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

The first element of the original array is the same as the first element of the XORed series.
For the remaining elements, we XOR each element of the XORed series with the element that comes before it. The result of this operation will be the corresponding element in the original array.
This works because the XOR operation effectively cancels out the previous XORs, leaving us with just the single original value.

Code Implementation

def find_original_array(prefix_xor_array):
    original_array = [0] * len(prefix_xor_array)

    # The first element is the same
    original_array[0] = prefix_xor_array[0]

    for index in range(1, len(prefix_xor_array)):
        # XOR with the previous element cancels out
        # the cumulative XOR, leaving the single value.
        original_array[index] = prefix_xor_array[index] ^ prefix_xor_array[index - 1]

    return original_array

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input array `pref`	Return an empty array if `pref` is null or empty, as there's no original array to derive.
Input array `pref` with a single element	Return an array containing only the first element of `pref`, as the original array would consist of just that element.
Integer overflow when calculating XOR	The XOR operation can potentially lead to integer overflow if the values in `pref` are very large; consider using larger data types or modular arithmetic if necessary depending on constraints.
Large input array `pref` to assess time and space complexity.	Ensure the solution has linear time and space complexity (O(n)), avoiding nested loops or data structures that grow excessively with input size.
All elements in `pref` are the same.	The original array will contain the first element of `pref` and the rest will be zeros, handled correctly by the iterative XOR difference.
Zero value in `pref` at any index	The XOR operation handles zeros correctly, so the logic should naturally account for it by producing the prior array value.
Negative numbers as values in the XOR result (if applicable/allowed in the language)	Check the programming language's handling of negative numbers in XOR operation and ensure correct output based on language specification.
Pref array is monotonically increasing or decreasing	Prefix XOR doesn't mandate increasing or decreasing properties, the iterative XOR-difference approach will still reconstruct the array correctly.