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

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

The Maximum Sum of Subsequence with No Twenty-Two Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no two consecutive elements in the subsequence can be 22.

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 denote this array as dp, where dp[i] represents the maximum sum of subsequences ending at index i.

We can initialize dp[0] as the first element of the sequence. Then, for each subsequent element at index i, we have two options:

1. Include the current element: In this case, we cannot include the previous element (i-1) in the subsequence, as it would violate the constraint of not having consecutive 22s. Therefore, the maximum sum of subsequences ending at index i would be dp[i-2] + sequence[i].

2. Exclude the current element: In this case, the maximum sum of subsequences ending at index i would be dp[i-1].

We can then update dp[i] as the maximum value between the two options mentioned above. Finally, the maximum sum of subsequences would be the maximum value in the dp array.

By following this approach and iterating through the sequence, we can solve the Maximum Sum of Subsequence with No Twenty-Two Consecutive Elements problem using dynamic programming.