# Maximum circular subarray sum

Given n numbers (both +ve and -ve), arranged in a circle, fnd the maximum sum of consecutive number.

Examples:

```Input: a[] = {8, -8, 9, -9, 10, -11, 12}
Output: 22 (12 + 8 - 8 + 9 - 9 + 10)

Input: a[] = {10, -3, -4, 7, 6, 5, -4, -1}
Output:  23 (7 + 6 + 5 - 4 -1 + 10)

Input: a[] = {-1, 40, -14, 7, 6, 5, -4, -1}
Output: 52 (7 + 6 + 5 - 4 - 1 - 1 + 40)```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Method 1 There can be two cases for the maximum sum:

• Case 1: The elements that contribute to the maximum sum are arranged such that no wrapping is there. Examples: {-10, 2, -1, 5}, {-2, 4, -1, 4, -1}. In this case, Kadane’s algorithm will produce the result.
• Case 2: The elements which contribute to the maximum sum are arranged such that wrapping is there. Examples: {10, -12, 11}, {12, -5, 4, -8, 11}. In this case, we change wrapping to non-wrapping. Let us see how. Wrapping of contributing elements implies non-wrapping of non-contributing elements, so find out the sum of non-contributing elements and subtract this sum from the total sum. To find out the sum of non contributing, invert the sign of each element and then run Kadane’s algorithm.
Our array is like a ring and we have to eliminate the maximum continuous negative that implies maximum continuous positive in the inverted arrays. Finally, we compare the sum obtained by both cases and return the maximum of the two sums.

Thanks to ashishdey0 for suggesting this solution.
Following are C/C++, Java and Python implementations of the above method.

## C++

 `// C++ program for maximum contiguous circular sum problem ` `#include ` `using` `namespace` `std; ` ` `  `// Standard Kadane's algorithm to ` `// find maximum subarray sum ` `int` `kadane(``int` `a[], ``int` `n); ` ` `  `// The function returns maximum ` `// circular contiguous sum in a[] ` `int` `maxCircularSum(``int` `a[], ``int` `n) ` `{ ` `    ``// Case 1: get the maximum sum using standard kadane' ` `    ``// s algorithm ` `    ``int` `max_kadane = kadane(a, n); ` ` `  `    ``// Case 2: Now find the maximum sum that includes ` `    ``// corner elements. ` `    ``int` `max_wrap = 0, i; ` `    ``for` `(i = 0; i < n; i++) { ` `        ``max_wrap += a[i]; ``// Calculate array-sum ` `        ``a[i] = -a[i]; ``// invert the array (change sign) ` `    ``} ` ` `  `    ``// max sum with corner elements will be: ` `    ``// array-sum - (-max subarray sum of inverted array) ` `    ``max_wrap = max_wrap + kadane(a, n); ` ` `  `    ``// The maximum circular sum will be maximum of two sums ` `    ``return` `(max_wrap > max_kadane) ? max_wrap : max_kadane; ` `} ` ` `  `// Standard Kadane's algorithm to find maximum subarray sum ` `// See https:// www.geeksforgeeks.org/archives/576 for details ` `int` `kadane(``int` `a[], ``int` `n) ` `{ ` `    ``int` `max_so_far = 0, max_ending_here = 0; ` `    ``int` `i; ` `    ``for` `(i = 0; i < n; i++) { ` `        ``max_ending_here = max_ending_here + a[i]; ` `        ``if` `(max_ending_here < 0) ` `            ``max_ending_here = 0; ` `        ``if` `(max_so_far < max_ending_here) ` `            ``max_so_far = max_ending_here; ` `    ``} ` `    ``return` `max_so_far; ` `} ` ` `  `/* Driver program to test maxCircularSum() */` `int` `main() ` `{ ` `    ``int` `a[] = { 11, 10, -20, 5, -3, -5, 8, -13, 10 }; ` `    ``int` `n = ``sizeof``(a) / ``sizeof``(a[0]); ` `    ``cout << ``"Maximum circular sum is "` `<< maxCircularSum(a, n) << endl; ` `    ``return` `0; ` `} ` ` `  `// This is code is contributed by rathbhupendra `

## C

 `// C program for maximum contiguous circular sum problem ` `#include ` ` `  `// Standard Kadane's algorithm to find maximum subarray ` `// sum ` `int` `kadane(``int` `a[], ``int` `n); ` ` `  `// The function returns maximum circular contiguous sum ` `// in a[] ` `int` `maxCircularSum(``int` `a[], ``int` `n) ` `{ ` `    ``// Case 1: get the maximum sum using standard kadane' ` `    ``// s algorithm ` `    ``int` `max_kadane = kadane(a, n); ` ` `  `    ``// Case 2: Now find the maximum sum that includes ` `    ``// corner elements. ` `    ``int` `max_wrap = 0, i; ` `    ``for` `(i = 0; i < n; i++) { ` `        ``max_wrap += a[i]; ``// Calculate array-sum ` `        ``a[i] = -a[i]; ``// invert the array (change sign) ` `    ``} ` ` `  `    ``// max sum with corner elements will be: ` `    ``// array-sum - (-max subarray sum of inverted array) ` `    ``max_wrap = max_wrap + kadane(a, n); ` ` `  `    ``// The maximum circular sum will be maximum of two sums ` `    ``return` `(max_wrap > max_kadane) ? max_wrap : max_kadane; ` `} ` ` `  `// Standard Kadane's algorithm to find maximum subarray sum ` `// See https:// www.geeksforgeeks.org/archives/576 for details ` `int` `kadane(``int` `a[], ``int` `n) ` `{ ` `    ``int` `max_so_far = 0, max_ending_here = 0; ` `    ``int` `i; ` `    ``for` `(i = 0; i < n; i++) { ` `        ``max_ending_here = max_ending_here + a[i]; ` `        ``if` `(max_ending_here < 0) ` `            ``max_ending_here = 0; ` `        ``if` `(max_so_far < max_ending_here) ` `            ``max_so_far = max_ending_here; ` `    ``} ` `    ``return` `max_so_far; ` `} ` ` `  `/* Driver program to test maxCircularSum() */` `int` `main() ` `{ ` `    ``int` `a[] = { 11, 10, -20, 5, -3, -5, 8, -13, 10 }; ` `    ``int` `n = ``sizeof``(a) / ``sizeof``(a[0]); ` `    ``printf``(``"Maximum circular sum is %dn"``, ` `           ``maxCircularSum(a, n)); ` `    ``return` `0; ` `} `

## Java

 `// Java program for maximum contiguous circular sum problem ` `import` `java.io.*; ` `import` `java.util.*; ` ` `  `class` `MaxCircularSum { ` `    ``// The function returns maximum circular contiguous sum ` `    ``// in a[] ` `    ``static` `int` `maxCircularSum(``int` `a[]) ` `    ``{ ` `        ``int` `n = a.length; ` ` `  `        ``// Case 1: get the maximum sum using standard kadane' ` `        ``// s algorithm ` `        ``int` `max_kadane = kadane(a); ` ` `  `        ``// Case 2: Now find the maximum sum that includes ` `        ``// corner elements. ` `        ``int` `max_wrap = ``0``; ` `        ``for` `(``int` `i = ``0``; i < n; i++) { ` `            ``max_wrap += a[i]; ``// Calculate array-sum ` `            ``a[i] = -a[i]; ``// invert the array (change sign) ` `        ``} ` ` `  `        ``// max sum with corner elements will be: ` `        ``// array-sum - (-max subarray sum of inverted array) ` `        ``max_wrap = max_wrap + kadane(a); ` ` `  `        ``// The maximum circular sum will be maximum of two sums ` `        ``return` `(max_wrap > max_kadane) ? max_wrap : max_kadane; ` `    ``} ` ` `  `    ``// Standard Kadane's algorithm to find maximum subarray sum ` `    ``// See https:// www.geeksforgeeks.org/archives/576 for details ` `    ``static` `int` `kadane(``int` `a[]) ` `    ``{ ` `        ``int` `n = a.length; ` `        ``int` `max_so_far = ``0``, max_ending_here = ``0``; ` `        ``for` `(``int` `i = ``0``; i < n; i++) { ` `            ``max_ending_here = max_ending_here + a[i]; ` `            ``if` `(max_ending_here < ``0``) ` `                ``max_ending_here = ``0``; ` `            ``if` `(max_so_far < max_ending_here) ` `                ``max_so_far = max_ending_here; ` `        ``} ` `        ``return` `max_so_far; ` `    ``} ` ` `  `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `a[] = { ``11``, ``10``, -``20``, ``5``, -``3``, -``5``, ``8``, -``13``, ``10` `}; ` `        ``System.out.println(``"Maximum circular sum is "` `+ maxCircularSum(a)); ` `    ``} ` `} ``/* This code is contributed by Devesh Agrawal*/`

## Python

 `# Python program for maximum contiguous circular sum problem ` ` `  `# Standard Kadane's algorithm to find maximum subarray sum ` `def` `kadane(a): ` `    ``n ``=` `len``(a) ` `    ``max_so_far ``=` `0` `    ``max_ending_here ``=` `0` `    ``for` `i ``in` `range``(``0``, n): ` `        ``max_ending_here ``=` `max_ending_here ``+` `a[i] ` `        ``if` `(max_ending_here < ``0``): ` `            ``max_ending_here ``=` `0` `        ``if` `(max_so_far < max_ending_here): ` `            ``max_so_far ``=` `max_ending_here ` `    ``return` `max_so_far ` ` `  `# The function returns maximum circular contiguous sum in ` `# a[] ` `def` `maxCircularSum(a): ` ` `  `    ``n ``=` `len``(a) ` ` `  `    ``# Case 1: get the maximum sum using standard kadane's ` `    ``# algorithm ` `    ``max_kadane ``=` `kadane(a) ` ` `  `    ``# Case 2: Now find the maximum sum that includes corner ` `    ``# elements. ` `    ``max_wrap ``=` `0` `    ``for` `i ``in` `range``(``0``, n): ` `        ``max_wrap ``+``=` `a[i] ` `        ``a[i] ``=` `-``a[i] ` ` `  `    ``# Max sum with corner elements will be: ` `    ``# array-sum - (-max subarray sum of inverted array) ` `    ``max_wrap ``=` `max_wrap ``+` `kadane(a) ` ` `  `    ``# The maximum circular sum will be maximum of two sums ` `    ``if` `max_wrap > max_kadane: ` `        ``return` `max_wrap ` `    ``else``: ` `        ``return` `max_kadane ` ` `  `# Driver function to test above function ` `a ``=` `[``11``, ``10``, ``-``20``, ``5``, ``-``3``, ``-``5``, ``8``, ``-``13``, ``10``] ` `print` `"Maximum circular sum is"``, maxCircularSum(a) ` ` `  `# This code is contributed by Devesh Agrawal `

## C#

 `// C# program for maximum contiguous ` `// circular sum problem ` `using` `System; ` ` `  `class` `MaxCircularSum { ` ` `  `    ``// The function returns maximum circular ` `    ``// contiguous sum in a[] ` `    ``static` `int` `maxCircularSum(``int``[] a) ` `    ``{ ` `        ``int` `n = a.Length; ` ` `  `        ``// Case 1: get the maximum sum using standard kadane' ` `        ``// s algorithm ` `        ``int` `max_kadane = kadane(a); ` ` `  `        ``// Case 2: Now find the maximum sum that includes ` `        ``// corner elements. ` `        ``int` `max_wrap = 0; ` `        ``for` `(``int` `i = 0; i < n; i++) { ` `            ``max_wrap += a[i]; ``// Calculate array-sum ` `            ``a[i] = -a[i]; ``// invert the array (change sign) ` `        ``} ` ` `  `        ``// max sum with corner elements will be: ` `        ``// array-sum - (-max subarray sum of inverted array) ` `        ``max_wrap = max_wrap + kadane(a); ` ` `  `        ``// The maximum circular sum will be maximum of two sums ` `        ``return` `(max_wrap > max_kadane) ? max_wrap : max_kadane; ` `    ``} ` ` `  `    ``// Standard Kadane's algorithm to find maximum subarray sum ` `    ``// See https:// www.geeksforgeeks.org/archives/576 for details ` `    ``static` `int` `kadane(``int``[] a) ` `    ``{ ` `        ``int` `n = a.Length; ` `        ``int` `max_so_far = 0, max_ending_here = 0; ` `        ``for` `(``int` `i = 0; i < n; i++) { ` `            ``max_ending_here = max_ending_here + a[i]; ` `            ``if` `(max_ending_here < 0) ` `                ``max_ending_here = 0; ` `            ``if` `(max_so_far < max_ending_here) ` `                ``max_so_far = max_ending_here; ` `        ``} ` `        ``return` `max_so_far; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int``[] a = { 11, 10, -20, 5, -3, -5, 8, -13, 10 }; ` ` `  `        ``Console.Write(``"Maximum circular sum is "` `+ maxCircularSum(a)); ` `    ``} ` `} ` ` `  `/* This code is contributed by vt_m*/`

## PHP

 ` ``\$max_kadane``)? ``\$max_wrap``: ``\$max_kadane``;  ` `}  ` ` `  `// Standard Kadane's algorithm to  ` `// find maximum subarray sum  ` `// See https://www.geeksforgeeks.org/archives/576 for details  ` `function` `kadane(``\$a``, ``\$n``)  ` `{  ` `    ``\$max_so_far` `= 0; ` `    ``\$max_ending_here` `= 0;  ` `    ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++)  ` `    ``{  ` `        ``\$max_ending_here` `= ``\$max_ending_here` `+``\$a``[``\$i``];  ` `        ``if` `(``\$max_ending_here` `< 0)  ` `            ``\$max_ending_here` `= 0;  ` `        ``if` `(``\$max_so_far` `< ``\$max_ending_here``)  ` `            ``\$max_so_far` `= ``\$max_ending_here``;  ` `    ``}  ` `    ``return` `\$max_so_far``;  ` `}  ` ` `  `    ``/* Driver code */` `    ``\$a` `= ``array``(11, 10, -20, 5, -3, -5, 8, -13, 10);  ` `    ``\$n` `= ``count``(``\$a``); ` `    ``echo` `"Maximum circular sum is "``. maxCircularSum(``\$a``, ``\$n``);  ` ` `  `// This code is contributed by rathbhupendra ` `?> `

Output:

`Maximum circular sum is 31`

Complexity Analysis:

• Time Complexity: O(n), where n is the number of elements in input array.
As only linear traversal of the array is needed.
• Auxiliary Space: O(1).
As no extra space is required.

Note that the above algorithm doesn’t work if all numbers are negative e.g., {-1, -2, -3}. It returns 0 in this case. This case can be handled by adding a pre-check to see if all the numbers are negative before running the above algorithm.

Method 2
Approach: In this method, modify Kadane’s algorithm to find a minimum contiguous subarray sum and the maximum contiguous subarray sum, then check for maximum value among the max_value and the value left after subtracting min_value from the total sum.

Algorithm

1. We will calculate the total sum of the given array.
2. We will declare variable curr_max, max_so_far, curr_min, min_so_far as the first value of the array.
3. Now we will use Kadane’s Algorithm to find maximum subarray sum and minimum subarray sum.
4. Check for all the values in the array:-
1. If min_so_far is equaled to sum, i.e. all values are negative, then we return max_so_far.
2. Else, we will calculate the maximum value of max_so_far and (sum – min_so_far) and return it.

C++ implementation of the above method is given below.

## C++

 `// C++ program for maximum contiguous circular sum problem ` `#include ` `using` `namespace` `std; ` ` `  `// The function returns maximum ` `// circular contiguous sum in a[] ` `int` `maxCircularSum(``int` `a[], ``int` `n) ` `{ ` `    ``// Corner Case ` `    ``if` `(n == 1) ` `        ``return` `a[0]; ` ` `  `    ``// Initialize sum variable which store total sum of the array. ` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``sum += a[i]; ` `    ``} ` ` `  `    ``// Initialize every variable with first value of array. ` `    ``int` `curr_max = a[0], max_so_far = a[0], curr_min = a[0], min_so_far = a[0]; ` ` `  `    ``// Concept of Kadane's Algorithm ` `    ``for` `(``int` `i = 1; i < n; i++) { ` `        ``// Kadane's Algorithm to find Maximum subarray sum. ` `        ``curr_max = max(curr_max + a[i], a[i]); ` `        ``max_so_far = max(max_so_far, curr_max); ` ` `  `        ``// Kadane's Algorithm to find Minimum subarray sum. ` `        ``curr_min = min(curr_min + a[i], a[i]); ` `        ``min_so_far = min(min_so_far, curr_min); ` `    ``} ` ` `  `    ``if` `(min_so_far == sum) ` `        ``return` `max_so_far; ` ` `  `    ``// returning the maximum value ` `    ``return` `max(max_so_far, sum - min_so_far); ` `} ` ` `  `/* Driver program to test maxCircularSum() */` `int` `main() ` `{ ` `    ``int` `a[] = { 11, 10, -20, 5, -3, -5, 8, -13, 10 }; ` `    ``int` `n = ``sizeof``(a) / ``sizeof``(a[0]); ` `    ``cout << ``"Maximum circular sum is "` `<< maxCircularSum(a, n) << endl; ` `    ``return` `0; ` `} `

Output:

`Maximum circular sum is 31`

Complexity Analysis:

• Time Complexity: O(n), where n is the number of elements in input array.
As only linear traversal of the array is needed.
• Auxiliary Space: O(1).
As no extra space is required.

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