The question is, "Given an array A of n integers, you need to find the maximum sum of any contiguous subarray. For instance, the maximum sum of any contiguous subarray in the array -1, 2, 3, -2, 5, -6, 7, -8 is 9 (which is the sum of the subarray 2, 3, -2, 5, 6, 7). Complete the following Dynamic Programming solution for the above problem:
DP[0] = A[0]
For i = 1 to n
DP[i] = max(A[i],____)"
This is the problem statement of Kadane's Algorithm.
To solve the problem of finding the maximum sum of any contiguous subarray using a dynamic programming approach, we can use the following logic:
Initialization:
Define an array DP where DP[i] will store the maximum sum of any contiguous subarray that ends at index i.
Initialize DP[0] to A[0] because the maximum sum of a subarray ending at the first element is just the first element itself.
Dynamic Programming Transition:
For each element A[i] (where i ranges from 1 to n-1), we need to decide whether to include it in the existing subarray (that ends at i-1) or start a new subarray from A[i].
Thus, DP[i] should be the maximum of:
A[i] itself (which means starting a new subarray from A[i])
DP[i-1] + A[i] (which means extending the subarray ending at i-1 to include A[i]).
So, the dynamic programming relation can be written as:
𝐷𝑃[𝑖]=max(𝐴[𝑖],𝐷𝑃[𝑖−1]+𝐴[𝑖])