# Maximum array sum with prefix and suffix multiplications with -1 allowed

• Difficulty Level : Hard
• 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 sumint 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 functionint main() {      int a[] = {-4, 2, 0, 5, 0};    int n = sizeof(a)/sizeof(a[0]);    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 sumdef 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 functiona = [-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)

My Personal Notes arrow_drop_up