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

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

The Maximum Sum of Subsequence with No Thirty-Four 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 thirty-four consecutive elements.

To solve this problem using dynamic programming, we can use an array to store the maximum sum of subsequences ending at each index. Let's call this array dp.

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

1. Include the current element in the subsequence: In this case, the maximum sum of the subsequence ending at index i is dp[i-2] + sequence[i]. We use dp[i-2] instead of dp[i-1] because we cannot have thirty-four consecutive elements.

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

We choose the maximum value between the two options and store it in dp[i]. Finally, the maximum sum of the subsequence with no thirty-four consecutive elements will be the maximum value in the dp array.

The dynamic programming approach allows us to efficiently solve this problem by breaking it down into smaller subproblems and reusing the solutions to those subproblems. By storing the maximum sum of subsequences ending at each index, we can avoid redundant calculations and find the overall maximum sum.