# Dynamic Programming | Set 15 (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

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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 C++ implementation of the above Dynamic Programming solution.

## C++

```/* Dynamic Programming implementation of longest bitonic subsequence problem */
#include<stdio.h>
#include<stdlib.h>

/* 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\n", lbs( arr, n ) );
return 0;
}
```

## Java

```/* Dynamic Programming implementation in Java for longest bitonic
subsequence problem */
import java.util.*;
import java.lang.*;
import java.io.*;

class LBS
{
/* 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]
*/
static 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;
}

public static void main (String[] args)
{
int arr[] = {0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5,
13, 3, 11, 7, 15};
int n = arr.length;
System.out.println("Length of LBS is "+ lbs( arr, n ));
}
}
```

## 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)

```

## C#

```/* Dynamic Programming implementation in
C# for longest bitonic subsequence problem */
using System;

class LBS {

/* 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]
*/
static 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 code
public static void Main()
{
int[] arr = { 0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5,
13, 3, 11, 7, 15 };
int n = arr.Length;
Console.WriteLine("Length of LBS is " + lbs(arr, n));
}
}

// This code is contributed by vt_m.

```

Output:

``` Length of LBS is 7
```

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