# Maximum Sum Decreasing Subsequence

• Difficulty Level : Easy
• Last Updated : 05 May, 2021

Given an array of N positive integers. The task is to find the sum of the 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

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This problem is a variation of the 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;``}` `// Driver 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;``    ``}``    ` `    ``// Driver 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.`

## Python3

 `# Python3 code to return the maximum sum``# of decreasing subsequence in arr[]` `# Function to return the maximum``# sum of decreasing subsequence``# in arr[]``def` `maxSumDS(arr, n):``    ` `    ``i, j, ``max` `=` `(``0``, ``0``, ``0``)``    ` `    ``MSDS``=``[``0` `for` `i ``in` `range``(n)]`` ` `    ``# Initialize msds values``    ``# for all indexes``    ``for` `i ``in` `range``(n):``        ``MSDS[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``                ``MSDS[i] < MSDS[j] ``+` `arr[i]):``                ``MSDS[i] ``=` `MSDS[j] ``+` `arr[i]`` ` `    ``# Pick maximum of all msds values``    ``for` `i ``in` `range``(n):``        ``if` `(``max` `< MSDS[i]):``            ``max` `=` `MSDS[i]` `    ``return` `max``    ` `if` `__name__ ``=``=` `"__main__"``:``    ` `    ``arr``=``[``5``, ``4``, ``100``, ``3``,``         ``2``, ``101``, ``1``]``    ``n``=``len``(arr)``    ``print``(``"Sum of maximum sum decreasing subsequence is: "``,``           ``maxSumDS(arr, n))` `# This code is contributed by Rutvik_56`

## 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;``    ``}``    ` `    ``// Driver 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

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## Javascript

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Output

`Sum of maximum sum decreasing subsequence is: 106`

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

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