What is the Maximum Sum Increasing Subsequence problem and how can it be solved using Dynamic Programming?

What is the Maximum Sum Increasing Subsequence problem and how can it be solved using Dynamic Programming?

The Maximum Sum Increasing Subsequence problem is a problem where we need to find the subsequence of a given sequence that has the maximum sum and is increasing.

To solve this problem using Dynamic Programming, we can follow these steps:

1. Define an array, let's call it "dp", of the same length as the given sequence. This array will store the maximum sum increasing subsequence ending at each index.

2. Initialize the "dp" array with the values of the given sequence.

3. Iterate through the given sequence starting from the second element. For each element, compare it with all the previous elements and if the current element is greater than the previous element, update the "dp" array at that index by adding the current element to the maximum sum ending at the previous index.

4. After iterating through the entire sequence, find the maximum value in the "dp" array. This will be the maximum sum increasing subsequence.

5. Additionally, we can also track the actual subsequence by storing the indices of the elements that contribute to the maximum sum in a separate array.

By following these steps, we can solve the Maximum Sum Increasing Subsequence problem using Dynamic Programming.