HelloKoding

Practical coding guides

Longest Common Subsequence

Problem

  • Given two strings, S of length m and T of length n
  • Write an algorithm to find the length of the longest common subsequence (LCS) of both S and T

Example

  • Input: given two strings “XMJYAUZ” and “MZJAWXU”
  • Expected output: 4 (the LCS is “MJAU”)

Approach: Bottom Up Dynamic Programming

  • Optimal substructure: L(i, j) = L(i-1, j-1) + 1 if S(i-1) == T(j-1) with L(i, j) is the length of the common subsequence at character S(i) and T(j)

For the above example, the LCS “MJAU” has the length at the last common character U of both S and T, L(6, 7), equals to the length at the immediate preceding common character A, L(5, 4), plus 1

  • Bottom up filling the 2D array L[m+1][n+1]. The length of the LCS is at L[m][n]

Implementation

DP_LongestCommonSubsequence.java

package com.hellokoding.algorithm;

public class DP_LongestCommonSubsequence {
    static int lengthOfLongestCommonSubsequence(String S, String T) {
        int m = S.length();
        int n = T.length();
        int[][] lengths = new int[m+1][n+1];

        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {
                if(S.charAt(i-1) == T.charAt(j-1)) {
                    lengths[i][j] = lengths[i-1][j-1] + 1;
                } else {
                    lengths[i][j] = Math.max(lengths[i-1][j], lengths[i][j-1]);
                }
            }
        }

        return lengths[m][n];
    }

    public static void main(String[] args) {
        System.out.println(lengthOfLongestCommonSubsequence("XMJYAUZ", "MZJAWXU"));
    }
}

Complexity

  • Time complexity: O(mn)
  • Space complexity: O(mn)
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