Dynamic Programming Questions
The Maximum Sum of Subsequence with No Forty-Five Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, with the constraint that no forty-five 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 make sure that the previous forty-five elements are not included in the subsequence. So, we iterate from index i-2 to i-46 (or until index 0, whichever comes first) and find the maximum sum of a subsequence ending at each of these indices. We add the maximum sum to the current element and update dp[i] if it is greater than the current maximum.
Finally, the maximum sum of a subsequence with no forty-five consecutive elements will be the maximum value in the dp array.
The time complexity of this dynamic programming solution is O(n), where n is the length of the given sequence.