<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 distance(vector<int>& a, vector<double>& b){
        return sqrt( (1.0 * a[0]-b[0]) * (1.0 * a[0]-b[0]) + (1.0 * a[1]-b[1]) * (1.0 *a[1] - b[1]));
    }

    double get_total_distance(vector<vector<int>>& positions, vector<double>& p){
        double sum = 0;
        for (auto pos : positions){
            sum += distance(pos, p);
        }
        return sum;
    }


    double getMinDistSum(vector<vector<int>>& positions) {
        double step = 100;
        double err = 1e-6; 
        vector<int> dirs = {1,0,-1,0,1};
        vector<double> p = {0,0};
        double ans = get_total_distance(positions,p);
        while(step > err){
            bool moved = false;
            for (int i = 0; i<4; i++){
                vector<double> next = {p[0] + step * dirs[i], p[1] + step * dirs[i+1]};
                double dist = get_total_distance(positions, next);
                if (dist < ans){
                    moved = true;
                    p = next;
                    ans = dist;
                    break;
                }
            }
            if (!moved){
                step /= 2;
            }
        }
        return ans;
    }
};
23600