Decode XORed Permutation

Decode XORed Permutation

What is this problem about?

The Decode XORed Permutation interview question involves a hidden permutation of the first n positive integers (where n is odd). You are given an encoded array where encoded[i] = perm[i] XOR perm[i+1]. Your goal is to reconstruct the original perm array. This Decode XORed Permutation coding problem is a mathematical puzzle based on the properties of the XOR operation.

Why is this asked in interviews?

Amazon and other firms use this to test a candidate's knowledge of Bit Manipulation and their ability to derive information from global properties. The key to the problem isn't local iteration, but realizing that you can calculate the XOR sum of the entire permutation and use it to find the first element. It evaluates mathematical intuition and algebraic manipulation of bits.

Algorithmic pattern used

This follows the Array, Bit Manipulation interview pattern.

1. Total XOR: Calculate the XOR sum of all numbers from 1 to n (let's call this totalXOR). Since perm is a permutation of 1 to n, totalXOR is the same as the XOR of all elements in perm.
2. Partial XOR: Calculate the XOR sum of all elements in the encoded array at odd indices: encoded[1], encoded[3], encoded[5]...
3. Identify First Element: Notice that encoded[1] = perm[1]^perm[2], encoded[3] = perm[3]^perm[4], etc. XORing these gives you the XOR sum of all elements except perm[0].
4. Solve: perm[0] = totalXOR XOR (encoded[1] ^ encoded[3] ^ ...)
5. Reconstruct: Once perm[0] is known, perm[i+1] = perm[i] XOR encoded[i].

Example explanation

n = 3. perm is a permutation of [1, 2, 3]. totalXOR = 1^2^3 = 0. encoded = [3, 1]. (where 3 = perm[0]^perm[1] and 1 = perm[1]^perm[2]).

1. XOR of encoded at odd indices: encoded[1] = 1. This is perm[1]^perm[2].
2. perm[0] = totalXOR XOR (perm[1]^perm[2]) = 0 ^ 1 = 1.
3. perm[1] = perm[0] ^ encoded[0] = 1 ^ 3 = 2.
4. perm[2] = perm[1] ^ encoded[1] = 2 ^ 1 = 3. Result: [1, 2, 3].

Common mistakes candidates make

• Trying to guess perm[0]: Attempting to iterate through all possible values for the first element, which is inefficient.
• Ignoring the "n is odd" constraint: Failing to use the specific mathematical advantage that an odd n provides for pairing up XOR terms.
• XOR Logic: Forgetting that A XOR B = C implies A XOR C = B.

Interview preparation tip

Memorize the properties of XOR: X XOR X = 0, X XOR 0 = X, and its associative/commutative nature. Most bit manipulation "magic" relies on these three rules.