Introduction

In this blog post, we will explore how to solve the LeetCode problem "2683" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

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

Explanation

Here's the solution to the problem:

• Core Idea: The problem is essentially asking whether a linear system of equations (over GF(2)) has a solution. The key observation is that the XOR sum of the derived array must equal 0 if a valid original array exists. This arises from the cyclic nature of the XOR operations.

• XOR Sum Check: Calculate the XOR sum of the derived array. If the sum is 0, then a solution exists. Otherwise, no solution exists.

• Why XOR Sum Works: Consider the XOR sum of the derived array: derived[0] ^ derived[1] ^ ... ^ derived[n-1]. Expanding this using the definition of derived, we get a series of XOR operations involving original elements. Due to the cyclic nature (the last derived element involves the first original element), each original element appears exactly twice in the expansion (once normally and once when calculating the next derived value or cyclically wrapping around), except for the last element, which also appears twice. Since x ^ x = 0, all the original elements cancel out if and only if the xor sum of derived is equal to zero. If the number of 1's in derived is even, it implies that a solution exists, regardless of the original array's size.

• Runtime & Storage Complexity:

  • Runtime: O(n) for calculating the XOR sum.
  • Storage: O(1) (constant space).

Code

    def doesValidArrayExist(derived):
    """
    Checks if a valid original array exists for the given derived array.

    Args:
        derived (list): A list of 0s and 1s representing the derived array.

    Returns:
        bool: True if a valid original array exists, False otherwise.
    """
    xor_sum = 0
    for num in derived:
        xor_sum ^= num
    return xor_sum == 0