Given an array of n positive integers. Write a program to find the sum of maximum sum subsequence of the given array such that the intgers in the subsequence are sorted in increasing order. For example, if input is {1, 101, 2, 3, 100, 4, 5}, then output should be 106 (1 + 2 + 3 + 100), if the input array is {3, 4, 5, 10}, then output should be 22 (3 + 4 + 5 + 10) and if the input array is {10, 5, 4, 3}, then output should be 10

Solution
This problem is a variation of standard Longest Increasing Subsequence (LIS) problem. We need a slight change in the Dynamic Programming solution of LIS problem. All we need to change is to use sum as a criteria instead of length of increasing subsequence.

Following are C implementations for Dynamic Programming solution of the problem.

C 
/* Dynamic Programming implementation of Maximum Sum Increasing
Subsequence (MSIS) problem */
#include<stdio.h>
 
/* maxSumIS() returns the maximum sum of increasing subsequence
    in arr[] of size n */
int maxSumIS( int arr[], int n )
{
    int i, j, max = 0;
    int msis[n];
 
    /* Initialize msis values for all indexes */
    for ( i = 0; i < n; i++ )
        msis[i] = arr[i];
 
    /* Compute maximum sum values in bottom up manner */
    for ( i = 1; i < n; i++ )
        for ( j = 0; j < i; j++ )
            if ( arr[i] > arr[j] && msis[i] < msis[j] + arr[i])
                msis[i] = msis[j] + arr[i];
 
    /* Pick maximum of all msis values */
    for ( i = 0; i < n; i++ )
        if ( max < msis[i] )
            max = msis[i];
 
    return max;
}
 
/* Driver program to test above function */
int main()
{
    int arr[] = {1, 101, 2, 3, 100, 4, 5};
    int n = sizeof(arr)/sizeof(arr[0]);
    printf("Sum of maximum sum increasing subsequence is %d\n",
           maxSumIS( arr, n ) );
    return 0;
}

Output:

Sum of maximum sum increasing subsequence is 106

Time Complexity: O(n^2)

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