<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 {
public:
    
    double Distance(std::vector<int> point, double x, double y) {
        return std::sqrt((x - point[0]) * (x - point[0]) + (y - point[1]) * (y - point[1]));
    }
    
    double getMinDistSum(vector<vector<int>>& points) {
        
        if (points.size() == 1) {
            return 0;
        }

        // Sample point should not be from sample points
        std::random_device rd;
        std::mt19937 gen(rd());
        std::uniform_real_distribution<> distrib(1.0, 10.0);

        double x = distrib(gen);
        double y = distrib(gen);

        double precision = 1e-7;

        double distance = 1e6;

        while (true) {
            double denominator = 0;
            double nx = 0, ny = 0;

            double new_distance = 0;
            for (auto& point: points) {
                auto d = Distance(point, x, y);
                new_distance += d;
                denominator += 1.0 / (d + precision);

                nx += point[0] / (d + precision);
                ny += point[1] / (d + precision);
            }

            nx /= denominator;
            ny /= denominator;

            x = nx;
            y = ny;

            if (abs(distance - new_distance) < precision) {
                break;
            }
            distance = new_distance;
        }
        
        return distance;
    }
};
13700