Understanding the Longest Common Subsequence (LCS)
The Longest Common Subsequence (LCS) problem is a classic computer science problem that involves finding the longest sequence of characters that appears in the same relative order in two or more sequences, but not necessarily contiguously. Unlike a "Longest Common Substring," which requires characters to be consecutive, a subsequence allows characters to be separated by others.
What is a Subsequence?
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. For example, "ACE" is a subsequence of "ABCDE", but "AEC" is not.
LCS vs. Longest Common Substring
- Longest Common Subsequence (LCS): Characters do not need to be consecutive. Example: LCS of "ABCBDAB" and "BDCABA" is "BDAB" (length 4).
- Longest Common Substring: Characters must be consecutive. Example: Longest Common Substring of "ABCBDAB" and "BDCABA" is "AB" (length 2).
Applications of LCS
The LCS algorithm has numerous practical applications across various fields:
- Bioinformatics: Used to compare DNA or protein sequences to find similarities, which can indicate evolutionary relationships or functional similarities.
- Diff Utilities: Tools like
diff(used in version control systems like Git) rely on LCS to identify the minimal set of changes (insertions, deletions) required to transform one file into another. - Plagiarism Detection: Can be adapted to compare documents and identify sections of text that are similar.
- Data Compression: Some compression algorithms use LCS principles.
How the LCS Calculator Works (Dynamic Programming)
This calculator uses a dynamic programming approach to find the LCS. It constructs a table where each cell dp[i][j] stores the length of the LCS of the first i characters of Sequence 1 and the first j characters of Sequence 2. The algorithm proceeds as follows:
- Initialize a 2D array (DP table) with dimensions
(length(Sequence1) + 1) x (length(Sequence2) + 1), filling the first row and column with zeros. - Iterate through the table:
- If the characters at the current positions in Sequence 1 and Sequence 2 match, then
dp[i][j] = 1 + dp[i-1][j-1](the LCS length increases by one, inheriting from the diagonal). - If the characters do not match, then
dp[i][j] = max(dp[i-1][j], dp[i][j-1])(the LCS length is the maximum of the LCS lengths obtained by excluding one character from either sequence).
- If the characters at the current positions in Sequence 1 and Sequence 2 match, then
- Once the table is filled, the value in the bottom-right cell
dp[m][n](wheremandnare the lengths of the sequences) represents the length of the LCS. - To reconstruct the actual LCS string, the algorithm backtracks through the filled DP table, following the path that led to the maximum values.
Example Calculation
Let's find the LCS of "ABCBDAB" and "BDCABA":
- Sequence 1: ABCBDAB
- Sequence 2: BDCABA
The calculator will determine that the Longest Common Subsequence is "BDAB" with a length of 4.
Try it yourself with different sequences!
Longest Common Subsequence Calculator
Enter two sequences (strings) to find their Longest Common Subsequence (LCS).
Results:
Longest Common Subsequence:
Length of LCS: