Dynamic Programming Questions
The Maximum Sum of Subsequence with No Three Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from an array or sequence, where no three consecutive elements are included in the sum.
To solve this problem using dynamic programming, we can use a bottom-up approach. We create an auxiliary array, dp, of the same size as the input array. The dp array will store the maximum sum of subsequences ending at each index.
We initialize the first three elements of the dp array with the corresponding elements from the input array. Then, for each subsequent element at index i, we have two options:
1. Exclude the current element: In this case, the maximum sum of subsequences ending at index i is the same as the maximum sum of subsequences ending at index i-1.
2. Include the current element: In this case, we need to exclude the previous two elements (i-2 and i-3) to ensure that no three consecutive elements are included. Therefore, the maximum sum of subsequences ending at index i is the sum of the current element and the maximum sum of subsequences ending at index i-2.
We update the dp array at each index i by taking the maximum of the two options mentioned above. Finally, the maximum sum of subsequences with no three consecutive elements will be the maximum value in the dp array.
The time complexity of this dynamic programming solution is O(n), where n is the size of the input array.