Longest Common Subsequence 

Definition

Given two strings, find the length of the longest subsequence present in both of them. A subsequence is a sequence that appears in the same relative order, but not necessarily contiguously.

• Example: “ACE” is a subsequence of “ABCDE”.

How it Works

We compare the two strings character by character.

1. Match: If characters match (e.g., ‘A’ == ‘A’), we add 1 to the result and move both pointers forward.
2. No Match: If they don’t match, we branch into two possibilities:

• Skip a character from string A.
• Skip a character from string B.
• Take the maximum of these two options.

Code Example (Java)

public class LCS_DP {

    public static int lcs(String X, String Y) {
        int m = X.length();
        int n = Y.length();

        // Create DP table
        int[][] dp = new int[m + 1][n + 1];

        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {

                if (X.charAt(i - 1) == Y.charAt(j - 1)) {
                    dp[i][j] = 1 + dp[i - 1][j - 1]; // Match found
                } else {
                    dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - 1]); // No match
                }
            }
        }

        return dp[m][n];
    }

    public static void main(String[] args) {
        String s1 = "AGGTAB";
        String s2 = "GXTXAYB";

        System.out.println("Length of LCS is " + lcs(s1, s2)); // Output: 4
    }
}

Complexity Analysis

• Time Complexity: O(M times N) where M and N are string lengths.
• Space Complexity: O(M times N) for the table.

Use Cases

• Diff Tools: Comparing two versions of a file (like Git).
• DNA Sequencing: Finding similarities between gene strands.