Problem 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. 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.
Key Insights
- The derived array is created by XORing adjacent elements of the original array, with the last element XORed with the first.
- The properties of XOR imply that knowing the derived values can help deduce possible values for the original array.
- Binary values (0 or 1) lead to specific constraints that can be verified to ascertain the existence of a valid original array.
Space and Time Complexity
Time Complexity: O(n)
Space Complexity: O(1)
Solution
To determine whether a valid original binary array exists based on the derived array, we can utilize the properties of XOR. The derived values create a system of equations based on their construction from the original array. The key observation is that if we know the value of the first element of the original array, we can compute all other elements iteratively. We can check the parity of the count of 1s in the derived array and infer whether an original array is possible.