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Maximize sum of all elements which are not a part of the Longest Increasing Subsequence
• Difficulty Level : Hard
• Last Updated : 23 Feb, 2021

Given an array arr[], the task is to find the maximum sum of all the elements which are not a part of the longest increasing subsequence.
Examples:

Input: arr[] = {4, 6, 1, 2, 3, 8}
Output: 10
Explanation:
Elements are 4 and 6
Input: arr[] = {5, 4, 3, 2, 1}
Output: 14
Explanation:
Elements are 5, 4, 3, 2

Approach:

• The idea is to find the longest increasing subsequence with the minimum sum and then subtract it from the sum of all elements.
• To do this we will use the concept of LIS using Dynamic Programming and store the sum along with the length of the subsequences and update the minimum sum accordingly.

Below is the implementation of the above approach.

## C++

 `// C++ program to find the Maximum sum of``// all elements which are not a part of``// longest increasing sub sequence` `#include ``using` `namespace` `std;` `// Function to find maximum sum``int` `findSum(``int``* arr, ``int` `n)``{``    ``int` `totalSum = 0;` `    ``// Find total sum of array``    ``for` `(``int` `i = 0; i < n; i++) {``        ``totalSum += arr[i];``    ``}` `    ``// Maintain a 2D array``    ``int` `dp[n];``    ``for` `(``int` `i = 0; i < n; i++) {``        ``dp[i] = 1;``        ``dp[i] = arr[i];``    ``}` `    ``// Update the dp array along``    ``// with sum in the second row``    ``for` `(``int` `i = 1; i < n; i++) {``        ``for` `(``int` `j = 0; j < i; j++) {``            ``if` `(arr[i] > arr[j]) {``                ``// In case of greater length``                ``// Update the length along``                ``// with sum``                ``if` `(dp[i] < dp[j] + 1) {``                    ``dp[i] = dp[j] + 1;``                    ``dp[i] = dp[j]``                               ``+ arr[i];``                ``}` `                ``// In case of equal length``                ``// find length update length``                ``// with minimum sum``                ``else` `if` `(dp[i]``                         ``== dp[j] + 1) {``                    ``dp[i]``                        ``= min(dp[i],``                              ``dp[j]``                                  ``+ arr[i]);``                ``}``            ``}``        ``}``    ``}``    ``int` `maxm = 0;``    ``int` `subtractSum = 0;` `    ``// Find the sum that need to``    ``// be subtracted from total sum``    ``for` `(``int` `i = 0; i < n; i++) {``        ``if` `(dp[i] > maxm) {``            ``maxm = dp[i];``            ``subtractSum = dp[i];``        ``}``        ``else` `if` `(dp[i] == maxm) {``            ``subtractSum = min(subtractSum,``                              ``dp[i]);``        ``}``    ``}` `    ``// Return the sum``    ``return` `totalSum - subtractSum;``}` `// Driver code``int` `main()``{``    ``int` `arr[] = { 4, 6, 1, 2, 3, 8 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);` `    ``cout << findSum(arr, n);` `    ``return` `0;``}`

## Java

 `// Java program to find the Maximum sum of``// all elements which are not a part of``// longest increasing sub sequence``class` `GFG{` `// Function to find maximum sum``static` `int` `findSum(``int` `[]arr, ``int` `n)``{``    ``int` `totalSum = ``0``;` `    ``// Find total sum of array``    ``for``(``int` `i = ``0``; i < n; i++)``    ``{``       ``totalSum += arr[i];``    ``}` `    ``// Maintain a 2D array``    ``int` `[][]dp = ``new` `int``[``2``][n];``    ``for``(``int` `i = ``0``; i < n; i++)``    ``{``       ``dp[``0``][i] = ``1``;``       ``dp[``1``][i] = arr[i];``    ``}` `    ``// Update the dp array along``    ``// with sum in the second row``    ``for``(``int` `i = ``1``; i < n; i++)``    ``{``       ``for``(``int` `j = ``0``; j < i; j++)``       ``{``          ``if` `(arr[i] > arr[j])``          ``{``              ` `              ``// In case of greater length``              ``// Update the length along``              ``// with sum``              ``if` `(dp[``0``][i] < dp[``0``][j] + ``1``)``              ``{``                  ``dp[``0``][i] = dp[``0``][j] + ``1``;``                  ``dp[``1``][i] = dp[``1``][j] + arr[i];``              ``}``              ` `              ``// In case of equal length``              ``// find length update length``              ``// with minimum sum``              ``else` `if` `(dp[``0``][i] == dp[``0``][j] + ``1``)``              ``{``                  ``dp[``1``][i] = Math.min(dp[``1``][i],``                                      ``dp[``1``][j] + arr[i]);``              ``}``          ``}``       ``}``    ``}``    ``int` `maxm = ``0``;``    ``int` `subtractSum = ``0``;` `    ``// Find the sum that need to``    ``// be subtracted from total sum``    ``for``(``int` `i = ``0``; i < n; i++)``    ``{``       ``if` `(dp[``0``][i] > maxm)``       ``{``           ``maxm = dp[``0``][i];``           ``subtractSum = dp[``1``][i];``       ``}``       ``else` `if` `(dp[``0``][i] == maxm)``       ``{``           ``subtractSum = Math.min(subtractSum, dp[``1``][i]);``       ``}``    ``}` `    ``// Return the sum``    ``return` `totalSum - subtractSum;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `arr[] = { ``4``, ``6``, ``1``, ``2``, ``3``, ``8` `};``    ``int` `n = arr.length;` `    ``System.out.print(findSum(arr, n));``}``}` `// This code is contributed by sapnasingh4991`

## Python3

 `# Python3 program to find the maximum sum ``# of all elements which are not a part of``# longest increasing sub sequence` `# Function to find maximum sum``def` `findSum(arr, n):` `    ``totalSum ``=` `0` `    ``# Find total sum of array``    ``for` `i ``in` `range``(n):``        ``totalSum ``+``=` `arr[i]` `    ``# Maintain a 2D array``    ``dp ``=` `[[``0``] ``*` `n ``for` `i ``in` `range``(``2``)]` `    ``for` `i ``in` `range``(n):``        ``dp[``0``][i] ``=` `1``        ``dp[``1``][i] ``=` `arr[i]` `    ``# Update the dp array along``    ``# with sum in the second row``    ``for` `i ``in` `range``(``1``, n):``        ``for` `j ``in` `range``(i):``            ``if` `(arr[i] > arr[j]):` `                ``# In case of greaer length``                ``# update the length along``                ``# with sum``                ``if` `(dp[``0``][i] < dp[``0``][j] ``+` `1``):``                    ``dp[``0``][i] ``=` `dp[``0``][j] ``+` `1``                    ``dp[``1``][i] ``=` `dp[``1``][j] ``+` `arr[i]` `                ``# In case of equal length``                ``# find length update length``                ``# with minimum sum``                ``elif` `(dp[``0``][i] ``=``=` `dp[``0``][j] ``+` `1``):``                    ``dp[``1``][i] ``=` `min``(dp[``1``][i],``                                   ``dp[``1``][j] ``+``                                     ``arr[i])` `    ``maxm ``=` `0``    ``subtractSum ``=` `0` `    ``# Find the sum that need to``    ``# be subtracted from total sum``    ``for` `i ``in` `range``(n):``        ``if` `(dp[``0``][i] > maxm):``            ``maxm ``=` `dp[``0``][i]``            ``subtractSum ``=` `dp[``1``][i]` `        ``elif` `(dp[``0``][i] ``=``=` `maxm):``            ``subtractSum ``=` `min``(subtractSum,``                              ``dp[``1``][i])` `    ``# Return the sum``    ``return` `totalSum ``-` `subtractSum` `# Driver code``arr ``=` `[ ``4``, ``6``, ``1``, ``2``, ``3``, ``8` `]``n ``=` `len``(arr)` `print``(findSum(arr, n))` `# This code is contributed by himanshu77`

## C#

 `// C# program to find the Maximum sum of``// all elements which are not a part of``// longest increasing sub sequence``using` `System;``class` `GFG{` `// Function to find maximum sum``static` `int` `findSum(``int` `[]arr, ``int` `n)``{``    ``int` `totalSum = 0;` `    ``// Find total sum of array``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ``totalSum += arr[i];``    ``}` `    ``// Maintain a 2D array``    ``int` `[,]dp = ``new` `int``[2, n];``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ``dp[0, i] = 1;``        ``dp[1, i] = arr[i];``    ``}` `    ``// Update the dp array along``    ``// with sum in the second row``    ``for``(``int` `i = 1; i < n; i++)``    ``{``        ``for``(``int` `j = 0; j < i; j++)``        ``{``            ``if` `(arr[i] > arr[j])``            ``{``                    ` `                ``// In case of greater length``                ``// Update the length along``                ``// with sum``                ``if` `(dp[0, i] < dp[0, j] + 1)``                ``{``                    ``dp[0, i] = dp[0, j] + 1;``                    ``dp[1, i] = dp[1, j] + arr[i];``                ``}``                    ` `                ``// In case of equal length``                ``// find length update length``                ``// with minimum sum``                ``else` `if` `(dp[0, i] == dp[0, j] + 1)``                ``{``                    ``dp[1, i] = Math.Min(dp[1, i],``                                        ``dp[1, j] + arr[i]);``                ``}``            ``}``        ``}``    ``}``    ``int` `maxm = 0;``    ``int` `subtractSum = 0;` `    ``// Find the sum that need to``    ``// be subtracted from total sum``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ``if` `(dp[0, i] > maxm)``        ``{``            ``maxm = dp[0, i];``            ``subtractSum = dp[1, i];``        ``}``        ``else` `if` `(dp[0, i] == maxm)``        ``{``            ``subtractSum = Math.Min(subtractSum, dp[1, i]);``        ``}``    ``}` `    ``// Return the sum``    ``return` `totalSum - subtractSum;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `[]arr = { 4, 6, 1, 2, 3, 8 };``    ``int` `n = arr.Length;` `    ``Console.Write(findSum(arr, n));``}``}` `// This code is contributed by sapnasingh4991`
Output:
`10`

Time Complexity: O(N2) where N is the length of the array arr[]
Auxiliary Space: O(N)

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