Minimum possible sum of absolute difference of pairs from given arrays

Given two arrays a[] and b[] of size N and M respectively (N < M), the task is to find the minimum possible sum of absolute difference of pairs formed by pairing each element of array a[] with an element of array b[]

Note: Each element of each array can be considered only once.

Examples:

Input: a[] = {2, 3, 5}, b[] = {1, 2, 3, 4, 5}
Output: 0
Explanation: Elements {2, 3, 5} in array a[] can be paired with {2, 3, 5} in array b[].
This will give a minimum absolute difference of 0.

Input: a[] = {1, 4, 5, 8}, b[] = {1, 3, 4, 6, 7}
Output: 2

Naive approach: The easiest way is to sort the arrays and try all the possible combinations to choose N elements from b[] and make pairs with a[].

Time Complexity: O(N*logN + M*logM + ^MC_N)
Auxiliary Space: O(1)

Efficient Approach: This problem can be efficiently solved by using the concept of dynamic programming using the following idea.

For each element at ith position of array a[], the jth element of b[] can either be used to form a pair with a[i] or not. 
So the overlapping subproblem property can be used to form a dp[][] array to store the minimum absolute difference till ith element of b[] and jth element of a[] is considered and can be reused for further calculations.

Follow the steps mentioned below to implement the above idea:

• First sort both arrays.
• Initialize a matrix dp[][] where dp[i][j] indicates the total minimum absolute difference till the index ith index of b[] and jth index of a[].
• While iterating through the index of the larger array b[], at each point of iterations there can be two cases:
  • Not taking the ith index into consideration of forming pair with minimum absolute difference sum. Then dp[i+1][j] = dp[i][j] as ith index is not considered and are moving to (i+1)th index directly.
  • Considering the ith index. So add the absolute difference between elements at both ith and jth index and then move to (i+1)th and (j+1)th index respectively. So total value is dp[i+1][j+1] = dp[i][j] + abs(a[j] - b[i]).
• The value at dp[M][N] is the required answer.

Below is the implementation of the above approach.

// C++ code to implement above approach

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

// Function to return the
// minimum absolute difference
int min_sum(int a[], int b[], int N, int M)
{
    // Sorting both the arrays
    sort(a, a + N);
    sort(b, b + M);

    int dp[M + 1][N + 1];

    // Initialising the dp to high value
    for (int i = 0; i <= M; i++) {
        for (int j = 0; j <= N; j++) {
            dp[i][j] = 1e9;
        }
    }
    dp[0][0] = 0;

    // Iterating through each element
    // of the larger array b
    for (int i = 0; i < M; i++) {
        
        // Case 1. Where we are not taking
        // the element at ith index
        for (int j = 0; j <= N; j++) {
            dp[i + 1][j] = dp[i][j];
        }

        // Case 2. When we have to take the
        // element at ith index
        for (int j = 0; j < N; j++) {
            dp[i + 1][j + 1]
                = min(dp[i + 1][j + 1],
                      dp[i][j] 
                      + abs(a[j] - b[i]));
        }
    }
    return dp[M][N];
}

// Driver code
int main()
{
    int a[] = { 1, 4, 5, 8 };
    int N = sizeof(a) / sizeof(a[0]);
  
    int b[] = { 1, 3, 4, 6, 7 };
    int M = sizeof(b) / sizeof(b[0]);
  
    // Function call
    cout << min_sum(a, b, N, M);
    return 0;
}

// Java program to implement above approach
import java.util.*;

class GFG {

  // Function to return the
  // minimum absolute difference
  static int min_sum(int[] a, int[] b, int N, int M)
  {
    
    // Sorting both the arrays
    Arrays.sort(a);
    Arrays.sort(b);

    int[][] dp = new int[M + 1][N + 1];

    // Initialising the dp to high value
    for (int i = 0; i <= M; i++) {
      for (int j = 0; j <= N; j++) {
        dp[i][j] = 1000000000;
      }
    }
    dp[0][0] = 0;

    // Iterating through each element
    // of the larger array b
    for (int i = 0; i < M; i++) {

      // Case 1. Where we are not taking
      // the element at ith index
      for (int j = 0; j <= N; j++) {
        dp[i + 1][j] = dp[i][j];
      }

      // Case 2. When we have to take the
      // element at ith index
      for (int j = 0; j < N; j++) {
        dp[i + 1][ j + 1]
          = Math.min(dp[i + 1][j + 1],
                     dp[i][j] 
                     + Math.abs(a[j] - b[i]));
      }
    }
    return dp[M][N];
  }

// Driver Code
public static void main(String args[])
{
    int[] a = { 1, 4, 5, 8 };
    int N = a.length;

    int[] b = { 1, 3, 4, 6, 7 };
    int M = b.length;

    // Function call
    System.out.print(min_sum(a, b, N, M));
}
}

// This code is contributed by code_hunt.

# python3 code to implement above approach

# Function to return the
# minimum absolute difference
def min_sum(a, b, N, M):

    # Sorting both the arrays
    a.sort()
    b.sort()

    dp = [[0 for _ in range(N+1)] for _ in range(M+1)]

    # Initialising the dp to high value
    for i in range(0, M + 1):
        for j in range(0, N+1):
            dp[i][j] = int(1e9)

    dp[0][0] = 0

    # Iterating through each element
    # of the larger array b
    for i in range(0, M):

        # Case 1. Where we are not taking
        # the element at ith index
        for j in range(0, N+1):
            dp[i + 1][j] = dp[i][j]

        # Case 2. When we have to take the
        # element at ith index
        for j in range(0, N):
            dp[i + 1][j + 1] = min(dp[i + 1][j + 1],
                                   dp[i][j]
                                   + abs(a[j] - b[i]))

    return dp[M][N]

# Driver code
if __name__ == "__main__":

    a = [1, 4, 5, 8]
    N = len(a)

    b = [1, 3, 4, 6, 7]
    M = len(b)

    # Function call
    print(min_sum(a, b, N, M))

    # This code is contributed by rakeshsahni

// C# code to implement above approach
using System;

public class GFG{

  // Function to return the
  // minimum absolute difference
  static int min_sum(int[] a, int[] b, int N, int M)
  {
    
    // Sorting both the arrays
    Array.Sort(a);
    Array.Sort(b);

    int[,] dp = new int[M + 1, N + 1];

    // Initialising the dp to high value
    for (int i = 0; i <= M; i++) {
      for (int j = 0; j <= N; j++) {
        dp[i,j] = 1000000000;
      }
    }
    dp[0,0] = 0;

    // Iterating through each element
    // of the larger array b
    for (int i = 0; i < M; i++) {

      // Case 1. Where we are not taking
      // the element at ith index
      for (int j = 0; j <= N; j++) {
        dp[i + 1, j] = dp[i, j];
      }

      // Case 2. When we have to take the
      // element at ith index
      for (int j = 0; j < N; j++) {
        dp[i + 1, j + 1]
          = Math.Min(dp[i + 1, j + 1],
                     dp[i, j] 
                     + Math.Abs(a[j] - b[i]));
      }
    }
    return dp[M, N];
  }

  // Driver code
  static public void Main (){

    int[] a = { 1, 4, 5, 8 };
    int N = a.Length;

    int[] b = { 1, 3, 4, 6, 7 };
    int M = b.Length;

    // Function call
    Console.Write(min_sum(a, b, N, M));
  }
}

// This code is contributed by hrithikgarg03188.

 <script>
        // JavaScript code for the above approach

// Function to return the
// minimum absolute difference
function min_sum(a,b,N, M)
{
    // Sorting both the arrays
    a.sort();
    b.sort();

    
    let dp = new Array(M+1);

    for(let i=0;i<dp.length;i++) {
        dp[i] = new Array(N+1)
    }

    // Initialising the dp to high value
    for (let i = 0; i <= M; i++) {
        for (let j = 0; j <= N; j++) {
            dp[i][j] = 1e9;
        }
    }
    dp[0][0] = 0;

    // Iterating through each element
    // of the larger array b
    for (let i = 0; i < M; i++) {
        
        // Case 1. Where we are not taking
        // the element at ith index
        for (let j = 0; j <= N; j++) {
            dp[i + 1][j] = dp[i][j];
        }

        // Case 2. When we have to take the
        // element at ith index
        for (let j = 0; j < N; j++) {
            dp[i + 1][j + 1]
                = Math.min(dp[i + 1][j + 1],
                      dp[i][j] 
                      + Math.abs(a[j] - b[i]));
        }
    }
    return dp[M][N];
}

// Driver code

    let a = [1, 4, 5, 8];
    let N = a.length;
  
    let b = [1, 3, 4, 6, 7 ];
    let M = b.length;
  
    // Function call
    document.write(min_sum(a, b, N, M));
   
     // This code is contributed by Potta Lokesh
    </script>

2

Time complexity: O(N * M log N)
Auxiliary Space: O(N * M)