Minimum Sum of Euclidean Distances to all given Points

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 distance` `double` `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 points` `double` `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 Code` `int` `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 approach` `class` `GFG{`   `// Function to calculate Euclidean distance` `static` `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 points` `static` `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 Code` `public` `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 approach` `from` `math ``import` `sqrt`   `# Function to calculate Euclidean distance` `def` `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 points` `def` `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 Code` `if` `__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 approach` `using` `System;` `class` `GFG{`   `// Function to calculate Euclidean distance` `static` `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 points` `static` `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 Code` `public` `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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