> For the complete documentation index, see [llms.txt](https://emory.gitbook.io/dsa-java/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://emory.gitbook.io/dsa-java/dynamic-programming/longest-common-subsequence.md).

# 10.3. Longest Common 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.

Given a string "ABCDE"

* Substring: {"A", "BC", "CDE", ...}
* Subsequence: {all substrings, "AC", "ACE", ...}
* Not subsequence: {"BA", "DAB", ...}

Longest common subsequence

* The longest subsequence commonly shared by multiple strings.
* e.g., “baal” is a LCS of “bilabial” and “balaclava”.

> Can there be more than one longest common subsequence?

Application

* Find the longest common subsequence in DNA (e.g., `GAATGTCCTTTCTCTAAGTCCTAAG`).

## Abstraction

Let us create the abstract class [`LCS`](https://github.com/emory-courses/dsa-java/blob/master/src/main/java/edu/emory/cs/dynamic/lcs/LCS.java):

```java
public abstract class LCS {
    /**
     * @param a the first string.
     * @param b the second string.
     * @return a longest common sequence of the two specific strings.
     */
    public String solve(String a, String b) {
        return solve(a.toCharArray(), b.toCharArray(), a.length() - 1, b.length() - 1);
    }

    /**
     * @param c the first array of characters.
     * @param d the second array of characters.
     * @param i the index of the character in {@code c} to be compared.
     * @param j the index of the character in {@code d} to be compared.
     * @return a longest common sequence of the two specific strings.
     */
    protected abstract String solve(char[] c, char[] d, int i, int j);
}
```

## Recursion

Let us create the [`LCSRecursive`](https://github.com/emory-courses/dsa-java/blob/master/src/main/java/edu/emory/cs/dynamic/lcs/LCSRecursive.java) inheriting `LCS`:

```java
public class LCSRecursive extends LCS {
    @Override
    protected String solve(char[] c, char[] d, int i, int j) {
        if (i < 0 || j < 0) return "";
        if (c[i] == d[j]) return solve(c, d, i - 1, j - 1) + c[i];

        String c1 = solve(c, d, i - 1, j);
        String d1 = solve(c, d, i, j - 1);
        return (c1.length() > d1.length()) ? c1 : d1;
    }
}
```

* `L4`: no character is left to compare for either string.
* `L5`: when two characters match, move onto `c[:i-1]` and `d[:j-1]`.&#x20;
* `L7`: gets the longest common subsequence between `c[:i-1]` and `d[:j]`.
* `L8`: gets the longest common subsequence between `c[:i]` and `d[:j-1]`.
* `L9`: returns the longest subsequence.

The followings demonstrate a recursive way of finding a LCS between two strings:

{% embed url="<https://www2.slideshare.net/jchoi7s/dynamic-programming-longest-common-subsequence-recurisve>" %}

## Dynamic Programming

Let us create the [`LCSDynamic`](https://github.com/emory-courses/dsa-java/blob/master/src/main/java/edu/emory/cs/dynamic/lcs/LCSDynamic.java) class inheriting `LCS`.

```java
public class LCSDynamic extends LCS {
    @Override
    protected String solve(char[] c, char[] d, int i, int j) {
        return solve(c, d, i, j, createTable(c, d));
    }

    /**
     * @param c the first string.
     * @param d the second string.
     * @return the dynamic table populated by estimating the # of LCSs in the grid of the two specific strings.
     */
    protected int[][] createTable(char[] c, char[] d) {
        final int N = c.length, M = d.length;
        int[][] table = new int[N][M];

        for (int i = 1; i < N; i++)
            for (int j = 1; j < M; j++)
                table[i][j] = (c[i] == d[j]) ? table[i - 1][j - 1] + 1 : Math.max(table[i - 1][j], table[i][j - 1]);

        return table;
    }
}
```

* `L4`: creates a dynamic table and passes it to the solver.
* `L12`: the dynamic table is pre-populated before any recursive calls.

The following show the dynamic table populated by the previous example:

{% embed url="<https://www2.slideshare.net/jchoi7s/dynamic-programming-longest-common-subsequence-dynamic-table>" %}

Let us define the `solve()` method:

```java
protected String solve(char[] c, char[] d, int i, int j, int[][] table) {
    if (i < 0 || j < 0) return "";
    if (c[i] == d[j])  return solve(c, d, i - 1, j - 1, table) + c[i];
    
    if (i == 0) return solve(c, d, i, j - 1, table);
    if (j == 0) return solve(c, d, i - 1, j, table);
    return (table[i - 1][j] > table[i][j - 1]) ? solve(c, d, i - 1, j, table) : solve(c, d, i, j - 1, table);
}
```

* `L2`: no character is left to compare for either string.
* `L3`: when two characters match, move onto `c[:i-1]` and `d[:j-1]`.&#x20;
* `L5`: gets the longest common subsequence between `c[:i]` and `d[:j-1]`.
* `L6`: gets the longest common subsequence between `c[:i-1]` and `d[:j]`.
* `L9`: returns the longest subsequence by looking up the values in the dynamic table.

The followings demonstrate how to find a LCS using the dynamic table:

{% embed url="<https://www2.slideshare.net/jchoi7s/dynamic-programming-longest-common-subsequence-dynamic-programming>" %}


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://emory.gitbook.io/dsa-java/dynamic-programming/longest-common-subsequence.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.