<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>>& positions) {
        struct Point{
            long double x{0.0};
            long double y{0.0};    
        };

        struct Square{
            Point topLeft{};
            long double length{0.0};

            constexpr Square() = default;

            Square(const Point& topLeft, long double length) : topLeft(topLeft), length(length){}

            Square getSubsquare(std::size_t i, std::size_t j) const{
                return Square(
                    {topLeft.x + length / 10 * i, topLeft.y + length / 10 * j},
                    length / 10
                );
            }

            Square getExpandedSquare() const{
                return Square ({topLeft.x-length/2, topLeft.y-length/2}, length*2);
            }

            Point getCenter() const{
                return {topLeft.x + length / 2, topLeft.y + length / 2};
            }
        };

        const auto getDistance = [&](Point p){
            long double distance = 0;
            for(auto position : positions){
                const long double dx = p.x - position[0];
                const long double dy = p.y - position[1];
                distance += std::sqrt(dx * dx + dy * dy);
            }

            return distance;
        };

        Square grid({0, 0}, 100);

        while(grid.length > 1e-5){
            long double minDist = std::numeric_limits<double>::max();
            Square minSquare;

            for(std::size_t i = 0; i < 10; i++){
                for(std::size_t j = 0; j < 10; j++){
                    const Square subSquare = grid.getSubsquare(i, j);
                    const long double dist = getDistance(subSquare.getCenter());

                    if(dist < minDist){
                        minSquare = subSquare;
                        minDist = dist;
                    }
                }
            }

            grid = minSquare.getExpandedSquare();
        }

        return getDistance(grid.getCenter());
    }
};
86500