Longest Decreasing Subsequence

Given an array of N integers, find the length of the longest subsequence of a given sequence such that all elements of the subsequence are sorted in strictly decreasing order.

Examples :

Input: arr[] = [15, 27, 14, 38, 63, 55, 46, 65, 85]
Output: 3
Explanation: The longest decreasing sub sequence is {63, 55, 46}



Input: arr[] = {50, 3, 10, 7, 40, 80}
Output: 3
Explanation: The longest decreasing subsequence is {50, 10, 7}

The problem can be solved using Dynamic Programming

Optimal Substructure:

Let arr[0..n-1] be the input array and lds[i] be the length of the LDS ending at index i such that arr[i] is the last element of the LDS.
Then, lds[i] can be recursively written as:

lds[i] = 1 + max( lds[j] ) where i > j > 0 and arr[j] > arr[i] or
lds[i] = 1, if no such j exists.

To find the LDS for a given array, we need to return max(lds[i]) where n > i > 0.

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// CPP program to find the length of the
// longest decreasing subsequence
#include <bits/stdc++.h>
using namespace std;
  
// Function that returns the length
// of the longest decreasing subsequence
int lds(int arr[], int n)
{
    int lds[n];
    int i, j, max = 0;
  
    // Initialize LDS with 1 for all index
    // The minimum LDS starting with any
    // element is always 1
    for (i = 0; i < n; i++)
        lds[i] = 1;
  
    // Compute LDS from every index
    // in bottom up manner
    for (i = 1; i < n; i++)
        for (j = 0; j < i; j++)
            if (arr[i] < arr[j] && lds[i] < lds[j] + 1)
                lds[i] = lds[j] + 1;
  
    // Select the maximum 
    // of all the LDS values
    for (i = 0; i < n; i++)
        if (max < lds[i])
            max = lds[i];
  
    // returns the length of the LDS
    return max;
}
// Driver Code
int main()
{
    int arr[] = { 15, 27, 14, 38, 63, 55, 46, 65, 85 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << "Length of LDS is " << lds(arr, n);
    return 0;
}

chevron_right






                        filter_none









Java














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// Java program to find the  


// length of the longest  


// decreasing subsequence 


import java.io.*; 


  


class GFG  


{ 


  


// Function that returns the  


// length of the longest  


// decreasing subsequence 


static int lds(int arr[], int n) 


{ 


    int lds[] = new int[n]; 


    int i, j, max = 0; 


  


    // Initialize LDS with 1  


    // for all index. The minimum  


    // LDS starting with any 


    // element is always 1 


    for (i = 0; i < n; i++) 


        lds[i] = 1; 


  


    // Compute LDS from every  


    // index in bottom up manner 


    for (i = 1; i < n; i++) 


        for (j = 0; j < i; j++) 


            if (arr[i] < arr[j] &&  


                         lds[i] < lds[j] + 1) 


                lds[i] = lds[j] + 1; 


  


    // Select the maximum  


    // of all the LDS values 


    for (i = 0; i < n; i++) 


        if (max < lds[i]) 


            max = lds[i]; 


  


    // returns the length 


    // of the LDS 


    return max; 


} 


// Driver Code 


public static void main (String[] args) 


{ 


    int arr[] = { 15, 27, 14, 38,  


                  63, 55, 46, 65, 85 }; 


    int n = arr.length; 


    System.out.print("Length of LDS is " +  


                             lds(arr, n)); 


} 


} 


  


// This code is contributed by anuj_67. 



















                        chevron_right







                        filter_none









Python 3














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















# Python 3 program to find the length of  


# the longest decreasing subsequence 


  


# Function that returns the length 


# of the longest decreasing subsequence 


def lds(arr, n): 


  


    lds = [0] * n 


    max = 0


  


    # Initialize LDS with 1 for all index 


    # The minimum LDS starting with any 


    # element is always 1 


    for i in range(n): 


        lds[i] = 1


  


    # Compute LDS from every index 


    # in bottom up manner 


    for i in range(1, n): 


        for j in range(i): 


            if (arr[i] < arr[j] and 


                lds[i] < lds[j] + 1): 


                lds[i] = lds[j] + 1


  


    # Select the maximum  


    # of all the LDS values 


    for i in range(n): 


        if (max < lds[i]): 


            max = lds[i] 


  


    # returns the length of the LDS 


    return max


  


# Driver Code 


if __name__ == "__main__": 


      


    arr = [ 15, 27, 14, 38,  


            63, 55, 46, 65, 85 ] 


    n = len(arr) 


    print("Length of LDS is", lds(arr, n)) 


  


# This code is contributed by ita_c 



















                        chevron_right







                        filter_none









C#














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// C# program to find the  


// length of the longest  


// decreasing subsequence 


using System; 


  


class GFG  


{ 


  


// Function that returns the  


// length of the longest  


// decreasing subsequence 


static int lds(int []arr, int n) 


{ 


    int []lds = new int[n]; 


    int i, j, max = 0; 


  


    // Initialize LDS with 1  


    // for all index. The minimum  


    // LDS starting with any 


    // element is always 1 


    for (i = 0; i < n; i++) 


        lds[i] = 1; 


  


    // Compute LDS from every  


    // index in bottom up manner 


    for (i = 1; i < n; i++) 


        for (j = 0; j < i; j++) 


            if (arr[i] < arr[j] &&  


                        lds[i] < lds[j] + 1) 


                lds[i] = lds[j] + 1; 


  


    // Select the maximum  


    // of all the LDS values 


    for (i = 0; i < n; i++) 


        if (max < lds[i]) 


            max = lds[i]; 


  


    // returns the length 


    // of the LDS 


    return max; 


} 


// Driver Code 


public static void Main () 


{ 


    int []arr = { 15, 27, 14, 38,  


                63, 55, 46, 65, 85 }; 


    int n = arr.Length; 


    Console.Write("Length of LDS is " +  


                          lds(arr, n)); 


} 


} 


  


// This code is contributed by anuj_67. 



















                        chevron_right







                        filter_none









PHP














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















<?php 


// PHP program to find the  


// length of the longest  


// decreasing subsequence 


  


  


// Function that returns the  


// length of the longest  


// decreasing subsequence 


function lds($arr, $n) 


{ 


    $lds = array(); 


    $i; $j; $max = 0; 


  


    // Initialize LDS with 1  


    // for all index The minimum 


    // LDS starting with any 


    // element is always 1 


    for ($i = 0; $i < $n; $i++) 


        $lds[$i] = 1; 


  


    // Compute LDS from every  


    // index in bottom up manner 


    for ($i = 1; $i < $n; $i++) 


        for ($j = 0; $j < $i; $j++) 


            if ($arr[$i] < $arr[$j] and 


                $lds[$i] < $lds[$j] + 1) 


                { 


                    $lds[$i] = $lds[$j] + 1; 


                } 


  


    // Select the maximum  


    // of all the LDS values 


    for ($i = 0; $i < $n; $i++) 


        if ($max < $lds[$i]) 


            $max = $lds[$i]; 


  


    // returns the length 


    // of the LDS 


    return $max; 


} 


  


// Driver Code 


$arr = array(15, 27, 14, 38, 63, 


             55, 46, 65, 85); 


$n = count($arr); 


echo "Length of LDS is " ,  


            lds($arr, $n); 


  


// This code is contributed by anuj_67. 


?> 



















                        chevron_right







                        filter_none











Output :


Length of LDS is 3





Time Complexity: O(n2)

Auxiliary Space: O(n)


Related Article: https://www.geeksforgeeks.org/longest-increasing-subsequence/



          
          
          
          

            

    

        My Personal Notes
        arrow_drop_up
    

    

        
        

            
            
        

    


Recommended Posts:
aganjali10
Check out this Author's contributed articles.
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on  the "Improve Article" button below.


Improved By :  vt_m, chitranayal