Find The Original Array of Prefix Xor

Find The Original Array of Prefix Xor

1. What is this problem about?

The Find The Original Array of Prefix Xor interview question is a bitwise logic challenge. You are given an array pref where each element pref[i] is the result of XORing all elements from an original array arr from index $0$ to $i$. Your task is to reconstruct the original array arr. This is the inverse operation of calculating a prefix XOR array.

2. Why is this asked in interviews?

Companies like Microsoft and Morgan Stanley ask the Find The Original Array of Prefix Xor coding problem to evaluate your understanding of the XOR operator's properties. Specifically, it tests if you know that XOR is its own inverse ($A \oplus B = C \implies A \oplus C = B$). It evaluations your ability to manipulate data bitwise and handle array transformations in linear time.

3. Algorithmic pattern used

This problem follows the Prefix Difference pattern using bitwise operations.

• The Relationship: By definition, pref[i] = arr[0] ^ arr[1] ^ ... ^ arr[i].
• The Deduction: Similarly, pref[i-1] = arr[0] ^ arr[1] ^ ... ^ arr[i-1].
• Reconstruction: To isolate arr[i], you can XOR the two prefix values: arr[i] = pref[i] ^ pref[i-1].
• Base Case: arr[0] = pref[0].

4. Example explanation

pref = [5, 2, 0, 3]

1. arr[0] = pref[0] = 5.
2. arr[1] = pref[1] ^ pref[0] = 2 ^ 5 = 7.
3. arr[2] = pref[2] ^ pref[1] = 0 ^ 2 = 2.
4. arr[3] = pref[3] ^ pref[2] = 3 ^ 0 = 3. Original Array arr = [5, 7, 2, 3]. Verify: $5 \oplus 7 = 2$, $5 \oplus 7 \oplus 2 = 0$, $5 \oplus 7 \oplus 2 \oplus 3 = 3$. Correct!

5. Common mistakes candidates make

• Using Subtraction: Thinking that because it's a "prefix sum" style problem, you should use subtraction. XOR is different from addition; you must use XOR to "subtract" a value.
• Extra Space: Creating multiple intermediate arrays when the reconstruction can be done in a single pass ($O(N)$ time and space).
• Looping backwards: While possible, reconstructing from the front is more intuitive and avoids index errors.

6. Interview preparation tip

Remember the three major properties of XOR: $A \oplus A = 0$, $A \oplus 0 = A$, and it is commutative/associative. These properties allow you to "cancel out" elements in a prefix chain, which is a common trick in Bit Manipulation interview patterns.