<p>You are given a positive integer <code>n</code> representing <code>n</code> cities numbered from <code>1</code> to <code>n</code>. You are also given a <strong>2D</strong> array <code>roads</code> where <code>roads[i] = [a<sub>i</sub>, b<sub>i</sub>, distance<sub>i</sub>]</code> indicates that there is a <strong>bidirectional </strong>road between cities <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code> with a distance equal to <code>distance<sub>i</sub></code>. The cities graph is not necessarily connected.</p>

<p>The <strong>score</strong> of a path between two cities is defined as the <strong>minimum </strong>distance of a road in this path.</p>

<p>Return <em>the <strong>minimum </strong>possible score of a path between cities </em><code>1</code><em> and </em><code>n</code>.</p>

<p><strong>Note</strong>:</p>

<ul>
	<li>A path is a sequence of roads between two cities.</li>
	<li>It is allowed for a path to contain the same road <strong>multiple</strong> times, and you can visit cities <code>1</code> and <code>n</code> multiple times along the path.</li>
	<li>The test cases are generated such that there is <strong>at least</strong> one path between <code>1</code> and <code>n</code>.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/12/graph11.png" style="width: 190px; height: 231px;" />
<pre>
<strong>Input:</strong> n = 4, roads = [[1,2,9],[2,3,6],[2,4,5],[1,4,7]]
<strong>Output:</strong> 5
<strong>Explanation:</strong> The path from city 1 to 4 with the minimum score is: 1 -&gt; 2 -&gt; 4. The score of this path is min(9,5) = 5.
It can be shown that no other path has less score.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/12/graph22.png" style="width: 190px; height: 231px;" />
<pre>
<strong>Input:</strong> n = 4, roads = [[1,2,2],[1,3,4],[3,4,7]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> The path from city 1 to 4 with the minimum score is: 1 -&gt; 2 -&gt; 1 -&gt; 3 -&gt; 4. The score of this path is min(2,2,4,7) = 2.
</pre>

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

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= roads.length &lt;= 10<sup>5</sup></code></li>
	<li><code>roads[i].length == 3</code></li>
	<li><code>1 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= n</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li><code>1 &lt;= distance<sub>i</sub> &lt;= 10<sup>4</sup></code></li>
	<li>There are no repeated edges.</li>
	<li>There is at least one path between <code>1</code> and <code>n</code>.</li>
</ul>


const int SIZE = 1e5 + 5;
const int INF = 1e9 + 7;

pair<int,int> pn[SIZE];
int sz[SIZE];

pair<int,int> find_set(int v, int d) {
    if (v == pn[v].first)
        return {v,min(d,pn[v].second)};

    pair<int,int> ret = find_set(pn[v].first, min(d,pn[v].second));
    return pn[v] = {ret.first, min(min(d,ret.second),pn[v].second)};
}

int union_sets(pair<int,int> a, pair<int,int> b) {
    a = find_set(a.first, a.second);
    b = find_set(b.first, b.second);
    int Mn = min(min(pn[b.first].second,b.second),min(pn[a.first].second,a.second));
    // if (a.first != b.first) {
        // if (sz[a.first] < sz[b.first])
            // swap(a, b);
        pn[b.first] = {a.first, Mn};
        pn[a.first] = pn[b.first];
        // sz[a.first] += sz[b.first];
        // sz[b.first] = sz[a.first];
    // }
    return Mn;
}

class Solution {
public:
    int minScore(int n, vector<vector<int>>& roads) {
        for(int i=1;i<=n;i++){
            pn[i]={i,INF};
            sz[i]=1;
        }
        int m=roads.size();
        int ans=INF;
        for(int i=0;i<m;i++){
            int a=roads[i][0];
            int b=roads[i][1];
            int c=roads[i][2];
            union_sets({a,c},{b,c});
        }
        ans=min(ans,union_sets(pn[1],pn[n]));
        return ans;
    }
};
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