Dynamic Programming Questions
The Longest Common Subsequence (LCS) problem is a classic computer science problem that involves finding the longest subsequence that is common to two given sequences. A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.
Dynamic Programming can be used to solve the LCS problem efficiently. The approach involves building a table to store the lengths of the longest common subsequences for all possible prefixes of the given sequences. By filling the table iteratively, we can find the length of the LCS.
The steps to solve the LCS problem using Dynamic Programming are as follows:
1. Create a table with dimensions (m+1) x (n+1), where m and n are the lengths of the two given sequences.
2. Initialize the first row and first column of the table with zeros.
3. Iterate through the table, row by row and column by column.
4. If the characters at the current positions in the sequences match, add 1 to the value in the table at the previous diagonal position (i.e., one row up and one column left).
5. If the characters do not match, take the maximum value from the adjacent positions (i.e., one row up or one column left) and store it in the current position.
6. After iterating through the entire table, the value in the bottom-right corner will represent the length of the LCS.
7. To find the actual LCS, start from the bottom-right corner and backtrack through the table. If the characters at the current positions in the sequences match, add the character to the LCS and move diagonally up and left. If the characters do not match, move either up or left, depending on the larger adjacent value.
8. Reverse the obtained LCS to get the correct order of characters.
By using this Dynamic Programming approach, we can efficiently solve the Longest Common Subsequence problem in a time complexity of O(mn), where m and n are the lengths of the given sequences.