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2683. Neighboring Bitwise XOR

Description

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

Solutions

Solution 1: Bit Manipulation

Let's assume the original binary array is $a$, and the derived array is $b$. Then, we have:

$$ b_0 = a_0 \oplus a_1 \ b_1 = a_1 \oplus a_2 \ \cdots \ b_{n-1} = a_{n-1} \oplus a_0 $$

Since the XOR operation is commutative and associative, we get:

$$ b_0 \oplus b_1 \oplus \cdots \oplus b_{n-1} = (a_0 \oplus a_1) \oplus (a_1 \oplus a_2) \oplus \cdots \oplus (a_{n-1} \oplus a_0) = 0 $$

Therefore, as long as the XOR sum of all elements in the derived array is $0$, there must exist an original binary array that meets the requirements.

The time complexity is $O(n)$, where $n$ is the length of the array. The space complexity is $O(1)$.

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class Solution:
    def doesValidArrayExist(self, derived: List[int]) -> bool:
        return reduce(xor, derived) == 0

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class Solution {
    public boolean doesValidArrayExist(int[] derived) {
        int s = 0;
        for (int x : derived) {
            s ^= x;
        }
        return s == 0;
    }
}

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class Solution {
public:
    bool doesValidArrayExist(vector<int>& derived) {
        int s = 0;
        for (int x : derived) {
            s ^= x;
        }
        return s == 0;
    }
};

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func doesValidArrayExist(derived []int) bool {
    s := 0
    for _, x := range derived {
        s ^= x
    }
    return s == 0
}

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function doesValidArrayExist(derived: number[]): boolean {
    return derived.reduce((acc, x) => acc ^ x) === 0;
}

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