Maximum Sum Circular Subarray

Problem Description

Given a circular integer array nums of length n, return the maximum possible sum of a non-empty subarray of nums. A circular array means the end of the array connects to the beginning of the array. Formally, the next element of nums[i] is nums[(i + 1) % n] and the previous element of nums[i] is nums[(i - 1 + n) % n]. A subarray may only include each element of the fixed buffer nums at most once.

Key Insights

• The maximum sum of a subarray can be found using Kadane's algorithm for linear arrays.
• For circular subarrays, the maximum sum can also be derived by considering the total sum of the array minus the minimum subarray sum.
• The maximum possible sum is the maximum between the maximum sum found using Kadane's algorithm and the total sum minus the minimum subarray sum.
• Special cases occur when all elements are negative, where the maximum sum will be the maximum single element.

Space and Time Complexity

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

Solution

To solve the problem, we can use the following approach:

1. First, apply Kadane's algorithm to find the maximum subarray sum for the non-circular case.
2. Then, calculate the total sum of the array.
3. Apply Kadane's algorithm again to find the minimum subarray sum.
4. The maximum circular subarray sum can be calculated as total_sum - minimum_subarray_sum.
5. Return the maximum value between the non-circular maximum subarray sum and the circular maximum subarray sum. Handle cases where all elements are negative.

Code Solutions