<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>

/**
Approach:
- we don't need to find the location, we just need to find that one exists at a given minimum distance to all customers
Theory:
- a location will exist  will exist if the circle around each customer with radius 


**/

static constexpr double NO_DISTANCE = 1000000;
static constexpr double THRESHOLD = 0.000001;
static const vector<int> dirs = {0, 0, -1, 1};

double calcDistance(vector<vector<int>>& positions, pair<double, double> coords) {
    double ans = 0;
    for (auto pos : positions) {
        ans += sqrt(pow(pos[0] - coords.first, 2) + pow(pos[1] - coords.second, 2));
    }

    return ans;
}

class Solution {
public:
    double getMinDistSum(vector<vector<int>>& positions) {
        double step = 50;
        pair<double, double> coords = {50, 50};
        double lastDistance = calcDistance(positions, coords);

        while (true) {
            pair<double, double> bestCoords = {-1, -1};
            double bestDistance = NO_DISTANCE;
            for (int dir = 0; dir < 4; ++dir) {
                pair<double, double> newCoords = {
                    coords.first + dirs[dir] * step,
                    coords.second + dirs[(dir + 2) % 4] * step
                };

                double newDistance = calcDistance(positions, newCoords);
                if (newDistance < lastDistance) {
                    bestCoords = newCoords;
                    bestDistance = newDistance;
                }
            }

            // No improvement, decrease step size
            if (bestDistance > lastDistance) {
                step /= 2;
                if (step < THRESHOLD) {
                    break;
                }
            }

            // Improvement, update and continue
            else {
                coords = bestCoords;
                lastDistance = bestDistance;
            }
        }
        
        return lastDistance;
    }
};
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