<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 {
    constexpr static array<pair<int, int>, 4> directions{{{0, 1}, {1, 0}, {0, -1}, {-1, 0}}};
    static constexpr auto distance = [] (const vector<vector<int>>& positions, const vector<double>& p) {
        static constexpr auto euclidean = [] (const vector<int>& lhs, const vector<double>& rhs) {
            return sqrt(pow(lhs.front() - rhs.front(), 2) + 
                        pow(lhs.back() - rhs.back(), 2));
        };
        double res = 0;
        for (const auto& position: positions)
            res += euclidean(position, p);
        return res;
    };
public:
    double getMinDistSum(vector<vector<int>>& positions) {
        double res = numeric_limits<int>::max();
        double step = 100;
        vector<double> start{0, 0};
        while (step > 0)
        {
            bool f = true;
            for (const auto& [x, y]: directions)
            {
                const vector<double> next{start.front() + step * x, start.back() + step * y};
                if (const double d = distance(positions, next); d < res)
                {
                    res = d;
                    start = next;
                    f = false;
                    break;
                }
            }
            if (f)
                step /= 2;
        }
          return res;
    }
};
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