<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:

// Helper function to calculate the sum of distances from the center to all positions
double getTotalDistance(const vector<vector<int>>& positions, double x, double y) {
    double totalDistance = 0.0;
    for (const auto& pos : positions) {
        totalDistance += sqrt(pow(x - pos[0], 2) + pow(y - pos[1], 2));
    }
    return totalDistance;
}

double getMinDistSum(vector<vector<int>>& positions) {
    double epsilon = 1e-7; // tolerance for convergence
    double step = 1.0; // initial step size
    double decay = 0.9; // step decay factor

    double x = 0.0, y = 0.0; // starting point: the average of all customer positions
    for (const auto& pos : positions) {
        x += pos[0];
        y += pos[1];
    }
    x /= positions.size();
    y /= positions.size();

    double prevDist = getTotalDistance(positions, x, y);
    
    while (step > epsilon) {
        bool foundBetter = false;
        
        // Try moving in four directions: up, down, left, right
        for (const auto& [dx, dy] : vector<pair<double, double>>{{step, 0}, {-step, 0}, {0, step}, {0, -step}}) {
            double newX = x + dx;
            double newY = y + dy;
            double newDist = getTotalDistance(positions, newX, newY);

            if (newDist < prevDist) {
                prevDist = newDist;
                x = newX;
                y = newY;
                foundBetter = true;
                break;
            }
        }

        if (!foundBetter) {
            // Reduce step size when no improvement found
            step *= decay;
        }
    }

    return prevDist;
}
};
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