<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 DisjointSet {
    vector<int> parent;
    vector<int> size;

public:
    DisjointSet(int n) {
        size = vector<int>(n, 1);
        parent = vector<int>(n);
        for (int i = 0; i < n; ++i)
            parent[i] = i;
    }

    int findRoot(int x) {
        if (parent[x] == x) return x;
        else return parent[x] = findRoot(parent[x]);
    }

    bool connected(int x, int y) {
        return findRoot(x) == findRoot(y);
    }

    void doUnion(int x, int y) {
        x = findRoot(x), y = findRoot(y);
        if (size[x] < size[y]) swap(x, y);
        parent[y] = x;
        size[x] += size[y];
    }
};

class Edge {
public:
    int u, v;
    int w;

    Edge(int u, int v, int w): u(u), v(v), w(w) {}
};

class comp {
public:
    bool operator() (const Edge &a, const Edge &b) const {
        if (a.w != b.w) return a.w > b.w;
        if (a.u != b.u) return a.u < b.u;
        return a.v < b.v;
    }
};

class Solution {

    int getDist(const vector<int> &a, const vector<int> &b) {
        return abs(a[0] - b[0]) + abs(a[1] - b[1]);
    }

    // vector<Edge> getMst(int n, vector<Edge> edges) {
    //     vector<Edge> res;

    //     sort(edges.begin(), edges.end(), comp());
    //     int treeSize = 0;
    //     DisjointSet ds(n);
    //     for (int i = 0; i < edges.size() && res.size() < n - 1; ++i) {
    //         int u = edges[i].u;
    //         int v = edges[i].v;
    //         int w = edges[i].w;
    //         if (!ds.connected(u, v)) {
    //             ds.doUnion(u, v);
    //             res.push_back(edges[i]);
    //         }
    //     }
    //     return res;
    // }

    vector<Edge> getMst(int n, vector<Edge> edges) {
        vector<Edge> res; 
        // build heap in linear time 
        priority_queue<Edge, vector<Edge>, comp> q(edges.begin(), edges.end());
        
        DisjointSet ds(n);
        while (q.size() && res.size() < n - 1) {
            Edge e = q.top();
            q.pop();
            if (!ds.connected(e.u, e.v)) {
                ds.doUnion(e.u, e.v);
                res.push_back(e);
            }
        }
        return res;
    }

public:
    int minCostConnectPoints(vector<vector<int>>& points) {
        int n = points.size();
        vector<Edge> edges;
        for (int u = 0; u < points.size(); ++u) {
            for (int v = u + 1; v < points.size(); ++v) {
                Edge e(u, v, getDist(points[u], points[v]));
                edges.push_back(e);
            }
        }      

        vector<Edge> mst = getMst(n, edges);
        int res = 0;
        for (const auto &e: mst) {
            res += e.w;
        }  
        return res;
    }
};
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