# Dynamic Programming | Set 14 (Maximum Sum Increasing Subsequence)

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

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

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/C++ and Python implementations for Dynamic Programming solution of the problem.

## C/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 %dn",
maxSumIS( arr, n ) );
return 0;
}
```

## Java

```/* Dynamic Programming Java implementation of Maximum Sum
Increasing Subsequence (MSIS) problem */
class MSIS
{
/* maxSumIS() returns the maximum sum of increasing
subsequence in arr[] of size n */
static int maxSumIS( int arr[], int n )
{
int i, j, max = 0;
int msis[] = new int[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 */
public static void main(String args[])
{
int arr[] = new int[]{1, 101, 2, 3, 100, 4, 5};
int n = arr.length;
System.out.println("Sum of maximum sum increasing "+
" subsequence is "+
maxSumIS( arr, n ) );
}
}
/* This code is contributed by Rajat Mishra */
```

## Python

```# Dynamic Programming bsed Python implementation of Maximum Sum Increasing
# Subsequence (MSIS) problem

# maxSumIS() returns the maximum sum of increasing subsequence in arr[] of
# size n
def maxSumIS(arr, n):
max = 0
msis = [0 for x in range(n)]

# Initialize msis values for all indexes
for i in range(n):
msis[i] = arr[i]

# Compute maximum sum values in bottom up manner
for i in range(1, n):
for j in range(i):
if arr[i] > arr[j] and msis[i] < msis[j] + arr[i]:
msis[i] = msis[j] + arr[i]

# Pick maximum of all msis values
for i in range(n):
if max < msis[i]:
max = msis[i]

return max

# Driver program to test above function
arr = [1, 101, 2, 3, 100, 4, 5]
n = len(arr)
print("Sum of maximum sum increasing subsequence is " +
str(maxSumIS(arr, n)))

# This code is contributed by Bhavya Jain
```

Output:

`Sum of maximum sum increasing subsequence is 106`

Time Complexity: O(n^2)

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