<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>

import numpy as np
class Solution:
    def getMinDistSum(self, positions: List[List[int]]) -> float:
        offset = 10
        x_min, y_min = positions[0]
        x_max, y_max = positions[0]
        x_best, y_best = 0, 0
        min_cost = float('inf')
        for x, y in positions:
            x_min = min(x, x_min)
            x_max = max(x, x_max)
            y_min = min(y, y_min)
            y_max = max(y, y_max)
        while offset >= 1e-5:
            for x in np.arange(x_min, x_max + offset, offset):
                for y in np.arange(y_min, y_max + offset, offset):
                    cost = 0
                    for position in positions:
                        cost += math.sqrt((x-position[0]) ** 2 + (y-position[1]) ** 2)
                    if cost < min_cost:
                        min_cost = cost
                        x_best, y_best = x, y
            
            x_min = x_best - offset
            y_min = y_best - offset
            x_max = x_best + offset
            y_max = y_best + offset
            offset /= 10

        return min_cost
                

29000