Dynamic Programming Questions Medium
In the context of the Maximum Subarray problem, the concept of optimal substructure refers to the property that an optimal solution to the problem can be constructed from optimal solutions to its subproblems.
The Maximum Subarray problem involves finding the contiguous subarray within a given array of integers that has the largest sum. To solve this problem using dynamic programming, we can utilize the concept of optimal substructure.
Let's consider an array A of size n. We can divide this array into two parts: the left subarray and the right subarray. The left subarray contains elements from index 0 to i, and the right subarray contains elements from index i+1 to n-1, where i is any index between 0 and n-1.
Now, let's assume that we have already solved the Maximum Subarray problem for both the left and right subarrays. The optimal solution for the entire array A can be obtained by combining the optimal solutions for the left and right subarrays.
There are three possible scenarios to consider when combining the solutions:
1. The maximum subarray lies entirely in the left subarray.
2. The maximum subarray lies entirely in the right subarray.
3. The maximum subarray crosses the boundary between the left and right subarrays.
In the first two scenarios, the optimal solution for the entire array A is simply the maximum subarray sum obtained from either the left or right subarray.
In the third scenario, the maximum subarray crosses the boundary between the left and right subarrays. In this case, we need to find the maximum subarray that includes elements from both the left and right subarrays. This can be done by finding the maximum subarray sum starting from the rightmost element of the left subarray and ending at the leftmost element of the right subarray.
By considering all possible scenarios and selecting the maximum subarray sum among them, we can obtain the optimal solution for the entire array A.
This property of optimal substructure allows us to solve the Maximum Subarray problem efficiently using dynamic programming. We can break down the problem into smaller subproblems, solve them independently, and then combine their solutions to obtain the optimal solution for the original problem.