Dynamic Programming Questions
The Maximum Sum of Subsequence with No Nine Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no nine consecutive elements are included in the subsequence.
To solve this problem using dynamic programming, we can use a bottom-up approach. We define an array dp[] to store the maximum sum at each index i of the given sequence.
The dynamic programming algorithm can be outlined as follows:
1. Initialize dp[] array with the same length as the given sequence, and set all elements to 0.
2. Iterate through each index i of the given sequence.
3. For each index i, calculate the maximum sum ending at index i by considering two cases:
a. If the current element at index i is 9, set dp[i] to 0, as we cannot include it in the subsequence.
b. Otherwise, set dp[i] to the maximum value between:
- The sum of the current element at index i and the maximum sum ending at the previous index (i-1) (dp[i-1]).
- The sum of the current element at index i and the maximum sum ending at the index before the previous index (i-2) (dp[i-2]).
4. After iterating through all indices, the maximum sum of the subsequence with no nine consecutive elements will be the maximum value in the dp[] array.
5. Return the maximum sum as the solution to the problem.
By using dynamic programming, we can efficiently solve the Maximum Sum of Subsequence with No Nine Consecutive Elements problem in a time complexity of O(n), where n is the length of the given sequence.