Dynamic Programming Questions Medium
To solve the Longest Alternating Subsequence problem using Dynamic Programming, we can follow the below steps:
1. Define the problem: The Longest Alternating Subsequence problem requires finding the length of the longest subsequence in an array such that the elements alternate between being greater and smaller than the previous element.
2. Identify the subproblems: In this case, the subproblem can be defined as finding the longest alternating subsequence ending at index i, where i is the current index in the array.
3. Formulate the recurrence relation: Let's define two arrays, increasing and decreasing, where increasing[i] represents the length of the longest increasing subsequence ending at index i, and decreasing[i] represents the length of the longest decreasing subsequence ending at index i.
The recurrence relation can be defined as follows:
- For each index i from 0 to n-1, where n is the length of the array:
- For j from 0 to i-1:
- If arr[i] > arr[j], update increasing[i] as max(increasing[i], decreasing[j] + 1).
- If arr[i] < arr[j], update decreasing[i] as max(decreasing[i], increasing[j] + 1).
4. Solve the subproblems: Iterate through the array and update the increasing and decreasing arrays using the recurrence relation defined in step 3.
5. Find the maximum length: After solving all the subproblems, the maximum length of the longest alternating subsequence can be found by taking the maximum value from the increasing and decreasing arrays.
6. Return the result: Return the maximum length as the solution to the Longest Alternating Subsequence problem.
By following these steps, we can use Dynamic Programming to efficiently solve the Longest Alternating Subsequence problem. The time complexity of this approach is O(n^2), where n is the length of the input array.