Dynamic Programming Questions
The Maximum Sum of Subsequence with No Adjacent Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from an array, where no two elements in the subsequence are adjacent.
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. Each element in the dp array represents the maximum sum of a subsequence ending at that index.
We initialize the first two elements of the dp array as follows:
- dp[0] = arr[0] (the first element of the input array)
- dp[1] = max(arr[0], arr[1]) (the maximum of the first two elements of the input array)
Then, for each subsequent element in the input array, we calculate the maximum sum of a subsequence ending at that index by considering two cases:
1. If we include the current element, the maximum sum would be the current element plus the maximum sum of a subsequence ending at the previous non-adjacent element (dp[i-2]).
2. If we exclude the current element, the maximum sum would be the maximum sum of a subsequence ending at the previous element (dp[i-1]).
We update the dp array accordingly:
- dp[i] = max(arr[i] + dp[i-2], dp[i-1])
Finally, the maximum sum of a subsequence with no adjacent elements would be the last element of the dp array:
- maxSum = dp[n-1] (where n is the size of the input array)
By following this approach, we can efficiently solve the Maximum Sum of Subsequence with No Adjacent Elements problem using dynamic programming.