<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) {
    constexpr double kErr = 1e-6;
    double currX = 50;
    double currY = 50;
    double ans = distSum(positions, currX, currY);
    double step = 1;

    while (step > kErr) {
      bool shouldDecreaseStep = true;
      for (const auto& [dx, dy] : vector<pair<double, double>>{
               {0, step}, {0, -step}, {step, 0}, {-step, 0}}) {
        const double x = currX + dx;
        const double y = currY + dy;
        const double newDistSum = distSum(positions, x, y);
        if (newDistSum < ans) {
          ans = newDistSum;
          currX = x;
          currY = y;
          shouldDecreaseStep = false;
        }
      }
      if (shouldDecreaseStep)
        step /= 10;
    }

    return ans;
  }

 private:
  double distSum(const vector<vector<int>>& positions, double a, double b) {
    double sum = 0;
    for (const vector<int>& p : positions)
      sum += sqrt(pow(a - p[0], 2) + pow(b - p[1], 2));
    return sum;
  }
};
10300