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

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

The Maximum Sum of Subsequence with No Four Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no four consecutive elements are included in the subsequence.

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 the first four elements of dp with the corresponding elements from the given sequence. Then, for each subsequent element at index i, we have two options:

1. Exclude the current element: In this case, the maximum sum of a subsequence ending at index i is equal to the maximum sum of a subsequence ending at index i-1, which is dp[i-1].

2. Include the current element: In this case, we cannot include the previous three elements (i-1, i-2, and i-3) in the subsequence. Therefore, the maximum sum of a subsequence ending at index i is equal to the current element plus the maximum sum of a subsequence ending at index i-4, which is dp[i-4].

We take the maximum value between the two options and store it in dp[i]. Finally, the maximum sum of a subsequence with no four consecutive elements is 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.