<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>

#define ld long double
class Solution {
public:
    double getMinDistSum(vector<vector<int>>& positions) {
        ld eps = 1e-7;
        vector<ld> x;
        vector<ld> y;
        for(int i=0; i<positions.size(); i++) {
            x.push_back(positions[i][0]);
            y.push_back(positions[i][1]);
        }

        ld dist = 50;
        ld cx = 50;
        ld cy = 50;
        vector<int> xm = {0,0,-1,1,0};
        vector<int> ym = {-1,1,0,0,0};
        vector<ld> val(5);
        vector<int> is(5);
        iota(is.begin(),is.end(),0);
        while(dist >= eps) {
            // cout<<cx<<" "<<cy<<"\n";
            for(int k=0; k<5; k++) {
                val[k] = 0;
                for(int i=0; i<positions.size(); i++) {
                    val[k] += get_dist(cx+xm[k]*dist,cy+ym[k]*dist,x[i],y[i]);
                }
            }
            sort(begin(is),end(is),[&](auto a, auto b) { return val[a] < val[b]; });
            // cout<<val[4]<<" "<<val[is[0]]<<"\n";;
            if(val[4] - eps <= val[is[0]]) {
                dist /= 2.0;
            }
            else {
                cx += xm[is[0]]*dist;
                cy += ym[is[0]]*dist;
            }
        }
        ld res = 0 ;
        for(int i=0; i<positions.size(); i++) {
            res += get_dist(cx,cy,x[i],y[i]);
        }
        return (double)res;
    }
    ld get_dist(ld x1, ld y1, ld x2, ld y2) {
        return sqrt((x2-x1)*(x2-x1)+(y2-y1)*(y2-y1));
    }
};
11500