Longest Common Subsequence

Preview

1. What is the definition of Longest Common Subsequence in computer science?
2. How does one calculate the length of the Longest Common Subsequence?
3. What is the difference between Longest Common Subsequence and Longest Common Substring?
4. How can one efficiently compute the Longest Common Subsequence between two strings?
5. In what practical applications can the Longest Common Subsequence problem be used?

Explain

Longest Common Subsequence (LCS) is a classical problem in Computer Science that seeks to find the longest subsequence that is common to two or more sequences.

A subsequence is any subset of a sequence that preserves the relative order of the elements in the sequence. For example, if we have the sequence “abcdefg”, the subsequences “aceg”, “bcd”, and “defg” are valid.

To find the LCS between two sequences, we need to identify all the subsequences that are common to both sequences, and then select the longest one. For example, if we have the sequences “ACADB” and “CBDA”, the longest common subsequence is “CBA”.

This problem is particularly relevant in fields such as bioinformatics, where we need to find the similarities and differences between DNA or protein sequences. Algorithms such as dynamic programming are commonly used to solve this problem in an efficient way.

Keypoint

1. A subsequence is a subset of characters in a given sequence, arranged in the same order as the original sequence.
2. A common subsequence is a subsequence that appears in two or more given sequences.
3. The longest common subsequence (LCS) is the subsequence of greatest length that appears in two or more given sequences.
4. The LCS problem involves finding the LCS of two or more given sequences.
5. The length of the LCS can be computed using dynamic programming, which involves building a matrix and filling it in a bottom-up manner.
6. The LCS matrix is built by comparing the characters of the two sequences and filling in values in the matrix based on whether or not there is a match.
7. The length of the LCS is the value in the bottom-right corner of the LCS matrix.
8. The actual LCS can be found by backtracking through the LCS matrix.
9. The time complexity of the LCS algorithm is O(mn), where m and n are the lengths of the input sequences.
10. The LCS problem has many applications in string manipulation, genetics, and bioinformatics.

Review

1. What is a Longest Common Subsequence (LCS)?

Answer: A LCS is a sequence of characters that are found in two or more given strings in the same relative order, but not necessarily contiguous or the same length.

2. What is the difference between LCS and Longest Common Substring (LCS)?

Answer: LCS is based on finding the longest string of characters that are common to two or more strings, while LCS is based on finding the longest subsequence of characters that are common to two or more strings.

3. How can you compute the LCS of two given strings in a dynamic programming approach?

Answer: You can compute the LCS of two strings using a dynamic programming approach that involves creating a matrix of the lengths of common substrings between the two strings, and then tracing back the matrix to find the LCS.

4. What is the time complexity of computing the LCS using the dynamic programming approach?

Answer: The time complexity of computing the LCS using the dynamic programming approach is O(mn), where m and n are the lengths of the two input strings.

5. How can you extend the LCS problem to include more than two input strings?

Answer: You can extend the LCS problem to include more than two input strings by creating a multi-dimensional matrix that includes one dimension for each input string, and then computing the LCS using a recursive formula that compares all possible common subsequences across all input strings.