# 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}

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++

 `// CPP program to find the length of the ` `// longest decreasing subsequence ` `#include ` `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); ` `    ``cout << ``"Length of LDS is "` `<< lds(arr, n); ` `    ``return` `0; ` `} `

## Java

 `// 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. `

## Python 3

 `# 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 `

## C#

 `// 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. `

## PHP

 ` `

Output :

```Length of LDS is 3
```

Time Complexity: O(n2)
Auxiliary Space: O(n)

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

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