Program to find Minkowski Distance

Given two arrays A[] and B[] as position vector of two points in n-dimensional space along with an integer p, the task is to calculate Minkowski Distance between these two points.

The Minkowski distance is a generalization of other distance measures, such as Euclidean and Manhattan distances, and is defined as:

D(A, B) = \left( \sum_{i=1}^{n} |A_i - B_i|^p \right)^{1/p}

where A and B are vectors representing points in the multidimensional space, n is the number of dimensions, and p is a positive constant known as the order parameter.

Examples:

Input: A[] = {1,2,3,4}, B[] = {5,6,7,8}, P = 3
Output: 6.340

Input: A = {1,2,3,4}, B[] = {5,6,7,8} P = 2
Output: 8

Approach:

Traverse using loop and calculate X as {(A₁ - B₁)^P + (A₂ - B₂)^P . . . (A_N - B_N)^P}. Then calculate Minkowski Distance as X^(1/P)

Step-by-step approach:

• Create a variable let say X.
• Run a loop and follow below mentioned steps under the scope of loop:
  • X += Power ((A_i - B_i), P)
• Calculate Z as (1/P)
• Return Power(X, Z)

Below is the implementation of the above approach:

// CPP code to implement the approach

#include <bits/stdc++.h>
using namespace std;

// Function to calculate Minkowski
double minkowski(const vector<double>& A,
                 const vector<double>& B, double P)
{
    // Variable to store value of X
    double X = 0.0;

    // Loop to calculate value of X
    for (size_t i = 0; i < A.size(); ++i) {
        X += pow(abs(A[i] - B[i]), P);
    }
    
     // Calculating Z as (1/P)
    double Z = 1.0 / P;

    // Returning X^(1/P) as X^Z
    return pow(X, Z);
}

int main()
{
    // Input vectors
    vector<double> A = { 1, 2, 3, 4 };
    vector<double> B = { 5, 6, 7, 8 };
    double P = 2.0;

    //Function_ call
    cout << minkowski(A, B, P) << endl;

    return 0;
}

import java.util.ArrayList;
import java.util.List;

public class MinkowskiDistance {

    // Function to calculate Minkowski
    private static double
    minkowski(List<Double> A, List<Double> B, double P)
    {
        // Variable to store value of X
        double X = 0.0;

        // Loop to calculate value of X
        for (int i = 0; i < A.size(); ++i) {
            X += Math.pow(Math.abs(A.get(i) - B.get(i)), P);
        }

        // Calculating Z as (1/P)
        double Z = 1.0 / P;

        // Returning X^(1/P) as X^Z
        return Math.pow(X, Z);
    }

    public static void main(String[] args)
    {
        // Input vectors
        List<Double> A = List.of(1.0, 2.0, 3.0, 4.0);
        List<Double> B = List.of(5.0, 6.0, 7.0, 8.0);
        double P = 2.0;

        // Function call
        System.out.println(minkowski(A, B, P));
    }
}

import math

# Function to calculate Minkowski
def minkowski(A, B, P):
    # Variable to store value of X
    X = 0.0

    # Loop to calculate value of X
    for i in range(len(A)):
        X += abs(A[i] - B[i]) ** P
    
    # Calculating Z as (1/P)
    Z = 1.0 / P

    # Returning X^(1/P) as X^Z
    return X ** Z

if __name__ == "__main__":
    # Input lists
    A = [1, 2, 3, 4]
    B = [5, 6, 7, 8]
    P = 2.0

    # Function call
    print(minkowski(A, B, P))

# This code is contributed by rambabuguphka

using System;
using System.Collections.Generic;

public class Program
{
    // Function to calculate Minkowski distance
    static double Minkowski(List<double> A, List<double> B, double P)
    {
        // Variable to store value of X
        double X = 0.0;

        // Loop to calculate value of X
        for (int i = 0; i < A.Count; ++i)
        {
            X += Math.Pow(Math.Abs(A[i] - B[i]), P);
        }

        // Calculating Z as (1/P)
        double Z = 1.0 / P;

        // Returning X^(1/P) as X^Z
        return Math.Pow(X, Z);
    }

    public static void Main()
    {
        // Input lists
        List<double> A = new List<double> { 1, 2, 3, 4 };
        List<double> B = new List<double> { 5, 6, 7, 8 };
        double P = 2.0;

        // Function call
        Console.WriteLine(Minkowski(A, B, P));
    }
}


// This code is contributed by akshitaguprzj3

// Function to calculate Minkowski
function minkowski(A, B, P) {
    // Variable to store value of X
    let X = 0.0;

    // Loop to calculate value of X
    for (let i = 0; i < A.length; ++i) {
        X += Math.pow(Math.abs(A[i] - B[i]), P);
    }

    // Calculating Z as (1/P)
    let Z = 1.0 / P;

    // Returning X^(1/P) as X^Z
    return Math.pow(X, Z);
}

// Input arrays
let A = [1, 2, 3, 4];
let B = [5, 6, 7, 8];
let P = 2.0;

// Function call
console.log(minkowski(A, B, P));

8

Time Complexity: O(n * log₂(p))
Auxiliary Space: O(1)

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• Minkowski distance in Python