<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 getMinDistSum(self, positions: List[List[int]]) -> float:
        directions = [(-1, 0), (1, 0), (0, 1), (0, -1)]
        factor = 100

        def euclidean_dist(center_x, center_y):
            total_dist = 0
            for position in positions:
                total_dist += sqrt((center_x - position[0])**2 + (center_y - position[1])**2)
            return total_dist

        curr_x = sum(x for x, _ in positions) / len(positions)
        curr_y = sum(y for _, y in positions) / len(positions)
        curr_dist = euclidean_dist(curr_x, curr_y)

        while factor > 1e-6:
            reduce_factor = True
            for dx, dy in directions:
                new_x = curr_x + dx * factor
                new_y = curr_y + dy * factor
                new_dist = euclidean_dist(new_x, new_y)
                if new_dist < curr_dist:
                    curr_dist = new_dist
                    curr_x = new_x
                    curr_y = new_y
                    reduce_factor = False
                    break
            if reduce_factor:
                factor /= 2.0

        return curr_dist
16900