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Maximum array sum with prefix and suffix multiplications with -1 allowed
• Difficulty Level : Medium
• Last Updated : 10 May, 2021

Given N elements (both positive and negative). Find the maximum sum, provided that the first operation is to take some prefix of the sequence and multiply all numbers in this prefix by -1. The second operation is to take some suffix and multiply all numbers in it by -1. The chosen prefix and suffix may intersect. What is the maximum total sum of the sequence that can be obtained by applying the described operations?

Examples:

```Input : -1 -2 -3
Output : 6
Explanation: Multiply prefix {-1, -2} with -1.
Multiply suffix {-3} with -1. We get total
sum as 1 + 2 + 3 = 6

Input : -1 10 -5 10 -2
Output : 18
Explanation: Multiply -1 with prefix {-1} and
multiply -1 with suffix {-2}. Elements after
multiplying {1, 10, -5, 10, 2} and sum is
1 + 10 -5 + 10 + 2 = 18.

Input: -4 2 0 5 0
Output:  11
Explanation: Multiply {-4} with -1. Do not
multiply anything in the suffix, so we get
{4, 2, 0, 5, 0} to get sum as 11. ```

If desired prefix and suffix intersect, then their common part is remaining with the initial sign, and therefore, this case is equivalent to the case when we take the same suffix and prefix, but without their common part.
We traverse from left to right and see if sum or -the sum is more at any step by multiplying -1 to it, and store the maximum of pre_sum and -pre_sum at any index, and continue this process for all elements.
Then we traverse from end to start, and check whose sum is more either the (prefix_sum at that index + negative sum) or the previous maximum that we obtained, if we find at any index the negative sum + prefix sum at that index appears to be more at any step, then we replace the ans to sum*(-1) + pre_sum.

## C++

 `// CPP program to find maximum array sum``// with multiplications of a prefix and a``// suffix with -1 allowed.``#include ``using` `namespace` `std;`` ` `// function to maximize the sum``int` `maximize(``int` `a[], ``int` `n)``{  ``    ``// stores the pre sum``    ``int` `presum[n];``     ` `    ``// to store sum from 0 to i``    ``int` `sum = 0;` `    ``// stores the maximum sum with``    ``// prefix multiplication with -1.``    ``int` `max_sum = 0;``     ` `    ``// traverse from 0 to n``    ``for` `(``int` `i = 0; i= 0; --i)``    ``{``        ``// stores the sum multiplied by (-1)``        ``g -= a[i];` `        ``// stores the max of ans and``        ``// presum + (-1*negative sum);``        ``ans = max(ans, g + presum[i]);``    ``}``     ` `    ``// returns answer``    ``return` `ans;``}` `// driver program to test the above function``int` `main() {`` ` `    ``int` `a[] = {-4, 2, 0, 5, 0};``    ``int` `n = ``sizeof``(a)/``sizeof``(a);``    ``cout << maximize(a, n);    ``    ``return` `0;``}`

## Java

 `// JAVA program to find maximum array sum``// with multiplications of a prefix and a``// suffix with -1 allowed.` `import` `java.math.*;``class` `GFG {``    ` `    ``// function to maximize the sum``    ``static` `int` `maximize(``int` `a[], ``int` `n)``    ``{  ``        ``// stores the pre sum``        ``int` `presum[] =``new` `int``[n];``          ` `        ``// to store sum from 0 to i``        ``int` `sum = ``0``;``     ` `        ``// stores the maximum sum with``        ``// prefix multiplication with -1.``        ``int` `max_sum = ``0``;``          ` `        ``// traverse from 0 to n``        ``for` `(``int` `i = ``0``; i= ``0``; --i)``        ``{``            ``// stores the sum multiplied by (-1)``            ``g -= a[i];``     ` `            ``// stores the max of ans and``            ``// presum + (-1*negative sum);``            ``ans = Math.max(ans, g + presum[i]);``        ``}``          ` `        ``// returns answer``        ``return` `ans;``    ``}``     ` `    ``// driver program to test the above function``    ``public` `static` `void` `main(String args[]) {``      ` `        ``int` `a[] = {-``4``, ``2``, ``0``, ``5``, ``0``};``        ``int` `n = a.length;``        ``System.out.println(maximize(a, n));    ``    ``}``}` `/*This code is contributed by Nikita Tiwari.*/`

## Python3

 `# Python 3 program to find maximum array``# sum with multiplications of a prefix``# and a suffix with -1 allowed.` `# function to maximize the sum``def` `maximize(a,n) :` `    ``# stores the pre sum``    ``presum ``=` `[``0``] ``*` `n``      ` `    ``# to store sum from 0 to i``    ``sm ``=` `0``    ` `    ``# stores the maximum sum with``    ``# prefix multiplication with -1.``    ``max_sum ``=` `0``    ` `    ``# traverse from 0 to n``    ``for` `i ``in` `range``(``0``,n) :` `        ``# calculate the presum``        ``presum[i] ``=` `max_sum``        ` `        ``# calculate sum``        ``max_sum  ``=``max_sum ``+` `a[i]``        ``sm ``=` `sm ``+` `a[i]``        ` `        ``max_sum  ``=` `max``(max_sum, ``-``sm)``    ` `      ` `    ``# Initialize answer.``    ``ans ``=` `max``(sm, max_sum)``    ` `    ``# traverse from back to start``    ``g ``=` `0``    ``for` `i ``in` `range``(n``-``1``,``-``1``,``-``1``) :``        ``# stores the sum multiplied by (-1)``        ``g ``=` `g ``-` `a[i]`` ` `        ``# stores the max of ans and``        ``# presum + (-1*negative sum);``        ``ans ``=` `max``(ans, g ``+` `presum[i])``    ` `    ``# returns answer``    ``return` `ans``    ` `# driver program to test the above function``a ``=` `[``-``4``, ``2``, ``0``, ``5``, ``0``]``n ``=` `len``(a)``print``(maximize(a, n))` `#This code is contributed by Nikita Tiwari.`

## C#

 `// C# program to find maximum array sum``// with multiplications of a prefix and a``// suffix with -1 allowed.``using` `System;` `class` `GFG``{``    ` `    ``// function to maximize the sum``    ``static` `int` `maximize(``int` `[]a, ``int` `n)``    ``{``        ``// stores the pre sum``        ``int` `[]presum =``new` `int``[n];``        ` `        ``// to store sum from 0 to i``        ``int` `sum = 0;``    ` `        ``// stores the maximum sum with``        ``// prefix multiplication with -1.``        ``int` `max_sum = 0;``        ` `        ``// traverse from 0 to n``        ``for` `(``int` `i = 0; i < n ; i++)``        ``{``            ``// calculate the presum``            ``presum[i] = max_sum ;``            ` `            ``// calculate sum``            ``max_sum += a[i];``            ``sum += a[i];``            ` `            ``max_sum = Math.Max(max_sum,``                               ``-sum);``        ``}``        ` `        ``// Initialize answer.``        ``int` `ans = Math.Max(sum, max_sum);``        ` `        ``// traverse from back to start``        ``int` `g = 0;``        ``for` `(``int` `i = n - 1; i >= 0; --i)``        ``{``            ``// stores the sum multiplied by (-1)``            ``g -= a[i];``    ` `            ``// stores the max of ans and``            ``// presum + (-1*negative sum);``            ``ans = Math.Max(ans, g + presum[i]);``        ``}``        ` `        ``// returns answer``        ``return` `ans;``    ``}``    ` `    ``// Driver Code``    ``public` `static` `void` `Main()``    ``{``    ` `        ``int` `[]a = {-4, 2, 0, 5, 0};``        ``int` `n = a.Length;``        ``Console.WriteLine(maximize(a, n));``    ``}``}` `// This code is contributed by vt_m.`

## PHP

 `= 0; --``\$i``)``    ``{``        ` `        ``// stores the sum``        ``// multiplied by (-1)``        ``\$g` `-= ``\$a``[``\$i``];` `        ``// stores the max of ans and``        ``// presum + (-1*negative sum);``        ``\$ans` `= max(``\$ans``, ``\$g` `+ ``\$presum``[``\$i``]);``    ``}``    ` `    ``// returns answer``    ``return` `\$ans``;``}` `    ``// Driver Code``    ``\$a` `= ``array``(-4, 2, 0, 5, 0);``    ``\$n` `= ``count``(``\$a``);``    ``echo` `maximize(``\$a``, ``\$n``);` `// This code is contributed by anuj_67.``?>`

## Javascript

 ``

Output:

`11`

Time complexity: O(n)

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