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# Printing Longest Bitonic Subsequence

The Longest Bitonic Subsequence problem is to find the longest subsequence of a given sequence such that it is first increasing and then decreasing. 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:  [1, 11, 2, 10, 4, 5, 2, 1]
Output: [1, 2, 10, 4, 2, 1] OR [1, 11, 10, 5, 2, 1]
OR [1, 2, 4, 5, 2, 1]

Input:  [12, 11, 40, 5, 3, 1]
Output: [12, 11, 5, 3, 1] OR [12, 40, 5, 3, 1]

Input:  [80, 60, 30, 40, 20, 10]
Output: [80, 60, 30, 20, 10] OR [80, 60, 40, 20, 10]```

In previous post, we have discussed about Longest Bitonic Subsequence problem. However, the post only covered code related to finding maximum sum of increasing subsequence, but not to the construction of subsequence. In this post, we will discuss how to construct Longest Bitonic Subsequence itself. Let arr[0..n-1] be the input array. We define vector LIS such that LIS[i] is itself is a vector that stores Longest Increasing Subsequence of arr[0..i] that ends with arr[i]. Therefore for an index i, LIS[i] can be recursively written as –

```LIS[0] = {arr[O]}
LIS[i] = {Max(LIS[j])} + arr[i] where j < i and arr[j] < arr[i]
= arr[i], if there is no such j```

We also define a vector LDS such that LDS[i] is itself is a vector that stores Longest Decreasing Subsequence of arr[i..n] that starts with arr[i]. Therefore for an index i, LDS[i] can be recursively written as –

```LDS[n] = {arr[n]}
LDS[i] = arr[i] + {Max(LDS[j])} where j > i and arr[j] < arr[i]
= arr[i], if there is no such j```

For example, for array [1 11 2 10 4 5 2 1],

```LIS[0]: 1
LIS[1]: 1 11
LIS[2]: 1 2
LIS[3]: 1 2 10
LIS[4]: 1 2 4
LIS[5]: 1 2 4 5
LIS[6]: 1 2
LIS[7]: 1```
```LDS[0]: 1
LDS[1]: 11 10 5 2 1
LDS[2]: 2 1
LDS[3]: 10 5 2 1
LDS[4]: 4 2 1
LDS[5]: 5 2 1
LDS[6]: 2 1
LDS[7]: 1```

Therefore, Longest Bitonic Subsequence can be

```LIS[1] + LDS[1] = [1 11 10 5 2 1]        OR
LIS[3] + LDS[3] = [1 2 10 5 2 1]        OR
LIS[5] + LDS[5] = [1 2 4 5 2 1]```

Below is the implementation of above idea –

## C++

 `/* Dynamic Programming solution to print Longest``   ``Bitonic Subsequence */``#include ``using` `namespace` `std;` `// Utility function to print Longest Bitonic``// Subsequence``void` `print(vector<``int``>& arr, ``int` `size)``{``    ``for``(``int` `i = 0; i < size; i++)``        ``cout << arr[i] << ``" "``;``}` `// Function to construct and print Longest``// Bitonic Subsequence``void` `printLBS(``int` `arr[], ``int` `n)``{``    ``// LIS[i] stores the length of the longest``    ``// increasing subsequence ending with arr[i]``    ``vector> LIS(n);` `    ``// initialize LIS[0] to arr[0]``    ``LIS[0].push_back(arr[0]);` `    ``// Compute LIS values from left to right``    ``for` `(``int` `i = 1; i < n; i++)``    ``{``        ``// for every j less than i``        ``for` `(``int` `j = 0; j < i; j++)``        ``{``            ``if` `((arr[j] < arr[i]) &&``                ``(LIS[j].size() > LIS[i].size()))``                ``LIS[i] = LIS[j];``        ``}``        ``LIS[i].push_back(arr[i]);``    ``}` `    ``/* LIS[i] now stores Maximum Increasing``       ``Subsequence of arr[0..i] that ends with``       ``arr[i] */` `    ``// LDS[i] stores the length of the longest``    ``// decreasing subsequence starting with arr[i]``    ``vector> LDS(n);` `    ``// initialize LDS[n-1] to arr[n-1]``    ``LDS[n - 1].push_back(arr[n - 1]);` `    ``// Compute LDS values from right to left``    ``for` `(``int` `i = n - 2; i >= 0; i--)``    ``{``        ``// for every j greater than i``        ``for` `(``int` `j = n - 1; j > i; j--)``        ``{``            ``if` `((arr[j] < arr[i]) &&``                ``(LDS[j].size() > LDS[i].size()))``                ``LDS[i] = LDS[j];``        ``}``        ``LDS[i].push_back(arr[i]);``    ``}` `    ``// reverse as vector as we're inserting at end``    ``for` `(``int` `i = 0; i < n; i++)``        ``reverse(LDS[i].begin(), LDS[i].end());` `    ``/* LDS[i] now stores Maximum Decreasing Subsequence``       ``of arr[i..n] that starts with arr[i] */` `    ``int` `max = 0;``    ``int` `maxIndex = -1;` `    ``for` `(``int` `i = 0; i < n; i++)``    ``{``        ``// Find maximum value of size of LIS[i] + size``        ``// of LDS[i] - 1``        ``if` `(LIS[i].size() + LDS[i].size() - 1 > max)``        ``{``            ``max = LIS[i].size() + LDS[i].size() - 1;``            ``maxIndex = i;``        ``}``    ``}` `    ``// print all but last element of LIS[maxIndex] vector``    ``print(LIS[maxIndex], LIS[maxIndex].size() - 1);` `    ``// print all elements of LDS[maxIndex] vector``    ``print(LDS[maxIndex], LDS[maxIndex].size());``}` `// Driver program``int` `main()``{``    ``int` `arr[] = { 1, 11, 2, 10, 4, 5, 2, 1 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``printLBS(arr, n);``    ``return` `0;``}`

## Java

 `/* Dynamic Programming solution to print Longest``Bitonic Subsequence */``import` `java.util.*;` `class` `GFG``{` `    ``// Utility function to print Longest Bitonic``    ``// Subsequence``    ``static` `void` `print(Vector arr, ``int` `size)``    ``{``        ``for` `(``int` `i = ``0``; i < size; i++)``            ``System.out.print(arr.elementAt(i) + ``" "``);``    ``}` `    ``// Function to construct and print Longest``    ``// Bitonic Subsequence``    ``static` `void` `printLBS(``int``[] arr, ``int` `n)``    ``{` `        ``// LIS[i] stores the length of the longest``        ``// increasing subsequence ending with arr[i]``        ``@SuppressWarnings``(``"unchecked"``)``        ``Vector[] LIS = ``new` `Vector[n];` `        ``for` `(``int` `i = ``0``; i < n; i++)``            ``LIS[i] = ``new` `Vector<>();` `        ``// initialize LIS[0] to arr[0]``        ``LIS[``0``].add(arr[``0``]);` `        ``// Compute LIS values from left to right``        ``for` `(``int` `i = ``1``; i < n; i++)``        ``{` `            ``// for every j less than i``            ``for` `(``int` `j = ``0``; j < i; j++)``            ``{` `                ``if` `((arr[i] > arr[j]) &&``                     ``LIS[j].size() > LIS[i].size())``                ``{``                    ``for` `(``int` `k : LIS[j])``                        ``if` `(!LIS[i].contains(k))``                            ``LIS[i].add(k);``                ``}``            ``}``            ``LIS[i].add(arr[i]);``        ``}` `        ``/*``        ``* LIS[i] now stores Maximum Increasing Subsequence``        ``* of arr[0..i] that ends with arr[i]``        ``*/` `        ``// LDS[i] stores the length of the longest``        ``// decreasing subsequence starting with arr[i]``        ``@SuppressWarnings``(``"unchecked"``)``        ``Vector[] LDS = ``new` `Vector[n];` `        ``for` `(``int` `i = ``0``; i < n; i++)``            ``LDS[i] = ``new` `Vector<>();` `        ``// initialize LDS[n-1] to arr[n-1]``        ``LDS[n - ``1``].add(arr[n - ``1``]);` `        ``// Compute LDS values from right to left``        ``for` `(``int` `i = n - ``2``; i >= ``0``; i--)``        ``{` `            ``// for every j greater than i``            ``for` `(``int` `j = n - ``1``; j > i; j--)``            ``{``                ``if` `(arr[j] < arr[i] &&``                    ``LDS[j].size() > LDS[i].size())``                    ``for` `(``int` `k : LDS[j])``                        ``if` `(!LDS[i].contains(k))``                            ``LDS[i].add(k);``            ``}``            ``LDS[i].add(arr[i]);``        ``}` `        ``// reverse as vector as we're inserting at end``        ``for` `(``int` `i = ``0``; i < n; i++)``            ``Collections.reverse(LDS[i]);` `        ``/*``        ``* LDS[i] now stores Maximum Decreasing Subsequence``        ``* of arr[i..n] that starts with arr[i]``        ``*/``        ``int` `max = ``0``;``        ``int` `maxIndex = -``1``;``        ``for` `(``int` `i = ``0``; i < n; i++)``        ``{` `            ``// Find maximum value of size of``            ``// LIS[i] + size of LDS[i] - 1``            ``if` `(LIS[i].size() + LDS[i].size() - ``1` `> max)``            ``{``                ``max = LIS[i].size() + LDS[i].size() - ``1``;``                ``maxIndex = i;``            ``}``        ``}` `        ``// print all but last element of LIS[maxIndex] vector``        ``print(LIS[maxIndex], LIS[maxIndex].size() - ``1``);` `        ``// print all elements of LDS[maxIndex] vector``        ``print(LDS[maxIndex], LDS[maxIndex].size());``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int``[] arr = { ``1``, ``11``, ``2``, ``10``, ``4``, ``5``, ``2``, ``1` `};``        ``int` `n = arr.length;` `        ``printLBS(arr, n);``    ``}``}` `// This code is contributed by``// sanjeev2552`

## Python3

 `# Dynamic Programming solution to print Longest``# Bitonic Subsequence`  `def` `_print(arr: ``list``, size: ``int``):``    ``for` `i ``in` `range``(size):``        ``print``(arr[i], end``=``" "``)`  `# Function to construct and print Longest``# Bitonic Subsequence``def` `printLBS(arr: ``list``, n: ``int``):` `    ``# LIS[i] stores the length of the longest``    ``# increasing subsequence ending with arr[i]``    ``LIS ``=` `[``0``] ``*` `n``    ``for` `i ``in` `range``(n):``        ``LIS[i] ``=` `[]` `    ``# initialize LIS[0] to arr[0]``    ``LIS[``0``].append(arr[``0``])` `    ``# Compute LIS values from left to right``    ``for` `i ``in` `range``(``1``, n):` `        ``# for every j less than i``        ``for` `j ``in` `range``(i):` `            ``if` `((arr[j] < arr[i]) ``and` `(``len``(LIS[j]) > ``len``(LIS[i]))):``                ``LIS[i] ``=` `LIS[j].copy()` `        ``LIS[i].append(arr[i])` `    ``# LIS[i] now stores Maximum Increasing``    ``# Subsequence of arr[0..i] that ends with``    ``# arr[i]` `    ``# LDS[i] stores the length of the longest``    ``# decreasing subsequence starting with arr[i]``    ``LDS ``=` `[``0``] ``*` `n``    ``for` `i ``in` `range``(n):``        ``LDS[i] ``=` `[]` `    ``# initialize LDS[n-1] to arr[n-1]``    ``LDS[n ``-` `1``].append(arr[n ``-` `1``])` `    ``# Compute LDS values from right to left``    ``for` `i ``in` `range``(n ``-` `2``, ``-``1``, ``-``1``):` `        ``# for every j greater than i``        ``for` `j ``in` `range``(n ``-` `1``, i, ``-``1``):` `            ``if` `((arr[j] < arr[i]) ``and` `(``len``(LDS[j]) > ``len``(LDS[i]))):``                ``LDS[i] ``=` `LDS[j].copy()` `        ``LDS[i].append(arr[i])` `    ``# reverse as vector as we're inserting at end``    ``for` `i ``in` `range``(n):``        ``LDS[i] ``=` `list``(``reversed``(LDS[i]))` `    ``# LDS[i] now stores Maximum Decreasing Subsequence``    ``# of arr[i..n] that starts with arr[i]` `    ``max` `=` `0``    ``maxIndex ``=` `-``1` `    ``for` `i ``in` `range``(n):` `        ``# Find maximum value of size of LIS[i] + size``        ``# of LDS[i] - 1``        ``if` `(``len``(LIS[i]) ``+` `len``(LDS[i]) ``-` `1` `> ``max``):` `            ``max` `=` `len``(LIS[i]) ``+` `len``(LDS[i]) ``-` `1``            ``maxIndex ``=` `i` `    ``# print all but last element of LIS[maxIndex] vector``    ``_print(LIS[maxIndex], ``len``(LIS[maxIndex]) ``-` `1``)` `    ``# print all elements of LDS[maxIndex] vector``    ``_print(LDS[maxIndex], ``len``(LDS[maxIndex]))`  `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``arr ``=` `[``1``, ``11``, ``2``, ``10``, ``4``, ``5``, ``2``, ``1``]``    ``n ``=` `len``(arr)` `    ``printLBS(arr, n)` `# This code is contributed by``# sanjeev2552`

## C#

 `/* Dynamic Programming solution to print longest``Bitonic Subsequence */``using` `System;``using` `System.Linq;``using` `System.Collections.Generic;` `class` `GFG``{` `    ``// Utility function to print longest Bitonic``    ``// Subsequence``    ``static` `void` `print(List<``int``> arr, ``int` `size)``    ``{``        ``for` `(``int` `i = 0; i < size; i++)``            ``Console.Write(arr[i] + ``" "``);``    ``}` `    ``// Function to construct and print longest``    ``// Bitonic Subsequence``    ``static` `void` `printLBS(``int``[] arr, ``int` `n)``    ``{` `        ``// LIS[i] stores the length of the longest``        ``// increasing subsequence ending with arr[i]``        ``List<``int``>[] LIS = ``new` `List<``int``>[n];` `        ``for` `(``int` `i = 0; i < n; i++)``            ``LIS[i] = ``new` `List<``int``>();` `        ``// initialize LIS[0] to arr[0]``        ``LIS[0].Add(arr[0]);` `        ``// Compute LIS values from left to right``        ``for` `(``int` `i = 1; i < n; i++)``        ``{` `            ``// for every j less than i``            ``for` `(``int` `j = 0; j < i; j++)``            ``{` `                ``if` `((arr[i] > arr[j]) &&``                    ``LIS[j].Count > LIS[i].Count)``                ``{``                    ``foreach` `(``int` `k ``in` `LIS[j])``                        ``if` `(!LIS[i].Contains(k))``                            ``LIS[i].Add(k);``                ``}``            ``}``            ``LIS[i].Add(arr[i]);``        ``}` `        ``/*``        ``* LIS[i] now stores Maximum Increasing Subsequence``        ``* of arr[0..i] that ends with arr[i]``        ``*/` `        ``// LDS[i] stores the length of the longest``        ``// decreasing subsequence starting with arr[i]``        ``List<``int``>[] LDS = ``new` `List<``int``>[n];` `        ``for` `(``int` `i = 0; i < n; i++)``            ``LDS[i] = ``new` `List<``int``>();` `        ``// initialize LDS[n-1] to arr[n-1]``        ``LDS[n - 1].Add(arr[n - 1]);` `        ``// Compute LDS values from right to left``        ``for` `(``int` `i = n - 2; i >= 0; i--)``        ``{` `            ``// for every j greater than i``            ``for` `(``int` `j = n - 1; j > i; j--)``            ``{``                ``if` `(arr[j] < arr[i] &&``                    ``LDS[j].Count > LDS[i].Count)``                    ``foreach` `(``int` `k ``in` `LDS[j])``                        ``if` `(!LDS[i].Contains(k))``                            ``LDS[i].Add(k);``            ``}``            ``LDS[i].Add(arr[i]);``        ``}` `        ``// reverse as vector as we're inserting at end``        ``for` `(``int` `i = 0; i < n; i++)``            ``LDS[i].Reverse();` `        ``/*``        ``* LDS[i] now stores Maximum Decreasing Subsequence``        ``* of arr[i..n] that starts with arr[i]``        ``*/``        ``int` `max = 0;    ``        ``int` `maxIndex = -1;``        ``for` `(``int` `i = 0; i < n; i++)``        ``{` `            ``// Find maximum value of size of``            ``// LIS[i] + size of LDS[i] - 1``            ``if` `(LIS[i].Count + LDS[i].Count - 1 > max)``            ``{``                ``max = LIS[i].Count + LDS[i].Count - 1;``                ``maxIndex = i;``            ``}``        ``}` `        ``// print all but last element of LIS[maxIndex] vector``        ``print(LIS[maxIndex], LIS[maxIndex].Count - 1);` `        ``// print all elements of LDS[maxIndex] vector``        ``print(LDS[maxIndex], LDS[maxIndex].Count);``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``int``[] arr = { 1, 11, 2, 10, 4, 5, 2, 1 };``        ``int` `n = arr.Length;` `        ``printLBS(arr, n);``    ``}``}` `// This code is contributed by PrinciRaj1992`

## Javascript

 `// Function to print the longest bitonic subsequence``function` `_print(arr, size) {``    ``for` `(let i = 0; i LIS[i].length) {``                ``LIS[i] = LIS[j].slice();``            ``}``        ``}``        ``LIS[i].push(arr[i]);``    ``}` `    ``// LIS[i] now stores the Maximum Increasing Subsequence of arr[0..i] that ends with arr[i]` `    ``// LDS[i] stores the length of the longest decreasing subsequence starting with arr[i]``    ``let LDS = ``new` `Array(n);``    ``for` `(let i = 0; i < n; i++) {``        ``LDS[i] = [];``    ``}` `    ``// initialize LDS[n-1] to arr[n-1]``    ``LDS[n - 1].push(arr[n - 1]);` `    ``// Compute LDS values from right to left``    ``for` `(let i = n - 2; i >= 0; i--) {``        ``// for every j greater than i``        ``for` `(let j = n - 1; j > i; j--) {``            ``if` `(arr[j] < arr[i] && LDS[j].length > LDS[i].length) {``                ``LDS[i] = LDS[j].slice();``            ``}``        ``}``        ``LDS[i].push(arr[i]);``    ``}` `    ``// reverse the LDS vector as we're inserting at the end``    ``for` `(let i = 0; i < n; i++) {``        ``LDS[i].reverse();``    ``}` `    ``// LDS[i] now stores the Maximum Decreasing Subsequence of arr[i..n] that starts with arr[i]` `    ``let max = 0;``    ``let maxIndex = -1;` `    ``for` `(let i = 0; i < n; i++) {``        ``// Find maximum value of size of LIS[i] + size of LDS[i] - 1``        ``if` `(LIS[i].length + LDS[i].length - 1 > max) {``            ``max = LIS[i].length + LDS[i].length - 1;``            ``maxIndex = i;``        ``}``    ``}` `    ``// print all but` `  ``// print all but last element of LIS[maxIndex] array``  ``_print(LIS[maxIndex].slice(0, -1), LIS[maxIndex].length - 1);` `  ``// print all elements of LDS[maxIndex] array``  ``_print(LDS[maxIndex], LDS[maxIndex].length);``}` `// Driver program``const arr = [1, 11, 2, 10, 4, 5, 2, 1];``const n = arr.length;` `printLBS(arr, n);`

Output:

`1 11 10 5 2 1`

Time complexity of above Dynamic Programming solution is O(n2). Auxiliary space used by the program is O(n2). This article is contributed by Aditya Goel. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.