Dynamic Programming Questions
The Maximum Sum of Subsequence with No Six Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no six consecutive elements are included in the subsequence.
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".
The dynamic programming approach involves iterating through the given sequence and updating the "dp" array at each index. At each index, we have two options:
1. Include the current element in the subsequence: In this case, we cannot include the previous five elements in the subsequence. So, the maximum sum up to the current index would be the sum of the current element and the maximum sum up to the index five positions before the current index (dp[i-6] if i >= 6).
2. Exclude the current element from the subsequence: In this case, the maximum sum up to the current index would be the same as the maximum sum up to the previous index (dp[i-1]).
We then choose the maximum value between the two options and update the "dp" array at the current index.
Finally, the maximum sum of the subsequence with no six consecutive elements would be the maximum value in the "dp" array.
Here is the pseudocode for solving the problem using dynamic programming:
1. Initialize an array dp of size n+1, where n is the length of the given sequence.
2. Set dp[0] = 0 and dp[1] = sequence[0].
3. Iterate from i = 2 to n:
- Set dp[i] = max(dp[i-1], sequence[i-1] + dp[i-6] if i >= 6).
4. Return the maximum value in the dp array.
By following this approach, we can efficiently solve the Maximum Sum of Subsequence with No Six Consecutive Elements problem using dynamic programming.