<p>A delivery company wants to build a new service center in a new city. The company knows the positions of all the customers in this city on a 2D-Map and wants to build the new center in a position such that <strong>the sum of the euclidean distances to all customers is minimum</strong>.</p>

<p>Given an array <code>positions</code> where <code>positions[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> is the position of the <code>ith</code> customer on the map, return <em>the minimum sum of the euclidean distances</em> to all customers.</p>

<p>In other words, you need to choose the position of the service center <code>[x<sub>centre</sub>, y<sub>centre</sub>]</code> such that the following formula is minimized:</p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/06/25/q4_edited.jpg" />
<p>Answers within <code>10<sup>-5</sup></code> of the actual value will be accepted.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/06/25/q4_e1.jpg" style="width: 377px; height: 362px;" />
<pre>
<strong>Input:</strong> positions = [[0,1],[1,0],[1,2],[2,1]]
<strong>Output:</strong> 4.00000
<strong>Explanation:</strong> As shown, you can see that choosing [x<sub>centre</sub>, y<sub>centre</sub>] = [1, 1] will make the distance to each customer = 1, the sum of all distances is 4 which is the minimum possible we can achieve.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/06/25/q4_e3.jpg" style="width: 419px; height: 419px;" />
<pre>
<strong>Input:</strong> positions = [[1,1],[3,3]]
<strong>Output:</strong> 2.82843
<strong>Explanation:</strong> The minimum possible sum of distances = sqrt(2) + sqrt(2) = 2.82843
</pre>

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

<ul>
	<li><code>1 &lt;= positions.length &lt;= 50</code></li>
	<li><code>positions[i].length == 2</code></li>
	<li><code>0 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 100</code></li>
</ul>

class Solution {
private:
    static constexpr double EPS = 1e-12;

    inline double sq(double x) {
        return x * x;
    }

    double distToAll(vector<vector<int>>& positions, double x, double y) {
        double d = 0;
        for (auto p : positions)
            d += sqrt(sq(x - p[0]) + sq(y - p[1]));
        return d;
    }

    void avg(vector<vector<int>>& positions, double &x, double &y) {
        x = 0, y = 0;
        for (int i = 0; i < positions.size(); ++i) {
            x += positions[i][0];
            y += positions[i][1];
        }
        x /= positions.size();
        y /= positions.size();
    }

public:
    double getMinDistSum(vector<vector<int>>& positions) {
        vector<double> weights(positions.size(), 1.0 / positions.size());
        double rad = 10;
        double x, y;
        double curr_dist;

        avg(positions, x, y);
        curr_dist = distToAll(positions, x, y);

        while (rad > EPS) {
            for (int ct = 0; ct < 10; ++ct) {
                cout << fixed << x << " " << y << endl;
                double x2 = x + (rand() - RAND_MAX / 2) / (double) (RAND_MAX / 2) * rad;
                double y2 = y + (rand() - RAND_MAX / 2) / (double) (RAND_MAX / 2) * rad;
                double dist = distToAll(positions, x2, y2);
                if (dist < curr_dist) {
                    x = x2, y = y2;
                    curr_dist = dist;
                }
            }

            rad *= 0.7;
        }

        return curr_dist;
    }
};
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