In the Longest Common Subsequence problem, what is the time complexity of the dy
Practice Questions
Q1
In the Longest Common Subsequence problem, what is the time complexity of the dynamic programming solution?
O(m + n)
O(m * n)
O(m^2 + n^2)
O(m^2 * n^2)
Questions & Step-by-Step Solutions
In the Longest Common Subsequence problem, what is the time complexity of the dynamic programming solution?
Step 1: Understand that the Longest Common Subsequence (LCS) problem involves finding the longest sequence that appears in the same order in both sequences, but not necessarily consecutively.
Step 2: Recognize that we use dynamic programming to solve this problem by breaking it down into smaller subproblems.
Step 3: Identify that we create a 2D table (array) to store the lengths of the longest common subsequences for different pairs of prefixes of the two sequences.
Step 4: Note that if the first sequence has length m and the second sequence has length n, the table will have m+1 rows and n+1 columns (to account for the empty subsequence).
Step 5: Realize that we fill this table by comparing characters from both sequences, which requires iterating through each character of both sequences.
Step 6: Understand that filling each cell in the table takes constant time, and since there are m rows and n columns, the total time taken is O(m * n).
Step 7: Conclude that the time complexity of the dynamic programming solution for the LCS problem is O(m * n).
