Dynamic Programming Questions
The Longest Increasing Subsequence (LIS) problem is a classic problem in computer science that involves finding the length of the longest subsequence of a given sequence that is strictly increasing.
Dynamic Programming can be used to solve the LIS problem efficiently. The basic idea is to use a dynamic programming table to store the lengths of the longest increasing subsequences ending at each position in the given sequence.
To solve the problem using dynamic programming, we can follow these steps:
1. Create an array dp of the same length as the given sequence, initialized with all 1s. This array will store the lengths of the longest increasing subsequences ending at each position.
2. Iterate through the given sequence from left to right. For each element at index i, iterate through all the previous elements from 0 to i-1.
3. For each previous element at index j, if the element at index i is greater than the element at index j, update dp[i] as the maximum of dp[i] and dp[j] + 1. This means that if the element at index i can be included in the increasing subsequence ending at index j, then it can also be included in the increasing subsequence ending at index i.
4. After iterating through all the elements, the maximum value in the dp array will represent the length of the longest increasing subsequence in the given sequence.
5. Additionally, if we want to find the actual subsequence, we can keep track of the indices of the elements that contribute to the longest increasing subsequence. We can start from the maximum value in the dp array and backtrack through the indices, adding the corresponding elements to the subsequence.
By using dynamic programming, we can solve the Longest Increasing Subsequence problem in O(n^2) time complexity, where n is the length of the given sequence.