<p>You are given an array <code>points</code> representing integer coordinates of some points on a 2D-plane, where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code>.</p>

<p>The cost of connecting two points <code>[x<sub>i</sub>, y<sub>i</sub>]</code> and <code>[x<sub>j</sub>, y<sub>j</sub>]</code> is the <strong>manhattan distance</strong> between them: <code>|x<sub>i</sub> - x<sub>j</sub>| + |y<sub>i</sub> - y<sub>j</sub>|</code>, where <code>|val|</code> denotes the absolute value of <code>val</code>.</p>

<p>Return <em>the minimum cost to make all points connected.</em> All points are connected if there is <strong>exactly one</strong> simple path between any two points.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/26/d.png" style="width: 214px; height: 268px;" />
<pre>
<strong>Input:</strong> points = [[0,0],[2,2],[3,10],[5,2],[7,0]]
<strong>Output:</strong> 20
<strong>Explanation:</strong> 
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/26/c.png" style="width: 214px; height: 268px;" />
We can connect the points as shown above to get the minimum cost of 20.
Notice that there is a unique path between every pair of points.
</pre>

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

<pre>
<strong>Input:</strong> points = [[3,12],[-2,5],[-4,1]]
<strong>Output:</strong> 18
</pre>

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

<ul>
	<li><code>1 &lt;= points.length &lt;= 1000</code></li>
	<li><code>-10<sup>6</sup> &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 10<sup>6</sup></code></li>
	<li>All pairs <code>(x<sub>i</sub>, y<sub>i</sub>)</code> are distinct.</li>
</ul>

class Solution {
public:
    vector<int> parent;
    vector<int> usize;
    vector<pair<int, int>> edges;
    vector<pair<int, int>> weights; // <w, eid>
    vector<int> costs;
    int _n;

    void uinit(){
        for (int i=0; i<_n; i++){
            parent[i] = i;
        }
    }

    int find(int id){
        if (parent[id] == id){
            return id;
        }
        return find(parent[id]);
    }

    int merge(int i1, int i2, int cost){
        int r1 = find(i1);
        int r2 = find(i2);
        if (r1 == r2){
            // don't merge
            return usize[r1];
        }
        if (usize[r1] < usize[r2]){
            int t = r1;
            r1 = r2;
            r2 = t;
        }
        // so the size of r1 should be > r2
        parent[r2] = r1;
        usize[r1] += usize[r2];
        costs[r1] += costs[r2] + cost;
        return usize[r1];
    }

    int minCostConnectPoints(vector<vector<int>>& points) {
        _n = points.size();
        parent.resize(_n, 0);
        uinit();
        usize.resize(_n, 1);
        costs.resize(_n, 0);
        int en = (_n*(_n-1))/2;
        edges.resize(en);
        weights.resize(en);
        // build the complete graph
        int eid = 0;
        for (int i=0; i<_n; i++){
            for (int j=(i+1); j<_n; j++){
                edges[eid] = make_pair(i, j);
                weights[eid] = make_pair(dist(points[i], points[j]), eid);
                // cout << i << " " << j << " " << dist(points[i], points[j]) << endl;
                eid++;
            }
        }
        // sort edges
        std::sort (weights.begin(), weights.end());
        // do union find
        int ret = 0;
        for (int i=0; i<en; i++){
            int wi = weights[i].first;
            int eid = weights[i].second;
            int fi = edges[eid].first;
            int ti = edges[eid].second;
            int size = merge(fi, ti, wi);
            // cout << fi << " " << ti << " " << wi << endl; 
            if (size == _n){
                int ri = find(fi);
                ret = costs[ri];
            }
        }
        return ret;
    }

    int dist(vector<int>& p1, vector<int>& p2){
        return abs(p1[0] - p2[0]) + abs(p1[1] - p2[1]);
    }
};
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