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Longest Increasing Subsequence using Longest Common Subsequence Algorithm
  • Difficulty Level : Medium
  • Last Updated : 24 Sep, 2020

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:
{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:
 

 

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.
Note: Array should have distinct elements otherwise it might give wrong result. For example in {1, 1, 1} we know the longest increasing subsequence(a1 < a2 < … ak) is of length 1, but if we try out this example in LIS using LCS method we would get 3 (because it finds the longest common subsequence). 
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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C#

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// C# implementation of above approach
using System;
using System.Collections.Generic;
 
class GFG
{
     
// Function to return the size of the
// longest increasing subsequence
static int LISusingLCS(List<int> seq)
{
    int n = seq.Count;
 
    // 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
    List<int> sortedseq = new List<int>(seq);
 
    sortedseq.Sort();
 
    // 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] = 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)
{
    List<int> sequence = new List<int>();
    sequence.Add(12);
    sequence.Add(34);
    sequence.Add(1);
    sequence.Add(5);
    sequence.Add(40);
    sequence.Add(80);
 
    Console.WriteLine(LISusingLCS(sequence));
}
}
 
// This code is contributed by 29AjayKumar

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Output: 

4

 

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

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