Dynamic Programming Questions
The Maximum Sum of Subsequence with No Ten Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no more than ten consecutive elements can be included in the subsequence.
To solve this problem using dynamic programming, we can use a bottom-up approach. We create an array, dp, of the same length as the given sequence, where dp[i] represents the maximum sum of a subsequence ending at index i.
We initialize dp[0] as the first element of the sequence. Then, for each subsequent element at index i, we have two options:
1. Exclude the current element: In this case, the maximum sum of a subsequence ending at index i is the same as the maximum sum of a subsequence ending at index i-1, which is dp[i-1].
2. Include the current element: In this case, we need to ensure that there are no more than ten consecutive elements in the subsequence. So, we need to consider the maximum sum of a subsequence ending at index i-2, i-3, ..., i-10 (if they exist). The maximum sum among these indices can be found using dp[i-2], dp[i-3], ..., dp[i-10]. Therefore, the maximum sum of a subsequence ending at index i is the current element plus the maximum sum among dp[i-2], dp[i-3], ..., dp[i-10].
Finally, the maximum sum of a subsequence with no ten consecutive elements can be found by taking the maximum value from dp.
The time complexity of this dynamic programming solution is O(n), where n is the length of the given sequence.