# Maximum Sum Decreasing Subsequence

Given an array of N positive integers. The task is to find the sum of maximum sum decreasing subsequence(MSDS) of the given array such that the integers in the subsequence are sorted in decreasing order.

Examples:

Input: arr[] = {5, 4, 100, 3, 2, 101, 1}
Output: 106
100 + 3 + 2 + 1 = 106

Input: arr[] = {10, 5, 4, 3}
Output: 22
10 + 5 + 4 + 3 = 22

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

This problem is a variation of Longest Decreasing Subsequence problem. The Optimal Substructure for the above problem will be:

Let arr[0..n-1] be the input array and MSDS[i] be the maximum sum of the MSDS ending at index i such that arr[i] is the last element of the MSDS.
Then, MSDS[i] can be written as:

MSDS[i] = a[i] + max( MSDS[j] ) where i > j > 0 and arr[j] > arr[i] or,
MSDS[i] = a[i], if no such j exists.

To find the MSDS for a given array, we need to return max(MSDS[i]) where n > i > 0.

Below is the implementation of the above approach:

## C++

 `// CPP code to return the maximum sum ` `// of decreasing subsequence in arr[] ` `#include ` `using` `namespace` `std; ` ` `  `// function to return the maximum ` `// sum of decreasing subsequence ` `// in arr[] ` `int` `maxSumDS(``int` `arr[], ``int` `n) ` `{ ` `    ``int` `i, j, max = 0; ` `    ``int` `MSDS[n]; ` ` `  `    ``// Initialize msds values ` `    ``// for all indexes ` `    ``for` `(i = 0; i < n; i++) ` `        ``MSDS[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] && MSDS[i] < MSDS[j] + arr[i]) ` `                ``MSDS[i] = MSDS[j] + arr[i]; ` ` `  `    ``// Pick maximum of all msds values ` `    ``for` `(i = 0; i < n; i++) ` `        ``if` `(max < MSDS[i]) ` `            ``max = MSDS[i]; ` ` `  `    ``return` `max; ` `} ` ` `  `// Drive Code ` `int` `main() ` `{ ` `    ``int` `arr[] = { 5, 4, 100, 3, 2, 101, 1 }; ` `     `  `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` ` `  `    ``cout << ``"Sum of maximum sum decreasing subsequence is: "` `         ``<< maxSumDS(arr, n); ` `    ``return` `0; ` `} `

## Java

 `// Java code to return the maximum sum ` `// of decreasing subsequence in arr[] ` `import` `java.io.*; ` `import` `java.lang.*; ` ` `  `class` `GfG { ` `     `  `    ``// function to return the maximum ` `    ``// sum of decreasing subsequence ` `    ``// in arr[] ` `    ``public` `static` `int` `maxSumDS(``int` `arr[], ``int` `n) ` `    ``{ ` `        ``int` `i, j, max = ``0``; ` `        ``int``[] MSDS = ``new` `int``[n]; ` `     `  `        ``// Initialize msds values ` `        ``// for all indexes ` `        ``for` `(i = ``0``; i < n; i++) ` `            ``MSDS[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] &&  ` `                    ``MSDS[i] < MSDS[j] + arr[i]) ` `                    ``MSDS[i] = MSDS[j] + arr[i]; ` `     `  `        ``// Pick maximum of all msds values ` `        ``for` `(i = ``0``; i < n; i++) ` `            ``if` `(max < MSDS[i]) ` `                ``max = MSDS[i]; ` `     `  `        ``return` `max; ` `    ``} ` `     `  `    ``// Drive Code ` `    ``public` `static` `void` `main(String argc[]) ` `    ``{ ` `        ``int` `arr[] = { ``5``, ``4``, ``100``, ``3``, ``2``, ``101``, ``1` `}; ` `         `  `        ``int` `n = ``7``; ` `     `  `        ``System.out.println(``"Sum of maximum sum"` `               ``+ ``" decreasing subsequence is: "` `                           ``+ maxSumDS(arr, n)); ` `    ``} ` `} ` ` `  `// This code os contributed by Sagar Shukla. `

## C#

 `// C# code to return the ` `// maximum sum of decreasing ` `// subsequence in arr[] ` `using` `System; ` ` `  `class` `GFG ` `{ ` `     `  `    ``// function to return the  ` `    ``// maximum sum of decreasing ` `    ``// subsequence in arr[] ` `    ``public` `static` `int` `maxSumDS(``int` `[]arr,  ` `                               ``int` `n) ` `    ``{ ` `        ``int` `i, j, max = 0; ` `        ``int``[] MSDS = ``new` `int``[n]; ` `     `  `        ``// Initialize msds values ` `        ``// for all indexes ` `        ``for` `(i = 0; i < n; i++) ` `            ``MSDS[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] &&  ` `                    ``MSDS[i] < MSDS[j] + arr[i]) ` `                    ``MSDS[i] = MSDS[j] + arr[i]; ` `     `  `        ``// Pick maximum of  ` `        ``// all msds values ` `        ``for` `(i = 0; i < n; i++) ` `            ``if` `(max < MSDS[i]) ` `                ``max = MSDS[i]; ` `     `  `        ``return` `max; ` `    ``} ` `     `  `    ``// Drive Code ` `    ``static` `public` `void` `Main () ` `    ``{ ` `        ``int` `[]arr = {5, 4, 100,  ` `                     ``3, 2, 101, 1}; ` `        ``int` `n = 7; ` `        ``Console.WriteLine(``"Sum of maximum sum"` `+  ` `                ``" decreasing subsequence is: "` `+  ` `                              ``maxSumDS(arr, n)); ` `    ``} ` `} ` ` `  `// This code is contributed by m_kit `

## PHP

 ` `

Output:

```Sum of maximum sum decreasing subsequence is: 106
```

Time complexity: O(N2)
Auxiliary Space: O(N)

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