Longest Increasing Subsequence using Longest Common Subsequence Algorithm

Given an array arr[] of N integers, the task is to find and print the Longest Increasing Subsequence.

Examples:

Input: arr[] = {12, 34, 1, 5, 40, 80}
Output: 4
{12, 34, 40, 80} and {1, 5, 40, 80} are the
longest increasing subsequences.



Input: arr[] = {10, 22, 9, 33, 21, 50, 41, 60, 80}
Output: 6

Prerequisite: LCS, LIS

Approach: The longest increasing subsequence of any sequence is the subsequence of the sorted sequence of itself. It can be solved using a Dynamic Programming approach. The approach is the same as the classical LCS problem but instead of the second sequence, given sequence is taken again in its sorted form.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the size of the
// longest increasing subsequence
int LISusingLCS(vector<int>& seq)
{
    int n = seq.size();
  
    // Create an 2D array of integer
    // for tabulation
    vector<vector<int> > L(n + 1, vector<int>(n + 1));
  
    // Take the second sequence as the sorted
    // sequence of the given sequence
    vector<int> sortedseq(seq);
  
    sort(sortedseq.begin(), sortedseq.end());
  
    // Classical Dynamic Programming algorithm
    // for Longest Common Subsequence
    for (int i = 0; i <= n; i++) {
        for (int j = 0; j <= n; j++) {
            if (i == 0 || j == 0)
                L[i][j] = 0;
  
            else if (seq[i - 1] == sortedseq[j - 1])
                L[i][j] = L[i - 1][j - 1] + 1;
  
            else
                L[i][j] = max(L[i - 1][j], L[i][j - 1]);
        }
    }
  
    // Return the ans
    return L[n][n];
}
  
// Driver code
int main()
{
  
    vector<int> sequence{ 12, 34, 1, 5, 40, 80 };
  
    cout << LISusingLCS(sequence) << endl;
  
    return 0;
}

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Java














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//Java implementation of above approach 


import java.util.*; 


  


class GFG 


{ 


      


// Function to return the size of the 


// longest increasing subsequence 


static int LISusingLCS(Vector<Integer> seq) 


{ 


    int n = seq.size(); 


  


    // Create an 2D array of integer 


    // for tabulation 


    int L[][] = new int [n + 1][n + 1]; 


  


    // Take the second sequence as the sorted 


    // sequence of the given sequence 


    Vector<Integer> sortedseq = new Vector<Integer>(seq); 


  


    Collections.sort(sortedseq); 


  


    // Classical Dynamic Programming algorithm 


    // for Longest Common Subsequence 


    for (int i = 0; i <= n; i++)  


    { 


        for (int j = 0; j <= n; j++)  


        { 


            if (i == 0 || j == 0) 


                L[i][j] = 0; 


  


            else if (seq.get(i - 1) == sortedseq.get(j - 1)) 


                L[i][j] = L[i - 1][j - 1] + 1; 


  


            else


                L[i][j] = Math.max(L[i - 1][j],  


                                   L[i][j - 1]); 


        } 


    } 


  


    // Return the ans 


    return L[n][n]; 


} 


  


// Driver code 


public static void main(String args[]) 


{ 


    Vector<Integer> sequence = new Vector<Integer>(); 


    sequence.add(12); 


    sequence.add(34); 


    sequence.add(1); 


    sequence.add(5); 


    sequence.add(40); 


    sequence.add(80); 


  


    System.out.println(LISusingLCS(sequence)); 


} 


} 


  


// This code is contributed by Arnab Kundu 



















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Python3














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# Python3 implementation of the approach 


  


# Function to return the size of the 


# longest increasing subsequence 


def LISusingLCS(seq): 


    n = len(seq) 


  


    # Create an 2D array of integer 


    # for tabulation 


    L = [[0 for i in range(n + 1)]  


            for i in range(n + 1)] 


      


    # Take the second sequence as the sorted 


    # sequence of the given sequence 


    sortedseq = sorted(seq) 


  


    # Classical Dynamic Programming algorithm 


    # for Longest Common Subsequence 


    for i in range(n + 1): 


        for j in range(n + 1): 


            if (i == 0 or j == 0): 


                L[i][j] = 0


  


            elif (seq[i - 1] == sortedseq[j - 1]): 


                L[i][j] = L[i - 1][j - 1] + 1


  


            else: 


                L[i][j] = max(L[i - 1][j],  


                              L[i][j - 1]) 


  


    # Return the ans 


    return L[n][n] 


  


# Driver code 


sequence = [12, 34, 1, 5, 40, 80] 


  


print(LISusingLCS(sequence)) 


  


# This code is contributed by mohit kumar 



















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Output:


4





Time Complexity: O(n2) where n is the length of the sequence.



          
          
          
          

            

    

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