How to Solve Find The Original Array of Prefix Xor

To solve the problem, we need to use the properties of bitwise XOR to recover the original array from the given prefix XOR array. By leveraging the relationship between the prefix XOR values, we can iteratively calculate each element of the original array. The solution involves handling the XOR operation effectively and understanding its relationship to the prefix array.

Problem Statement

You are given an integer array pref of size n. The task is to find and return the array arr of size n that satisfies the following condition: For all indices i, pref[i] = arr[0] ^ arr[1] ^ ... ^ arr[i]. Note that ^ denotes the bitwise XOR operation, and it is guaranteed that the answer will be unique.

For example, if pref = [5,2,0,3,1], the correct output array is [5,7,2,3,2] because the following holds: pref[0] = 5, pref[1] = 5 ^ 7 = 2, pref[2] = 5 ^ 7 ^ 2 = 0, pref[3] = 5 ^ 7 ^ 2 ^ 3 = 3, and pref[4] = 5 ^ 7 ^ 2 ^ 3 ^ 2 = 1.

Examples

Example 1

Input: pref = [5,2,0,3,1]

Output: [5,7,2,3,2]

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]

We have pref[0] = arr[0] = 13.

Constraints

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

Solution Approach

Bitwise XOR and Prefix Relationship

To reconstruct the original array, start by setting arr[0] = pref[0]. For subsequent elements, use the relation arr[i] = pref[i] ^ pref[i-1] to calculate each element iteratively. This takes advantage of the prefix XOR property.

Iterative Calculation

By processing the prefix array from the second element onwards, we can efficiently derive the original array. Each element is derived from the XOR of the current and previous prefix values, which allows for an optimal solution.

Optimization Considerations

The time and space complexities can be kept at O(n) by only requiring a single pass through the array and storing the result directly in the output array.

Complexity Analysis

Both the time and space complexity of the solution are O(n), where n is the length of the input array pref. We only require one pass through the array and store the result in-place, making the solution space-efficient.

What Interviewers Usually Probe

• Can the candidate describe the properties of XOR and how it can be used to reverse the prefix array?
• Do they understand the iterative nature of the problem and can they apply the XOR operation correctly?
• Can they optimize the solution to handle large inputs efficiently?

Common Pitfalls or Variants

Common pitfalls

• Misunderstanding the XOR properties and attempting a more complex solution than necessary.
• Failing to handle edge cases where the array length is minimal or contains zeros.
• Overcomplicating the solution by introducing unnecessary operations or data structures.

Follow-up variants

• What if the prefix XOR array is given in reverse order? Can you adjust the approach?
• What happens if the array size is much larger? How would you ensure your solution is still efficient?
• Could a more complex data structure be used for a different variant of this problem?

How GhostInterview Helps

• GhostInterview provides a detailed explanation of XOR properties and how they are applied in this specific problem, helping users understand the solution approach step by step.
• By walking through each part of the problem, GhostInterview highlights common pitfalls and helps solidify the concepts needed to solve similar problems in interviews.
• With a focus on optimization, GhostInterview ensures that users can confidently apply efficient techniques, especially when dealing with large datasets.

Topic Pages

• Array
• Bit Manipulation

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FAQ

What is the pattern used in the 'Find The Original Array of Prefix Xor' problem?

This problem uses the pattern of 'Array plus Bit Manipulation', specifically utilizing the XOR operation to reverse the prefix XOR array.

How do I efficiently solve this problem for large inputs?

The key to solving this problem efficiently is to keep the time complexity to O(n) by calculating the elements iteratively with XOR, and ensuring the space complexity is also O(n).

Can this approach be used for other XOR-related problems?

Yes, understanding how to reverse or manipulate XOR sequences is a fundamental technique in many bit manipulation problems, particularly when dealing with cumulative or prefix operations.

What happens if the input prefix array contains zeros?

The solution works even if the prefix array contains zeros, as XOR with zero does not alter the value. The algorithm will correctly handle such cases.

Is there any alternative to the XOR approach?

While the XOR approach is optimal for this problem, alternative methods such as dynamic programming or brute force would likely be less efficient and are not recommended.