23. Unique Number II

The problem can be found at the following link: Question Link

Problem Description

Given an array arr[] of size 2*N + 2, where 2*N elements appear in pairs and two elements appear only once, your task is to find those two distinct unique numbers and return them in increasing order.

Examples

Example 1:

Input:

arr[] = [1, 2, 3, 2, 1, 4]

Output:

[3, 4]

Explanation:

3 and 4 occur exactly once in the array. All other elements appear in pairs.

Example 2:

Input:

arr[] = [2, 1, 3, 2]

Output:

[1, 3]

Explanation:

1 and 3 occur only once. 2 appears twice.

Example 3:

Input:

arr[] = [2, 1, 3, 3]

Output:

[1, 2]

Explanation:

1 and 2 occur once. 3 appears twice.

Constraints:

• $(2 \leq \text{arr.size()} \leq 10^6)$

• $(1 \leq \text{arr}[i] \leq 5 \times 10^6)$

• arr.size() is even

My Approach

XOR Partition Method

This is the most efficient and clever approach using bit manipulation.

Algorithm Steps:

1. XOR all elements → result is XOR of the two unique numbers: x = a ^ b.

2. Find the rightmost set bit in x.

3. Partition the array into two groups based on this bit.

4. XOR each group → you get a and b separately.

5. Return the numbers in increasing order.

Time and Auxiliary Space Complexity

• Expected Time Complexity: O(n), as we iterate through the array a constant number of times.

• Expected Auxiliary Space Complexity: O(1), as we only use a constant number of variables.

Code (C++)

class Solution {
  public:
    vector<int> singleNum(vector<int>& A) {
        int x = 0, a = 0, b = 0;
        for (int n : A) x ^= n;
        for (int n : A) (n & (x & -x) ? a : b) ^= n;
        return a < b ? vector<int>{a, b} : vector<int>{b, a};
    }
};

⚡ Alternative Approaches

📊 2️⃣ Hash Map Frequency Count

Algorithm Steps:

1. Traverse the array and count frequencies using a hash map.

2. Collect the two numbers that appear exactly once.

class Solution {
  public:
    vector<int> singleNum(vector<int>& a) {
        unordered_map<int, int> freq;
        for (int x : a) freq[x]++;
        vector<int> res;
        for (auto it = freq.begin(); it != freq.end(); ++it)
            if (it->second == 1) res.push_back(it->first);
        sort(res.begin(), res.end());
        return res;
    }
};

📝 Complexity Analysis:

• Time Complexity: O(n log n) (due to final sorting)

• Space Complexity: O(n)

✅ Why This Approach?

Simple and works with generalized inputs — even if frequencies are not exactly two.

📊 3️⃣ Sorting and Pair Skipping

Algorithm Steps:

1. Sort the array.

2. Compare elements in pairs. Push elements that do not match with their pair.

class Solution {
  public:
    vector<int> singleNum(vector<int>& a) {
        sort(a.begin(), a.end());
        vector<int> res;
        int i = 0, n = a.size();
        while (i < n - 1) {
            if (a[i] != a[i + 1]) {
                res.push_back(a[i++]);
            } else {
                i += 2;
            }
        }
        if (res.size() < 2) res.push_back(a[n - 1]);
        sort(res.begin(), res.end());
        return res;
    }
};

📝 Complexity Analysis:

• Time Complexity: O(n log n)

• Space Complexity: O(1) (excluding result storage)

✅ Why This Approach?

No extra data structures used beyond sorting. Best when space is limited.

🆚 Comparison of Approaches

Approach	⏱️ Time Complexity	🗂️ Space Complexity	✅ Pros	⚠️ Cons
XOR Partition	🟢 O(n)	🟢 O(1)	Fastest, elegant, minimal space	Works only with exactly two unique elements
Hash Map Frequency	🟢 O(n)	🔴 O(n)	Simple, handles arbitrary frequencies	More memory used
Sorting + Pairing	🔴 O(n log n)	🟢 O(1)	No extra space, good for sorted data	Slower due to sorting

✅ Best Choice?

Scenario	Recommended Approach
✅ Exactly 2 unique elements, rest in pairs	🥇 XOR Partition
✅ Frequencies may vary	🥈 Hash Map Frequency
✅ Limited space, sorting is acceptable	🥉 Sorting + Pair Check

🔹 Overall Best: XOR Partition, optimal in both time and space. 🔹 Best for flexible scenarios: Hash Map.

Code (Java)

class Solution {
    public int[] singleNum(int[] arr) {
        int x = 0, a = 0, b = 0;
        for (int n : arr) x ^= n;
        for (int n : arr) if ((n & (x & -x)) != 0) a ^= n; else b ^= n;
        return a < b ? new int[]{a, b} : new int[]{b, a};
    }
}

Code (Python)

class Solution:
    def singleNum(self, arr):
        x = 0
        for n in arr: x ^= n
        a = b = 0
        for n in arr:
            (a, b) = (a ^ n, b) if n & (x & -x) else (a, b ^ n)
        return [a, b] if a < b else [b, a]

Contribution and Support:

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