<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)
    {
        auto centerPoint = getPoint( positions );
        double distSum = 0;

        for( auto pos : positions )
        {
            distSum += getDist( pos[0], pos[1], centerPoint.first, centerPoint.second );
        }

        return distSum;
    }

    pair<double, double> getPoint( vector<vector<int>>& positions)
    {
        double X = 0, Y = 0;
        int size = positions.size();

        for( auto pos: positions )
        {
            X += pos[0];
            Y += pos[1];
        }

        X /= size;
        Y /= size;

        int round = 1000;
        double eps = 1e-8;
        double x0 = X;
        double y0 = Y;

        while( round > 0 )
        {
            round--;
            double sumX = 0;
            double sumY = 0;
            double denom = 0;

            for( auto pos : positions )
            {
                double currentDist = getDist(pos[0], pos[1], X, Y);
                if( currentDist == 0 )
                {
                    return pair<double, double>( X, Y );
                }

                double weight = 1 / currentDist;
                sumX += weight * pos[0];
                sumY += weight * pos[1];
                denom += weight;
            }

            X = sumX / denom;
            Y = sumY / denom;
            if( getDist( x0, y0, X, Y ) <= eps )
            {
                return pair<double, double>( X, Y );
            }
        }

        return pair<double, double>( X, Y );
    }

    double getDist( double x0, double y0, double x1, double y1 )
    {
        double diffX = x1 - x0;
        double diffY = y1 - y0;

        return sqrt( (diffX * diffX) + (diffY * diffY) );
    }
};
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