<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 getMinDistSum(vector<vector<int>>& pos) {
    double left = 100, bottom = 100, right = 0, top = 0;
    for (auto &p : pos) {
        left = min(left, (double)p[0]);
        bottom = min(bottom, (double)p[1]);
        right = max(right, (double)p[0]);
        top = max(top, (double)p[1]);
    }
    double res = DBL_MAX, res_x = 0, res_y = 0;
    for (double delta = 10; delta >= 0.00001; delta /= 10) {
        for (double x = left; x <= right; x += delta)
            for (double y = bottom; y <= top; y += delta) {
                double d = 0;
                for (auto &p : pos)
                    d += sqrt((p[0] - x) * (p[0] - x) + (p[1] - y) * (p[1] - y));
                if (res > d) {
                    res = d;
                    res_x = x;
                    res_y = y;
                }
            }
        left = res_x - delta;
        bottom = res_y - delta;
        right = res_x + delta * 2;
        top = res_y + delta * 2;
    }
    return res == DBL_MAX ? 0 : res;
}
};
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