Dynamic Programming Questions
The Maximum Sum of Subsequence with No Eleven Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no two consecutive elements in the subsequence can sum up to eleven.
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, i.e., dp[i] = dp[i-1].
2. Include the current element: In this case, we need to ensure that the previous element (at index i-1) is not included in the subsequence. If the previous element is included, the sum would be eleven, violating the given condition. Therefore, if the previous element is not included, the maximum sum of a subsequence ending at index i is the sum of the current element and the maximum sum of a subsequence ending at index i-2, i.e., dp[i] = sequence[i] + dp[i-2].
Finally, the maximum sum of a subsequence with no eleven consecutive elements would be the maximum value in the dp array.
By iteratively applying these steps for each element in the sequence, we can find the maximum sum of the desired subsequence using dynamic programming.