Maximize the Sum of a Subsequence from an Array based on given conditions

Given an array a[] consisting of N integers, the task is to perform the following operations: 
 

• Select a subsequence and for every p^th element of the subsequence, calculate the product p * a[i].
• Calculate the sum of the calculated values of p * a[i].
• The subsequence should be selected such that it maximizes the desired sum.

Examples:

Input: N = 3, a[] = {-1, 3, 4} 
Output: 17 
Explanation: 
The subsequence {-1, 3, 4} maximizes the sum = 1(-1) + 2(3) + 3(4) = 17
Input: N = 5, a[] = {-1, -9, 0, 5, -7} 
Output: 14 
Explanation: 
The subsequence {-1, 0, 5} maximizes the sum = 1(-1) + 2(0) + 3(5) = 14

Naive Approach: The simplest approach to solve the problem is to generate all possible subsequences from the array and calculate the sum for each subsequence. Finally, find the maximum sum. 
Time Complexity: O(N³) 
Auxiliary Space: O(N)

Efficient Approach: The above approach can be optimized by using Dynamic Programming. Follow the steps below to solve the problem:

• For every element, two possibilities exist, that is, either the element is part of the subsequence or is not.
• Initialize a dp[][] matrix, where dp[i][j] stores the maximum of: 
  
  • Sum generated by selecting a[i] as the j^th element of the subsequence, that is:

    a[i] * j + maximumSum(j + 1, i + 1)

• Sum generated by not selecting a[i] as the j^th element of the subsequence, that is:

  maximumSum(j, i + 1)

  Therefore, the recurrence relation is:

  dp[i][j] = max(a[i] * j + maximumSum(j + 1, i + 1), maximumSum(j, i + 1))

• Keep updating the dp[][] table by considering the above conditions for each array element and print the maximum sum possible from the entire array.

Below is the implementation of the above approach:

// C++ program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;
const int N = 6;

// Function to select the array elements to
// maximize the sum of the selected elements
int maximumSum(int a[], int count, 
               int index, int n, 
               int dp[N][N])
{
    // If the entire array
    // is solved
    if (index == n)
        return 0;

    // Memoized subproblem
    if (dp[index][count] != -1)
        return dp[index][count];

    // Calculate sum considering the
    // current element in the subsequence
    int take_element = a[index] * count + 
                         maximumSum(a, count + 1, 
                                  index + 1, n, dp);

    // Calculate sum without considering the
    // current element in the subsequence
    int dont_take = maximumSum(a, count, 
                               index + 1, n, dp);

    // Update the maximum of the above sums
    // in the dp[][] table
    return dp[index][count] = max(take_element, 
                                  dont_take);
}

// Driver Code
int main()
{
    int n = 5;
    int a[] = { -1, -9, 0, 5, -7 };

    // Initialize the dp array
    int dp[N][N];
    memset(dp, -1, sizeof(dp));

    cout << (maximumSum(a, 1, 0, n, dp));
}

// This code is contributed by Rajput-Ji

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

public class GFG {

    // Function to select the array elements to
    // maximize the sum of the selected elements
    public static int maximumSum(int[] a, int count,
                                 int index, int n,
                                 int[][] dp)
    {
        // If the entire array
        // is solved
        if (index == n)
            return 0;

        // Memoized subproblem
        if (dp[index][count] != -1)
            return dp[index][count];

        // Calculate sum considering the
        // current element in the subsequence
        int take_element
            = a[index] * count
              + maximumSum(a, count + 1,
                           index + 1, n, dp);

        // Calculate sum without considering the
        // current element in the subsequence
        int dont_take
            = maximumSum(a, count, index + 1, n, dp);

        // Update the maximum of the above sums
        // in the dp[][] table
        return dp[index][count]
            = Math.max(take_element, dont_take);
    }

    // Driver Code
    public static void main(String args[])
    {
        int n = 5;
        int a[] = { -1, -9, 0, 5, -7 };

        // Initialize the dp array
        int dp[][] = new int[n + 1][n + 1];
        for (int i[] : dp)
            Arrays.fill(i, -1);

        System.out.println(maximumSum(a, 1, 0, n, dp));
    }
}

# Python3 program to implement
# the above approach

# Function to select the array elements to
# maximize the sum of the selected elements
def maximumSum(a, count, index, n, dp):

    # If the entire array
    # is solved
    if(index == n):
        return 0

    # Memoized subproblem
    if(dp[index][count] != -1):
        return dp[index][count]

    # Calculate sum considering the
    # current element in the subsequence
    take_element = (a[index] * count + 
                 maximumSum(a, count + 1,
                               index + 1, 
                               n, dp))

    # Calculate sum without considering the
    # current element in the subsequence
    dont_take = maximumSum(a, count, 
                           index + 1, n, dp)

    # Update the maximum of the above sums
    # in the dp[][] table
    dp[index][count] = max(take_element,
                           dont_take)
                           
    return dp[index][count]

# Driver Code
n = 5
a = [ -1, -9, 0, 5, -7 ]

# Initialize the dp array
dp = [[-1 for x in range(n + 1)]
          for y in range(n + 1)]

# Function call
print(maximumSum(a, 1, 0, n, dp))

# This code is contributed by Shivam Singh

// C# program to implement
// the above approach
using System;

class GFG{

// Function to select the array elements to
// maximize the sum of the selected elements
public static int maximumSum(int[] a, int count,
                             int index, int n,
                             int[,] dp)
{
    
    // If the entire array
    // is solved
    if (index == n)
        return 0;

    // Memoized subproblem
    if (dp[index, count] != -1)
        return dp[index, count];

    // Calculate sum considering the
    // current element in the subsequence
    int take_element = a[index] * count + 
                       maximumSum(a, count + 1,
                                     index + 1,
                                     n, dp);

    // Calculate sum without considering the
    // current element in the subsequence
    int dont_take = maximumSum(a, count, 
                               index + 1, n, dp);

    // Update the maximum of the above sums
    // in the [,]dp table
    return dp[index, count] = Math.Max(take_element, 
                                       dont_take);
}

// Driver Code
public static void Main(String []args)
{
    int n = 5;
    int []a = { -1, -9, 0, 5, -7 };

    // Initialize the dp array
    int [,]dp = new int[n + 1, n + 1];
    for(int i = 0; i < n + 1; i++)
    {
        for(int j = 0; j < n + 1; j++)
        {
            dp[i, j] = -1;
        }
    }
    Console.WriteLine(maximumSum(a, 1, 0, n, dp));
}
}

// This code is contributed by PrinciRaj1992

<script>

// JavaScript program to implement
// the above approach
var N = 6;

// Function to select the array elements to
// maximize the sum of the selected elements
function maximumSum(a, count, index, n, dp)
{
    // If the entire array
    // is solved
    if (index == n)
        return 0;

    // Memoized subproblem
    if (dp[index][count] != -1)
        return dp[index][count];

    // Calculate sum considering the
    // current element in the subsequence
    var take_element = a[index] * count + 
                         maximumSum(a, count + 1, 
                                  index + 1, n, dp);

    // Calculate sum without considering the
    // current element in the subsequence
    var dont_take = maximumSum(a, count, 
                               index + 1, n, dp);

    // Update the maximum of the above sums
    // in the dp[][] table
    return dp[index][count] = Math.max(take_element, 
                                  dont_take);
}

// Driver Code
var n = 5;
var a = [ -1, -9, 0, 5, -7];
// Initialize the dp array
var dp = Array.from(Array(N), ()=>Array(N).fill(-1));
document.write(maximumSum(a, 1, 0, n, dp));


</script>

14

Time Complexity: O(N²) 
Auxiliary Space: O(N²)