Hammered distance between N points in a 2-D plane

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Given n number of point in 2-d plane followed by Xi, Yi describing n points. The task is to calculate the hammered distance of n points.
Note: Hammered distance is the sum of the square of the shortest distance between every pair of the point.

Examples:

Input: n = 3
0 1
0 0
1 0
Output: 4

Input: n = 4
1 0
2 0
3 0
4 0
Output: 20

Basic Approach:As we have to find out sum of square of shortest distance among all the pairs.So, we can take every possible pair and calculate the sum of square of distance.

// Pseudo code to find hammered-distance using above approach.
//this will store hammered distance
Distance=0
for(int i=0;i<n;i++)
{
    for(int j=i+1;j<n;j++)
    {
         //shortest distance between point i and j.
         Distance+=(x[i]-x[j])^2+(y[i]-y[j])^2
     }
}

Its time complexity will be O(n^2).
Efficient Approach: This problem can be solved in time complexity of O(N).

$\begin{document} $$Sum=\sum_{i=1}^{n} \sum_{j=1}^{i-1} \left(X_j-X_i \right)^2+\left(Y_j-Y_i \right)^2$$ We can solve separtely for x and y coordinates. For X: $$Sum_x=\sum_{i=1}^{n} \sum_{j=1}^{i-1} \left(X_j-X_i \right)^2$$ $$Sum_x= \sum_{i=1}^{n} \sum_{j=1}^{i-1} \left(X_j^2+X_i^2 -2 \cdot X_i \cdot X_j \right)$$ Now expand the summation part, We can write this equation as- $$Sum_x=\sum_{i=1}^{n} \left( (i-1)*X_i^2+\sum_{j=1}^{i-1}X_j^2-2 \cdot X_i\cdot \sum_{j=1}^{i-1} X_j \right)$$ $\sum_{j=1}^{i-1}X_j$ This is commulative sum of square of points upto i-1.So, this can be calculated in linear time. Similarly, This can also be calculated in linear time. $2 \cdot X_i\cdot \sum_{j=1}^{i-1} X_j$ \end{document}$

Below is the implementation of above approach:

C++

// C++ implementation of above approach 
#include <bits/stdc++.h> 
#define ll long long int 
using namespace std; 
 
// Function calculate cumulative sum 
// of x, y, x^2, y^2 coordinates. 
void cumm(vector<ll>& x, vector<ll>& y, 
        vector<ll>& cummx, vector<ll>& cummy, 
        vector<ll>& cummx2, vector<ll>& cummy2, ll n) 
{ 
    for (int i = 1; i <= n; i++) { 
        cummx[i] = cummx[i - 1] + x[i]; 
        cummy[i] = cummy[i - 1] + y[i]; 
        cummx2[i] = cummx2[i - 1] + x[i] * x[i]; 
        cummy2[i] = cummy2[i - 1] + y[i] * y[i]; 
    } 
} 
 
// Function ot calculate the hammered distance 
int calHammeredDistance(int n, vector<ll>& x, vector<ll>& y) 
{ 
    // cummx contains cumulative sum of x 
    // cummy contains cumulative sum of y 
    vector<ll> cummx(n + 1, 0), cummy(n + 1, 0); 
 
    // cummx2 contains cumulative sum of x^2 
    // cummy2 contains cumulative sum of y^2 
    vector<ll> cummx2(n + 1, 0), cummy2(n + 1, 0); 
 
    // calculate cumulative of x 
    //, y, x^2, y^2, because these terms 
    // required in formula to reduce complexity. 
 
    // this function calculate all required terms. 
    cumm(x, y, cummx, cummy, cummx2, cummy2, n); 
 
    // hdx calculate hammer distance for x coordinate 
    // hdy calculate hammer distance for y coordinate 
    ll hdx = 0, hdy = 0; 
 
    for (int i = 1; i <= n; i++) { 
 
        // came from formula describe in explanation 
        hdx += (i - 1) * x[i] * x[i] + cummx2[i - 1] 
            - 2 * x[i] * cummx[i - 1]; 
 
        // came from formula describe in explanation 
        hdy += (i - 1) * y[i] * y[i] + cummy2[i - 1] 
            - 2 * y[i] * cummy[i - 1]; 
    } 
 
    // total is the sum of both x and y. 
    ll total = hdx + hdy; 
    return total; 
} 
 
// Driver code 
int main() 
{ 
    // number of points 
    int n = 3; 
 
    // x contains the x coordinates 
    // y contains the y coordinates 
    //and converting the size to n+1
    vector<ll> x = {0, 0, 1, 0};
    vector<ll> y = {1, 0, 0, 0};
 
    cout << calHammeredDistance(n, x, y); 
 
    return 0; 
} 

Java

// Java implementation of above approach
 
 
class GFG{
  
// Function calculate cumulative sum
// of x, y, x^2, y^2 coordinates.
static void cumm(int [] x, int [] y,
        int [] cummx, int [] cummy,
        int [] cummx2, int [] cummy2, int n)
{
    for (int i = 1; i <= n; i++) {
        cummx[i] = cummx[i - 1] + x[i];
        cummy[i] = cummy[i - 1] + y[i];
        cummx2[i] = cummx2[i - 1] + x[i] * x[i];
        cummy2[i] = cummy2[i - 1] + y[i] * y[i];
    }
}
  
// Function ot calculate the hammered distance
static int calHammeredDistance(int n, int [] x, int [] y)
{
    // cummx contains cumulative sum of x
    // cummy contains cumulative sum of y
    int []cummx = new int[n + 1];
    int []cummy = new int[n + 1];
  
    // cummx2 contains cumulative sum of x^2
    // cummy2 contains cumulative sum of y^2
    int []cummx2 = new int[n + 1];
    int []cummy2 = new int[n + 1];
  
    // calculate cumulative of x
    //, y, x^2, y^2, because these terms
    // required in formula to reduce complexity.
  
    // this function calculate all required terms.
    cumm(x, y, cummx, cummy, cummx2, cummy2, n);
  
    // hdx calculate hammer distance for x coordinate
    // hdy calculate hammer distance for y coordinate
    int hdx = 0, hdy = 0;
  
    for (int i = 1; i <= n; i++) {
  
        // came from formula describe in explanation
        hdx += (i - 1) * x[i] * x[i] + cummx2[i - 1]
               - 2 * x[i] * cummx[i - 1];
  
        // came from formula describe in explanation
        hdy += (i - 1) * y[i] * y[i] + cummy2[i - 1]
               - 2 * y[i] * cummy[i - 1];
    }
  
    // total is the sum of both x and y.
    int total = hdx + hdy;
    return total;
}
  
// Driver code
public static void main(String[] args)
{
    // number of points
    int n = 3;
  
    // x contains the x coordinates
    // y contains the y coordinates
    int []x = new int[n + 1];
    int []y = new int[n + 1];
    x[2] = 1;
    y[0] = 1;
  
    System.out.print(calHammeredDistance(n, x, y));
  
}
}
 
// This code contributed by Rajput-Ji

Python3

# Python3 implementation of the 
# above approach 
 
# Function calculate cumulative sum 
# of x, y, x^2, y^2 coordinates. 
def cumm(x, y, cummx, cummy, 
               cummx2, cummy2, n): 
 
    for i in range(1, n+1): 
        cummx[i] = cummx[i - 1] + x[i] 
        cummy[i] = cummy[i - 1] + y[i] 
        cummx2[i] = cummx2[i - 1] + x[i] * x[i] 
        cummy2[i] = cummy2[i - 1] + y[i] * y[i] 
 
# Function ot calculate the 
# hammered distance 
def calHammeredDistance(n, x, y): 
 
    # cummx contains cumulative sum of x 
    # cummy contains cumulative sum of y 
    cummx = [0] * (n + 1)
    cummy = [0] * (n + 1) 
 
    # cummx2 contains cumulative sum of x^2 
    # cummy2 contains cumulative sum of y^2 
    cummx2 = [0] * (n + 1)
    cummy2 = [0] * (n + 1) 
 
    # calculate cumulative of x , y, x^2, y^2, 
    # because these terms are required in the
    # formula to reduce complexity. 
 
    # This function calculate all required terms. 
    cumm(x, y, cummx, cummy, cummx2, cummy2, n) 
 
    # hdx calculate hammer distance for x coordinate 
    # hdy calculate hammer distance for y coordinate 
    hdx, hdy = 0, 0
 
    for i in range(1, n + 1): 
 
        # came from formula describe in explanation 
        hdx += ((i - 1) * x[i] * x[i] + cummx2[i - 1] -
                             2 * x[i] * cummx[i - 1])
 
        # came from formula describe in explanation 
        hdy += ((i - 1) * y[i] * y[i] + cummy2[i - 1] -
                             2 * y[i] * cummy[i - 1])
     
    # total is the sum of both x and y. 
    total = hdx + hdy 
    return total 
 
# Driver Code
if __name__ == "__main__":
 
    # number of points 
    n = 3
 
    # x contains the x coordinates 
    # y contains the y coordinates 
    x = [0, 0, 1, 0] 
    y = [1, 0, 0, 0] 
 
    print(calHammeredDistance(n, x, y)) 
 
# This code is contributed by Rituraj Jain

C#

// C# implementation of above approach
using System;
 
class GFG{
   
// Function calculate cumulative sum
// of x, y, x^2, y^2 coordinates.
static void cumm(int [] x, int [] y,
        int [] cummx, int [] cummy,
        int [] cummx2, int [] cummy2, int n)
{
    for (int i = 1; i <= n; i++) {
        cummx[i] = cummx[i - 1] + x[i];
        cummy[i] = cummy[i - 1] + y[i];
        cummx2[i] = cummx2[i - 1] + x[i] * x[i];
        cummy2[i] = cummy2[i - 1] + y[i] * y[i];
    }
}
   
// Function ot calculate the hammered distance
static int calHammeredDistance(int n, int [] x, int [] y)
{
    // cummx contains cumulative sum of x
    // cummy contains cumulative sum of y
    int []cummx = new int[n + 1];
    int []cummy = new int[n + 1];
   
    // cummx2 contains cumulative sum of x^2
    // cummy2 contains cumulative sum of y^2
    int []cummx2 = new int[n + 1];
    int []cummy2 = new int[n + 1];
   
    // calculate cumulative of x
    //, y, x^2, y^2, because these terms
    // required in formula to reduce complexity.
   
    // this function calculate all required terms.
    cumm(x, y, cummx, cummy, cummx2, cummy2, n);
   
    // hdx calculate hammer distance for x coordinate
    // hdy calculate hammer distance for y coordinate
    int hdx = 0, hdy = 0;
   
    for (int i = 1; i <= n; i++) {
   
        // came from formula describe in explanation
        hdx += (i - 1) * x[i] * x[i] + cummx2[i - 1]
               - 2 * x[i] * cummx[i - 1];
   
        // came from formula describe in explanation
        hdy += (i - 1) * y[i] * y[i] + cummy2[i - 1]
               - 2 * y[i] * cummy[i - 1];
    }
   
    // total is the sum of both x and y.
    int total = hdx + hdy;
    return total;
}
   
// Driver code
public static void Main(String[] args)
{
    // number of points
    int n = 3;
   
    // x contains the x coordinates
    // y contains the y coordinates
    int []x = new int[n + 1];
    int []y = new int[n + 1];
    x[2] = 1;
    y[0] = 1;
   
    Console.Write(calHammeredDistance(n, x, y));  
}
}
 
// This code is contributed by PrinciRaj1992

Javascript

<script>
      // JavaScript implementation of above approach
      // Function calculate cumulative sum
      // of x, y, x^2, y^2 coordinates.
      function cumm(x, y, cummx, cummy, cummx2, cummy2, n) {
        for (var i = 1; i <= n; i++) {
          cummx[i] = cummx[i - 1] + x[i];
          cummy[i] = cummy[i - 1] + y[i];
          cummx2[i] = cummx2[i - 1] + x[i] * x[i];
          cummy2[i] = cummy2[i - 1] + y[i] * y[i];
        }
      }
 
      // Function ot calculate the hammered distance
      function calHammeredDistance(n, x, y) {
        // cummx contains cumulative sum of x
        // cummy contains cumulative sum of y
        var cummy = new Array(n + 1).fill(0);
        var cummx = new Array(n + 1).fill(0);
 
        // cummx2 contains cumulative sum of x^2
        // cummy2 contains cumulative sum of y^2
        var cummx2 = new Array(n + 1).fill(0);
        var cummy2 = new Array(n + 1).fill(0);
 
        // calculate cumulative of x
        //, y, x^2, y^2, because these terms
        // required in formula to reduce complexity.
 
        // this function calculate all required terms.
        cumm(x, y, cummx, cummy, cummx2, cummy2, n);
 
        // hdx calculate hammer distance for x coordinate
        // hdy calculate hammer distance for y coordinate
        var hdx = 0,
          hdy = 0;
 
        for (var i = 1; i <= n; i++) {
          // came from formula describe in explanation
          hdx +=
            (i - 1) * x[i] * x[i] + cummx2[i - 1] - 2 * x[i] * cummx[i - 1];
 
          // came from formula describe in explanation
          hdy +=
            (i - 1) * y[i] * y[i] + cummy2[i - 1] - 2 * y[i] * cummy[i - 1];
        }
 
        // total is the sum of both x and y.
        var total = hdx + hdy;
        return total;
      }
 
      // Driver code
      // number of points
      var n = 3;
 
      // x contains the x coordinates
      // y contains the y coordinates
      var x = new Array(n + 1).fill(0);
      var y = new Array(n + 1).fill(0);
      x[2] = 1;
      y[0] = 1;
 
      document.write(calHammeredDistance(n, x, y));
    </script>

Output

Time Complexity: O(n)
Auxiliary Space: O(n)

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