<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

error = 1e-13
max_iterations = int(1e9)


class Solution:
    def getMinDistSum(self, positions: List[List[int]]) -> float:
        def gradient(v):
            """
            :param v: 2D coordinate [x, y]
            return array of three scalars [f, dx, dy]
            """
            dv = v - data
            ds = np.sqrt(np.sum(dv ** 2, axis=1))
            expanded = ds[:, np.newaxis]
            dp = np.dot(weight, np.divide(dv, expanded, out=np.zeros_like(dv), where=expanded != 0))
            return dp

        def descend(derivatives):
            tentatives = (steps[:, np.newaxis] * derivatives[np.newaxis, :])
            newv = v[np.newaxis, :] - tentatives[:, np.newaxis]
            dv = newv - data[np.newaxis, :]
            ds = np.dot(np.sqrt(np.sum(dv ** 2, axis=-1)), weight)
            min_index = np.argmin(ds)
            return ds[min_index], newv[min_index, :, :]

        data, weight = zip(*tuple(Counter([tuple(e) for e in positions]).items()))
        data = np.array(data)
        weight = np.array(weight)
        steps = np.array([0,1e-8, .00001, .0001, .001, .01, .03, .1, .3, 1, 3, 10])
        v = np.sum(data, axis=0) / len(data)
        result, _ = descend(np.zeros((2,)))
        for _ in range(max_iterations):
            derivatives = gradient(v)
            new_result, v = descend(derivatives)
            if result - new_result < error:
                return result
            result = new_result

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