#2683

Medium

LeetCode Problem Workspace

Neighboring Bitwise XOR

Determine if a binary array original exists that produces the given derived array using neighboring bitwise XOR logic.

Problem Statement

You are given a 0-indexed binary array derived of length n, created by computing the bitwise XOR of adjacent elements from an unknown original binary array of the same length. For each index i in range [0, n - 1], derived[i] equals original[i] XOR original[(i + 1) % n].

Your task is to determine whether there exists at least one binary array original such that applying the neighboring XOR rule produces the given derived array. Return true if such an original array exists, or false otherwise.

Examples

Example 1

Input: derived = [1,1,0]

Output: true

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

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

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

Solution Approach

Use XOR Properties

Calculate the cumulative XOR of all values in derived. Since each original element appears twice in the XOR computations, the cumulative XOR must be zero for a valid original array to exist. This observation reduces the problem to a single linear pass.

Iterate Efficiently

Loop through the derived array once, maintaining a running XOR sum. If the final XOR sum is zero, return true; otherwise, return false. This approach uses O(n) time and O(1) space, matching the problem's constraints and leveraging bit manipulation insight.

Handle Edge Cases

For arrays of length 1, derived[0] must be 0 for a valid original array, since the element XOR itself is zero. Always consider the circular property of derived when applying XOR checks to ensure correctness for all lengths.

Complexity Analysis

Metric	Value
Time	O(n)
Space	O(1)

Time complexity is O(n) because each element is visited exactly once. Space complexity is O(1) as no extra arrays are needed; only a running XOR accumulator is required.

What Interviewers Usually Probe

Watch for understanding that each original element participates in two XOR operations.
Check if candidate handles circular array indexing correctly.
Ensure the solution does not attempt exhaustive original array construction, which is unnecessary.

Common Pitfalls or Variants

Common pitfalls

Forgetting that XORing the same element twice cancels it out in cumulative XOR checks.
Neglecting the circular property when computing derived[n-1] XOR original[0].
Attempting to reconstruct original array explicitly, which wastes time and space.

Follow-up variants

Instead of binary arrays, consider arrays with small integer values, still using neighboring XOR constraints.
Allow derived arrays to have unknown entries represented by -1 and check if any valid original exists.
Compute the number of possible original arrays that can generate the given derived array.

FAQ

What is the key insight for solving Neighboring Bitwise XOR?

The main insight is that each original element appears in exactly two XOR operations, so the cumulative XOR of derived must be zero for a valid original array.

Can I reconstruct the original array to check?

Explicit reconstruction is unnecessary; a single pass cumulative XOR check suffices and is more efficient.

How does array length affect the solution?

For length 1, derived[0] must be zero. Otherwise, the standard cumulative XOR check works for any n up to 10^5.

Does the circular property matter?

Yes, derived is defined circularly, so derived[n-1] = original[n-1] XOR original[0]; forgetting this leads to incorrect results.

Which pattern does this problem exemplify?

This is a classic Array plus Bit Manipulation pattern where XOR properties are used to validate constraints efficiently.

terminal

Solution

Solution 1: Bit Manipulation

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

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

Continue Practicing

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