Problem Description

The task is to find the maximum sum of a non-empty subarray within a given circular integer array, nums. A circular array means that the array is conceptually connected end-to-end, so the elements wrap around. For example, in such an array, the next element of the last element is the first element of the array, and the previous element of the first element is the last element of the array.

The critical challenge here is accounting for the circular nature of the array, which allows for subarrays that cross the end of the array and start again at the beginning. The maximum sum could come from a subarray in the middle of the array, a subarray that includes elements from both ends of the array, or even the entire array if all elements are positive.

The subarray we are trying to find should not include the same element more than once, meaning we can't wrap around the circular array more than once when selecting our subarray elements.

Intuition

To solve the maximum subarray problem for a circular array, we must consider two cases:

1. The maximum sum subarray is similar to the one found in a non-circular array (Kadane's algorithm is useful here).
2. The maximum sum subarray is the result of wrapping around the circular array.

For the first case, we use Kadane's algorithm to find the maximum sum subarray that does not wrap around. This is done by iterating through the array while maintaining the maximum sum ending at each index (denoted as f1 in the code), and the maximum sum so far (denoted as s1).

For the second case, to handle subarrays that wrap, consider that the answer could be the total sum of the array minus the minimum sum subarray. This is akin to "selecting" the rest of the array that doesn't include the minimum sum subarray. To find the minimum sum subarray, we modify Kadane's algorithm to keep track of the minimum sum ending at each index (denoted as f2), and the minimum sum so far (denoted as s2).

At the end, the maximum possible sum for the case when the subarray wraps around is computed as the total sum of the array minus the minimum sum subarray (sum(nums) - s2).

The edge case to consider is when all numbers in the array are negative. In this situation, Kadane's algorithm yields a subarray sum which is the maximum negative number (i.e., the least negative), and thus is the maximum sum we can achieve without wrapping around. Subtracting any subarray would result in a smaller sum, so we return the maximum subarray sum found without wrapping in this case.

Putting it all together, the final answer is the larger of the two possibilities: the maximum subarray sum found without wrapping (using Kadane's algorithm) or the total array sum minus the minimum subarray sum, unless all numbers are negative, in which case we return the maximum subarray sum without wrapping.

Learn more about Queue, Divide and Conquer, Dynamic Programming and Monotonic Queue patterns.

Solution Approach

The solution approach involves implementing two variations of Kadane's algorithm once for finding the maximum subarray sum, and another time for finding the minimum subarray sum. We then combine the results of these variations to account for the circular nature of the array.

Here's how we approach it step by step:

1. Initialization: First, we initialize four variables s1, s2, f1, and f2 with the value of the first element in the array. s1 will hold the maximum subarray sum found so far, s2 the minimum subarray sum, f1 the current maximum subarray sum ending at the current index, and f2 the current minimum subarray sum ending at the current index.

2. Iterate Over Array: Start iterating the array from the second element because the first element is used for initialization.

3. Kadane’s Algorithm for Maximum Sum: For each element num, we calculate the current maximum ending here (f1) as the maximum of num and num + f1 (i.e., either start a new subarray from current element or add the current element to the existing subarray). We then update the overall maximum (s1) with the maximum of itself and f1.

4. Kadane’s Algorithm for Minimum Sum: Likewise, we calculate the current minimum ending here (f2) as the minimum of num and num + f2 (i.e., either start a new subarray from current element or add the current element to the existing negative subarray). We update the overall minimum (s2) with the minimum of itself and f2.

5. Check for All Negative Elements: After iterating through the array, we check if the maximum subarray sum found (s1) is less than or equal to 0. If true, then all the elements are negative, and s1 is our answer as no subarray wrapping can result in a higher sum.

6. Find Maximum Sum Considering Circular Wrap: Otherwise, we find the maximum of s1 and sum(nums) - s2. This accounts for the possibility that the maximum sum subarray might be wrapping over the circular array. We subtract s2 (the minimum sum subarray) from the total sum of the array, which effectively gives us the sum of the subarray that wraps around.

7. Return the Result: The maximum of these two values gives us our answer, which is the maximum sum of a non-empty subarray for the circular array.

The final Python function looks like:

1class Solution:
2    def maxSubarraySumCircular(self, nums: List[int]) -> int:
3        s1 = s2 = f1 = f2 = nums[0]
4        for num in nums[1:]:
5            f1 = num + max(f1, 0)
6            f2 = num + min(f2, 0)
7            s1 = max(s1, f1)
8            s2 = min(s2, f2)
9        return s1 if s1 <= 0 else max(s1, sum(nums) - s2)

In this code snippet:

• max(f1, 0) and min(f2, 0) are used to decide whether to start a new subarray or to continue with the current subarray.
• sum(nums) is called only once to improve efficiency, just before the comparison of sums, as it could be computationally expensive for large arrays to call it repeatedly.

Example Walkthrough

Let's illustrate the solution approach using a small example with the circular array nums = [5, -3, 5].

1. Initialization: We initialize s1, s2, f1, and f2 with the value of the first element in the array. So, s1 = s2 = f1 = f2 = 5.

2. Iterate Over Array: Start iterating from the second element -3.

3. Kadane’s Algorithm for Maximum Sum:

   • For -3: f1 = max(-3, 5 - 3) = max(-3, 2) = 2, so s1 = max(5, 2) = 5.
   • For 5 (last element): f1 = max(5, 5 + 2) = max(5, 7) = 7, which is a wrap-around as we count 5 from both ends of the array. s1 is updated to max(5, 7) = 7.

4. Kadane’s Algorithm for Minimum Sum:

   • For -3: f2 = min(-3, 5 - 3) = min(-3, 2) = -3, so s2 becomes min(5, -3) = -3.
   • For 5: f2 = min(5, 5 - 3) = min(5, 2) = 2, and s2 remains at -3.

5. Check for All Negative Elements: Since the maximum subarray sum s1 is greater than 0, not all elements are negative.

6. Find Maximum Sum Considering Circular Wrap: We find max(7, sum([5, -3, 5]) - -3). The sum of nums is 7, and subtracting s2 (which is -3), we get 7 - (-3) = 10.

7. Return the Result: The answer is the maximum of 7 and 10, which is 10. So the maximum sum of a non-empty subarray for the given circular array nums is 10.

This walkthrough clearly demonstrates how the algorithm incorporates circularity by considering the inverse of the minimum subarray sum to find the potential maximum in a wrapping scenario. It also shows how to handle non-wrap-around scenarios using Kadane’s algorithm directly for the maximum subarray sum.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n), where n is the length of the input list nums. This is because the code iterates through all the elements of the array exactly once to calculate the maximum subarray sum for both the non-circular and circular cases.

The space complexity of the code is O(1) since it only uses a constant amount of extra space for variables s1, s2, f1, f2, and some temporary variables to perform the calculations, regardless of the size of the input list.

Learn more about how to find time and space complexity quickly using problem constraints.