<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-10;

    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;
    }

public:
    double getMinDistSum(vector<vector<int>>& positions) {
        double xc = 50, yc = 50, l = 50;
        double x[9], y[9];
        while (l > EPS) {
            x[0] = xc - l, y[0] = yc - l;
            x[1] = xc - l, y[1] = yc + l;
            x[2] = xc + l, y[2] = yc - l;
            x[3] = xc + l, y[3] = yc + l;
            x[4] = xc, x[4] = yc;
            x[5] = xc - l, y[5] = yc;
            x[6] = xc + l, y[6] = yc;
            x[7] = xc, y[7] = yc - l;
            x[8] = xc, y[8] = yc + l;
            
            double min_dist = 1e9;
            int nxt = 0;
            for (int i = 0; i < 9; ++i) {
                double d = distToAll(positions, x[i], y[i]);
                if (d < min_dist) {
                    min_dist = d;
                    nxt = i;
                }
            }

            xc = x[nxt], yc = y[nxt];

            l *= 2.0/3.0;
        }

        return distToAll(positions, xc, yc);
    }
};
51900