Longest Common Subsequence with at most k changes allowed

Given two sequence P and Q of numbers. The task is to find Longest Common Subsequence of two sequences if we are allowed to change at most k element in first sequence to any value.

Examples:

Input : P = { 8, 3 }
        Q = { 1, 3 }
        K = 1
Output : 2
If we change first element of first
sequence from 8 to 1, both sequences 
become same.

Input : P = { 1, 2, 3, 4, 5 }
        Q = { 5, 3, 1, 4, 2 }
        K = 1
Output : 3
By changing first element of first
sequence to 5 to get the LCS ( 5, 3, 4 }.

The idea is to use Dynamic Programming. Define a 3D matrix dp[][][], where dp[i][j][k] defines the Longest Common Subsequence for the first i numbers of first array, first j number of second array when we are allowed to change at max k number in the first array. 

Therefore, recursion will look like 

If P[i] != Q[j], 
    dp[i][j][k] = max(dp[i - 1][j][k], 
                      dp[i][j - 1][k], 
                      dp[i - 1][j - 1][k - 1] + 1) 
If P[i] == Q[j], 
    dp[i][j][k] = max(dp[i - 1][j][k], 
                      dp[i][j - 1][k], 
                      dp[i - 1][j - 1][k] + 1)

Below is the implementation of this approach:

C++

// CPP program to find LCS of two arrays with
// k changes allowed in first array.
#include <bits/stdc++.h>
using namespace std;
#define MAX 10

// Return LCS with at most k changes allowed.
int lcs(int dp[MAX][MAX][MAX], int arr1[], int n,
                       int arr2[], int m, int k)
{
    // If at most changes is less than 0.
    if (k < 0)
        return -1e7;

    // If any of two array is over.
    if (n < 0 || m < 0)
        return 0;

    // Making a reference variable to dp[n][m][k]
    int& ans = dp[n][m][k];

    // If value is already calculated, return
    // that value.
    if (ans != -1)
        return ans;

    // calculating LCS with no changes made.
    ans = max(lcs(dp, arr1, n - 1, arr2, m, k),
              lcs(dp, arr1, n, arr2, m - 1, k));

    // calculating LCS when array element are same.
    if (arr1[n-1] == arr2[m-1])
        ans = max(ans, 1 + lcs(dp, arr1, n - 1,
                                arr2, m - 1, k));

    // calculating LCS with changes made.
    ans = max(ans, 1 + lcs(dp, arr1, n - 1,
                          arr2, m - 1, k - 1));

    return ans;
}

// Driven Program
int main()
{
    int k = 1;
    int arr1[] = { 1, 2, 3, 4, 5 };
    int arr2[] = { 5, 3, 1, 4, 2 };
    int n = sizeof(arr1) / sizeof(arr1[0]);
    int m = sizeof(arr2) / sizeof(arr2[0]);

    int dp[MAX][MAX][MAX];
    memset(dp, -1, sizeof(dp));

    cout << lcs(dp, arr1, n, arr2, m, k) << endl;

    return 0;
}

Java

// Java program to find LCS of two arrays with
// k changes allowed in first array.
import java.util.*;
import java.io.*;

class GFG
{
    static int MAX = 10;

    // Return LCS with at most k changes allowed.
    static int lcs(int[][][] dp, int[] arr1,
                   int n, int[] arr2, int m, int k)
    {

        // If at most changes is less than 0.
        if (k < 0)
            return -10000000;

        // If any of two array is over.
        if (n < 0 || m < 0)
            return 0;

        // Making a reference variable to dp[n][m][k]
        int ans = dp[n][m][k];

        // If value is already calculated, return
        // that value.
        if (ans != -1)
            return ans;

        try
        {

            // calculating LCS with no changes made.
            ans = Math.max(lcs(dp, arr1, n - 1, arr2, m, k),
                           lcs(dp, arr1, n, arr2, m - 1, k));

            // calculating LCS when array element are same.
            if (arr1[n - 1] == arr2[m - 1])
                ans = Math.max(ans, 1 + lcs(dp, arr1, n - 1,
                                                arr2, m - 1, k));

            // calculating LCS with changes made.
            ans = Math.max(ans, 1 + lcs(dp, arr1, n - 1,
                                            arr2, m - 1, k - 1));
        } catch (Exception e) { }
          // Storing the value in dp.
          dp[n][m][k] = ans;
        return ans;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int k = 1;
        int[] arr1 = { 1, 2, 3, 4, 5 };
        int[] arr2 = { 5, 3, 1, 4, 2 };
        int n = arr1.length;
        int m = arr2.length;

        int[][][] dp = new int[MAX][MAX][MAX];
        for (int i = 0; i < MAX; i++)
            for (int j = 0; j < MAX; j++)
                for (int l = 0; l < MAX; l++)
                    dp[i][j][l] = -1;

        System.out.println(lcs(dp, arr1, n, arr2, m, k));
    }
}

// This code is contributed by
// krishnat208026

Python3

# Python3 program to find LCS of two arrays
# with k changes allowed in the first array.
MAX = 10

# Return LCS with at most k changes allowed.
def lcs(dp, arr1, n, arr2, m, k):

    # If at most changes is less than 0.
    if k < 0:
        return -(10 ** 7)

    # If any of two array is over.
    if n < 0 or m < 0:
        return 0

    # Making a reference variable to dp[n][m][k]
    ans = dp[n][m][k]

    # If value is already calculated,
    # return that value.
    if ans != -1:
        return ans

    # calculating LCS with no changes made.
    ans = max(lcs(dp, arr1, n - 1, arr2, m, k),
            lcs(dp, arr1, n, arr2, m - 1, k))

    # calculating LCS when array element are same.
    if arr1[n-1] == arr2[m-1]:
        ans = max(ans, 1 + lcs(dp, arr1, n - 1,
                                arr2, m - 1, k))

    # calculating LCS with changes made.
    ans = max(ans, lcs(dp, arr1, n - 1,
                        arr2, m - 1, k - 1))

    dp[n][m][k]=ans
    return dp[n][m][k]

# Driven Program
if __name__ == "__main__":

    k = 1
    arr1 = [1, 2, 3, 4, 5]
    arr2 = [5, 3, 1, 4, 2]
    n = len(arr1)
    m = len(arr2)

    dp = [[[-1 for i in range(MAX)] for j in range(MAX)] for k in range(MAX)]

    print(lcs(dp, arr1, n, arr2, m, k))

# This code is contributed by Rituraj Jain

C#

// C# program to find LCS of two arrays with
// k changes allowed in first array.
using System;

class GFG
{
    static int MAX = 10;

    // Return LCS with at most
    // k changes allowed.
    static int lcs(int[,,] dp, int[] arr1,
                        int n, int[] arr2,
                        int m, int k)
    {

        // If at most changes is less than 0.
        if (k < 0)
            return -10000000;

        // If any of two array is over.
        if (n < 0 || m < 0)
            return 0;

        // Making a reference variable
        // to dp[n,m,k]
        int ans = dp[n, m, k];

        // If value is already calculated,
        // return that value.
        if (ans != -1)
            return ans;

        try
        {

            // calculating LCS with no changes made.
            ans = Math.Max(lcs(dp, arr1, n - 1,
                                   arr2, m, k),
                           lcs(dp, arr1, n,
                                   arr2, m - 1, k));

            // calculating LCS when
            // array element are same.
            if (arr1[n - 1] == arr2[m - 1])
                ans = Math.Max(ans, 1 +
                           lcs(dp, arr1, n - 1,
                                   arr2, m - 1, k));

            // calculating LCS with changes made.
            ans = Math.Max(ans, 1 +
                       lcs(dp, arr1, n - 1,
                               arr2, m - 1, k - 1));
        } catch (Exception e) { }
        return ans;
    }

    // Driver Code
    public static void Main(String[] args)
    {
        int k = 1;
        int[] arr1 = { 1, 2, 3, 4, 5 };
        int[] arr2 = { 5, 3, 1, 4, 2 };
        int n = arr1.Length;
        int m = arr2.Length;

        int[,,] dp = new int[MAX, MAX, MAX];
        for (int i = 0; i < MAX; i++)
            for (int j = 0; j < MAX; j++)
                for (int l = 0; l < MAX; l++)
                    dp[i, j, l] = -1;

        Console.WriteLine(lcs(dp, arr1, n,
                                  arr2, m, k));
    }
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

// Javascript program to find LCS of two
// arrays with k changes allowed in
// first array.
let MAX = 10;

// Return LCS with at most k changes allowed.
function lcs(dp, arr1, n, arr2, m, k)
{

    // If at most changes is less than 0.
    if (k < 0)
        return -10000000;

    // If any of two array is over.
    if (n < 0 || m < 0)
        return 0;

    // Making a reference variable
    // to dp[n][m][k]
    let ans = dp;

    // If value is already calculated,
    // return that value.
    if (ans != -1)
        return ans;

    try
    {

        // Calculating LCS with no changes made.
        ans = Math.max(lcs(dp, arr1, n - 1, arr2, m, k),
                       lcs(dp, arr1, n, arr2, m - 1, k));

        // Calculating LCS when array element are same.
        if (arr1[n - 1] == arr2[m - 1])
            ans = Math.max(ans, 1 + lcs(dp, arr1, n - 1,
                                            arr2, m - 1, k));

        // Calculating LCS with changes made.
        ans = Math.max(ans, 1 + lcs(dp, arr1, n - 1,
                                        arr2, m - 1, k - 1));
    } catch (e) { }
    return ans;
}

// Driver Code
let k = 0;
let arr1 = [ 1, 2, 3, 4, 5 ];
let arr2 = [ 5, 3, 1, 4, 2 ];
let n = arr1.length;
let m = arr2.length;

let dp = new Array(MAX);
for(let i = 0; i < MAX; i++)
    for(let j = 0; j < MAX; j++)
        for(let l = 0; l < MAX; l++)
            dp = -1;

document.write(lcs(dp, arr1, n, arr2, m, k));

// This code is contributed by shivanisinghss2110

</script>

Output

4

Time Complexity: O(N*M*K).
Auxiliary Space: O(MAX³)

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