Dynamic Programming Questions
The Maximum Sum of Subsequence with No Twenty-Nine 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 be equal to 29.
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 assume the given sequence is stored in an array called "sequence" of size n.
We initialize two arrays: "dp" of size n to store the maximum sum of subsequences up to index i, and "exclude" of size n to store the maximum sum of subsequences up to index i excluding the element at index i.
We start by initializing the first two elements of the "dp" array as follows:
dp[0] = sequence[0] (as the maximum sum of a subsequence with only one element is the element itself)
dp[1] = max(sequence[0], sequence[1]) (as we cannot have two consecutive elements equal to 29)
Next, we iterate through the remaining elements of the sequence from index 2 to n-1. For each index i, we update the "dp" and "exclude" arrays as follows:
1. If the element at index i is equal to 29, we set dp[i] = exclude[i-1] (as we cannot include the element at index i in the subsequence).
2. If the element at index i is not equal to 29, we set dp[i] = max(dp[i-1], exclude[i-1] + sequence[i]) (as we can either include the element at index i or exclude it, taking the maximum sum).
Finally, the maximum sum of the subsequence with no twenty-nine consecutive elements will be the maximum value in the "dp" array.
By using dynamic programming, we avoid redundant calculations and solve the problem efficiently in O(n) time complexity.