Dynamic Programming Questions
The Maximum Sum of Subsequence with No Thirty-Nine 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 subsequence should contain 39 consecutive elements with the value of 39.
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 assume the given sequence is stored in an array called "sequence" of size n.
We initialize two arrays, "dp" and "no39", both of size n. The "dp" array will store the maximum sum of subsequences up to the current index, while the "no39" array will store the maximum sum of subsequences up to the current index without including the 39th consecutive element.
We start by initializing the first element of both arrays as the first element of the sequence:
dp[0] = sequence[0]
no39[0] = sequence[0]
Then, we iterate through the remaining elements of the sequence, updating the "dp" and "no39" arrays as follows:
For each index i from 1 to n-1:
1. If the current element is 39, we set dp[i] = no39[i-1] + sequence[i] since we cannot include the current element in the subsequence.
2. If the current element is not 39, we set dp[i] = max(dp[i-1], no39[i-1]) + sequence[i] since we can either include or exclude the current element in the subsequence.
3. We set no39[i] = max(dp[i-1], no39[i-1]) to ensure that we do not include the 39th consecutive element in the subsequence.
Finally, the maximum sum of the subsequence with no thirty-nine 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 size of the given sequence.