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# Python Program to Find the Longest Bitonic Subsequence

Given an array arr[0 … n-1] containing n positive integers, a subsequence of arr[] is called Bitonic if it is first increasing, then decreasing. Write a function that takes an array as argument and returns the length of the longest bitonic subsequence.
A sequence, sorted in increasing order is considered Bitonic with the decreasing part as empty. Similarly, decreasing order sequence is considered Bitonic with the increasing part as empty.
Examples:

```Input arr[] = {1, 11, 2, 10, 4, 5, 2, 1};
Output: 6 (A Longest Bitonic Subsequence of length 6 is 1, 2, 10, 4, 2, 1)

Input arr[] = {12, 11, 40, 5, 3, 1}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 12, 11, 5, 3, 1)

Input arr[] = {80, 60, 30, 40, 20, 10}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 80, 60, 30, 20, 10)```

Source: Microsoft Interview Question

Solution
This problem is a variation of standard Longest Increasing Subsequence (LIS) problem. Let the input array be arr[] of length n. We need to construct two arrays lis[] and lds[] using Dynamic Programming solution of LIS problem. lis[i] stores the length of the Longest Increasing subsequence ending with arr[i]. lds[i] stores the length of the longest Decreasing subsequence starting from arr[i]. Finally, we need to return the max value of lis[i] + lds[i] – 1 where i is from 0 to n-1.
Following is the implementation of the above Dynamic Programming solution.

## Python

 `# Dynamic Programming implementation of longest bitonic subsequence problem``"""``lbs() returns the length of the Longest Bitonic Subsequence in``arr[] of size n. The function mainly creates two temporary arrays``lis[] and lds[] and returns the maximum lis[i] + lds[i] - 1.`` ` `lis[i] ==> Longest Increasing subsequence ending with arr[i]``lds[i] ==> Longest decreasing subsequence starting with arr[i]``"""`` ` `def` `lbs(arr):``    ``n ``=` `len``(arr)`` ` ` ` `    ``# allocate memory for LIS[] and initialize LIS values as 1``    ``# for all indexes``    ``lis ``=` `[``1` `for` `i ``in` `range``(n``+``1``)]`` ` `    ``# Compute LIS values from left to right``    ``for` `i ``in` `range``(``1` `, n):``        ``for` `j ``in` `range``(``0` `, i):``            ``if` `((arr[i] > arr[j]) ``and` `(lis[i] < lis[j] ``+``1``)):``                ``lis[i] ``=` `lis[j] ``+` `1`` ` `    ``# allocate memory for LDS and initialize LDS values for``    ``# all indexes``    ``lds ``=` `[``1` `for` `i ``in` `range``(n``+``1``)]``     ` `    ``# Compute LDS values from right to left``    ``for` `i ``in` `reversed``(``range``(n``-``1``)): ``#loop from n-2 downto 0``        ``for` `j ``in` `reversed``(``range``(i``-``1` `,n)): ``#loop from n-1 downto i-1``            ``if``(arr[i] > arr[j] ``and` `lds[i] < lds[j] ``+` `1``):``                ``lds[i] ``=` `lds[j] ``+` `1` ` ` ` ` `    ``# Return the maximum value of (lis[i] + lds[i] - 1)``    ``maximum ``=` `lis[``0``] ``+` `lds[``0``] ``-` `1``    ``for` `i ``in` `range``(``1` `, n):``        ``maximum ``=` `max``((lis[i] ``+` `lds[i]``-``1``), maximum)``     ` `    ``return` `maximum`` ` `# Driver program to test the above function``arr ``=`  `[``0` `, ``8` `, ``4``, ``12``, ``2``, ``10` `, ``6` `, ``14` `, ``1` `, ``9` `, ``5` `, ``13``,``        ``3``, ``11` `, ``7` `, ``15``]``print` `"Length of LBS is"``,lbs(arr)`` ` `# This code is contributed by Nikhil Kumar Singh(nickzuck_007)`

Output:

` Length of LBS is 7`

Time Complexity: O(n^2)
Auxiliary Space: O(n)

Please refer complete article on Longest Bitonic Subsequence | DP-15 for more details!