Dynamic Programming Questions
The Maximum Sum of Subsequence with No Eight Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no eight 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 ending at each position. Let's denote this array as dp.
The dynamic programming approach involves iterating through the given sequence and updating the dp array at each position. At each position, we have two options:
1. Exclude the current element: In this case, the maximum sum of the subsequence ending at the current position would be the same as the maximum sum of the subsequence ending at the previous position. Therefore, we can set dp[i] = dp[i-1].
2. Include the current element: In this case, we need to make sure that the current element is not part of any eight consecutive elements. To achieve this, we can check if the element at position i-8 is included in the subsequence. If it is, we cannot include the current element. Otherwise, we can include it and update dp[i] as dp[i-1] + sequence[i].
After iterating through the entire sequence, the maximum sum of the subsequence with no eight consecutive elements can be obtained by finding 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.