What is the Maximum Sum of Subsequence with No Thirty-Three Consecutive Elements problem and how can it be solved using Dynamic Programming?

What is the Maximum Sum of Subsequence with No Thirty-Three Consecutive Elements problem and how can it be solved using Dynamic Programming?

The Maximum Sum of Subsequence with No Thirty-Three Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no three consecutive elements in the subsequence can sum up to 33.

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. If the sum of the current element and the previous two elements (dp[i-2]) is less than 33, we can include the current element in the subsequence. In this case, dp[i] will be the sum of the current element and the maximum sum of a subsequence ending at index i-2 (dp[i-2]).

2. If the sum of the current element and the previous two elements (dp[i-2]) is greater than or equal to 33, we cannot include the current element in the subsequence. In this case, dp[i] will be the maximum sum of a subsequence ending at index i-1 (dp[i-1]).

Finally, the maximum sum of a subsequence with no thirty-three 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.