Maximum sum combination from two arrays

Given two arrays arr1[] and arr2[] each of size N. The task is to choose some elements from both the arrays such that no two elements have the same index and no two consecutive numbers can be selected from a single array. Find the maximum sum possible of above-chosen numbers.

Examples:

Input : arr1[] = {9, 3, 5, 7, 3}, arr2[] = {5, 8, 1, 4, 5}
Output : 29
Select first, third and fivth element from the first array.
Select the second and fourth element from the second array.

Input : arr1[] = {1, 2, 9}, arr2[] = {10, 1, 1}
Output : 19
Select last element from the first array and first element from the second array.

Approach :
This problem is based on dynamic programming.

  • Let dp(i, 1) be the maximum sum of the newly selected elements if the last element was taken from the position(i-1, 1).
  • dp(i, 2) is the same but the last element taken has the position (i-1, 2)
  • dp(i, 3) the same but we didn’t take any element from position i-1
  • .

    Recursion realtions are :

    dp(i, 1)=max(dp (i – 1, 2) + arr(i, 1), dp(i – 1, 3) + arr(i, 1), arr(i, 1) );
    dp(i, 2)=max(dp(i – 1, 1) + arr(i, 2 ), dp(i – 1, 3) + arr (i, 2), arr(i, 2));
    dp(i, 3)=max(dp(i- 1, 1), dp( i-1, 2) ).

    We don’t actually need dp( i, 3), if we update dp(i, 1) as max(dp(i, 1), dp(i-1, 1)) and dp(i, 2) as max(dp(i, 2), dp(i-1, 2)).

    Thus, dp(i, j) is the maximum total sum of the elements that are selected if the last element was taken from the position (i-1, 1) or less. The same with dp(i, 2). Therefore the answer to the above problem is max(dp(n, 1), dp(n, 2)).

    Below is the implementation of the above approach :

    C++

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    // CPP program to maximum sum 
    // combination from two arrays
    #include <bits/stdc++.h>
    using namespace std;
      
      
    // Function to maximum sum 
    // combination from two arrays
    int Max_Sum(int arr1[], int arr2[], int n)
    {
        // To store dp value
        int dp[n][2];
          
        // For loop to calculate the value of dp
        for (int i = 0; i < n; i++) 
        {
            if(i==0)
            {
                dp[i][0] = arr1[i];
                dp[i][1] = arr2[i];
                continue;
            }
              
           dp[i][0] = max(dp[i - 1][0], dp[i - 1][1] + arr1[i]);
           dp[i][1] = max(dp[i - 1][1], dp[i - 1][0] + arr2[i]);
        }
          
        // Return the required answer
        return max(dp[n-1][0], dp[n-1][1]);
    }
      
    // Driver code
    int main()
    {
        int arr1[] = {9, 3, 5, 7, 3};
        int arr2[] = {5, 8, 1, 4, 5};
      
        int n = sizeof(arr1) / sizeof(arr1[0]);
          
        // Function call
        cout << Max_Sum(arr1, arr2, n);
      
        return 0;
    }

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    Java

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    // Java program to maximum sum 
    // combination from two arrays
    class GFG
    {
      
    // Function to maximum sum 
    // combination from two arrays
    static int Max_Sum(int arr1[], 
                       int arr2[], int n)
    {
        // To store dp value
        int [][]dp = new int[n][2];
          
        // For loop to calculate the value of dp
        for (int i = 0; i < n; i++) 
        {
            if(i == 0)
            {
                dp[i][0] = arr1[i];
                dp[i][1] = arr2[i];
                continue;
            }
              
            dp[i][0] = Math.max(dp[i - 1][0], 
                                dp[i - 1][1] + arr1[i]);
            dp[i][1] = Math.max(dp[i - 1][1],   
                                dp[i - 1][0] + arr2[i]);
        }
          
        // Return the required answer
        return Math.max(dp[n - 1][0],
                        dp[n - 1][1]);
    }
      
    // Driver code
    public static void main(String[] args) 
    {
        int arr1[] = {9, 3, 5, 7, 3};
        int arr2[] = {5, 8, 1, 4, 5};
      
        int n = arr1.length;
          
        // Function call
        System.out.println(Max_Sum(arr1, arr2, n));
    }
    }
      
    // This code is contributed
    // by PrinciRaj1992 

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    C#

    // C# program to maximum sum
    // combination from two arrays
    using System;

    class GFG
    {

    // Function to maximum sum
    // combination from two arrays
    static int Max_Sum(int []arr1,
    int []arr2, int n)
    {
    // To store dp value
    int [,]dp = new int[n, 2];

    // For loop to calculate the value of dp
    for (int i = 0; i < n; i++) { if(i == 0) { dp[i, 0] = arr1[i]; dp[i, 1] = arr2[i]; continue; } dp[i, 0] = Math.Max(dp[i - 1, 0], dp[i - 1, 1] + arr1[i]); dp[i, 1] = Math.Max(dp[i - 1, 1], dp[i - 1, 0] + arr2[i]); } // Return the required answer return Math.Max(dp[n - 1, 0], dp[n - 1, 1]); } // Driver code public static void Main() { int []arr1 = {9, 3, 5, 7, 3}; int []arr2 = {5, 8, 1, 4, 5}; int n = arr1.Length; // Function call Console.WriteLine(Max_Sum(arr1, arr2, n)); } } // This code is contributed // by anuj_67.. [tabbyending]

    Output:

    29
    

    Time Complexity: O(N)



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    Improved By : princiraj1992, vt_m