# Minimum Sum of Euclidean Distances to all given Points

• Difficulty Level : Medium
• Last Updated : 11 May, 2022

Given a matrix mat[][] consisting of N pairs of the form {x, y} each denoting coordinates of N points, the task is to find the minimum sum of the Euclidean distances to all points.

Examples:

Input: mat[][] = { { 0, 1}, { 1, 0 }, { 1, 2 }, { 2, 1 }}
Output: 4
Explanation:
Average of the set of points, i.e. Centroid = ((0+1+1+2)/4, (1+0+2+1)/4) = (1, 1).
Euclidean distance of each point from the centroid are {1, 1, 1, 1}
Sum of all distances = 1 + 1 + 1 + 1 = 4
Input: mat[][] = { { 1, 1}, { 3, 3 }}
Output: 2.82843

Approach:
Since the task is to minimize the Euclidean Distance to all points, the idea is to calculate the Median of all the points. Geometric Median generalizes the concept of median to higher dimensions

Follow the steps below to solve the problem:

• Calculate the centroid of all the given coordinates, by getting the average of the points.
• Find the Euclidean distance of all points from the centroid.
• Calculate the sum of these distance and print as the answer.

Below is the implementation of above approach:

## C++

 // C++ Program to implement// the above approach#include using namespace std; // Function to calculate Euclidean distancedouble find(double x, double y,            vector >& p){     double mind = 0;    for (int i = 0; i < p.size(); i++) {         double a = p[i][0], b = p[i][1];        mind += sqrt((x - a) * (x - a)                     + (y - b) * (y - b));    }     return mind;} // Function to calculate the minimum sum// of the euclidean distances to all pointsdouble getMinDistSum(vector >& p){     // Calculate the centroid    double x = 0, y = 0;    for (int i = 0; i < p.size(); i++) {        x += p[i][0];        y += p[i][1];    }    x = x / p.size();    y = y / p.size();     // Calculate distance of all    // points    double mind = find(x, y, p);     return mind;} // Driver Codeint main(){     // Initializing the points    vector > vec        = { { 0, 1 }, { 1, 0 }, { 1, 2 }, { 2, 1 } };     double d = getMinDistSum(vec);    cout << d << endl;     return 0;}

## Java

 // Java program to implement// the above approachclass GFG{ // Function to calculate Euclidean distancestatic double find(double x, double y,                   int [][] p){    double mind = 0;         for(int i = 0; i < p.length; i++)    {        double a = p[i][0], b = p[i][1];        mind += Math.sqrt((x - a) * (x - a) +                          (y - b) * (y - b));    }    return mind;} // Function to calculate the minimum sum// of the euclidean distances to all pointsstatic double getMinDistSum(int [][]p){         // Calculate the centroid    double x = 0, y = 0;    for(int i = 0; i < p.length; i++)    {        x += p[i][0];        y += p[i][1];    }         x = x / p.length;    y = y / p.length;     // Calculate distance of all    // points    double mind = find(x, y, p);     return mind;} // Driver Codepublic static void main(String[] args){         // Initializing the points    int [][]vec = { { 0, 1 }, { 1, 0 },                    { 1, 2 }, { 2, 1 } };     double d = getMinDistSum(vec);         System.out.print(d + "\n");}} // This code is contributed by Amit Katiyar

## Python3

 # Python3 program to implement# the above approachfrom math import sqrt # Function to calculate Euclidean distancedef find(x, y, p):     mind = 0    for i in range(len(p)):        a = p[i][0]        b = p[i][1]        mind += sqrt((x - a) * (x - a) +                     (y - b) * (y - b))                          return mind # Function to calculate the minimum sum# of the euclidean distances to all pointsdef getMinDistSum(p):     # Calculate the centroid    x = 0    y = 0         for i in range(len(p)):        x += p[i][0]        y += p[i][1]             x = x // len(p)    y = y // len(p)     # Calculate distance of all    # points    mind = find(x, y, p)     return mind # Driver Codeif __name__ == '__main__':     # Initializing the points    vec = [ [ 0, 1 ], [ 1, 0 ],            [ 1, 2 ], [ 2, 1 ] ]     d = getMinDistSum(vec)    print(int(d)) # This code is contributed by mohit kumar 29

## C#

 // C# program to implement// the above approachusing System;class GFG{ // Function to calculate Euclidean distancestatic double find(double x, double y,                   int [,] p){    double mind = 0;         for(int i = 0; i < p.GetLength(0); i++)    {        double a = p[i,0], b = p[i,1];        mind += Math.Sqrt((x - a) * (x - a) +                          (y - b) * (y - b));    }    return mind;} // Function to calculate the minimum sum// of the euclidean distances to all pointsstatic double getMinDistSum(int [,]p){         // Calculate the centroid    double x = 0, y = 0;    for(int i = 0; i < p.GetLength(0); i++)    {        x += p[i,0];        y += p[i,1];    }         x = x / p.Length;    y = y / p.Length;     // Calculate distance of all    // points    double mind = find(x, y, p);     return mind;} // Driver Codepublic static void Main(String[] args){         // Initializing the points    int [,]vec = { { 0, 1 }, { 1, 0 },                    { 1, 2 }, { 2, 1 } };     int d = (int)getMinDistSum(vec);         Console.Write(d + "\n");}} // This code is contributed by Rohit_ranjan

## Javascript



Output:

4

Time Complexity: O(N)
Auxiliary Space: O(1)

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