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

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

The Maximum Sum of Subsequence with No Fifteen Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, where no consecutive elements in the subsequence can have a difference of fifteen.

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 assume the given sequence is stored in an array called "arr" of size n.

We initialize two arrays, "include" and "exclude," both of size n, where "include[i]" represents the maximum sum of subsequences ending at index i and including the element at index i, and "exclude[i]" represents the maximum sum of subsequences ending at index i and excluding the element at index i.

We can define the following recurrence relation to calculate the maximum sum at each index:

include[i] = arr[i] + exclude[i-1] (if i > 0 and arr[i] - arr[i-1] != 15)
exclude[i] = max(include[i-1], exclude[i-1])

The maximum sum of subsequences with no fifteen consecutive elements will be the maximum value between include[n-1] and exclude[n-1].

By iterating through the array from left to right and applying the above recurrence relation, we can calculate the maximum sum of subsequences with no fifteen consecutive elements efficiently using dynamic programming.