Dynamic Programming Questions Medium
In the context of the Maximum Subarray Sum Circular problem, the concept of optimal substructure refers to the property that the optimal solution to the problem can be constructed from the optimal solutions of its subproblems.
The Maximum Subarray Sum Circular problem involves finding the maximum sum of a subarray within a circular array. This means that the subarray can wrap around to the beginning of the array. For example, in the array [1, -2, 3, -2], the maximum subarray sum would be 3 + (-2) + 1 = 2, as the subarray wraps around from the end to the beginning.
To solve this problem using dynamic programming, we can break it down into smaller subproblems. Let's consider an array A of size n. We can define a subproblem as finding the maximum subarray sum ending at index i, denoted as max_ending_at(i). This subproblem can be solved by considering two cases:
1. The maximum subarray sum ending at index i is within the original array A: In this case, max_ending_at(i) would be the maximum of A[i] and A[i] + max_ending_at(i-1). This means that we either start a new subarray at index i or extend the subarray from the previous index.
2. The maximum subarray sum ending at index i wraps around to the beginning of the array: In this case, max_ending_at(i) would be the maximum of A[i] and A[i] + max_ending_at(i-1) - min_ending_at(i-1), where min_ending_at(i-1) represents the minimum subarray sum ending at index i-1. This means that we either start a new subarray at index i or extend the subarray from the previous index, taking into account the minimum subarray sum.
By solving these subproblems for each index i, we can find the maximum subarray sum circular problem by taking the maximum value among all max_ending_at(i) values.
The optimal substructure property comes into play when solving these subproblems. The optimal solution to the problem can be constructed by considering the optimal solutions of its subproblems. In this case, the maximum subarray sum circular problem can be solved by finding the maximum value among all max_ending_at(i) values, which are the optimal solutions to the subproblems.
In summary, the concept of optimal substructure in the context of the Maximum Subarray Sum Circular problem means that the optimal solution to the problem can be constructed from the optimal solutions of its subproblems, which involve finding the maximum subarray sum ending at each index i.