Dynamic Programming Questions
The Maximum Sum of Subsequence with No Thirty-Seven Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no subsequence should contain 37 consecutive elements.
To solve this problem using dynamic programming, we can use an array to store the maximum sum of subsequences up to a certain index. Let's call this array dp.
We initialize dp[0] as the first element of the sequence, and dp[1] as the maximum value between the first and second elements. For all other indices i, we calculate dp[i] as the maximum value between the sum of the current element and the maximum sum of the subsequence ending at the previous element (dp[i-1]), and the sum of the current element and the maximum sum of the subsequence ending two elements before (dp[i-2]).
However, if the current element is 37, we skip it and calculate dp[i] as the maximum sum of the subsequence ending at the previous element (dp[i-1]).
Finally, the maximum sum of the subsequence with no thirty-seven consecutive elements can be found by taking the maximum value from the dp array.
Here is the dynamic programming algorithm to solve this problem:
1. Initialize dp[0] as the first element of the sequence.
2. Initialize dp[1] as the maximum value between the first and second elements.
3. For i = 2 to n (where n is the length of the sequence):
- If the current element is 37, set dp[i] = dp[i-1].
- Otherwise, set dp[i] as the maximum value between (current element + dp[i-1]) and (current element + dp[i-2]).
4. Find the maximum value from the dp array.
5. Return the maximum sum of the subsequence with no thirty-seven consecutive elements.
By using this dynamic programming approach, we can efficiently solve the Maximum Sum of Subsequence with No Thirty-Seven Consecutive Elements problem.