<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):
        n = len(positions)  # Number of positions
        center_x = sum(pos[0] for pos in positions) / n
        center_y = sum(pos[1] for pos in positions) / n

        decay_factor = 0.999
        convergence_threshold = 1e-6
        learning_rate = 0.5

        while True:
            gradient_x = 0
            gradient_y = 0
            total_distance = 0

            for pos in positions:
                delta_x = center_x - pos[0]
                delta_y = center_y - pos[1]
                distance = np.sqrt(delta_x ** 2 + delta_y ** 2)
                distance = distance + 1e-8  # Small value to avoid division by zero
                gradient_x += delta_x / distance
                gradient_y += delta_y / distance
                total_distance += distance

            step_x = gradient_x * learning_rate
            step_y = gradient_y * learning_rate

            if abs(step_x) <= convergence_threshold and abs(step_y) <= convergence_threshold:
                return total_distance

            center_x -= step_x
            center_y -= step_y
            learning_rate *= decay_factor

28100