# Maximum sum obtained by dividing Array into several subarrays as per given conditions

Given an array arr[] of size N, the task is to calculate the maximum sum that can be obtained by dividing the array into several subarrays(possibly one), where each subarray starting at index i and ending at index j (j>=i) contributes arr[j]-arr[i] to the sum.

Examples:

Input: arr[]= {1, 5, 3}, N=3
Output:
4
Explanation: The array can be divided into 2 subarrays:

• {1, 5} -> sum contributed by the subarray = 5-1 = 4
• {3} -> sum contributed by the subarray = 3-3 = 0

Therefore, the answer is 4.(It can be shown that there is no other way of dividing this array in multiple subarrays such that the answer is greater than 4).

Input: arr[] = {6, 2, 1}, N=3
Output:
0

Naive Approach: The naive approach is to consider all possible ways of dividing arr into 1 or more subarrays and calculating the maximum sum obtained for each.

Time Complexity: O(N*2N)
Auxiliary Space: O(1)

Observation: The observations necessary to solve the problem are below:

1. The array should be divided into several(possibly one) subarrays such that each subarray is the longest increasing subarray. For example, if arr[]={3,5,7,9,1}, it is optimal to consider {3,5,7,9} as a subarray as it would contribute 9-3=6 to the sum. Breaking it up further decreases the sum which is not optimal.
2. Every element of a non-increasing subarray should be considered as single element subarrays so that they contribute 0 to the sum. Otherwise, they would be contributing a negative value. For example, if arr[i]>arr[i+1], it is optimal to consider two subarrays of length 1 containing arr[i] and arr[i+1] separately, so that they contribute (arr[i]-arr[i]) +(arr[i+1]-arr[i+1]) =0 to the answer. If they were considered together, they would contribute arr[i+1]-arr[i] which is a negative number, thus decreasing the sum.

Efficient Approach: Follow the steps below to solve the problem:

1. Initialize a variable Sum to 0.
2. Traverse arr from 1 to N-1, using the variable i, and do the following:
1. If arr[i]>arr[i-1], add arr[i]-arr[i-1] to Sum. This works because the sum of differences of adjacent elements in a sorted array is equal to the difference of the elements at extreme ends. Here, only the increasing subarrays are considered as arr[i]>arr[i-1].

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ` `using` `namespace` `std;`   `// Function to find the required answer` `int` `maximumSum(``int` `arr[], ``int` `N)` `{` `    ``// Stores maximum sum` `    ``int` `Sum = 0;` `    ``for` `(``int` `i = 1; i < N; i++) {`   `        ``// Adding the difference of elements at ends of` `        ``// increasing subarray to the answer` `        ``if` `(arr[i] > arr[i - 1])` `            ``Sum += (arr[i] - arr[i - 1]);` `    ``}` `    ``return` `Sum;` `}` `// Driver Code` `int` `main()` `{` `    ``// Input` `    ``int` `arr[] = { 1, 5, 3 };` `    ``int` `N = (``sizeof``(arr) / (``sizeof``(arr)));`   `    ``// Function calling` `    ``cout << maximumSum(arr, N);` `    ``return` `0;` `}`

## Java

 `// Java program for the above approach` `import` `java.io.*;`   `class` `GFG{` `    `  `// Function to find the required answer` `public` `static` `int` `maximumSum(``int` `arr[], ``int` `N)` `{` `    `  `    ``// Stores maximum sum` `    ``int` `Sum = ``0``;` `    ``for``(``int` `i = ``1``; i < N; i++) ` `    ``{` `        `  `        ``// Adding the difference of elements at ends ` `        ``// of increasing subarray to the answer` `        ``if` `(arr[i] > arr[i - ``1``])` `            ``Sum += (arr[i] - arr[i - ``1``]);` `    ``}` `    ``return` `Sum;` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `    `  `    ``// Input` `    ``int` `arr[] = { ``1``, ``5``, ``3` `};` `    ``int` `N = arr.length;`   `    ``// Function calling` `    ``System.out.println(maximumSum(arr, N));` `}` `}`   `// This code is contributed by Potta Lokesh`

## Python3

 `# Python program for the above approach`   `# Function to find the required answer` `def` `maximumSum(arr, N):` `  `  `    ``# Stores maximum sum` `    ``Sum` `=` `0``;` `    ``for` `i ``in` `range``(``1``,N):` `      `  `        ``# Adding the difference of elements at ends of` `        ``# increasing subarray to the answer` `        ``if` `(arr[i] > arr[i ``-` `1``]):` `            ``Sum` `+``=` `(arr[i] ``-` `arr[i ``-` `1``])` `    `  `    ``return` `Sum``;` `    `  `# Driver Code`   `#Input` `arr ``=` `[ ``1``, ``5``, ``3` `];` `N ``=` `len``(arr)`   `# Function calling` `print``(maximumSum(arr, N));`   `# This code is contributed by SoumikMondal`

## C#

 `// C# program for the above approach` `using` `System;` `using` `System.Collections.Generic;`   `class` `GFG{`   `// Function to find the required answer` `static` `int` `maximumSum(``int` `[]arr, ``int` `N)` `{` `    `  `    ``// Stores maximum sum` `    ``int` `Sum = 0;` `    ``for``(``int` `i = 1; i < N; i++) ` `    ``{` `        `  `        ``// Adding the difference of elements at` `        ``// ends of increasing subarray to the answer` `        ``if` `(arr[i] > arr[i - 1])` `            ``Sum += (arr[i] - arr[i - 1]);` `    ``}` `    ``return` `Sum;` `}`   `// Driver Code` `public` `static` `void` `Main()` `{` `    `  `    ``// Input` `    ``int` `[]arr = { 1, 5, 3 };` `    ``int` `N = arr.Length;`   `    ``// Function calling` `    ``Console.Write(maximumSum(arr, N));` `}` `}`   `// This code is contributed by SURENDRA_GANGWAR`

## Javascript

 ``

Output

`4`

Time Complexity: O(N)
Auxiliary Space: O(1)

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