PHP Program for Maximum Circular Subarray Sum

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

Examples: 

Input: Arr = [ 8, -8, 9, -9, 10, -11, 12 ]
Output: 22 (12 + 8 - 8 + 9 - 9 + 10)

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

Input: Arr = [ -1, 40, -14, 7, 6, 5, -4, -1 ]
Output: 52 (7 + 6 + 5 - 4 - 1 - 1 + 40)

Approach: 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 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-contributions, 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 in both cases and return the maximum of the two sums.

The following are implementations of the above method. 

<?php
// PHP program for maximum
// contiguous circular sum problem

// The function returns maximum
// circular contiguous sum $a[]
function maxCircularSum($a, $n) {
  
    // Case 1: get the maximum sum
    // using standard kadane' s algorithm
    $max_kadane = kadane($a, $n);

    // Case 2: Now find the maximum
    // sum that includes corner elements.
    $max_wrap = 0;
    for ($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
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 = [11, 10, -20, 5, -3, -5, 8, -13, 10];
$n = count($a);
echo "Maximum circular sum is " . maxCircularSum($a, $n);

?>

Maximum circular sum is 31

Complexity Analysis

• Time Complexity: O(n), where n is the number of elements in the 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.

Please refer complete article on Maximum circular subarray sum for more details!