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In the Longest Common Subsequence problem, what is the time complexity of the dy

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Question: In the Longest Common Subsequence problem, what is the time complexity of the dynamic programming solution?

Options:

  1. O(m + n)
  2. O(m * n)
  3. O(m^2 + n^2)
  4. O(m^2 * n^2)

Correct Answer: O(m * n)

Solution: 

The time complexity of the dynamic programming solution for the Longest Common Subsequence problem is O(m * n), where m and n are the lengths of the two sequences.

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?
  1. O(m + n)
  2. O(m * n)
  3. O(m^2 + n^2)
  4. 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).
  • Dynamic Programming – A method for solving complex problems by breaking them down into simpler subproblems, storing the results of these subproblems to avoid redundant computations.
  • Longest Common Subsequence (LCS) – A classic problem in computer science that involves finding the longest subsequence present in two sequences, where a subsequence is a sequence that appears in the same relative order but not necessarily consecutively.
  • Time Complexity – A computational complexity that describes the amount of time it takes to run an algorithm as a function of the length of the input.
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