<p>You are given two integer arrays, <code>source</code> and <code>target</code>, both of length <code>n</code>. You are also given an array <code>allowedSwaps</code> where each <code>allowedSwaps[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> indicates that you are allowed to swap the elements at index <code>a<sub>i</sub></code> and index <code>b<sub>i</sub></code> <strong>(0-indexed)</strong> of array <code>source</code>. Note that you can swap elements at a specific pair of indices <strong>multiple</strong> times and in <strong>any</strong> order.</p>

<p>The <strong>Hamming distance</strong> of two arrays of the same length, <code>source</code> and <code>target</code>, is the number of positions where the elements are different. Formally, it is the number of indices <code>i</code> for <code>0 &lt;= i &lt;= n-1</code> where <code>source[i] != target[i]</code> <strong>(0-indexed)</strong>.</p>

<p>Return <em>the <strong>minimum Hamming distance</strong> of </em><code>source</code><em> and </em><code>target</code><em> after performing <strong>any</strong> amount of swap operations on array </em><code>source</code><em>.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> source = [1,2,3,4], target = [2,1,4,5], allowedSwaps = [[0,1],[2,3]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> source can be transformed the following way:
- Swap indices 0 and 1: source = [<u>2</u>,<u>1</u>,3,4]
- Swap indices 2 and 3: source = [2,1,<u>4</u>,<u>3</u>]
The Hamming distance of source and target is 1 as they differ in 1 position: index 3.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> source = [1,2,3,4], target = [1,3,2,4], allowedSwaps = []
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are no allowed swaps.
The Hamming distance of source and target is 2 as they differ in 2 positions: index 1 and index 2.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> source = [5,1,2,4,3], target = [1,5,4,2,3], allowedSwaps = [[0,4],[4,2],[1,3],[1,4]]
<strong>Output:</strong> 0
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>n == source.length == target.length</code></li>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= source[i], target[i] &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= allowedSwaps.length &lt;= 10<sup>5</sup></code></li>
	<li><code>allowedSwaps[i].length == 2</code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= n - 1</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
</ul>

class Solution {
public:
    int minimumHammingDistance(vector<int>& source, vector<int>& target, vector<vector<int>>& allowedSwaps) {
        int n = source.size();
        vector<int> P(n);
        vector<int> R(n, 1);
        for (int i = 0; i < n; i++) {
            P[i] = i;
        }
        for (auto& s : allowedSwaps) {
            setUnion(s[0], s[1], P, R);
        }
        unordered_map<int,vector<int>> M;
        for (int i = 0; i < n; i++) {
            M[getSet(i, P)].push_back(i);
        }
        int res = n;
        for(auto& p : M) {
            vector<int> &V = p.second;
            vector<int> V1;
            vector<int> V2;
            for (auto& idx : V) {
                V1.push_back(source[idx]);
                V2.push_back(target[idx]);
            }
            sort(V1.begin(), V1.end());
            sort(V2.begin(), V2.end());
            int idx = 0;
            for (int v : V1) {
                while (idx < V2.size() && V2[idx] < v) {
                    idx++;
                }
                if (idx < V2.size() && v == V2[idx]) {
                    idx++;
                    res--;
                }
            }
        }
        return res;
    }

    int getSet(int v, vector<int>& P) {
        if (P[v] == v) {
            return v;
        }
        return P[v] = getSet(P[v], P);
    }   

    void setUnion(int v1, int v2, vector<int>& P, vector<int>& R) {
        int p1 = getSet(v1, P);
        int p2 = getSet(v2, P);
        if (p1 == p2) {
            return;
        }
        if (R[p1] < R[p2]) {
            swap(p1, p2);
        }
        R[p1] += R[p2];
        P[p2] = p1;
    }
};
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