Dynamic Programming Questions
The Maximum Sum of Subsequence with No Thirteen 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 sum up to thirteen.
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 the first two elements of the "dp" array as follows:
- dp[0] = 0 (since there are no elements in the subsequence)
- dp[1] = max(0, sequence[1]) (the maximum sum of a subsequence ending at index 1 is either 0 or the value at index 1)
Then, for each index i starting from 2, we calculate the maximum sum of subsequences ending at index i using the following recurrence relation:
- dp[i] = max(dp[i-1], dp[i-2] + sequence[i])
The first part of the recurrence relation (dp[i-1]) represents the maximum sum of subsequences ending at index i-1, which means we exclude the current element at index i.
The second part of the recurrence relation (dp[i-2] + sequence[i]) represents the maximum sum of subsequences ending at index i-2, plus the current element at index i. This ensures that there are no consecutive elements summing up to thirteen.
Finally, the maximum sum of subsequences with no thirteen consecutive elements can be found by taking the maximum value from the "dp" array.
By using dynamic programming and the above recurrence relation, we can efficiently solve the Maximum Sum of Subsequence with No Thirteen Consecutive Elements problem.