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Longest Common Subsequence

Longest Common Subsequence: Here, we are going to learn about the solution of Longest Common Subsequence – which is an interview coding questions featured in any rounds of top companies. Submitted by Radib Kar, on February 20, 2020

Description

This question has been featured in interview rounds of Amazon, MakeMyTrip, VMWare etc.

Problem statement

Given two strings str1 and str2, find length of the longest common sub-sequence between them

    Let the strings be 
    str1="includehelp"
    str2="letsinclude"

    Output will be:
    Longest common sub-sequence length is 7
    The longest common sub-sequence is: "include"

The output is given above where the longest common sub-sequences is in same colour.

Solution Approach:

The problem can be solved in a brute-force way. By generating all sub-sequences and checking them whether equal or not. Finally taking the longest common subsequence. But undoubtedly this is not at all computable since generating all sub-sequence is itself exponential and then permutations for checking any two sub-sequences.

The recursive way to solve is

Let,

        l1 = Length of the first string,str1
        l2 = Length of the second string,str2

f(l1,l2) = Longest common subsequence length for string lengths l1 & l2 

Now,

Think of the following example,

Say first string is: x₁ x₂ ... x_l1

And the second string is: y₁ y₂ ... y_l2

Say,

Then obviously we need to find LCS for the remaining part of string and then add 1 for this character match

Else

Maximum of two case

1. LCS of the first string leaving character and second string
2. LCS of the first string and second string leaving character

Now, we need to recur down to 0. So,

Where base cases are,

If you generate this recursion tree, it will generate many overlapping sub-problems and thus, we need to reduce the re-computing. That’s why we need to convert it into dynamic programming where we will store the output of the sub-problems and we will use it to compute bigger sub-problems.

Converting to Dynamic programing

    1)  Initialize dp[l1+1][l2+1]  to 0
    2)  Convert the base case of recursion:
            for i=0 to l1
                dp[i][0]=0;
            for i=0 to l2
                dp[0][i]=0;

    3)  Fill the DP table as per recursion.
        for i=1 to l1    //i be the subproblem length for str1
            for j=1 to l2 //j be the subproblem length for str2
                if(str1[i-1]==str2[j-1]) //xl1==yl2
                    dp[i][j]=dp[i-1][j-1]+1;
                else
                    dp[i][j]=max(dp[i-1][j],dp[i][j-1]);
            end for
        end for  
    4)  The final output will be dp[l1][l2]

C++ Implementation

#include <bits/stdc++.h>
using namespace std;

int max(int a, int b)
{
    return (a > b) ? a : b;
}

int LCS(string str1, string str2)
{
    int l1 = str1.length();
    int l2 = str2.length();

    int dp[l1 + 1][l2 + 1];

    for (int i = 0; i <= l1; i++)
        dp[i][0] = 0;
    for (int i = 0; i <= l2; i++)
        dp[0][i] = 0;

    for (int i = 1; i <= l1; i++) {
        for (int j = 1; j <= l2; j++) {
            if (str1[i - 1] == str2[j - 1])
                dp[i][j] = dp[i - 1][j - 1] + 1;
            else
                dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]);
        }
    }

    return dp[l1][l2];
}
int main()
{
    string str1, str2;

    cout << "Enter first string\n";
    cin >> str1;
    cout << "Enter Second string\n";
    cin >> str2;

    cout << "Longest Common sub-sequence length is: " << LCS(str1, str2) << endl;

    return 0;
}

Output

Enter first string
includehelp
Enter Second string
letsincludeus
Longest Common sub-sequence length is: 7