<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 Point{
    public:
        double x;
        double y;
};


class Solution {
public:
    double getSumDist(Point p, vector<Point> pointList){
        double dist = 0;
        for(int i=0;i<pointList.size();i++){
            dist += sqrt((p.x-pointList[i].x)*(p.x-pointList[i].x) + (p.y-pointList[i].y)*(p.y-pointList[i].y));
        }

        return dist;
    }

    double getMinDistSum(vector<vector<int>>& positions) {
        vector<Point> pList;

        for(int i=0;i<positions.size();i++){
            Point p;
            p.x = positions[i][0];
            p.y = positions[i][1];
            pList.push_back(p);
        }

        vector<pair<int, int>> grad = {{-1,0}, {1,0}, {0, 1}, {0, -1}}; 
        double diff = 100;
        bool found = false;
        Point curP;

        for(int i=0;i<pList.size();i++) curP.x+= pList[i].x, curP.y+= pList[i].y;
        curP.x/=pList.size();
        curP.y/=pList.size();
        double curDist = getSumDist(curP, pList);

        while(diff> (1e-7)){
            // cout<<diff<<endl;
            // cout<<(diff> (10^(-6)))<<endl;
            bool reduce = true;
            for(int i=0;i<grad.size();i++){
                Point temp;
                double tempDist;
                temp.x  = curP.x + diff * grad[i].first;
                temp.y = curP.y + diff * grad[i].second;
                tempDist = getSumDist(temp, pList);    
                if(tempDist < curDist){
                    curDist = tempDist;
                    curP = temp;
                    reduce = false;
                    break;
                }
            }
            if(reduce){
                diff = diff/2;
            }
            
        }

        return curDist;
    }
};
17300