<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:

    def dist(self, position0: List[int], position1: List[int]) -> float:
        return ((position0[0] - position1[0]) ** 2 +
                (position0[1] - position1[1]) ** 2) ** 0.5

    def getMinDistSum(self, positions: List[List[int]]) -> float:
        n = len(positions)
        prev_x, curr_x = 200, 0
        prev_y, curr_y = 200, 0
        for x, y in positions:
            curr_x += 1.0 * x / n
            curr_y += 1.0 * y / n
        while self.dist([prev_x, prev_y], [curr_x, curr_y]) > 10 ** -8:
            num_x, den_x = 0, 0
            num_y, den_y = 0, 0
            for x, y in positions:
                d = self.dist([x, y], [curr_x, curr_y])
                if d == 0: continue
                num_x += x / d
                den_x += 1 / d
                num_y += y / d
                den_y += 1 / d
            prev_x = curr_x
            prev_y = curr_y
            if den_x == 0 and den_y == 0: break
            if den_x != 0: curr_x = num_x / den_x
            if den_y != 0: curr_y = num_y / den_y
        minDist = 0
        for x, y in positions: minDist += self.dist([x, y], [curr_x, curr_y])
        return minDist
16400