<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 {
public:
    double dist(const vector<double>& center, const vector<int>& aim) {
        double dx = center[0] - aim[0];
        double dy = center[1] - aim[1];
        return sqrt(dx * dx + dy * dy);
    }
    
    double distsum(const vector<double>& center, const vector<vector<int>>& positions) {
        double ans = 0;
        for (const auto& x : positions) {
            ans += dist(center, x);
        }
        return ans;
    }
    
    double getMinDistSum(vector<vector<int>>& positions) {
        vector<double> center(2, 0);
        for (const auto& pos : positions) {
            center[0] += pos[0];
            center[1] += pos[1];
        }
        center[0] /= positions.size();
        center[1] /= positions.size();
        
        double oridis = distsum(center, positions);
        const double precision = 1e-7;
        double step = 1;
        const int max_iterations = 100000;
        
        vector<double> directions = {0, 1, 0, -1, 1, 0, -1, 0, 1, 1, -1, -1, 1, -1, -1, 1};
        
        for (int iter = 0; iter < max_iterations; ++iter) {
            bool improved = false;
            for (int i = 0; i < 8; ++i) {
                vector<double> new_center = {
                    center[0] + directions[i*2] * step,
                    center[1] + directions[i*2+1] * step
                };
                double new_dis = distsum(new_center, positions);
                if (new_dis < oridis) {
                    center = new_center;
                    oridis = new_dis;
                    improved = true;
                    break;
                }
            }
            if (!improved) {
                if (step < precision) break;
                step /= 2;
            }
        }
        
        return oridis;
    }
};
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