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Maximum equilibrium sum in an array

• Difficulty Level : Easy

Given an array arr[]. Find the maximum value of prefix sum which is also suffix sum for index i in arr[].

Examples :

```Input : arr[] = {-1, 2, 3, 0, 3, 2, -1}
Output : 4
Prefix sum of arr[0..3] =
Suffix sum of arr[3..6]

Input : arr[] = {-2, 5, 3, 1, 2, 6, -4, 2}
Output : 7
Prefix sum of arr[0..3] =
Suffix sum of arr[3..7]```

A Simple Solution is to one by one check the given condition (prefix sum equal to suffix sum) for every element and returns the element that satisfies the given condition with maximum value.

C++

 `// CPP program to find ` `// maximum equilibrium sum.` `#include ` `using` `namespace` `std;`   `// Function to find ` `// maximum equilibrium sum.` `int` `findMaxSum(``int` `arr[], ``int` `n)` `{` `    ``int` `res = INT_MIN;` `    ``for` `(``int` `i = 0; i < n; i++)` `    ``{` `    ``int` `prefix_sum = arr[i];` `    ``for` `(``int` `j = 0; j < i; j++)` `        ``prefix_sum += arr[j];`   `    ``int` `suffix_sum = arr[i];` `    ``for` `(``int` `j = n - 1; j > i; j--)` `        ``suffix_sum += arr[j];`   `    ``if` `(prefix_sum == suffix_sum)` `        ``res = max(res, prefix_sum);` `    ``}` `    ``return` `res;` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `arr[] = {-2, 5, 3, 1, ` `                  ``2, 6, -4, 2 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``cout << findMaxSum(arr, n);` `    ``return` `0;` `}`

Java

 `// java program to find maximum` `// equilibrium sum.` `import` `java.io.*;`   `class` `GFG {` `    `  `    ``// Function to find maximum ` `    ``// equilibrium sum.` `    ``static` `int` `findMaxSum(``int` `[]arr, ``int` `n)` `    ``{` `        ``int` `res = Integer.MIN_VALUE;` `        `  `        ``for` `(``int` `i = ``0``; i < n; i++)` `        ``{` `            ``int` `prefix_sum = arr[i];` `            `  `            ``for` `(``int` `j = ``0``; j < i; j++)` `                ``prefix_sum += arr[j];` `        `  `            ``int` `suffix_sum = arr[i];` `            `  `            ``for` `(``int` `j = n - ``1``; j > i; j--)` `                ``suffix_sum += arr[j];` `        `  `            ``if` `(prefix_sum == suffix_sum)` `                ``res = Math.max(res, prefix_sum);` `        ``}` `        `  `        ``return` `res;` `    ``}` `    `  `    ``// Driver Code` `    ``public` `static` `void` `main (String[] args)` `    ``{` `        ``int` `arr[] = {-``2``, ``5``, ``3``, ``1``, ``2``, ``6``, -``4``, ``2` `};` `        ``int` `n = arr.length;` `        ``System.out.println(findMaxSum(arr, n));` `    ``}` `}`   `// This code is contributed by anuj_67.`

Python3

 `# Python 3 program to find maximum ` `# equilibrium sum.` `import` `sys`   `# Function to find maximum equilibrium sum.` `def` `findMaxSum(arr, n):` `    ``res ``=` `-``sys.maxsize ``-` `1` `    ``for` `i ``in` `range``(n):` `        ``prefix_sum ``=` `arr[i]` `        ``for` `j ``in` `range``(i):` `            ``prefix_sum ``+``=` `arr[j]`   `        ``suffix_sum ``=` `arr[i]` `        ``j ``=` `n ``-` `1` `        ``while``(j > i):` `            ``suffix_sum ``+``=` `arr[j]` `            ``j ``-``=` `1` `        ``if` `(prefix_sum ``=``=` `suffix_sum):` `            ``res ``=` `max``(res, prefix_sum)`   `    ``return` `res`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `    ``arr ``=` `[``-``2``, ``5``, ``3``, ``1``, ``2``, ``6``, ``-``4``, ``2``]` `    ``n ``=` `len``(arr)` `    ``print``(findMaxSum(arr, n))`   `# This code is contributed by` `# Surendra_Gangwar`

C#

 `// C# program to find maximum` `// equilibrium sum.` `using` `System;`   `class` `GFG {` `    `  `    ``// Function to find maximum ` `    ``// equilibrium sum.` `    ``static` `int` `findMaxSum(``int` `[]arr, ``int` `n)` `    ``{` `        ``int` `res = ``int``.MinValue;` `        `  `        ``for` `(``int` `i = 0; i < n; i++)` `        ``{` `            ``int` `prefix_sum = arr[i];` `            `  `            ``for` `(``int` `j = 0; j < i; j++)` `                ``prefix_sum += arr[j];` `        `  `            ``int` `suffix_sum = arr[i];` `            `  `            ``for` `(``int` `j = n - 1; j > i; j--)` `                ``suffix_sum += arr[j];` `        `  `            ``if` `(prefix_sum == suffix_sum)` `                ``res = Math.Max(res, prefix_sum);` `        ``}` `        `  `        ``return` `res;` `    ``}` `    `  `    ``// Driver Code` `    ``public` `static` `void` `Main ()` `    ``{` `        ``int` `[]arr = {-2, 5, 3, 1, 2, 6, -4, 2 };` `        ``int` `n = arr.Length;` `        ``Console.WriteLine(findMaxSum(arr, n));` `    ``}` `}`   `// This code is contributed by anuj_67.`

PHP

 ` ``\$i``; ``\$j``--)` `        ``\$suffix_sum` `+= ``\$arr``[``\$j``];`   `    ``if` `(``\$prefix_sum` `== ``\$suffix_sum``)` `        ``\$res` `= max(``\$res``, ``\$prefix_sum``);` `    ``}` `    ``return` `\$res``;` `}`   `// Driver Code` `\$arr` `= ``array``(-2, 5, 3, 1,` `              ``2, 6, -4, 2 );` `\$n` `= ``count``(``\$arr``);` `echo` `findMaxSum(``\$arr``, ``\$n``);`   `// This code is contributed by anuj_67.` `?>`

Javascript

 ``

Output :

`7`

Time Complexity: O(n2
Auxiliary Space: O(n)

A Better Approach is to traverse the array and store prefix sum for each index in array presum[], in which presum[i] stores sum of subarray arr[0..i]. Do another traversal of the array and store suffix sum in another array suffsum[], in which suffsum[i] stores sum of subarray arr[i..n-1]. After this for each index check if presum[i] is equal to suffsum[i] and if they are equal then compare their value with the overall maximum so far.

C++

 `// CPP program to find ` `// maximum equilibrium sum.` `#include ` `using` `namespace` `std;`   `// Function to find maximum` `// equilibrium sum.` `int` `findMaxSum(``int` `arr[], ``int` `n)` `{` `    ``// Array to store prefix sum.` `    ``int` `preSum[n];`   `    ``// Array to store suffix sum.` `    ``int` `suffSum[n];`   `    ``// Variable to store maximum sum.` `    ``int` `ans = INT_MIN;`   `    ``// Calculate prefix sum.` `    ``preSum[0] = arr[0];` `    ``for` `(``int` `i = 1; i < n; i++) ` `        ``preSum[i] = preSum[i - 1] + arr[i]; `   `    ``// Calculate suffix sum and compare` `    ``// it with prefix sum. Update ans` `    ``// accordingly.` `    ``suffSum[n - 1] = arr[n - 1];` `    ``if` `(preSum[n - 1] == suffSum[n - 1])` `        ``ans = max(ans, preSum[n - 1]);` `        `  `    ``for` `(``int` `i = n - 2; i >= 0; i--) ` `    ``{` `        ``suffSum[i] = suffSum[i + 1] + arr[i];` `        ``if` `(suffSum[i] == preSum[i]) ` `            ``ans = max(ans, preSum[i]);     ` `    ``}`   `    ``return` `ans;` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `arr[] = { -2, 5, 3, 1,` `                   ``2, 6, -4, 2 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``cout << findMaxSum(arr, n);` `    ``return` `0;` `}`

Java

 `// Java program to find maximum equilibrium sum.` `import` `java.io.*;`   `public` `class` `GFG {` `    `    `    ``// Function to find maximum` `    ``// equilibrium sum.` `    ``static` `int` `findMaxSum(``int` `[]arr, ``int` `n)` `    ``{` `        `  `        ``// Array to store prefix sum.` `        ``int` `[]preSum = ``new` `int``[n];` `    `  `        ``// Array to store suffix sum.` `        ``int` `[]suffSum = ``new` `int``[n];` `    `  `        ``// Variable to store maximum sum.` `        ``int` `ans = Integer.MIN_VALUE;` `    `  `        ``// Calculate prefix sum.` `        ``preSum[``0``] = arr[``0``];` `        ``for` `(``int` `i = ``1``; i < n; i++) ` `            ``preSum[i] = preSum[i - ``1``] + arr[i]; ` `    `  `        ``// Calculate suffix sum and compare` `        ``// it with prefix sum. Update ans` `        ``// accordingly.` `        ``suffSum[n - ``1``] = arr[n - ``1``];` `        `  `        ``if` `(preSum[n - ``1``] == suffSum[n - ``1``])` `            ``ans = Math.max(ans, preSum[n - ``1``]);` `            `  `        ``for` `(``int` `i = n - ``2``; i >= ``0``; i--) ` `        ``{` `            ``suffSum[i] = suffSum[i + ``1``] + arr[i];` `            `  `            ``if` `(suffSum[i] == preSum[i]) ` `                ``ans = Math.max(ans, preSum[i]); ` `        ``}` `    `  `        ``return` `ans;` `    ``}` `    `  `    ``// Driver Code` `    ``static` `public` `void` `main (String[] args)` `    ``{` `        ``int` `[]arr = { -``2``, ``5``, ``3``, ``1``, ``2``, ``6``, -``4``, ``2` `};` `        ``int` `n = arr.length;` `        `  `        ``System.out.println( findMaxSum(arr, n));` `    ``}` `}`   `// This code is contributed by anuj_67`

Python3

 `# Python3 program to find` `# maximum equilibrium sum.`   `# Function to find maximum` `# equilibrium sum.` `def` `findMaxSum(arr, n):`   `    ``# Array to store prefix sum.` `    ``preSum ``=` `[``0` `for` `i ``in` `range``(n)]`   `    ``# Array to store suffix sum.` `    ``suffSum ``=` `[``0` `for` `i ``in` `range``(n)]`   `    ``# Variable to store maximum sum.` `    ``ans ``=` `-``10000000`   `    ``# Calculate prefix sum.` `    ``preSum[``0``] ``=` `arr[``0``]` `    `  `    ``for` `i ``in` `range``(``1``, n):` `    `  `        ``preSum[i] ``=` `preSum[i ``-` `1``] ``+` `arr[i]`   `    ``# Calculate suffix sum and compare` `    ``# it with prefix sum. Update ans` `    ``# accordingly.` `    ``suffSum[n ``-` `1``] ``=` `arr[n ``-` `1``]` `    ``if` `(preSum[n ``-` `1``] ``=``=` `suffSum[n ``-` `1``]):` `        ``ans ``=` `max``(ans, preSum[n ``-` `1``])` `     `  `    ``for` `i ``in` `range``(n ``-` `2``, ``-``1``, ``-``1``):` `        ``suffSum[i] ``=` `suffSum[i ``+` `1``] ``+` `arr[i]` `        ``if` `(suffSum[i] ``=``=` `preSum[i]):` `            ``ans ``=` `max``(ans, preSum[i])` `    `  `    ``return` `ans`   `# Driver Code` `if` `__name__``=``=``'__main__'``:`   `    ``arr ``=` `[``-``2``, ``5``, ``3``, ``1``,``2``, ``6``, ``-``4``, ``2``]` `    ``n ``=` `len``(arr)` `    ``print``(findMaxSum(arr, n))` `    `  `# This code is contributed by pratham76`

C#

 `// C# program to find maximum equilibrium sum.` `using` `System;`   `public` `class` `GFG {` `    `    `    ``// Function to find maximum` `    ``// equilibrium sum.` `    ``static` `int` `findMaxSum(``int` `[]arr, ``int` `n)` `    ``{` `        `  `        ``// Array to store prefix sum.` `        ``int` `[]preSum = ``new` `int``[n];` `    `  `        ``// Array to store suffix sum.` `        ``int` `[]suffSum = ``new` `int``[n];` `    `  `        ``// Variable to store maximum sum.` `        ``int` `ans = ``int``.MinValue;` `    `  `        ``// Calculate prefix sum.` `        ``preSum[0] = arr[0];` `        ``for` `(``int` `i = 1; i < n; i++) ` `            ``preSum[i] = preSum[i - 1] + arr[i]; ` `    `  `        ``// Calculate suffix sum and compare` `        ``// it with prefix sum. Update ans` `        ``// accordingly.` `        ``suffSum[n - 1] = arr[n - 1];` `        `  `        ``if` `(preSum[n - 1] == suffSum[n - 1])` `            ``ans = Math.Max(ans, preSum[n - 1]);` `            `  `        ``for` `(``int` `i = n - 2; i >= 0; i--) ` `        ``{` `            ``suffSum[i] = suffSum[i + 1] + arr[i];` `            `  `            ``if` `(suffSum[i] == preSum[i]) ` `                ``ans = Math.Max(ans, preSum[i]); ` `        ``}` `    `  `        ``return` `ans;` `    ``}` `    `  `    ``// Driver Code` `    ``static` `public` `void` `Main ()` `    ``{` `        ``int` `[]arr = { -2, 5, 3, 1, 2, 6, -4, 2 };` `        ``int` `n = arr.Length;` `        `  `        ``Console.WriteLine( findMaxSum(arr, n));` `    ``}` `}`   `// This code is contributed by anuj_67`

PHP

 `= 0; ``\$i``--) ` `    ``{` `        ``\$suffSum``[``\$i``] = ``\$suffSum``[``\$i` `+ 1] + ``\$arr``[``\$i``];` `        ``if` `(``\$suffSum``[``\$i``] == ``\$preSum``[``\$i``]) ` `            ``\$ans` `= max(``\$ans``, ``\$preSum``[``\$i``]); ` `    ``}`   `    ``return` `\$ans``;` `}`   `// Driver Code` `\$arr` `= ``array``( -2, 5, 3, 1, 2, 6, -4, 2 );` `\$n` `= sizeof(``\$arr``);` `echo` `findMaxSum(``\$arr``, ``\$n``);`   `// This code is contributed by ajit.` `?>`

Javascript

 ``

Output:

`7`

Time Complexity: O(n)
Auxiliary Space: O(n)

Further Optimization :
We can avoid the use of extra space by first computing the total sum, then using it to find the current prefix and suffix sums.

Steps to solve this problem:

1. Initialize two variables sum and prefix_sum to 0.

2. sum is the sum of all elements of the arr array, calculated using the accumulate function. prefix_sum is used to keep track of the sum of the elements of the subarray.

3. Initialize a variable res to INT_MIN, which will store the maximum sum of the subarray such that the sum of the elements of the subarray is equal to the sum of the elements of the rest of the array.

4. Loop through the array arr from index 0 to index n-1.

5. For each iteration of the loop, increment the value of prefix_sum by the current element of the arr array.

6. If prefix_sum is equal to sum, update the value of res to be the maximum of res and prefix_sum.

7. Decrement the value of sum by the current element of the arr array.

8. Repeat steps 5-7 for all elements of the arr array.

9. After the loop, return the value of res.

Implementation:

C++

 `// CPP program to find` `// maximum equilibrium sum.` `#include ` `using` `namespace` `std;`   `// Function to find ` `// maximum equilibrium sum.` `int` `findMaxSum(``int` `arr[], ``int` `n)` `{` `    ``int` `sum = accumulate(arr, arr + n, 0);` `    ``int` `prefix_sum = 0, res = INT_MIN;` `    ``for` `(``int` `i = 0; i < n; i++)` `    ``{` `    ``prefix_sum += arr[i]; ` `    ``if` `(prefix_sum == sum)` `        ``res = max(res, prefix_sum); ` `    ``sum -= arr[i];` `    ``}` `    ``return` `res;` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `arr[] = { -2, 5, 3, 1, ` `                   ``2, 6, -4, 2 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``cout << findMaxSum(arr, n);` `    ``return` `0;` `}`

Java

 `// Java program to find maximum equilibrium` `// sum.` `import` `java.lang.Math.*;` `import` `java.util.stream.*;`   `class` `GFG {` `    `  `    ``// Function to find maximum equilibrium` `    ``// sum.` `    ``static` `int` `findMaxSum(``int` `arr[], ``int` `n)` `    ``{` `        ``int` `sum = IntStream.of(arr).sum();` `        ``int` `prefix_sum = ``0``,` `        ``res = Integer.MIN_VALUE;` `        `  `        ``for` `(``int` `i = ``0``; i < n; i++)` `        ``{` `            ``prefix_sum += arr[i]; ` `            `  `            ``if` `(prefix_sum == sum)` `                ``res = Math.max(res, prefix_sum); ` `            ``sum -= arr[i];` `        ``}` `        `  `        ``return` `res;` `    ``}` `    `  `    ``// Driver Code` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``int` `arr[] = { -``2``, ``5``, ``3``, ``1``, ` `                    ``2``, ``6``, -``4``, ``2` `};` `        ``int` `n = arr.length;` `        ``System.out.print(findMaxSum(arr, n));` `    ``}` `}`   `// This code is contributed by Smitha.`

Python3

 `# Python3 program to find` `# maximum equilibrium sum.` `import` `sys`   `# Function to find ` `# maximum equilibrium sum. ` `def` `findMaxSum(arr,n):` `    `  `    ``ss ``=` `sum``(arr)` `    ``prefix_sum ``=` `0` `    ``res ``=` `-``sys.maxsize` `    `  `    ``for` `i ``in` `range``(n):` `        ``prefix_sum ``+``=` `arr[i]` `        `  `        ``if` `prefix_sum ``=``=` `ss:` `            ``res ``=` `max``(res, prefix_sum); ` `            `  `        ``ss ``-``=` `arr[i];` `        `  `    ``return` `res` ` `  `# Driver code   ` `if` `__name__``=``=``"__main__"``:` `    `  `    ``arr ``=` `[ ``-``2``, ``5``, ``3``, ``1``, ` `             ``2``, ``6``, ``-``4``, ``2` `]` `    ``n ``=` `len``(arr)` `    `  `    ``print``(findMaxSum(arr, n))`   `# This code is contributed by rutvik_56`

C#

 `// C# program to find maximum equilibrium sum.` `using` `System.Linq;` `using` `System;`   `class` `GFG {` `    `  `    ``static` `int` `Add(``int` `x, ``int` `y) { ` `        ``return` `x + y; ` `    ``} ` `    `  `    ``// Function to find maximum equilibrium` `    ``// sum.` `    ``static` `int` `findMaxSum(``int` `[]arr, ``int` `n)` `    ``{` `        ``int` `sum = arr.Aggregate(func:Add);` `        ``int` `prefix_sum = 0,` `        ``res = ``int``.MinValue;` `        `  `        ``for` `(``int` `i = 0; i < n; i++)` `        ``{` `            ``prefix_sum += arr[i]; ` `            `  `            ``if` `(prefix_sum == sum)` `                ``res = Math.Max(res, prefix_sum); ` `            ``sum -= arr[i];` `        ``}` `        `  `        ``return` `res;` `    ``}` `    `  `    ``// Driver Code` `    ``public` `static` `void` `Main()` `    ``{` `        ``int` `[]arr = { -2, 5, 3, 1, ` `                    ``2, 6, -4, 2 };` `        ``int` `n = arr.Length;` `        ``Console.Write(findMaxSum(arr, n));` `    ``}` `}`   `// This code is contributed by Smitha.`

Javascript

 ``

Output :

`7`

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

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