# C Program to Find the Longest Bitonic Subsequence

• Last Updated : 21 Dec, 2021

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.

## C

 `/* Dynamic Programming implementation of longest bitonic subsequence problem */``#include``#include`` ` `/* 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]``*/``int` `lbs( ``int` `arr[], ``int` `n )``{``   ``int` `i, j;`` ` `   ``/* Allocate memory for LIS[] and initialize LIS values as 1 for``      ``all indexes */``   ``int` `*lis = ``new` `int``[n];``   ``for` `(i = 0; i < n; i++)``      ``lis[i] = 1;`` ` `   ``/* Compute LIS values from left to right */``   ``for` `(i = 1; i < n; i++)``      ``for` `(j = 0; j < i; j++)``         ``if` `(arr[i] > arr[j] && lis[i] < lis[j] + 1)``            ``lis[i] = lis[j] + 1;`` ` `   ``/* Allocate memory for lds and initialize LDS values for``      ``all indexes */``   ``int` `*lds = ``new` `int` `[n];``   ``for` `(i = 0; i < n; i++)``      ``lds[i] = 1;`` ` `   ``/* Compute LDS values from right to left */``   ``for` `(i = n-2; i >= 0; i--)``      ``for` `(j = n-1; j > i; j--)``         ``if` `(arr[i] > arr[j] && lds[i] < lds[j] + 1)``            ``lds[i] = lds[j] + 1;`` ` ` ` `   ``/* Return the maximum value of lis[i] + lds[i] - 1*/``   ``int` `max = lis[0] + lds[0] - 1;``   ``for` `(i = 1; i < n; i++)``     ``if` `(lis[i] + lds[i] - 1 > max)``         ``max = lis[i] + lds[i] - 1;``   ``return` `max;``}`` ` `/* Driver program to test above function */``int` `main()``{``  ``int` `arr[] = {0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5,``              ``13, 3, 11, 7, 15};``  ``int` `n = ``sizeof``(arr)/``sizeof``(arr[0]);``  ``printf``("Length of LBS is %d``", lbs( arr, n ) );``  ``return` `0;``}`

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!

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